WIND EFFECTS ON
STRUCTURES
WIND EFFECTS ON
STRUCTURES
Fundamentals and Applications
to Design
Third Edition
EMIL SIMIU
Nlsr Fellow, Building and Fire Research Laboratory, National lnstitute of
Standards and Technology, Gaithersburg, Maryland
ROBERT H. SCANLAN
Vicw ol'C--hicago with Standard Oil Company (lncliana) huiltlilrpi lr('iu ('('ntr'r (Alt lritet'ts:
Pcrkins and Will, and Edward Durell Stone and Associirtt.s)
Professor, Department of Civil Engineering, The Johns Hopkins
University, Baltimore, Maryland
"
';:-l:nEltA]ltr
DO pOtsTO
i:acuicjacle de Engenharra
BIBLI' TECA
Hr.'
:3
oata
I?
6o't
.1lL *,{-
I tg
A Wiley-lnterscience publication
JOHN WILEY & SONS, INC.
New York / Chichester / Brisbano / Toronto
/
Singapore
Soave sia il vento
tranquilla sia I'onda
ed ogni elemento
tranquillo risponda
ai nostri desir!
Cosi Fan Tutte
Act l, Scene V (Terzettino)
This text is printed on acid-free paper
Copyright
All
O
1996 by John
Wiley & Sons, Inc.
rights reserved. Published simultaneously in Canada.
Reproduction or translation of any part of this work beyond
that permitted by Section 107 or 108 of the 1976 United
States Copyright Act without the permission of the copyright
owner is unlawful. Requests for permission or further
information should be addressed to the Permissions Department,
John Wiley & Sons, Inc., 605 Third Avenue, New York, NY
For Devra, Erica, and Michael paul.
t0r58-0012.
This publication is designed to provide accurate and
authoritative information in regard to the subject
matter covered. It is sold with the understanding that
the publisher is not engaged in rendering legal, accounting,
or other prcfcssional services. If legal advice or other
expert assistance is required, the services of a competent
prof'essional person should be sought.
Library of Congre ss Cataloging-in-Publication Data:
Simiu, Emil.
Wind effects on structures: fundamentals and applications to design
/ Emil Simiri, Robert H. Scanlan. - 3rd ed.
p.
ciTl.
Includes index.
ISBN 0471-12157-6 (cloth : alk. paper)
l.
Wind-pressure. 2. Buildings-Aerodynamics. 3. Wind resistant
design.
I.
Scanlan, Robert
H. II. Title.
TA654.5.S55 1996
624.1'76-dc2o
Printed in the United States of America
1098765432
96-5238
PREFACE
'f
'lrt' tlrird edition
of Wind Effects of Stuctures reflects the many developments
tlrrrt occurred during the last decade in the wind engineering field. The text has
lrt'en rcvised, updated, and augmented to include new information and/or refcrrnccs on, among other topics, windstorm damage and insurance, hurricane
rrricnrrnctcorology, aerodynamics of circular cylinders for large Reynolds numlrt'r's, c<lmputational fluid dynamics, tail-limited extreme value modeling of
rrorrtnrpical storm and hurricane winds by "peaks over threshold" methods,
t'rn;rirical aeroelastic models, progress and limitations in wind tunnel modeling,
tLrrrrping of flexible buildings, across-wind and torsional effects on tall struc-
Iule:;, low-rise buildings, behavior
of roofing, power lines,
and wind load
l:rt'lors. The material on suspended-span structures has been almost completely
rt'wlillcn. A new chapter on standards has been added, which includes a useful
rt'lt'rcncc to a diskette, appended to this book, containing an interactive comlrrrtt'r vcrsion of the ASCE 7-95 Standard provisions for wind loads.*f
Wc thank the many colleagues who used our book in their professional
prirt'ticc or as a primary text for teaching wind engineering and gave us the
lx'rrclit of their comments. Special thanks go to Dr. R. D. Marshall, whose
t'xpelicncc and judgment, particularly in the areas of wind tunnel-modeling
rrrrtl winrl-loading codification, are reflected in several portions of this book,
'll Il. St'lnlan was rcsponsiblc lirr rcwriting Sccrs. 5.3.3, 6.1.1, 6.5, 6.6, and 13.1. E. Simiu
rv;t: tt'slxrttsiblc lirr all othcr rcvisions antl ltltlitions. His contributions to this book are made in
lrr: privitlc c:tllrcity ancl cio not ncccssruily lrl)lcsont thc position of the U.S. Department of
( onrnr'rcc ol ol tlrc Nirlional lnslilllle ol Slirntl:rnls irrrtl 'l'cchnokrgy.
rllrr'tliskt:l1c's witttl loutlittg srtllw:trt', tt'lttrxlrrtcrl
lirrrrr ltcl. l7-5, is in rhc public rkrrnain ancl
r: r()l iul AS('li llrrlllic:rliorr ol tkrr'ttttrr.ll
vll
vial
l'llllA(,1
I() llll
lllllll)ll)lll()l'l
ol 'l'okyo, wll(r llirirrs(irkirrgly
ilu(l {o l)rrrli'ssor .l . Krrttrllr ol lltt' [ )trrvcrsily
tlut a nutnbcr ol'
.t,c.t c,l (lrt, tcx( ol' tlrc sr:t'oiltl crliti0rr rrrrtl krrrrlly lxrintccl
(h:tttk rtttt'txlittlrs, charlcs Schmicg, Ira
typogrirlllriclrl ornrrs. wc irlso wislt lo
tir,*irt y, arrtl Nancy Lin, krr lhcir-hcllllirl cooPt.r'rrtion._.
was
the Russian translation irl'oLrr trixrk, a Chinese translation
reit
while
welcomed
'.lkrwing
b! Tongli University Prcss, Shanghai' We
publishcd
norms'
the latter';disregard of intemational copyright
.
[..tting
Etuttl-
Stvtu
RoeBnr H. ScnNl,qN
Rockville, MaryLand
Baltimore, Maryland
Mav 1996
PREFACE TO THE SECOND EDITION
lrr tlrc alnrost ten years that have elapsed since the writing of Wind Elfects on
.\tt'ttt'torrcs a number of significant advances have occurred in the wind engiIrr.e ring lield. These include the development of the following: improved microrrrt'tcrlrological models, particularly for atmospheric flows over the ocean, which
:rle ol'interest in the design of offshore structures; procedures for the estimation
ol e:xtrcme winds from short records; new information on the modeling of
r'xlrcnlc winds in hurricane- and tornado-prone regions; improved procedures
lor r:stitnating the along-wind response of structures; new procedures for estirrurting the across-wind and torsional response of tall buildings and the acrossrvirxl rcsponse of towers and stacks; simple and probabilistically rigorous meth,rrls lirr taking wind directionality effects into account in design; practical pro, t'rlrrrcs fbr the risk-consistent design of cladding for wind loads, which make
rt lxrssible to achieve more economical designs for any given safety level, or
s:rlr.r clcsigns for any given cost; methods for estimating the response of offshore
:itru('lurcs to wind in the presence of current and waves; and new information
t,n wirxl cffects on various types of structure, including trussed frameworks,
lrylrrbolic cooling towers, and semisubmersible platforms.
'l'hc tcxt has been expanded to reflect these and other advances. It now
rrrt'lrrtlcs fivc new chapters, as well as a new appendix that is intended to provide
tlrt.r'cirtlcr with a brief introduction to modern structural reliability concepts.
'Ilrc original chaptcr on wincl ttttttrc:ls wlts substantially revised, and much new
rrr;rtt'r'iirl was aclclcd to thc ollrrrl t'lt:tP(cr.s, plrrlicularly those on the atmospheric
lrrrlrtlirry laycr, cxtrcrttc wirttl t'liltrrrlokrgy, blull'body aerodynamics, aerot.l:rslic ltlrcrxlntir, tall builtlings, lrrrtl (orrrlrtLr ellecls. Most of the new material
r'orrsists ol'prlrclicirl tlcsigrr irtlotttr:tltotr ;tutl trtt'lltrxls. As in thc first edition, a
tx
I'lll ln(il
l() llll
:;l (:()Nl )ll)l
ll{)l!
consistcnl cllirrt lurs lrr't'rt nt:rrlt'l() l)()lnl ottl rrttrl tlist'ttss (ltr: ttttccrtrtitttics,
limitatiorrs, rrrrtl crnrls irrlrt'r't'rrl irr vrrriorrs rllrllr, rrrt'llrtxls. luttl lcclrrriqttcs.
Thc aulhors woulcl likt'l() ('xl)r'('ss lltt'rr wlrrnr:rpptccirrtiott lo I)r. I{. D.
Marshall, who initiatcd antl tlcvclolletl tlrc wintl cngirrccring prograrn at thc
National Bureau of Standanls; I)r. N. lsyrrnrov ol'tlrc Univcrsity of Westcrn
Ontario, fbr contributions to Scct. 9.-l; rrrrtl l'rolcssor l). A. Recd of the University of Washington, forcontribu(ions to ('hrrp(cr ll. Spccial thanks are also
due, for valuable comments and criticisrrr, t<l l)rol'cssor E. A. Arens ol'the
University of California at Berkeley; Dr. R. l. Basu of H. G. Enginccring,
Inc.; Professor O. Ditlevsen of the Engineering Academy of Denmark; Dr.
B. R. Ellingwood of the National Bureau of Standards; Professor Y. Fu.jino ol
the University of Tokyo; Dr. M. P. Gaus of the George Washington Univcrsity;
Dr. P. S. Jackson of the University of Auckland; Dr. F. Mahmoodi of thc 3M
Company; and Professor B. J. Vickery of the University of Western Ontario.
However, the responsibility for any errors or omissions lies solcly with thc
authors. We also wish to thank our Editor, E. W. Smethurst, Editorial Supervisor, Balwan R. Singh, Designer, Lee Davidson, and Production Supcrvisor,
Linda Shapiro, all of John Wiley & Sons, and Technical Editorof the Russian
translation (1984), Dr. B. E. Maslov.
The references to the authors' affiliations are for purposes of identification
only. The book is not a U.S. Government publication, and the views exprcssed
do not necessarily represent those of the U.S. Govemment or any ol' its agcncies.
Etr,tlt- Stnaru
Ronp.nr
Rtx'kvilla, Mur.ylund
H.
Sr',rur.a.N
PREFACE TO THE FIRST EDITION
l'lrc wind loading of.civil engineering structures involves,
in certain cases,
t'rnsiderable complexities that must be taken into
account in order to achieve
s;rl'c and serviceable.designs. Examples of
wind engineering problems that
.r:quire special attention include: the dynamic .".poni"
of tati structures; the
pcrformance of exterior glass and curtain walls, particularly
in high_rise uuito_
i'gs; the serviceability of pedestrian areas in clrtain types
or t'uilt environ_
rrcnts, the oscillations and flutter ofsuspension bridges;
it.
u.tion oftornadoes
rrr nuclear power plants; the estimation of the piobability of
occurence of
('x(reme winds at a given site.
Motivated by the need to provide rational descriptions
of the phenomena
rrv<llved and to develop. appropriate analytical
and design tools, a vast spet'i.lized literature-not always easily accessible-has
emlrged in the last two
tlcc.ades. wind Effects on structure^s is an attempt
to preseni a synthesis of the
rrrrin trends of this literature in the form of a texi
designed for use by advanced
strdcnts of engineering,and by practicing structural
lngineers and architects.
l'hc tcxt devleops its chosen
independentry anal as often as possible,
_topics
llirrn fundamental principles. In addition, extensive
references are provided to
:r widc range
of primary sources.
'l'he level of preparation
assumctr .r' rhc rcader corresponds
tr that of thc bachelor's degrec irr st'it'rt.c .r cngineering. a approximately
effort
lr:rs bccn rnadc to avoid unncr'cssiuily t'l;rlxrrr(c
"onrirtent
rnathematical
formulations.
Silnplc n<ltions o1'probahirity rlrt.'ry. sr;rrrsr it.s :rrrtl thc
theory of random pro_
t't'sscs ctllpl<lyccl in tlloclcrlt wittrl t'ttlirrrt't'rirr11 :uurlysis
havc been prescnted in
lrpltt:ntliccs, irr which iltluitivc itl)p1ry.1q lrt.:, lr;rvt. lrct.rr slr-6ngly
clnphlrsizcl.
'l'lrc lilsl
ol'llrc lcxl tlist'rrs:,t.:,
1xrt1
t'lirlr:r(ologit';rl :rspt't'ls
ol'llrc wnlrl
trrr.tr.nrolo;r11.:tl, rrrit.lrirrc,tcgr.gl69iclrl, lrnrl
r.trlnorrrrrt.lrl
lllrl luc ol.inlt,t.t.sl irr wilrrl
xt
xll
plrEl ACt
r() llll
t
llisll
FDltloN
engineering. 'l'hc scconcl part proscnls l)itsic (:l(:tttL:ltts ol'acnrclyttittttic:s, structural dynamics, and aoroclasticity, lillkrwr:d by applications to thr: clcsign of
various types of Structures and structural tnctttbcrs. Scparate chaptcrs are devoted to a discussion of wind-induced discotttlir( in and around buildings, and
to assessments of the wind tunnel as a design tool.
Wind engineering is a new and rapidly developing field. Cunent procedures
for estimating wind effects, and the information on which they are based, should
therefore not be regarded as definitive. It is the authors' strong feeling that
areas of unceftainty must be carefully defined, and that the limitations inherent
in current procedures must be stated clearly. This has been done throughout
the text.
The division of responsibility for the work has been as follows: E. Simiu
has written Chapters 1-3, 5, 7,9-ll, and the Appendices, and R. H. Scanlan
*
has written Chapters 4, 6, and 8. The authors have, however, shared editorship
on all parts of the text.
exchange
and extensive critical
extended to the following persons who read
thanks
are
sincere
The authors'
porlions of the manuscript and offered valuable criticisms: Professor H' A.
Fanofsky, Pennsylvania state University; Dr. N. J. Cook, Building Research
Establishment, U.K.; Dr. J. F. Costello, U.S. Nuclear Regulatory Commission; Dr. H. L. Crutcher, National Climatic Center, National Oceanic and
Atmospheric Administration; Dr. J. J. Filliben, statistical Engineering Laboratory, National Bureau of standards; Dr. J. c. R. Hunt, Cambridge university, U.K.; Dr. G. E. Mattingly, Institute forBasic Standards, National Bureau
of Standards; Dr. J. M. Mitchell, Environmental Data Service, National Oceanic
and Atmospheric Administration; Dr. R. N. Wright, center for Building Technology, National Bureau of Standards; and Professor J. T. P. Yao, Purdue
University. All of them should share the recognition for the many improvemcnts their comments brought about. The responsibility for all errors or imperfections rests, however, wholly with the authors. Many thanks are also due
to Devra Simiu and Robert N. Scanlan for careful reading and editing of the
text, and to Mrs. Sue Murray, Mrs. Rebecca Hocker, and Mrs. Nora Scanlan
for their capable typing effort. The authors also wish to express their indebtedness to the late R. S. Woolson, Editor, J. Frances Tindall, Editorial Supervisor, Joel L. Bromberg, Editorial Assistant, and Debbie Oppenheimer and
Sandra
CONTENTS
INTRODUCTION
PART
1
2
3
Evn
I
SrvItu
RoeEnr
H.
ScaNI-aN
Washington, D.C.
Princeton, New Jersey
June,1977
*Chaptcrs 4, (1, ilnrl ll ol llrc lirst ctlilioil trrrrr:spond in lltt'sctotttl irrtrl llrinl oditions to Clhilptcrs
4, 6, antl 13. l,9r'llrc st't.orrtl t.tlition, l{. ll. Scunltn lrirs tevist'rl lltc r'ltirplcr ott witttl lttttncls. antl
Ii. Sirrriu lrls bcrlr n.s;xrrrrihh'lor tlrt'rrllrr'r'r'cvisirttts ittxl ttrltliliorts l() tltc t('xl.
THE ATMOSPHERE
ATMOSPHERIC CIRCULATIONS
5
THE ATMOSPHERIC BOUNDARY LAYER
33
EXTREME WIND CLIMATOLOGY
9'1
PART
Winkler, Production Supervisors, all of John Wiley & Sons.
A
B
WIND LOADS AND THEIR EFFECTS ON
STRUCTURES
Fundamentals
4
5
6
7
B
BLUFF-BODY AERODYNAMICS
135
STRUCTURAL DYNAMICS
195
AEROELASTIC PHENOMENA
216
WIND TUNNELS
WIND DIRECTIONALITY EFFFC IS
273
308
xlll
xiv
ll
(,(
'N
il
N
tl
Applications to Design
9
BUILDINGS: WIND LOADS, STRUCILJHAL RESPONSE'
AND DESIGN OF CLADDING ANt] TIOOFING
1O SLENDER TOWERS AND STACKS WII'H
CIRCULAR
CROSS SECTION
11 HYPERBOLIC COOLING TOWERS
12 TRUSSED FRAMEWORKS AND PLATE GIRDERS
13 SUSPENDED-SPAN BRIDGES, TENSION STRUCTURES,
14
15
327
383
404
420
AND POWER LINES
446
OFFSHORE STRUCTURES
487
INTRODUCTION
WIND.INDUCED DISCOMFORT IN AND AROUND
511
BUILDINGS
16 TORNADO EFFECTS
17 STANDARD PROVISIONS FOR WIND LOADING
APPENDIX
APPENDIX
APPENDIX
576
l'lrc rlcvclopment of modem materials
and construction techniques has resulted
of a new generation of structures that
are often, to a degree
rrrrkrr'wn in the past, remarkably
flexible, low in damping, and light in
weight.
srrt'' structures, as welr as uu.iou, nou.r
typ", of rigid
A1 ELEMENTS OF PROBABILITY THEORY AND
APPLICATIONS
APPENDIX
551
42 RANDOM PROCESSES
43 ELEMENTS OF STRUCTURAL RELIABILITY
A4 PRESSURE COEFFICIENTS FOR BUILDINGS
AND STRUCTURES
rrr rhc cmergence
591
629
structures, exhibit an
rrrt're'scd susceptibility ro rhe action
of wina. acirdt;;,r,1;has become
rr('(('ssilry to develop tools enabling
the designer to
effects with
;r lrig'cr degree of refinement than-was
".ti*ai"'*ind
previ,ously ,"quiJ.
wini
rs rlr. discipline that has evolved,
p.imarityau.ing ilr"iurt-e."d;
"ngrn"".rng
rsu! uvvsuwD' fiom
efforts
;rrrrctl a[ developing such tools.
It is the task of the engineer to ensure that the
performance of structures
:'rr'jt:crcd to the action of wind will
be adequate during their anticipated rife
lr.rrr rhc standpoint oftroth^structural
safety and
t'r(,, rhc designer needs information regarding serviceibilitv. io'u"tieve this
(r) tt" winJ'environment,
tlrt' re l.tirn between that environment
12;
arid the iorces it induces on,h" r,*",u.",
;rrrrl ('1) thc behavior of the structure
under the action of these forces.
643
665
INDEX
676
ABOUT THE DISK
684
rHE WIND ENVIRONMENT
lrrlr.rrr(i<ln <ln the wincr cnvinrnrrcnt needed
in design includes elements de_
r,u'cd lhrrn
.mctconrlogy, nticr()nlcte()tl)l()!,y, and climatology.
Merc.nrl'gy pnlvirrc.s lr trcsr.r'iPr i.r,,,,,,i'"^prunation
of the basic f.eatures of
'rlrttosltltcrr-ic lklws. Such ll.rrlrrt.s rrrirv lrt.ol.c.gnsirlcrabl"
;j;;ifi.;nce fiom a
.'lttttlttt':tl tlcsigrr vic:wP.irrl.
li,r't'rrrrrrIlt'. irr llrc t.usc.l,rr",nrnu,ir, the
prcs_
t'ttt t' ol'ir rcgiorr ol' low irlrnosplrt,r,,
1,,,.r,n,,,.,. ;tl lllc cctllcl- <ll. lhcl slrlrrrr is lr
l:rt'tr''l'rrrrj.r irrrp.rrirrrr'e
r
irr tir,, ,r,.,,,j.,,
,r
rrrrr.rt.lrr.rx)w(rr prirrrrs.
lN
il
i( )t
)l,o
l l( )N
MiCrornCtcgnrftlgy lrltr.lrrlrls (p rlt'sr'rilrc llrr' rlt'lltrlt'rl sltlt('ltllc ol ltltttospllt't ic
flows near the gnruncl. 'l'oltics ol rlirt't'l ('()n('('ltt {o lltc sttrtcltrrltl tlcsigrrt't'ilt
clude the variation of nrcan spcorls willr heip,lrt irltovt: gtttttlttl, tlrc tle:sct-illtitlll
of atmospheric turbulence, and thc tlcl'rctttltttcc ol' (ltc tttcitrt spectls ancl tll'
PART A
turbulence upon roughness of terrain.
Climatology, as applied to the wind cnvirotrtttcttt, is ctlnccrncd with thc
prediction of wind conditions at given geographical locations. Probability statements on future wind speeds may be conveniently summarizcd in wind maps,
such as are currently included in various building codes.
WIND-INDUCED FORCES ON STRUCTURES
A structure immersed in a given flow field is subjected to aerodynamic forces
that, in general, may be estimated using available results of aerodynamic theory
and experiments. However, if the environmental conditions or the properties
of the structure are unusual, it may be necessary to conduct special wind tunnel
tests.
Aerodynamic forces include drag (along-wind) forces, which act in the direction of the mean flow, and lift (across-wind) forces, which act perpendicularly to that direction. If the distance between the elastic center of the structure
and the aerodynamic center (i.e., the point of application of the aerodynamic
force) is large, the structure is also subjected to torsional moments that may
significantly affect the structural design.
STRUCTURAL RESPONSE TO WIND LOADS
Because the aerodynamic forces are dependent on time, the methods of structural dynamics may have to be employed to determine the response. Furthermore, the random character of this dependence requires that elements of the
theory of random vibrations be applied to the analysis. In certain cases. it may
be necessary to perform an aeroelastic analysis, that is, a study
ofthe interaction
between the aerodynamic and the inertial, damping, and elastic forces, with
the purpose of investigating the aerodynamic stability of the structure.
From the foregoing it is seen that the design of modern structures subiected
to wind loads requires the use of information and methods derived from a broad
spectrum of disciplines. It will not be suggested here that complete answers to
the questions involved exist at the present time. However, considerable progreSS has been made toward an understanding of some of these questions. As a
result, procedures and techniqucs have been devclopccl lhat have significantly
improved the designcr's irhility to ostinratc thc cll'ccts ol wirttl l-rorn thc standpoint of both strcrrgtlr ltrrtl scrvicr:lrbility. lt is lht: itittt ol tlris lcxt to prescnt
thesc proccdurcs llul tt't'lrrrit;rrt's, to llrovirlc lltr-: lrirt'kgnrttrrrl trtlttcrial rccluircd
firr rrntlcrsllrrrtlirrg, llrcir r':rliorr:rlc, rrrul lo cxrttttittt' t'rilit':rlly llrt'ir clrplrhilitics irs
wcll rrs tlrt'ir lirtritlrli()ttri it,'i th'si1'.rr (txrls.
THE ATMOSPHERE
CHAPTER
1
ATMOSPHERIC CIRCULATIONS
Wirrrl, or the motion of air with respect to the surface of the earth, is fundarrrt'ntally caused by variable solar heating of the earth's atmosphere. It is initrirlt:tl, in a more immediate sense, by differences of pressure between points
ol t't;ual elevation. Such differences may be brought about by thermodynamic
;rrrrl rncchanical phenomena that occur in the atmosphere nonuniformly both in
tilil(' irnd space.
'l'hc energy required for the occurrence of these phenomena is provided by
tlrt' sun in the form of radiated heat. While the sun is the original source, the
.,()rr'('c of energy most directly influential upon the atmosphere is the surface
,l thc carth. Indeed, the atmosphere is to a large extent transparent to the solar
rrtli:rtion incident over the earth, much in the same way as the glass roof of a
I'rccnltouse. That portion of the solar radiation that is not reflected or scattered
lr:rt'k into space may therefore be assumed to be absorbed almost entirely by
tlrc crrfth. The earth, upon being heated, will emit energy in the form of terrcslrirrl radiation, the characteristic wave lengths of which are long (of the order
ol l0p) compared to those of heat radiated by the sun. The atmosphere, which
r' lrrlgcly transparent to solar but not to terrestrial radiation, absorbs the heat
r:rtlilrtccl by the earth and re-emits some of it toward the ground.
I.1
ATMOSPHERIC THERMODYNAMICS
1.1.1 Temperature of the Atmosphere
'lir illtrstlutc thc nrlc ol'thc lctttgx'r'lrltul'tlistribution in the atmosphere in the
pttxlttt'liott ol'wincls, a sirtrplilit'rl tttrxlt'l tll'atrnosphcric circulation will be
n
lMolit't il illo ollrot,l All()Nti
I I AIM( )r;t,ilt ilt(; il il ttM()llyNAMt(il;
prescntcd. In this rnoclcl thc cll'ccls tll'tltc vctlicitl vrtriittion tll'itir tcttlpcraturc,
of the humiclity ol'thc air, ol'lho nrlirliorr ol lltc eirrllt, artd ol'l'riction will be
ignored, ancl the surfhcc of thc carlh will bc rrssrttttctl to ho unilorm and smooth.
It will be recalled that thc axis ol'nrllliou ol tlrc carlh is inclined at approximately 66'30' to the plane of its orbit an)rrrKl tltc sun (planc of the ecliptic).
Therefore, the average annual intcnsity ol'stllitr ladiation and, consequently,
the intensity of terrestrial radiation and thc lcnlpcralurc of the atmosphere will
be higher in the equatorial than in the polar rcgions. To explain the circulation
pattem that arises as a result of this tempcraturc difl'crence, Humphreys Il-l]
proposed the following ideal experiment (Fig. l.l.l).
Assume that the tanks A and B are filled with fluid of uniform temperature
up to level a and that tubes 1 and 2 are closed. If the temperature of the fluid
in A is raised while the temperature in B is maintained constant, the fluid in
A will expand and reach the level b. The expansion entails no change in the
total weight of the fluid contained in A. The pressure at c remains therefore
unchanged, and if tube 2were opened, there would be no flow between A and
B. If tube I is opened, however, fluid will flow from A to B, on account of
the difference of head (b - a).Consequently, at level c the pressure in A will
decrease while the pressure in B will increase. Upon opening tube 2, fluid will
now flow through it from B to A. The circulation thus developed will continue
as long as the temperature difference between A and B is maintained.
If tanks A and B are replaced conceptually by the column of air abovc the
equator and above the pole, it can be seen that, in the absence of other efl'ects,
W0r rn
FIGURE 1.1.2. Simplified model of atmospheric circulation.
rrn atmospheric
circulation would be developed that could be represented as in
iig. 1.1.2. In reality, the circulation of the atmosphere is vastly complicated
by the factors neglected in the above model. The effect of these factors will
f
bc discussed later in this chapter.
The temperature of the atmosphere is determined by the following processes
lt-2, I-3, l-4, l-5,
l-61:
o Solar and terrestrial radiation, as discussed previously in this
o Radiation in the atmosphere.
o Compression or expansion of the air.
r
chapter.
Molecular and eddy conduction.
o Evaporation
and condensation of water vapor.
1.1.2 Radiation in the Atmosphere
As a conceptual aid, consider the action of the following model. The heat
rrrtliated by the surface of the earth is absorbed by the layerof airimmediately
:rlrovc the ground (or the surface of the ocean) and reradiated by this layer in
two parts, one going downward and one going upward. The latter is absorbed
Iry the next higher layer of air and again reradiated downward and upward.
'l'lrc transport of heat through radiation in the atmosphere, according to this
t'onccptual model, is represented in Fig. 1.1.3.
1.1.3 Compression and
Alrrrosphcric pressurc is pnrtlucctl by tlrr-: wcight of the overlying air. A small
(or particlc) of clry irir rrurvirrg vr.rtit'lrlly thus experiences a change of
ptl'srittro to which thcro ctlrtcsltottrls lt cltirnp,e ol'lcrnpcrature. To determine the
l;rltcr. lho oc;uation ol'stirle lirr'1x'r'lt'r'l llits('s iul(l thc lirst law <lf thcrrnodynamics
rrr:rss
l.l.l, ('in'ulittiotr l)irllcnr tluc to tcmpcrattrtc tlillercrtcc bctween two coltrl'llrritl. lirrrrrr /'/ry,rlr',r tl tlrt Ait hy W. .1. lltrtrrplttcys. ('opyright 1929, l94O
hy W..l . llrrntlrlrreys. llst,rl wrllr lrcrrrrissiort ol Mc(irrrw llill lixrk ('otrtpany.
FIGURIT
r.rrrrns
Expansion
lrlc ttsctl:
n lM():;l
,lll lrl{i (:llr(:t,t Ail{}t..ti:
II
Ilcirl rir.lrillrll
Irt(,
!lr.r r.
lriltlr'r
--f
--
l,'l(;tlRl,l 1.1.3. Transport of
heat through radiation in the atmosphere.
pu-RT
dq:c,flT*pdu
In these expressions p is the pressure,
(l.
r. r)
(1.1.2)
the specific volume, R the spccific
er
air, Z the absolute temperature, dq the amount of heat transferred to the particle, and c, the specific heat at constant volume.
Differentiating the first relation and substituting the quantity p der thus obconstant for dry
tained in the second relation, there results
dq:(c,,+R)dT*udp
(r.r.3)
Comparing this relation with
dq
:
(l .t.41
crdT
which cxpresses the first law of thermodynamics in the particular case of
an
n l M(
'r,t
,ilt ilt{ . I ilt
nM(
)t
)yNn Mt(.1
kttowtt lrs Poissorr'r, or llrt'rlry rrtli:rbalic c(pr:rrr()n. lior tlly ;lt, ll/t.,, o.llitJ.
A lirrrriliar e,x:ttttplr'ol tlrt'e'llecl ol'prcssrrrc r'llrnp,c orr llrt' lcrrrPcrirltrrt: is llrt'
lrt':rtirrg ol'cornprt'ssr'tl :ur irr lr tirc pultrp.
ll', in thc itltttospltr:t't'. tlrc verlical rnotiorr ol'irrr rrir'pirrccl is sullicicntly
rrrpid, thc hcat cxcltiurgc ol'that parcel with its cnvir'onnrcnt tnay bc consiclcrcd
t. bc negligiblc and thc assumption dq : 0 is appnrximatcly correct. lt then
Iolkrws from Poisson's equation that since ascending air experiences a pressure
rltrcrcase, its temperature will also decrease. The temperature drop of adiabatit'rrlly ascending dry air is known as the dry adiabatic lapse rate and is approximately 1"C/100 meters in the earth's atmosphere.
consider a small mass of dry air at position 1 (Fig. 1.1.4). Its elevation and
l('nrperature are h1 and z', respectively. If the particle moves vertically upward
irt some reasonable speed, its temperature change will effectively be adiabatic,
regardless of the lapse rate (temperature variation with height above ground)
Plcvailing in the atmosphere. At position 2, while the temperature of the amlricnt air is 22, the temperature of the element of air mass is T) : T,
- (hz /r,)'y,,, where 7o is the adiabatic lapse rate. Since the pressure of the element
rrrrtl of the ambient air will be the same, it follows from the equation of state
tlrirt to the temperature difference T5 - T, there corresponds a difference of
tlt:rrsity between the element of air and the ambient air. This generates a buoyrrrrcy force that, if rz I Ti, acts upwards and thus moves the element farther
;rway from its initial position (superadiabatic lapse rate, as in Fig. 1.1.4), or,
tl 'l'2 ) Tl, acts downwards, thus tending to retum the particle to its initial
lxrsition. The stratification of the atmosphere is said to be unstable in the first
r'rrsc and stable in the second. If T2: Ti,that is, if the lapse rate prevailing
rrr llre atmosphere is adiabatic, the stratification is said to be neutral.
A simple example of the stable stratification of fluids is provided by a layer
.l water underlying a layer of oil, while the opposite (unstable) case would
lurvc the water above the oil.
isobaric (constant pressure) process (co is the specific heat at constant pressure),
it is easy to see that c,, * R : cr. It is therefore possible to write, if the
equation of state is used once more,
dq
Processes for which
dq
:
:
cpdT
-
dn
RT -:_
p
(l.l.s)
2(h2,
r;)
I:
Lapse rate prevailing
in the atmosphere
O
are referred to as adiabatic. For such processes,
II
the previous relation becomes
qI_4oo _o
T
cpP
(1. 1.6)
which, alicr irr(cgltrtion, yicltls tlrc cquation
,1,
't',,
(';,)""
(l l7)
lrl(ltlRl'l I l..l= l,rl,i.(
r,rt(.s
Adiabatic lapse rate
1
0
AI
MO:lt,l tf
ntc otttct,l
A t t( )Nr;
\ lM( )t;t,nt nt(
1.1.4 Molecular and Eddy Conductlon
1.1.5 Condensation and Evaporation of Water Vapor
The pressure of moist air is, according to Dalton's law, equal to the sum p of
the partial pressure e of the water vapor and that of the dry air, p - e. It has
bccn established experimentally that if the pressure e exceeds some value E,
known as the saturation vapor pressure, condensation of the excess moisture
will occur, and that the saturation pressure E increases exponentially as the
tcmpcrature of the moist air increases.
An elementary mass of ascending unsaturated moist air (i.e., for which
< l) will experience
a temperature drop that can be shown to be essentially
equal to the dry adiabatic lapse rate. As the element ascends and its temperature
decreases, its saturation pressure will also decrease. If the element reaches a
level at which the ratio e/E becomes unity, condensation will normally occur.
Above this level, water vapor contained in the air element will continue to
condense. In the process, heat of condensation is released. This is equal to the
heat that was originally required to change the phase of water from liquid to
vapor, that is, the latent heat of vaporization stored in the vapor.
The heat of condensation contributes to the mechanical work involved in
the expansion of an ascending particle, which before saturation was performed
only at the expense of the internal energy. The temperature drop of the saturated
adiabatically ascending element of air is therefore slower than for dry or moist
unsaturated air (Fig. 1.1.5). By furnishing energy that increases the temperature
\ to,r,
Mt(;ti
typcs
ol winds.
1.2
ATMOSPHERIC HYDRODYNAMICS
'l'hc motion of an elementary mass
of air is determined by Newton's second
l:rw
DF
:
ma
(1.2.1)
where m is the mass, a is the acceleration, and D F is the sum of forces acting
the elementary mass of air. It is the purpose of this section to briefly describe
'rr
tho forces F and some of their effects upon the motion
of air.
1-2.1 The Horizontal Pressure Gradient Force
('onsider an infinitesimal volume of air dx dy dz (Fie.
r.2.r), and let the mean
l)rcssures acting on the lower and upper face be p and p + (0pl0z) d1, respec_
tivcly. In the absence of forces other than pressures, the net vertical force
rrcting on the volume dx dy dz will be
-(0pl0z) dx dy da, or -)pllzperunit
volume. similarly, the net forces per unit volume acting in the and y direction
"r
will be denoted -\pl0x and -0pl0y, respectively. The resultant of these forces
is called the horizontal pressure gradient and is denoted
-0pl0n, where n is
thc normal to some contour of constant horizontal pressure. The horizontal
l)rcssure gradient is the driving force that initiates the horizontal motion of air.
lapse rate
llll,) l. l.-5. l illtt'(s ol' corttlcrtsirlion ulxrr
II
parliclc: witlr n.spcr'l lo wlrat it woultl hirvc lrccrr urrtkrr tlrv arliabirtit.
cottclititlns, tltc hcitl ttl cttttdcttsation hclps sul)lxrt1 corrvct.liorr ol thc 1ir t9
lrighcr levcls ol'thc ltlntospltcrc. 'l'his lactor is irrrpollirrr( irr tlrc gcncsis ol'cqlain
{or saturation)
\diabatic
l,'l( ;t
)t)yNn
ol'a
Molecular conduction is a dill'usion pft)ccss thlt cll'ccls a translbr ol'hcat. It is
achieved through the motion of individual rnoloculcs and is negligiblc insofar
as atmospheric processes are concerncd. Hddy hoat conduction involves the
transfer of heat by actual movement of air in which hcat is stored.
e/E
, I tytrn(
Lrlrsc nrlc
l.'l(;tlltll 1.2.1. Vcrtic:rl Irr':,:.rrr.s
()r irr e lcrrL)nrrrry
nrirss
.l
uir.
12
n
lMolil'l ll lil(; cllt(;l/ln ll()N:i
I 1' n I M{ t!;t't il !tl(
I
tyl)t
l( )l )vNn Mt(
tli
l3
Htglr llrcsltrtrr
An
I
o,r".,,.,, ol
pressurc gradrent
I-IGURE 1.2.3. Apparcnt rnotion ol'an air particle due to
the earth's rotation.
Low presure
FIGURE 1.2.2. Direction of prcssurc gradient fbrce.
t';rstcrly wind). In the Southern Hemisphere the reverse of these statements is
llllc.
'l'hc nct lirrce per unit mass exerted by the horizontal pressure gradient, (l/p)
t)1tl\n, is oltcn referred to as the pressure gradient force (p is the air density.)
Air subjccted solely to the action of pressure gradient forces will move from
rcgions of high pressure to regions of low pressure. The direction of the pressure gradient force is indicated in Fig. 1.2.2, in which the isobars (lines contained in the same horizontal plane and connecting points of equal pressure)
are also shown.
ll' the Coriolis parameter is defined
f:2llsind
F,.:2m(v x a)
(r.2.3)
wlrc:rc d is the latitude of the point considered, it follows that the Coriolis force
;rcting per unit of mass in a plane (P) parallel to the surface of the earth (Fig.
I 2.4) on an element of air moving in such a plane with velocity v relative to
tlre carth will have the magnitude
1.2.2 The Deviating Force Due to the Earth's Rotation
F,
If defined with respect to an absolute frame of reference, the motion of a particle
not subjected to the action of an external force will follow a straight line. To
an observer on the rotating earth, however, the path described by the particle
will appear curved. The deviation of the particle motion from a straight line
fixed with respect to the rotating earth may be attributed to an apparent force,
the Coriolis force, the vector expression of which is [1-7]
as
f
lrc values of
f
are given
:
(1.2.4)
mfu
in Table 1.2. 1 as functions of latitude.
(t.2.2)
where m is the mass of the particle, or is the angular velocity vector of the
earth, and v is the velocity of the particle relative to a coordinate system rotating
with the earth. Fc is perpendicular to crr and to v, is directed according to the
vector multiplication (right-hand) rule, and has the magnitude 2nlc,rl lvl sin a,
where a is the angle between o and v.
Let N (Fig. I .2 .3) be the norlh pole, and consider an element of air moving
in a straight line in space along the direction NP. If the motion starts from N
at time t : O, at time / the particle arrives aI P, and the position of the meridian
along which the motion started is NP'. To an observer on the earth, it appears
that the element is deflected westward by an amount P'P.
It can thus be seen that, in thc Northern Hemispherc, owing to the rotation
of the earth, a wind initially dircctcd along a mcridian vcors to the right of its
initial dircction; (lurl is, il tlirr:ctctl northwarcl il vc:crs lowurtl thc oast (bccomcs
a wcstcrly wintl). ll tlilct'tctl sottlltwatd il vccrs lowrrnl llrc wcst (bccorncs an
fll(;lllll,l 1.2.,1. l 'r,ntlr,lr.trl:, ol tlrt'lrltrlion vccl()r
(').
14
n
lM()l;l'l ll lllc ()ll l(;t,l
n
ll()Nl;
I
;'
n I M(
):
il ,t
lt ||t( il\ I rlt( )l ryNAMt(:l
TABLE 1.2.I. Corirllis l)ararnclcr.
f:2asinb
-')
Latitude
0
l-ittitudo
(rlcg)
(s
(deg)
0
0.1211 x l0
-50
a
.f -- 2a sin $
(r
10
0.2533
()( )
r5
0.3'175
()-5
t.1947
r.2630
r.3218
20
0.49ttu
() (rI(r4
10
I .3705
5
l5
)
t( )
(
l'r
O
,lo
.ll
5.5
/:l().1
lJ
l(r\
15
1.4087
tt( )
t.4363
1{-5
o (n/1
9o
')
1.1172 x
10-4
1.4629
1.4584
I Oll\
High
1.2.3 The Frictionless Wind
Balance
At sullicicrrtly grcat hcights, the
eft'ects on the wind due
to friction along the
ground bccome negligible, and the horizontal motion of air relative to the
surface of the earth is determined, in unaccelerated flow, by the balance among
the pressure gradient, the Coriolis, and the centrifugal force.
The effect of the forces acting on an elementary mass of air is shown in
Fig. 1.2.5 (the mass is assumed to be in the Northern Hemisphere). If the
particle started to move in the direction of the pressure gradient force (denoted
P), it would be deflected by the Coriolis force F.o @ig. I .2.5a). The particle
would then move in the direction of the resultant of P and F..o, shown as
direction II in Fig. 1.2.5b. The deflecting force would now become F,.6, to
which there would correspond a new direction of motion (direction III in Fig.
1.2.5b). When a steady state is reached, the wind flows along the isobars as
shown in Fig. 1.2.5c.
The isobars in Fig. 1.2.5 are depicted as straight, which means that in the
is no centrifugal force. However, in the more general case
of a curved isobar centrifugal forces will be involved. This case is taken up
case shown there
below.
The steady velocity for which a balance between the pressure gradient force
and the Coriolis force alone obtains is called the geostrophic wind velociry G
and is related to the pressure gradient by the equation
Low pressure
(b)
rvlrcrc P is the magnitude of the vector
,,
.,,
'Q!!'
pl'
(t.2.6)
P,/is
the Coriolis parameter, and p is
tlrt'lrir density.
I l' the isobars are curved (Fig. 1 .2 .6) , the force P as well as the centrifugal
I'r.cc C will act on the elementary mass of air in the direction normal to the
r:,.birrs, and the resulting steady wind will again flow along the isobars. Its
High pressure
ll'rr\:tion Of
wrrrrl (in the
I
(1.2.s)
(c)
FIGURE 1.2.5. Frictionless wind balance in geostrophic flow.
N,rr
2u:Csin6:P:-'do/dn
p
pressure
wind (in the
tltorn
Northern
l"rrst)here)
Itl(llllll,l
High pressure
Direction of
Heqrisphere)
Low preisurc
(cycl<xric circulirlr0il)
Low pressure
{anticyclonic circulation)
(n)
(h)
1.2.(r. lrlictiorrlt.ss rvrrrrl
lr;rl;rrri
r
ttr ( y( l()ni(' lurtl lrrrlit'yr'lorrit' llow
16
n
tM()t;l'l tt Ittc otti(;lll n ll()Nl;
A I Mil'.t'l
il ltt(
I
li
I rl
to| ryNn
Ml( ::
l7
velocity results fiom the relations
yl : ,
"r
vn,f -r
_
dp/dn
Irl',rlltoil!l!'t('
(1.2.7)
(;r,r(il,nt
p
the mass of air is in the Northern Hemisphere, the positive or the
is used according as the circulation is cyclonic (around a center
sign
negative
6fl<tw prcssurc) or anticyclonic (around a center of high pressure), and where
r is tlrc nrrlirrs ol'crtrvalurc of thc airtra.icctory.* The velocity I/r,is calledthe
.tittttlit,ttt rt,itttl tt,lttt'ilv; it is ct;ttal ltl thc geostrophic wind velocity in the
p;rrtrt rrlrrr t:rst' irr wlrit'lr lltc ('tlrviltlllc tll' (hc isobars iS zerO. If the radiuS of
where,
wrnrl luvtl
if
-layer depth)
r'llr\';ll1!(' rs ltrvlt'. trr lllt' Nol'tltt'r-rt I lt:rrtisphcrc
l';,,
lirr cyckrrric wirrtls,
'{ 'l',':: - ({)')''
(1.2.8)
FIGURE 1.2.7. The atmospheric boundary layer.
atrd
vr,: t{,
-le)'
-;#)"
o
2s)
for anticyclonic winds. The sign of the radicals is given by the condition that
Vc,: Owhendp/dn: 0. It follows from the expressions fot Vr, that for the
same values of r, f and dp/dn, anticyclonic winds are weaker than cyclonic
winds [1-1, p. 121].
The foregoing discussion explains Buys-Ballot's /aw, which states: If, in the
Northem Hemisphere, a person stands with his back to the wind, the high
pressure will be on his right and the low pressure will be on his left. In the
Southern Hemisphere the reverse is true.
1.2.4 Effects of Friction
The surface of the earth exerts upon the moving air a horizontal drag force,
the effect of which is to retard the flow. The effect of this force upon the flow
decreases as the height above ground increases and, as indicated previously,
It is the wind regime within the boundary layer of the
atmosphere that is of
'lrr.ct interest to the designer of civil enginlering structures. The questions of
tlrt' boundary-layer height, of the variati,cn of wind
speed and direction with
lr.rrht above ground, and of the turbulence structure
within the boundary layer
.ut' therefbre discussed in more detail in Chapter
2.
It will be noted here that unrike the gradient wind velocity,
the steady_state
r'irrrl velocity within the boundary layer crosses
the isobars. consider a geo_
''t.rphic flow (i.e., a flow in which the isobars
may be assumed to be straight)
'rrrtl the balance of the forces acting on particles A and, B, which
move hori_
z'rrrirlly within irs boundary rayer (Fig. t.z.s;. If ,4 (Fig.
r.2.ga) is at a higher
l('vcl than B (Fig. 1.2.8b), its speed u and (by
virtue oithe relation F, : mfu)
rts ('.riolis force will be largerthan
those orb. tre deviation;;gr;
between
tlrt'wind direction and the isobars will therefore be smaller
" higher
fir
ttre
tl;rs(cr) particle. The angle o will be zero at the gradient
level and will reach
rts.rrraximum value os nearthe ground. In
the Northern Hemisphere the wind
r.ltrcity in the boundary rayer may thus be represented
by a spiral, as in Fig.
l.).9.
becomes negligible above a height 6 known as the height of the boundary layer
High pressure
of the atmosphere. Above this height the frictionless wind balance is established, and the wind flows with the gradient wind velocity along the isobars.
The atmosphere abovc thc boundary layer is called the free atmosphere (Fig.
1.2."t).
F" (Coriolis force)
High
pressure
Low
l)ressure
S (fricliorr lorr:r')
A
l)[oction
P {Jlressrrrc r;r;rlrr.rrt lorr
r.)
It()lt()rr
*Strictly sllctrkirrl'., llrc r:rtlirrs ol ( lr v:r1lrt' ol lllc triricct()t] rtr:ry tlillt'r lirrltt lltc radius ol'curvaturc
ll llrc isglr:rr'.'llrc tlilltrclr,r.rrlrV lrc lreliletlctl, ltowrrvt'1. il tt r':rtt lx':tssttlttctl th:rl lhc wintl llrlw
is :rllptrrx irttitlt'ly slt':trly
Low
l)t{:ssrlr(:
(r)
lf
l( ll lltl,) l.2.ll.
llrl:rrrt'c ol lor, {.., rr
(h)
tlrr ,tt IItr':,1)lt('t
i(. lroulrtlltt.y l:tyt.r
18
ATMOSPHERTC ctRCUl
r;r
AroNsi
AIM()i;l't
r nt(: M()lt()Nli
1g
ol itttttosphcric t'tttttltltrttts itl s()tnc shorl tittre irltcl llrc t.ollecli6rr el'llrr-r rlirlir.
'l'hc calculatctl vitlttcs ol lhc six variablcs
obllirrctl by irrtcgnrtiorr can (hcrr lrr.:
ttsctl as initial corttlitiotts lilr a l'urthcr inlcgnrtiorr;;11rp. 'l'his succcssivc
apPfilxitnation pK)(:L!ss is lhc lrasis of numcrical wclllrcl procliction tcchniquos
lllirl came into bcing lirllowing the increasecl availability ol- observationsrncluding, more reccntly, observations obtained by satellites (Fig. 1.3.1)-and
tlrc: clevelopment of modern electronic computers.
Atrnospheric motions may be described as superpositions of interdependent
lkrws characterized by scales ranging from approximately one millimeter to
llr.usands of kilometers. To analyze such motions, it is convenient to classifz
llrr:rn according to their horizontal scale. In meteorology three main groups of
rrtrnospheric scales are commonly defined: microscale, mesoscale, and synoptic
st'irlc. According to the classification of Il-g], the synoptic scale includes mo_
tions with characteristic dimensions exceeding 500 kmor so and time scales
ol two days or more. The microscale includes motions with characteristic dirrrc:nsions of less than 20 km or so and time scales of less than one hour.
The
nrcsoscale is defined by dimensions and periods between those characteristic
ol' rnicroscale and synoptic scale.
FIGURE 1.2.9. Wind velocity spiral in the atmospheric
boundary layer.
In the case of a cyclonic storrn (or flow around a center of low pressure),
will cross the isobars toward the center. The air will
near the ground, the wind
thus slowly converge and ascend. If the low-level convergence exceeds highlevel divergence, the mass and weight of the air column at the center of the
storm gradually increase and therefore the inward-directed pressure gradient
force decreases. As a result of such a decrease, the center of low pressure is
dissipated and filling is said to occur.
In the case of an anticyclone, the wind near the ground will cross the isobars
away from the center of high pressure. In the lower portions of a high, if lowlevel divergence exceeds high-level convergence, the atmosphere will tend to
spread out and sink, and dissipation of the center will occur.
1.3
ATMOSPHERIC MOTIONS
Most atmospheric pmcesses can be described in terms of the quantities briefly
discussed in the prcccding sections: wind velocity (i.e., horizontal and vertical
wind), pressurc, tcmpcrature , density, and moisture. The behavior of these six
quantities is govcrncrl by six cquations: the equation of state, the first law of
thermodynanrics. tlrc ctluirtirtns <ll'continuity of mass and moisture, and the
horizontal antl vcrlicll ccprations ol'tnotion. Proviclccl that an adequate data
basc cxists, llrr:sc r:r;rrirlit)ns clur bc intcgratcd to
yicltl it t;tutrttitative dcscription
lrlOtIRli 1.3.1.
(
Srrtcllitc vicw
ol
lrrrrrrt'irrrt' lrili
)ccanic ;rrrtl Alrrrosplrcric Ailtrrtrrisl rrrl rnrr )
Se pl
.
113.
1974 (coLrnosy National
20
n
lMosl'l ll lll(; (;lilc(ll All()N:;
I:r AIM{}!,t ,tililt(;M()Il(iNl;
2l
;rtttl ltlws itt sttttttttct. ()wirrg lrl tltc vlrsl llrtttl rturss ol llrt.Asi;rrr r'onlirrt.rrl,
lll()llso()ll cllccts irl.r'(l('v('lolx'rl ttxrsl strorrgly irr Asitr. wlrcrc llrcy lurvt: tr t'olt
Nortlr Polc
Polar easterlies
Polar front
sitlcnrhlr: inl]rrcltct' orr lltc sr:lrsonal charrgcs
,z .z .z
t/
tt
t/
t/
l,'l(,llil{l,l 1..1.,1. lltt'llrrlrrr rrrt'ritliorrirl r'ircttluliott rnodel. After General Meterology
lry ll l{. llyt'r:. ('opyrill,lrl l()17, l(4,1 hy llrc Mc(iraw-Hill BookCompany, Inc. Used
wrllr 1rt'nnissiort ol Mt ( ilrrw I lill lltxrk ('olrtpittty.
ol welrllrt.r
l)itllcnrsi.
Hurricanes. -lr<lpiclrl cycklncs are storms that dcrivc all thcir cnorgy
I'r<lrn thc
llrtt:nl heat relcascd by tho condcnsation of watcr vapor and <lriginatc, gcncrally,
lrt'lwccn the 5 and 20 latitude circles. Their diamctcrs arc usually of the order
,rl' scvcral hundred kilometers. The depth of the atmosphere involved is of the
orrlcr of ten kilometers. Hurricanes are defined as tropical cyclones with surface
wirrd vefocities exceeding about l2o kmlhr. Spacecraft views of hurricanes are
slrown ln Figs. 1.3.1 and 1.3.3.
Hurricanes (known as typhoons in the Far East and cyclones in the region
ol Australia and the Indian Ocean) occur most frequently during the late sum-
1.3.1 The General Circulation
The combined effects of the earth's rotation and of friction break the thermal
circulation cell of Fig. l.l .2 into a pattern that consists basically of three
circulation cells as represented in Fig. 1.3.2 [l-2]. The theoretical pattem is
compatible with the existence (at sea level) of a high pressure belt at the horse
latitudes and of a low pressure belt at the polar front.
In reality, the tricellular meridional circulation model is complicated by
seasonal and by geographical effects. Seasonal effects consist of variation in
position and intensity of the pressure belts and are caused by the annual march
of the sun north and south of the equator. Geographical effects are caused by
the difference in physical properties and by the uneven distribution of water
and land over the globe.
In summer, because the ocean surface warns up more slowly than the land,
the air is colder over the ocean than over land. Just as in Fig. 1.1. I fluid flows
in tube 2 from the colder to the warmer tank, air near the surface will be driven
in summer by a pressure gradient force directed from the ocean toward the
land. On the other hand, in winter the air is colder over land and the oceans
become heat sources.
1.3.2 Thermally Direct Secondary Circulations: Monsoons and
Hurricanes
Secondary circulations are said to be of the thermally clircct typc if the centers
of high or low pressure (i.e., the highs or the lows) aroutttl which thcy develop
are formed by heating or cooling of the lower alttros;rhe rc.
Monsoor.rs itrc scirsorral winrls tlurt lirrrrr r'ells ol (lrc gcncral circ:ulalion irntl tlcvr:krp rrrtrttntl llrclrrurlly prrxlttt'etl t'ottltttr'ttlitl Irighs irt wiltlcr
Monsoons,
;1*W*d,*
lrl(;llltli l.-f,1. Ilrrrlit'ilrc (illulvr ir, .,(.r.u l)v llrt' Apollo
( )t
t':rrrit' lrntl A( rrlrsplrt'rit' Atlrr
rirrr:,1 r ;rl inrr
I
crcw (cotrrtcsy Nlrliorrirl
n
IM():ll'l il lil(; (;llr(;t,l
n
ll()Nli
at
o
f,i o@
>@
o:-
_CO
c -O
oz
I
":-
z
ON
O
oc
>o
"'I
where strong uptlrllts i1i. l)t('sc1l , scl)itfttti()rl ol'lltt'lrottttrl;rly l;rycl-lllily occur.
According to (illrlrlutr :rrrtl llrrtlson ll l ll :rrr r'xlttcssiott ol (lrc lirl'nr
I tllt
1t
,lt
(
','
1,,
l)
\ /{,,,t'
I
/ I
ti",
r
(r.3.r)
P
:l vs.!:Ee
.l> g5 C3n-1i-.
q V " h.:q
-o
9b'u<=<
lnor ancl oarly autumn months (August-September on the Northem Hemisphere,
February-March on the Southern Hemisphere), except in the Northern lndian
Ocean. Hurricanes normally travel as whole entities at speeds of 5 to 50 km/
hr. The mean directions of hurricane motions are shown in Fig. 1.3.4. It is
noted, however, that individual hurricanes may follow unusual, indccd erratic,
paths. World tropical cyclone statistics are presented in Fig. 1.3.5 ll-91. Data
on tropical cyclones reaching the United States coastline are presented in some
detail in Sect. 3.3. For detailed basic information on hurricanes, scc Il-10]
and [3-62].
As seen in a vertical plane section, the structure of a hurricanc in the mature
stage consists of five main regions, represented schematically in Fig. 1.3.6, in
which approximate dimensions are also shown. Region I consists of a nrughly
circular, relatively dry core of calm or light winds, calk:cl thc cyc, around
which the storm is centered. The air rises slowly near thc pcritnctcr of thc eye
and settles in its center. Region II consists of a vortox in which warm, moist
air is convected at high altitudes (by the thermodynantic Incchanism discussed
in Sect. 1.1) and forms tall convective clouds. Conrlcttsulion of water vapor
occurs as the moist air rises, and this results in intcnsc lirinlirll and thc rclease
of vast amounts of latent heat. It has been estimatctl lhitt lltc cttrrtlcnsation heat
energy released by a hurricanc in one hour may bc ctltrivltleltl lo thc clcctrical
t)1. 'l'lrc 1ir l'lows out
energy used in the cntirc Unitcd States in onc wct:k ll
of region II into an outllow ltrycr (rcgion lll). ln lcgiorr lV llrt' llow is vortcxtlike
and settles vcry sl<lwly iltlo tllc lltlunclaty lltyct'tcgiolt V. llclow rcgion II,
S R;
f tslS-r
:HSits;i
ug.EaEa
so o= vo
oo 9=
!'3:;t;
I;6r-9f,
lrl( ll ll{l,l 1..1.:1. Mcrrrr tlircctions ol' hurricane motions [1-9].
I
1:
I
o
o
-!biEa
i-\i
f
u9Qoo<
j
t:
Oh
ci-ri
>^
b
X'6 o
q
I
)
ii,;
bs>
€
c^
> o 6
<i@
<a >5
l?
l=
l"
lt;
i:
l;
lri
I
i
ai
o@
'io
9 *-
O
3-g a
i;o
i:
\
6
oU.
oc:
.g o?
O a6
di";
i1a
lr
(.)
E
3>f
a
cc:
- rf
-O
l^
l3
lo
,9
l:l>
{ l,'i
:U@
L@
o-
o
6
o
:
:+c:
Y!r
F:I
b
^
'll
crooo
,o
rj
ra
r
z
.cj
; oi@
; uo
-O-
tn'j
Ud
;
v
;
qo
OO
(llllotlr t,xl,rl[,r.r^V
o
o.
(.)
:@
z
3=3
>l'3P
is
l;\
;
..
o f,
14
p
24
n lM()..il,l
ll lil(; (;llt(;(,1
n
ll()Nli
I r
A I M{
il ilt(;
,:,t,t
M(
)
ll( )Nl;
25
h (km)
T6
xI
l0
"lt
I
.t
l
)ll
'l'
tt
lil(lllltl,l
kilr-t*;E;
.-- !
t
#Sr
(krrr)
r.i*si',,.
: .'4&''.,
1..1.(r. Slrrrcturc ()l' a hurricane.
is thc pressure that is approached as the radius r + @, p0 is the
pressure at the center of the hurricane eye, (pn - pdlp is assumed independent
of height, and Ra is twice the radius of maximum dp/dr, is fairly representative
of typical hurricane pressure fields. If this description of the pressure gradient
field is used, the gradient wind velocity field results from the expression of the
gradient wind (valid for cyclonic winds) derived in Sect. 1.2. There results
from this expression that the gradient velocity reaches a maximum at a radius
of the order of R.. From this radius the velocity decreases rapidly to zero at
the center of the eye, and more slowly to the relatively small values that obtain
at large distances from the center.
While the gradient wind velocity is directed along the isobars (see Fig.
1.2.5c), in the boundary layer the wind velocity has a radial component directed
toward the low pressures, as was shown in Sect. l.2.It is this component that
effects the inflow of the warm moist air at the ocean surface into region II,
thereby maintaining the supply of energy of the storm. Over land the dissipative
effect of friction increases, while the supply of energy in the form of warm
moist air tends to be cut off. As a result tropical storms over land usually fill
up within a few days at most.
The destructive effects of hurricanes are considerable and are due to the
direct action of the wind-which may reach peak surface velocities of 250 km/
hr or more-and, usually to an even larger extent, to the massive piling up of
water by the wind known as storm surge, together with flooding by heavy
in which
f,ii "-e- q'
',lE-* k
p6r
rainfall (Fig. 1 .3.7).ln 1992 Hurricane Andrew alone caused damage estimated
at more than $20 billion I l-121.
Arctic Hurricanes. An'tic' lrrrrr.icltncs tlcsignulc polirr krws with a sylnlnctrical
cl6utl signittrrr.t'iurtl winrls ol ;rl lt';rst.lO rrt/s (5ll krrots), wlriclr rcgulitrly cxccccl
llrr: corrvenliorlrl llrrr'slroltl lol lrrrn'it':utc lirt'cc witttls. ;rrtrl irt wlriclr lltrxcs <ll
.::a'!4
:fS-*;*
.J..''r
Ed
{*$
sw
L*il
1ffi,j+
i$iffi
l.'l(;uRE 1.3.7. Hurricane damage, Mississippi (courtesy National oceanic
and At-
rr
rospheric Administration).
lrr:at at the sea surface are largely responsible for the
structure and the mainl('nance of the storm Il-131.
1.3.3 The Extratropical
Cyctone
srrch circulations are produccrr cirrrcr hy rhc mcchanical action
of mountain
lrltrricrs on large-scale atmospltt'r'ic crrrrcrrls, ol by the interaction
of air masses
:rlong fionts. An examplc rll'tlltrttirgt't'rruserl by an cxtratropical
storm is shown
Iiig. 1.3.8.
Air tnasscs arc charitclct'izcrl lry rt'lrrtrvt'ly rrrrilirlrr physical propcrties over
lrolizontal tlislirnccs colrrllrllrlrlt'ln 111,.,1,,,,,.,rsiorrs ol'.ra"un,
rlr contincnts.
rrr
'l'lle ir physicitl
PtrrPt:t'tics lttc;tcr;ttttr'rl ur tlr(' :,our'( (. regiorr arrtl nr:ty 5c rr16rlilicrl
rlttl'irtg strbscrlrrcrrl ll'irvt'l ol llrr.;rrr rrr;r.,r, Arr rrr:rsst,s lrury be:
r.llrssilicrl, lrc
26
nl
Mo:itllil iltc (;ilrot,l
A
lloNl;
I.r AIM{r:il,lilIil(
27
t\,4o111 ;11,
Wrrrrr
Wtttttt +
- C*""ll
-/Cold
Warnr lrcrrl slopr:
Colrl lrorrl slo;tc
Ir'l(;tJlll,l 1.3.9. Warm and coltl lirrrrt skrps.
rltrcntly ahead of cold fronts squall lines develop that may be associated with
lruge thunderstorms and with tornadoes. The disturbance of the temperature,
vclocity, or pressure gradient field may cause wavelike perturbations on the
lhrnt that propagate as waves in a continuous medium. Major disturbances may
('iruse waves whose amplitudes increase with time and develop into
intense
vortices. The formation and development of the most intense large-scale cirt'ulation in middle latitudes, the extratropical cyclones, is connected with such
rrrrstable waves occurring predominantly along a front. on the average, the
t'xtratropical cyclones move eastward with velocities of the order of 20 km/hr
irr summer and 50 km/hr in winter.
1.3.4 Local Winds
'l'hc influence of small-scale local
winds on the general circulation is negligible.
ll.wever, their intensity may sometimes be considerable and in certain cases
f.ovcrn the design of buildings or structures.
Foehn winds. Air ffowing across a mountain ridge is forced by the mountain
If the air ascends to sufficiently great heights, condensation and
prccipitation due to adiabatic cooling will occur on the windward side. After
lurving thus lost most of its initial water-vapor content, the air passes over the
t |cst and is forced to descend. Consequent adiabatic compression results in
lrigh temperatures of the dry descending air. An example of a
foehn wind is
skrpe to rise.
FIGURE 1.3.8. Damage caused by winter storm, Fire Island, New York, March 7,
1962 (courtesy National Oceanic and Atmospheric Administration).
cording to the source region, into three main groups: arctic, polar, and tropical;
each of these may in turn be divided into continental and maritime. Continental
polar air, for example, is dry and cold, whereas maritime tropical air is rnoist
and warm.
Transition zones between air masses are called frontal zones. The variation
of the physical properties of the atmosphere across frontal zones being fairly
rapid, the latter may be idealized as surfaces of discontinuity known as f-rontal
surfaces. The intersection of a frontal surlace with a surface of equal elevation
with respect to the sea level is called a front.
srrggested
in Fig. 1.3.10.
_,f;-l_'
-5"C
,"::i\,(
) -.,.
,,r' )
r-3,ooo
,1
,/
ilt&"
The equilibrium slope of the front between two air masses can be calculated
approximately on the basis of simple hydrostatic considerations and varies
normally between l/50 and l/400.
A front is rclcrrcd to as a <'old.f'ront or as a warm .frrn.l (Fig. 1 .3.9) according
as it movcs in thc tlircctiott <tl'thc wanlcr tlr coltlcr irir. Gcncrally, a warm
l-r<lnt rrrovcs slowly irntl is rro( rrssociir(crl wilh violclrl wt'itlhc:r conclitions. On
llrtr othcr hturtl, ir t'olrl lirrrrl ciur nrovc: rapirlly ltttrl r';tttsc s('v(:r'(: wcitlhcr. Irrc-
(
l^
.L\-
V
)_
"',)
+15"C
l,'l(ll lltl'l
t..1,
ll).
lror.lrrr rvirrtl.
m
28
AIMosPHlBtcciltculAlloNS
In the United States intense and highly turbulent winds of the fbehn typc,
called chinook winds, develop on the slopes of the Rocky Mountains. In winter
l .r
\
l)rroction ol movement
\-\
FIGURE 1.3.11. section through a thundersrorm in the marure
stage Il-14].
tial speeds of tornadoes have been estimated to be of
the orderof 350 km/hr,
but the possibility that some may actuaily be
considerably higher has not been
nrlcd out.
Tornadoes are observed as funnel_shaped clouds (Fig.
I .3.12). The tangen_
(i.l speeds are probably highest at the iunnel edge
uid d.op off toward the
ccnter and with increasing distance outside the funnel.
falling water is evaporated in the underlying atmosphere that is thus cooled
and therefore sinks. The cold downdraft spreads over the ground in the manner
of a wall jet (i.e., a flow caused by a jet impinging on a wall) and produces
squally winds. This stage in the life cycle of a thunderstorm associated with
strong downdrafts usually lasts from 5 to 30 min and is called the moture stage
ll-141. As the energy supplied by the updraft is depleted, dissipation of the
thunderstorm occurs. A schematic vertical cross section through a thunderstorm
cell in the mature stage is shown in Fig. I .3 . I 1 . Characteristic of thunderstorms
is the sharp wind speed increase, known as Jirst gasl, which is associated with
the passage of the discontinuity zone between the cold downdraft and the
surrounding air.
Tornadoes. Tornadoes contain the most powerful of all winds t1-151. A torof a vortex of air, typically of the order of 300 m in diameter,
that develops within a severe thunderstorm and moves with respect to the
ground with spceds of the order of 30-100 km/hr in a path, approximately 15
krl long, clircclcrl prcdorninanlly toward the northcast. Thc maximum tangen-
29
2
The Bora. The adiabatic heating during the clescent of a very cold mass of
1ir that has passcrl ovcr il rn<lunlain hitrricr tlr it plitloittt tltily nol hc sullicient
to t'llrrrgc it irrlo it wiu'nr wintl rll'tltt'loclttt ly1rc. As tltc still crlltl air falls
grirvrl;rlror!irllv irrlo llrt'wrrlrrrcr rcgton ou lltt'lr'r sirlc, ils polr'rrlilrl cncrgy is
I ptiliFtlr-rl itrll kittr.lir. 11!lrlrgV Wttrrll nl r.rlrr.rrrt. rnlcrrsi(y tuit.y lhus bc proe,l. r ltrltnr leti:+:rl lr1, g_ii,ilr t'l l5(l ,r(Xl ktrr/lrt sr'rntirlr'tl hy pcriotls ol'calm.
'lu'
Wlttrl;q ol tlte !rrrtrt ty|e rrr r ur llt rttr-ilri rvltr'!t' tt slcr'p slopc scpitratcs a cold
Flitlt,:Il ltlrll !l tr.taal lr!!t!i! Airrnrrp- llrr-lreil kttowtt lrorit wittds arc thosc that
rrl r ur il ial:-;rlr- erirrl f'liiilrr- uir lllr: irrlllrlir:;l crursl ol tlrc Adriatic.
Thunderstorrns. A necessary condition for the occurrence of thundcrstoms
is thc fbrmation of tall convective clouds produced by the upward motion of
woffi, moist air. The motion may be started by thermal instability or by the
presence of mountain slopes or of a front. Thunderstorms are classified accordingly as thermally convective, orographic, and frontal.
If condensation of the water vapor contained in the ascending air produces
heavy precipitation, viscous drag forces exerted by the rain on the air through
which it falls contribute to the initiation of a strong downdraft. Part of the
)t;t,ilt iltc M()il()Nli
krn
chinook winds are notable for bringing sudden high temperature rises, with
unusually rapid dissipation of local snow.
Jel FJtetI Wltlcl*. I lrt- ir'l rllrt'l t'urrsisls ol irrr int'rcasc in wind intcnsity due
ln tupngrirplrtr rrl r orrliplrirrlrorrs llrirl pirrtlttt't lr ('onvcrgcnce of streamlincs. The
Intrrti;rl rvirrrl ol llrr'lowct ltlrtittc Virllery irt soutltcrn France is a wcll-known
r'rirnrplt' ol ir lxrnr winrl inlcnsilictl by .yot cllbcts.
n tM(
nado consists
l|l(;tjRlJ 1.3.12- 'lirrttitrlo ltttttt('l (utllrl('ry N:rtiorr:rl()t'crrrric irntl Atrrlrsphcric
Arlrnin-
islrilli()lt).
30
ATMOSPHERIC CIRCULATIONS
ilrIIntNot:
31
FIGURE 1.3.13. Balance of forces in tornado vorrex.
Slttt
lrtlrt=',.
t' {ltt' t'r'trltilirliirl lott'cs itt llrt' lolrurtlo v()t1cx l:u' cxcccrl lhc Coriolis
llrt'l;tllr'r ttt;tv lrt'ttt'1',lrt'lrrl;ur(l llr('1ir;rtlrrlrl wirrrl t.t;rurlir)n (scc Sect.
I .'l rri:rt lrt- rr,t illr.rr :r',
tllt
I
tlt, t' l
lirtitt llti
,1,
(1..r.2)
lt
Ltlrrrit ili, lll{. r yr ll,,ltn1rllir ru'rrrrl vt'lot r(y, r'is tlrc rurliirl rlistance
rillr-t nl ll!r. t Iilr=r, l, t', lltr, ;rll (l('rrsr{y, tntl tllthlr is l[c prcssure
IrLiIIIFIrI it|ilit!' IItr' t;trIttt',
I rirrl | 11' I I I t, \\'lurlr r('l)r('r.t'rrl:; rlrc lon.t.s lrc(irtg on a
l)lrrticlc in a
lilllt;lrlo \'iltlt'\, tl r:ttt lrt' rr'r'rr llritl llrt' ;rrt'sstn' ilt lt t<lrnad<l (locfcilr,ics loward
ll5 ( ('lllt'l Iltr' rllllt'tt'trt't' lrt'lwt't'n llrt' plt'ssrrlc a( thc ccntcr iyxl at a f'ew
Irrrrrrlrt'rl lt't'r lr,rrr rlrt' t'cnrt:r' .l rlre v()11cx .ury bc as high as 0. I ol'
one
irlrrrospltt'n', or lrlxrul J(XX) llsl'.
'l'.rrrrtlrr's lr:rvc rrls. hccrr rcp.r1cd, although
much less fiequcntly rhan in
thc Unitcrl Statcs, in Australia, wcstern Europe, India, and Japan. -l.ornadoes
that occur in Japan are known as totsumaki.s. Typical diameteri lilr {atsumakis
are of the order of 50 m. Their forward speeds are of the ordcr o1'40--50 km/
hr; the average length of their paths, which are directed gencrally toward the
northeast, is about 3 km and their maximum tangential .speecls arc probably
about 200 km/hr t1-161.
The destructive effects of tornadoes on buildings are illustrate<t by Fig.
1,.
r
I"IGURE 1.3.14. Tomado damage in Rochester,
Indiana (courtesy prof.essor
Koehler, Ball State University).
u.
F.
1.3.14.
REFERENCES
ADDENDUM: LOSSES DUE TO W|ND STORMS
l-l
wind storms are the largest single cause of economic and insured losses due
to natural disasters, well ahead ofearthquakes and floods
[1-17, 1-lg]. In the
l-2
l-3
w'
J' Humphreys, physics of the Air, McGraw-Hilr,
New york, 1g40 (reprint,
Dover. New york. I964).
H. R. Byers, General Meteorology, McGraw_Hill,
New york, 1944.
G' J' Haltinerand F. L. Martin, Dynamicnr and physicar
Meteororogy,McGrawHill, New york, 19-57.
united States, between 1986 and 1993 hurricanes and tornadoes caused about
$41 billion in insured catastrophic losses, compared with $6.1g billion for all
othernatural hazards combined [1-18, p.4], hurricanes being the largest con-
l-4 L' T' Matveev,
caused $10
l--5
l-6 M'
tributor to the losses [1-19].
losses
[
In Europe, in 1990 alone, four winter storms
billion in insured losses, and an estimated $15 billion in economic
-20
to
l-221.
Fhrt,st2s t..f
rrtt'Atttr-,s1ilttrc,'167-513g0,
u.s. Department of
commerce, Nationar Tcc:rrrrit'rrr rr'rr.rrrrrri.' scrvicc,
springfield, va.
A. Miller. Mctutnil.1i.v, (,lurr.lr.s 1,, Mt.rril, (..1'rrrbus,
OH, I971.
Ncihurgcr,.t. (;. ririrr;icr.;uur W r) rr.rurcr., (/rttrcrsruntrirtlg
tht A!n._
I.)nt,innncttl, W. ll. l;rt.r.trr;rrr. S;rrr l;l;urr.ist.o,
s1;haric
1973.
32
AtM()t;t't
l-l
l-8
l-9
l-10
1-l
I
l-12
l nt(: (ilt(;ut n il()Nli
H. Goldstcin, Chssit'ul Mcclrttttit',t, Atltlison Weslcy, Ncw Yolk, 1950.
F. Fiedlerand H. A. Panof.sky, "Atrrxrsphct'ic Scalcs and Spccrral Gaps," Ilull.
Am. Meteorol. Soc., 51 (Dec. 1970), I I 14,l I 19.
Hurricane, U.S. Department of Commercc, ESSA/PI 670009, 1969.
R. A. Anthes, Tropical Cyclones: Their Evolution, Structure and Effects,
Monograph No. 41, Am. Meteorol. Soc., Boston, 1982.
H. E. Graham and G. N. Hudson, Surface llinds Near the Center of Hurricanes
(and Other Cyclones), National Hurricane Research Project. Report No. 39,
U.S. Department of Commerce, Washington, DC, 1960.
R. D. Marshall, Wind Load Provisions of the Manufactured Home Construction
urul Safety Standards: A Review and Recommendations for Improvemer?/, NIS'l'lR -5 189, National Institute of Standards and Technology, Gaithersburg, MD,
I
993.
I
1.1 S. Busingcr, "Arctic Hurricanes," Am. Scientist, 79 (1991), l8-33.
l-14 'l'ltuntle rstorm, Report of the Thunderstolm Project, U.S. Department of Com-
l-15
rncrcc, Washington, DC, 1949.
E. Kessler, "Tomadoes," Bull. Am. Meteorol. Soc.,
1-16 H. Ishizaki et al. "Disasters
5l (Oct. 1970),
1-18 D. D. Mclean, Chairman's Report
to the Annual Meeting, First Annual Meeting
of Insurance Institute for Property Loss Reduction, Seattle, October 12, 1994.
1-19 A. C. Boissonade and S. K. Gunturi, "A Knowledge-based Computer System
for Financial Wind Risk Management," Proceedings, Computing in Civil En-
l-22
LAYER
Caused by Severe Local Storms
Science Reviews, 18 (1993), 120-125.
I-21
THE ATMOSPHERIC BOUNDARY
926-936.
in Japan," Bull.
Diaster Prev. Res. lesf., Kyoto University, 20 (March l97l),227-243.
l-11 G. A. Berz, "Global Warming and the Insurance Industry," Interdisciplinary
l-20
CHAPTER 2
gineering (K. Khozeimeh, ed.), Am. Soc. Civil Engineers, New York, 1994.
G. Berz and K. Conrad, "Stormy Weather: The Mounting Windst<lrm Risk and
Consequences for the Insurance Industry," Ecodecision, April 1994, pp. 65-68.
Winter Storms in Europe-Analysis of 1990 Losst,s arul l;'utur( L()ss Potential,
Munich Reinsurance Company, D-80791 Munich, 1993.
Windstorm-New Loss Dimensions of a Natural llultnl, Munich llcinsurance
Company, D-80791 Munich, 1990.
As was indicated in chapter l, the Earth's surface
exerts on the moving air a
horizontal drag force, whose effect is to retard
the flow. This effect is diffused
by turbulent mixing throughout a region referred
to as the ot_orp,lr"rf, bound_
ary layer. The depth of the boundary rayer normally ranges
in the case of
neutrally stratified flows from a few hundred meters
to sl"ueJ kilometers,
depending upon wind intensity, roughness of
terrain, una ungt" of latitude.
within the boundary rayer, the wind speed increases with
elev?tion; its magnitude at the top of the boundary layeris often referred
to as the gracrient speed.
Outside the boundary layer, that is, in the f.""
ut_orpt;;", ;# wind flows
approximately with the gradient speed along the
isobars.
This chapter is devoted to the study of aspects of
atmospheric boundarylayer flow that are of interest in structural design.
The theoretical and experi_
mental results presented include descriptions of
irean wind profileq the relation
between wind speeds in different roughn"r,
regimes, and the structure of atrnospheric turbulence. Since the structural engineer
is concerned fimarlty wittr
the effect of strong winds, unress otherwise noted
it will be assumed in the
lbllowing that the flow is neutrally stratified. The justification
of this assumption.is that' in strong winds, mechanical turbulence*
dominates the heat convection by far, so that thorough turbulent mixing
tends to p.odu." neutrar
stratification, .iust as in a shailow raycr .f incompressible
fluid mixing tends to
produce an isothermar state. Ars., sircc
wincr speeds are considerably lower
than the specd of sound, inc<lrrr1'l'cssirririry rrury
bc assumed in the study of the
clynamics of thc flow.
'r'A tltrirlilitlivc rk:sclipliolr
ol-llrc
rrrt.clrrrrtrr.:rl ltrrlrrrh,rrrr.plrt.rr6rrrt,rr6rr
is prcsclrlul i1 Sccl.,1..l.
33
34
2.1
il[ AtMr)til'ilt ilt(: rr()uNt)nny rnyt
Irovl Itl.itNti |(Jt
tl
In
Il()Ni;
35
vt'tlical vltriittiott ol lltr'ltot izotttrtl l)rL:ssur('1ll=lrrlrt'rrl tlt'1x.rrrls rr;xrrr llrc horilior llrc l)urposc ol tlris it.xt, it will bc srrllicicnt kl
t'ottsidcrorrly lklws irr wlrit'lr llrc horizontal dc:nsily gllrtlicnl is rrogligiblc (c.g.,
l):rft)tft)pic l1ows; c.g., scc l2-2 1). In this casc rlrr: lrolizorrtul prcssurc gradient
tlocs not vary with lrciglrt irnd l.hus has, througlrotrl lhc lroundary layer, the
srrure magnitude as at tlro top of the boundary layor:
GOVERNING EQUATIONS
zorrtal dcnsity gr':rrlit'rrl.
2.1.1 Equations of Mean Motion
The motion of the atmosphere is governed by the tundamental cquations of
continuum mechanics that include the equation of continuity-a consequence
of the principle of mass conseryation-and the equations of balance of momenta, that is, Newton's second law. These equations must be supplemented
by phenomenological relations, that is, empirical relations that describe the
spcci(ic rcsponse to external effects of the continuous medium considered. (In
llrc cirsc ol'a lincarly elastic body, for example, the phenomenological relations
t'orrsist ol'lltc: so-citllccl Hookc's law.)
ll tlrc t'r;rrrrtiorr ol'conlinuity and the equation of balance of momenta are
irvcllrp.t'tl with rcsllcct to litttc:, ancl if'tcrms that can be shown to be negligible
;rlt'rlroplrctl ll 1.2 21" thi: lirlkrwing cquations describing the mean motion in
llrt' llrrrrrtllrty lrrye l ol tltt: rtlttursphcrc arQ obtained:
U
0u IV }tl
dx
I
0v
AV AV
t/- 0x+ v-dv t
ll)
W+L
il;
w
dV
02,
I
0n
p 6x
-fv*:*:o
l0n
l0r.
t.fU- -;r:0
r-+
p dz
pdy
l0o
;;,r*8:o
AU AV AW
E* i,r+E:o
H: ,lrn,, *Y7
whcre
z'
is the gradient velocity,
(2.t.s)
r is the radius of curvature of the isobars,
Eq. 1.2.7).If the geostrophic
:urtl n is the direction of the gradient wind (see
rrlrproximation may be applied, it follows from Eq. 1.2.6 that
l0n :
-:
pdx
fv"
(2.1.6a)
(2rr)
l0n
-+:-fu"
pdy
(2.r.2)
wlrcre U, and Vr are the components of the geostrophic velocity G along the
(2.1.6b)
r and y axes.
(2'l'3)
(2.t.4)
where U, V, and W are the mean velocity components along the axes x, y, and
z of a Cartesian system of coordinates, whose z axis is vertical; p, p,f, and g
are the mean pressure, the air density, the Coriolis parameter, and the acceleration of gravity, respectively; and r, and r,, are shear stresses in the x and y
directions, respectively. The x axis is selected, for convenience, to coincide
with the direction of the shear stress at the surface, denoted rs (Fig. 2.1.1).
It can be seen, by differentiatingEq.2.l .3 with respect to,r or y, that the
The boundary conditions may be stated as follows: at the ground surface
lcvel the velocity vanishes, while at an elevation from the ground equal to the
lroundary-layer thickness, the shear stresses vanish and the wind flows with the
liurdient velocity.
2.1.2
Mean Velocity Field Closure
'lir
solve the equations of mean motion, it is necessary that phenomenological
t:lations (also referred to as closure relations) be assumed defining the stresses
t,, and r,. A well-known assumption [2-l] is that an eddy viscosity Kand a
rnixing length L may be defined such that
ru
:
pK(x.
AU
!. z) ^
(2.1.7a)
c,Z.
llll,l :.1. 1 "
( 'oorrlirrrrlt'
rrrts
\,-..
:
pK(x.
K(x, y, z)
:
1.21,, v,
j,
z)
.,
AV
(2.r.7b)
^
dz.
l(ur:)'
*
(Yu,)'1"
(2.1.8)
'lrc usc ol' ljc;s. 2.1.1 -2.1 .tl in t'orr jrrnt'ti'n with l,)qs. 2.1.1-2.1.4 is rcfcrred
Io its lhc tttcatt vckrcity licltl t'krsrrrt' lrr lir;s. 2.1.7 cithcr lho ultly viscosity
ot'lltc ntixittg lcrrglh lir:kl rrrrrsl lrt. ,.tpr.t rlit.rl.
'f
Itl( il
r,,
36
ilil
2.1.3
n
:r:'
lM():;t't il nt(; lt()t,Nt)nt ty
tull
ANvi llll llYl't
r()r rr rlirNil()ril,/()NrAr
ry
il{rMr}rir Nr
,t,:,il()w
37
Mean Turbulent Field Closure
From the equations of balance of momenta fbr the mean motion, thc lirllowing
equation may be derived (e.g., see [2-31):
la
lul3x
r,
*
e).,h(9. u,\:,/q'\l)l- liA
;al
+ve.n). c:0
..,
* a
l
oI)
r,,3V)
(2.1.e)
rvlrt'rt. llrt'lr;rrs irrrlit'lrlt'irvcllrging willr rcspcct to time, u, tt, w are turbulent
lrlor rly llrrclrrrtliutrs irr llrr r, t,, ;: tlirct.tions, rtspcctively.
0.5
Iu I /' I r,'')"'
llrr rt'srrll;rrrl llrrtlturlrrrft vt.lrxily, /,'is llrt.llrrctrurtirrg prssurc, and e is the
t;tlc of ctt('tf.v rltsstlt:tltott l)('r unil rrurss. lit;rr:rliorr 2.1.9 is rclbrrccl to as the
r:;
tttrltttlttt! /.ittt'tit' .'tt.'t,q.\' r'rlttrtliott rrrrtl c'xPlt'sscs llrt. blrllurcc ol'turbulcnt cnergy
ittlvcctiott (llle (rrttls itt lltc lit'st brlrt'kel ). pnrtlrrcliorr (thc lcrr.ns in thc second
brackct), clillirsion, antl tlissipirtiorr. 'l'hc rrsc ol lxl. 2.1.9 and attendant phenomenological rclations-in c<lnjuncti<ln witlr llqs. 2. l.l-2.1.4 is rcf'crred to
as the mean turbulent field closurc. Phcnorncrrokrgical dcscriptions of the quantities involved in Eq. 2.1 .9 havc becn attorrrptccl by various authors lz-4,2-5,
2-61 . Successful predictions of boundary-laycr characteristics based on Eq.
2.1.9 and various phenomenological descriptions have been reported in the
literature [2-7], although differences of opinion with regard to the relative merits
of these descriptions still exist.
In particular, the mean turbulent field closure appears to be advantageous in
the study of three-dimensional boundary-layer flows. Following [2-g] and
12-91, [2-10] proposed the relations
{rf,+ r2,lt'':
/p's2\
*
' \;
z
)
porq,
Tu
(2.1.10)
t.1..)
/v\
: q'n''^^.to'taz
1;1
(qr)r,,
L
--
LaQ/6)
:
'''
)Vl0z
|Ul6z
(2.t .1t)
(2.t.t2)
(2.1.13)
in which ar = 0.16, 6 is the boundary-layer thickness, and e,. is the resultanl
velocity at the edgc of thc boundary layer (or thc gr:rtlicnt vclocity in atmosphcric filrw).
1.O
v/6
FIGURE 2.1,2. Empirical functions. From J. F. Nash, .,The calculation
of rhree_
I)imensional rurbulent Boundary Layers in Incompressible Flow,"
J. Fluiel Mech.,37
(1969), Cambridge University press, New york, p.
629.
In the case of the mean turbulent field closure in which Eqs. 2. r.g-2.r.r3
are used, the empirical functions that have to be specified are the
diffusion
l'unctions a2(yl6), and the dissipation length z7(y/6). Reference
2-r0 proposes
lbr these functions the form represented infrg.'Z.t.Z.
2.1.4 Second-Order Closure
'fhe second-order closure consists in supplementing
the equations of balance
rf momenta and of continuity by the Riynolds
which govem the
behavior of the stress tensor components and are"qirations,
dlrived from first principles
12-111. Reynolds equations contain unknown terms, including triple velocity
correlations, for which suitable phenomenological relation, ,rruit be
sought. To
rbtain such relations, the method of invariant modeling has been proposed,
which is based upon the following requirements. The
-od"l"d terms must: (r)
cxhibit the tensor and symmetry properties of the original terms in
Reynolds
cquations, (2) be dimensionally correct, (3) be invariani under
a Galilean translbrmation, that is, a translation of the coordinate axes, (4) satis$z
all the general
conservation laws [2-1 r, z-rz]. The second-order closure has
been applied, for
cxample, to the study of the flow structure in the boundary layer
near a sudden
change of surface roughness 12-131.
2.2
MEAN VELOCITY PROFILES IN HORIZONTALLY
HOMOGENEOUS FLOW
It lnay bc assurnccl that in littgc st'lrlc sl()lnl:i, within a horizontal sitc
ol unilirrm
(lttghncss ovcr a sullicicrrlly lltrgt lt'lt lr ;r rr'p,ion
cxisls ovcr w6ich rho ll.w is
ilil AtM()lipltilil(; t()t,Nt)Aily tAyl il
3B
i':' Ml AN
FIGURE 2.2.1. Growth of a two-dimensional boundary layer along a flat plate.
horizontally homogeneous. The existence of horizontally homogeneous atmospheric flows is supported by observations and distinguishes atmospheric
boundary layers from two-dimensional boundary rayers such as occur along flat
plates. Indeed, it is known that in the latter case the flow in the boundary layer
is decelerated by the horizontal stresses, so that the boundary-layer thickness
grows as shown in Fig. 2.2.1 . rn atmospheric boundary layers, however, the
horizontal pressure gradient-which, below the gradient height, is only partly
balanced by the coriolis force (Fig. 1.2.8)-"re-energizes" the fluid and counteracts boundaryJayer growth. Horizontal homogeneity of the flow is thus
maintained 12-141.
Under equilibrium conditions, in horizontally homogeneous flow Eqs. 2.1.1
and2.l .2, in which Eqs. 2.1.6 are used, become
. - v: !0"
p.f az
(2.2.ta)
u--u:-!0"
pfaz
"
(2.2.tb)
v^
2.2.1 The Ekman
2.2,2 The Turbulent
I
6Gt1 -
v:4ct
t2
where
a : (.l.l2K)v2
*
e-oz(cos az
l2
2l and 12-151.
A different type of
approach was recently developed in l2-l4l in which,
lrrther than resorting to a mean velocity field, closure is based on similarity
t'onsiderations analogous to those used in the theory of two-dimensional boundirry layer flows. In this approach the boundary layer is divided into two regions,
ir surface layer and an outer layer. It is logical to assert that the surface shear
r,, rnust depend upon the flow velocity at some small distance z from the ground,
thc roughness ofthe terrain (i.e., a roughness length zo), and the density p of
thc air. Thus rs may be expressed as a function F of these quantities:
-
sin az.)l
(2.2.2a)
*
sin uz.)l
(2.2.2b)
:
F(Ui
*
Vi, z, zo, p)
(2.2.3)
and j are unit vectors in the x and y directions, respectively.
r'onvenient to write Eq. 2.2.3 in nondimensional form as
where
i
ui+vj : (:)
/, \(0/
It
is
(2.2.4)
U4.
where the quantity
U*:
(?)"'
(2.2.s)
is known as the shear velocity ol' lhc lkrw and.ll is some function of the ratio
2.2.4 is a lirnrt 1vl' 1111: vve:ll-known "law of the wall" and describcs thc flow in thc surlitc:g litycr.
In thc outcr laycr it can hcr sirrrillrrly irsscl'lc(l that the reduction of velocity
l(t/,i + V*.)) * (Ui + lz.i)l rrl lu'rglrt ,'ttrtrsi tlcpond upon the surface shear
r1y, the: hr:ight to wlrich thrr cllt't'l ol llrc witll sttt:ss hits diffused in the flow,
;:/;1y. Equation
e-"Z(cos az.
Ekman Layer
ro
model the shear stresses are represented by Eqs. 2.1.7 andif,
in addition, it is assumed that the eddy viscosity is constant, thJmodel obtained
is called the Ekman spiral. Equations 2.2.1 thenbecome a system with constant
coefficients. With the boundary conditions U : V: 0 for z : O and U :
U* V : Vrfor z: oo, the solution of the system is
39
Mcteorologists have attempted to solve Eqs. 2.2.1 using assumptions on the
vlriation of eddy viscosity with height that are more plausible than the assurnption of constancy. A survey of corresponding solutions can be found in
Spirat
:
t{'{ tiy t,il()l ll ll; lN ll()lll.1()NlAl ly ll(,Mrxil Nl ()(,li lt()w
Ucluations 2.2.2. wlrrclr tk:scribc tltc Ilkntittt s;rit'irl. irtl' rcl)lcric:tltctl sclter
rrratically in liig. 1.2.(). 'l'lrc ap,rocrnont ol' lhr:su ctlttitliotts witlt obscrvrrlitttts
lrirs bccn fountl to be rrnsirtislactory, howcvol'. liot cxruttplc, whilc acc<lrding
to Eqs. 2.2.2 lhc irngle rr,, hctwcen thc surlacc strc:ss 11 antl thc geostrophic
wind direction (trigs. 2.1.1 and 1.2.9) is 45o, obscrvations indicate that, in
blrotropic flows, dcpcnding chiefly upon roughncss ol'tcrrain, this angle may
rrrrrge between approximately 6" and 30'. The causc of the discrepancies is
thc assumption, mathematically convenient but physically incorrect, that the
t,tldy viscosity does not depend on height.
If in the above
U
Vt
40
nil AtM()t;t'ilI tl(i
tx)(,Nt)nt
ty tAyt l
:':, Ml nN vt tr){ ili t,n()l ll Il; lN ll()lll,/()Nlnl ly ill )M(t(it Nt ()t |; |()w
that is, thc boundary-laycr thickncss 6, arrrl thc tlursity p ol'thc air. 'l'hc cxprossion of this dependence in nondimensional lirrrrr is known as thc '.vckrcity
defect law":
ui+vj
u4
_ u|i + v|i .,
(;)
u4
v.i
-.1
*.
(;) (*)l
t
(2.2.7)
f{0 :1ln g'lkyi
(2.2.8)
fzG):gngr/!i+fj
B and k are constants. Substituting Eqs. 2.2.g
2.2.6, respectively,
where
U4
ui+vi
U4
i ('";
Usi
+
:
0 at z
:
anct
(2.2.tt)
k\
u.^ : 16
- ln
-
tt4 k
'i) ' . f i
(2.2.12)
2.0
vu--4
k
(usi + vs:)r az
[ ,.r, (#)
o,
:
:
, t /,
-t--tt
trt * vi -
:
!t*i
(2.2.16)
where the integration is carried out over the boundary-layer depth. Since the
lrrrlk of the mass transport takes place in those parts of the boundary layer
whcre Eq. 2.2.6 holds-which include the overlap part of the surface layer
tlrrwn presumably to a very small height-the velocity profile in Eq. 2.2.16
rrrry be approximately described by Eq. 2.2.6. rf Eq.2.2.15 is now substituted
irrl<r Eq. 2.2.6 andEq.2.2.5 is used, the left-hand side of 8q.2.2.16 becomes
2.2.9 in Eqs. 2.2.7
(2.2.10)
VRj
(2.2.1s)
f
6, it follows that
J
(2.2.t3)
T [,u, ot
const
apl
(2.2.16a)
'l'hat is, Eq. 2.2.16 is verified and the validity
of Eq. 2.2.15 is established
l),-141. Equation 2.2.14 may then be written as
o:lu' . (^n-
If Eqs. 2-2.10 and2.2.l 1 are now equated in the overlap region, there result
llq:
(2.2.14)
(2.2.e)
+ r'.e) i
t't4
t/.'
whcre c is a constant. To prove this relation, let Eqs. 2.2.1a and 2.2.1b be
rrrultiplied by the unit vectors j and i, respectively. From the expressions thus
rrlrtained, and remembering that r, : r0, r, : O at the surface and that r, :
/,,
the two functions must be logarithms tz-16, 2-ril. The requirements of the
problem at hand will be satisfied if f1 and f2 are defined as [2-14]
Ui+Vi
,
d^Ux
-._
irrrrrr tlrc lirnn ol' L<trs.
and
,
ttt
(nt+rn'"\ "
;,,,/ [
\
It can further be shown that the boundary-laycr thickness 6 may be expressed
2.2.6 and 2.2.7, and the condition that their right-hand
sidcs bc cqual in the overlap region, it follows that a multiplying factor inside
the function 11 must be equivalent to an additive quantity outside the function
f2. rn the case of the analogous two-dimensional problem, it is well-known that
f
(;
lts
Il it is postulated that a gradual change occurs from conditions near the
gnrrrnrl to contlilions in the outer layer, it may be assumed that a region of
ovcr'f ir1r e xists in which b<llh laws are valid. Let Eq. 2.2.4 be written in the
lorttt
ll
lhrrrr wlriclt thert. lirlIrws
(2.2.6)
where f2 is some function to be defined.
ui t
41
o)')''' T
(2.2.17)
llquation 2.2.17 was oblairrctl irrrlt'pcrrtlcntly in [2-18] and [2-5]. The derivrrtion o1'12-51 is bascd <lrr tlrt'lrrrlrrrlr.rrl crrcrgy equation and the assumption
rrl'ir rnixing lcngth pnrporlionlrl to .'. 'l'lrc tlrurntities A and B arc univcrsal
('()nslanls. lironr lho arr:rlysis ol olrst'rv;r(r('ns il wrrs liluncl that 4.3 < 1l < 5.3
irrrtl
0 < A < 2.tl12-14.215, I l!'1. .) l{). ). )O.221.2-22,2-2'3.2-241. On
ilt lltt' u,urrl lrrrrrrr'l ;rrrtl irr lhc: irtrrroslthcn., llrc wt.ll
(lre llitsis ol'cxpcritttt:ttls
THE ATMOSPHERIC BOUNDAIIY LAYER
l,;'
known von Kdrmin's constant is generally assumcd to bc ft = 0.4.r,
coefficient c in Eq. 2.2.15 is of the order of 0.25-0.3 \Z-ZO, 2-261.
Mt AN Vl
l{x.ily
t,t t()t
iltsi
tN il()nt./()NtAt ty llttYl
11
,; Nt
()ljti ll()W
43
4.O
3.O
r) Z.3ti
crn
2.O
2.2.3 The Logarithmic
Law
Equation 2.2.1O may be written
as
_
,9
I
It(2.):!r*n1
o
(2.2.18)
?.o
ll'l), rvlrt'rc .r rr llrt'lrt'ip.lrl rrlrovt'tlrc srrrlircc, z, is the roughness length,
rrtttl l/( I tr' lltt' t!rr';ttt lvttttl s1x't'rl. lir;rurlion l.l.lli is known as thc logarithmic
I
o.+
tt
It
o.2
)
it tr'
lillr'i{rittt'l('ornluliir;rl tt'sr'irlt lr lurs cstlrblishccl thirl thc hcight above
gtrttttttl .'r ltp t{r t!lttr lr lttl ,r ,r tll nr;ry lrc:tsstrrrrt.tl lo lrr: irpproxirnatcly valid,
i:; rlr'Ilirrrl lrv lltr' Ir'lirlrrrrr
llr=r r=ttl
i1
0.1
o.08
o.o6
u*= O.147 m/s
ze= O.OO9
1 cm
o.04
0.5 0.75 1.0 1.25
(2.2.19)
,"'.,'
0.6
6
a
(l
1.0
o.B
1.5
VELOCITY (m/s)
whcrc b is a corrstlrrt. lhc onlcr ol'rrragnitutlc ol'which is 0.015-0.03
12-26,
2.271. As notcd in 12-261,l.c1.2.2.19 oxprcsscs the fact, well-kn.wn from
laboratory experiments-including cxpcriments conducted in rotating wind tunnels [2-28, p. 148, 2-29]-that the logarithmic layer extends to some fiaction
(of the order of lo%) of the boundary layer depth 6 (see Fig. 2.2.2). Figure
2.2.3 [2-30] represents averages of 14 mean wind profiles (average mean speed
at 9.1 m above ground u(9.1) : 5.3 m/s) measured in nearry neutral flow near
Dallas, Texas. It is seen that for the profiles of Fig. 2.2.3 the logarithmic law
provides a good description of the data up to at least 100 m elevation. This is
in agreement with Eq. 2.2.19.Indeed, for U(9.1) : 5.3 m/s, zs : 0.03 m,
f = 0.77 x 10-4 Qable 1.2.1), and b = 0.022, Eqs. 2.2.1g arnd'2.2.19 yield
zr = 100 m. Note in Figs. 2.2.2 and2.2.3 that the use of the logarithmic law
for heights exceeding z7 is conservative from a structural design viewpoint.
Equation 2.2.19 may also be shown to follow from the assumption that, in
the region 0 1 z 1 21, the shear stress r, differs little from the surface stress
rs (see, for example,I2-1, p. a89l), and the component Izof the velocity is
small. Integration of Eq. 2.2.la over the height z7 yields
ru
:
ro
* ,f
I',
(vc
-
V) dz
=
ro
*
pf
Vrzl
Publishing Company.
ltXlnlo;
zo=3
cr
-Ulu*=
o 50F<(T9.1-T320)<60F
(2.2.20a)
*The acttral valuc ol ,/< has in roconl ycars bccomc
thc objcct ol sorlc ilchatc 12-251. Hgwcvcr,
cllculalions ol inlcrcst in t:rrgirrccring upplications dcscrihcrl irr tlris lcx{ urt rxrt allcctctl signilit'lrrrlly by lht' irtlrr:rl vlrlrrt. ol (.
FIGURE 2.2.2. Mean wind profile as
measured in a rotating wind tunnel
12-291. Copyright @ l9j5 by D. Reidel
10
l,'l(JIllll,l 2.2.J. Avcrugc ol l,l
1?
t4
t(; 1r1 20
ll/rt.
22 24
,l,cxlrs.
nt(.iut \\,lt(l ;rrolrlt. rr.t.otrlt.rl rrrlrr. l)irlllrs.
Allcr.
'l'lrLrillcl.:ttttl II (). l,lrP1x" "Wttrrl
;u!11 l('nrlx'rrrulr. l,r'olilc ('hlrr.rrt.lt'r.islit.s
liirrrr
()lrst'r'vitliorrs olr rr l,:l(X) li 'lirr.vr'r."
./ 11t1tl At,.t .l (l()1y1 y, l()() .t0(r, Alilt.r.it.;rrr
Mr'lt'on rlolt it'lrl Sot, it'l y.
It II
llll
n
lM()l;l'l ll lilo lr()llNl)nl rY lAYl lr
:,:, Ml nN vt trI tt\
I,t i()t
It1; tN lt()l il./()Nlnt t\ ilirM{)(,t
'l'Altl,lt
lpJ vrz,l
:
2.2.1. Vnlucs ol'Sur.ljrt, ltorrghlrcss l,t,rrglh (11)
Crrclliciclrls lirr V:u.ious 'l,y;x.s ol' ,l.rrrains
(2.2.20b)
qru
Type of Surlircc
where 4 is a small number. Using Eqs. 2.2.5 and2.2.13,
lt
is slrowrr in
12
2(rl irntl l2-3
rlu2*
fvg
ll
nk
,u4
:.fB'*:Dj
that thc logarithmic law holds, for practical
l)urlr(rs('s. t'vt'rr lrt'yontl lrciglrts lrt wlrich r7 is of the order of 30%.
Il. lor r'r:rrrrplt'. / l() 'r st't' '. Il - 30 m/s at l0 m above ground, zo :
(l (f1 rrr (()lx'r l('rr:rirr). :rrul /r 0.02, il lillltlws then from Bqs.2.2.18 and
,l(X) rrr. lrr llrr.'t'irsc ol'stnrng winds, the validity of the
Io1';1,r,1r"r'. 1;rw rrp 1o t'lt'v:rliorts rll'tltc ortlcr of 200 m has been confirmed by
nrr':rsur('nr('nls rt'por1t'tl irr l2 12l lrrrd l2-331, as well as by observations at Sale
('r'rrrrlit'kl l2 .i-5 1 rrnirlyz.ccl in 12-221.
f J .l-lf rrrrrl
()rr:rct'orrnt rll (hc lirrito hcight of the roughness elements, the following
cnrpirical urodilication of'Eq. 2.2.18 is required 12-361. The quantity z, rather
than dcnoting hcight above ground, is defined as
=
Densely built-up suburbs, townsb
Centers of large cities,
il
(2.2.22)
LJ
"Reference [2-38].
"Values of eo to be used in conjunction with the assumption
{t
where 11 is the general roof-top level.
Typical values ofzs forvarious types ofterrain, and the corresponding values
of the surface drag coelilcients (defined as
.:I n I'
|
ln
110/zo)
5.2-7 .6
7
.6-13.0
90-100
20-40
28.0-30.0
80-120
200-300
25.1-3s.6
10.5-15.4
61.8- 1 10.4
oi fuii-scare
data,
The surface drag coefficient r (Eq. 2.2.23)
for windflow over water surjhces
upon wind speed. on the tasis of a large numbe,
or
1"pr9r
-"urur.ments,
the following empirical relations were proposed
for the range 4 < u(r') <
20 mls [2-431:
r : 5.1 x l0-4 [U(10)]046
r : lo-a [7.5 + 0.67U(10)]
:l
(2.2.24a)
(2.2.24b)
I
where u(10) is the mean wind speed in m/s
at r0 m above the mean water
level. According to [2-44], tor UltO; ) 20
m/s or so l< is constanr.
A more recent evaluation of existing measurements led to
the expression
proposed in [2-45] for wind speeds U(10)
up ro 40 m/s:
r:0.00r-s
I
[r
I
+
exp
( tl0!!.5)l
t..5.
/l
' .0.00104
e.2.2s)
il utlO) : 20 nr/s. ir irrr.ws liirrrr l:qs. 2.2.25 ancr 2.2.23
that
2.5 X l0 I ancl 7,, : 0.3-5 c.rrr. ll t.lrn lrt.vt.l.ilit:tl lhlrl
,
crr<lrs irr the cstirnation
o1'wincl spoctls cluc ttl tlnccr-lrtittlrt's :rss.t irrt'<l
witlr tlillcrcrrccs lrr'.ng [Jqs.
2'2'21t' 2'2'241't, antl 2.2.25 ittt' ittsil'trrlrt;rrt Atkliliorurl
inlirnrrirli., .rr lhc
surlirct'tlrlrg lirr-wirrtr fl.w.vt.r'lrrr',,,.,',,,, ,. prr.r;t.rrrt.rr
irr l?,r{rl :rrrtr lr rr.2 l
For examplc.
z0 is expressed
4.1-4.7
: 0 [2_42].
given in 12-421.
(2.2.23)
in meters) are given in Table 2.2.1 [2-39,2-40,2-41,
2-421. Table 2.2.I also incluclcs suggested values o1'2,' lirr built-up terrain. The
determination <ll'rcproscn(irtivc wintl pnrfilcs in brrill rrp tcrrain is gcncrally
rlillicLrlt on acc()unl ol lot'rrl llow irrlllrrxrgcncitics (c.9.,llrosc associlr(crl wilh
in which
2,1
:
tl
(2.2.22a)
2-3
wake effects). For this reason values of
zo in built_up terrain may differ con_
siderably from experiment to experiment. ihe
values listed in Table2.z.l are
intended for use in structural engineering calculations
in .on;un.tion with the
assumption Za
0- They are based on a careful analysis
where z, is the height above ground and za is a length known as the zero plane
displacement 12-371. The quantity z will be referred to as the effective height.
The flow parameters ze and z./ are determined empirically and are functions of
the nature, height, and distribution of the roughness elements [2-38]. The
roughness length z6 is a measure of the eddy size at the ground. It is suggested
in [2-33] that reasonable values of the zero plane displacement in cities may
be obtained using the formula
,r:H-?
1.9-3.4
3.4-5.2
l0-30
per l0 m2; Z+
12 m l2*4}l\
Sparsely builrup surburbsb
t.2,1.9
t.9-2.9
t-4
4-to
Palmetto
il
1-1R
I 0.6
0.l-l
Pine forest(mean height of trees: 15 m; one
tree
.'.' lt) tlrrt .:1
l0tr<
0.
Mown grass (-0.01 m)
Low grass, steppe
Fallow field
High grass
I
45
iurl ol.Srrll:rt.t. l)r:rg
0.ol o.I
Snow surface
\2.2.21)
tl; tt()w
(t'nr)
Sand"
zt:
Nt ()t
x
46
ltE AlMosplt tirc llouNunny
InyFR
Ml AN Vl
According to 12-1331, the influence on thc wavcs on thc wind prolilc appcars
to be restricted to elevations below three wave heights; in this zone wind speeds
are lower than indicated by the logarithmic profile.
2.2.4 The Power
Law
Historically the first representation of the mean wind profile in horizontally
horrrogcncous lcrrain has been the powcr law, proposed
l/(.',,r)
{/(;,,,1
in l9l6 by
/-.\"
('"'
12-471:
(2.2.26)
}
\ r ,.'/
rrlrerr-,* irr;ur r'\lr(llt(=ttl rL'|t'tttlt'irl rrlxrrr rorrglrrrr.ss 0l lctrrain ancl ;*l and zsz
lt=lir rlr. lir. tprltl,, irl rlt,t. f, rorrrrr I
Itt l.t .lHl il lii rrfiirltiit(=rl ( l) tlrirt lltr'powcr l:rw lroltls wilh consllrrrt cxponent
rr ttp ll lltr= gtiulir'ltl lrr'igltl ri;rttrl (,t)llrirl i rtst'll rs ir lrrrrcliolt ol'rv alonc. The
fi1ril nl llrt'rit' rri:iiiittlf tlttr; lrrrIltr.r llt:tl
r
"l;" (;)
(2.2.27)
lor
11y
1,11111
ll
l:; tN ilOlll,/()NtAt ty ltilM0rit l.ll (,t |; Ilrw
2.2.5 Relation botwecn Wind Speeds in Dlfferent Roughness
Regimes
Considcr tw<l acliaccnl lcrrt'ltitts, cach of'unilirrrrr nrrrglrrrcss iurrl ol'srrllicirrrrtly
large fetch. Lct tlrc nlttgltttoss lcngths fbr thc lwo lcl'ruins bc rlcnolctl by 1111
and 20, and assume that z9r ( zs. The retardation ol'thc llow hy surl'acc l'riction
will be more effective over the rougher terrain; thcrclirrc, if the geostrophic
speed is the same over both sites, at equal elevations the mean wind speeds
will be lower over the rougher site. A schematic representation of the respective
wind profiles is shown inFig.2.2.4.
The profiles of Fig. 2.2.4 suggest the following procedure for relating wind
speeds in different roughness regimes. To calculate the wind speed U(z* z6)
over the rougher terrain if the speed U(24, zo) is known, Eq. 2.2.27 is applied
to each profile; then the quantity G is eliminated from the two relations thus
obtained, and
u(2,,
zs):
(6)*"'(?)"
" (r(,,,,,0,)
t'llt'r'l tur crrginccling sirrrplilicltion of the
boundary-laycr clcptlr clcscri;rtiorr givcn by l;.t1.2.2.1-5. Vllucs ol'D and a
recommendcd for dcsign l)url)oscs irr l2-4ttl ancl l2-491 arc slrown in Table
2.2.2. Yalues of 6 (in opcn tr:rrairt artrl ccntcrs ol' largc citics) sintillr to those
given in Table 2.2.2 were pK)poscd in 193,5 by Pagon [2-.5t), p. 7441. The
ASCE 7-95 Standard 12-1391 is bascd on the valucs ol' a arrtl d given in
12-491. However, [2-1391 uses 3-s gust speeds instead ol'fastosr-rnilc speeds,
and the power law exponents are adjusted accordingly-scc 'I'ablc 2.2.2.
known, it follows from Eq. 2.2.18
Currently, the logarithmic law is regarded by meteor<lkrgists as a superior
of strong wind profiles in the lower atmosphcrc 12-26,2-51,
2-52, 2-53, 2-54, 2-551.
representation
TABLE 2.2.2. Yalaes of 6 and
Coastal
Reference
2-48
2-49
2-139*
*lirr
3-s grrsls
Areas
c, Recommended
Open
10
l/il
5
12-491,
[2-1391
Centers of
Large
Suburban
Terrain
and
Terrain
Cities
6
0
6
o
(m)
(m)
(m)
(m)
0. r6
l/
in [2-48],
213
2ll
l7
t 19.5
275
274
274
0.28
400
14.5
366
U1
366
I
0.40
U3
U5
(2.2.28)
where cv(zo),6(ae) and ot(zo),6(261) correspond to the roughness lengths Zs and
Equation2.2.28 was proposed in [2-48] and will be referred
to as the power law model.
Recently, an alternative procedure has been proposed that is based on results
of both theoretical and experimental studies I2-22l.If the speed U(zd, zs) is
Zs1, respectively.
'l-ho socott(l itssttttt;rliott lcl)r'cscnls rrr
47
520
457
457
l,'l(lllltl,l
2.1..1. Wrrrrl lt.lot rty lrlrliles
48
lltt AIM()lil'ilt tit(; tr()t,Nt)nny tnyt n
:';' Ml ANVI lr)rllt
U(2,,r,:,rr)
''
U*t:-
(2.2.2e)
2.5 ln(zrlzor)
where the notation of Eq. 2.2.22 is used. Applying now Eq. 2.2.29 to the two
profiles represented in Fig. 2.2.4 and eliminating G,
[r'* ('".* - o)'),.:lu' * (,"
o)'l''' u*,
^^,-
rrr)
2.5u*
a*.
lnk
(2.2.31)
l.l.19, l.l. 10, rrrrtl 2.2.31 will be ref'erred to as the similarity model.
As lrrrs bct'rr slrown in 12 22l, tl"rc unccrtainty with regard to the exact values
of llrc corrstiurts ,4 urrcl /i in tlq. 2.2.30 turns out to be of little consequence
insolirr as cstinratcs ol'wind speeds in the lower atmosphere are concemed.
With possible errors of the order of 3% or less, it may be assumed A : 1.4
and B : 4.7. Also, the dependence of the results on z* andl is insignificant
and may be neglected. For practical purposes, therefore, the ratios u*lu*1 may
be calculated simply as functions of the roughness lengths Z1y and zor. The
dependence of u*/u*1 upon Ze and zor can be represented by the relation [2-56]
l'irlrurtitrrrs
z r 0.0706
/Zo\
*1
-:t-l
\zor ,/
u
12-421.
The application of the similarity model will now be illustrated by a numerical
example. The data used in the example were obtained by measurernents in and
near London and were reported in [2-33]. At Heathrow, Z,t : 0.08 m, 2,,, =
0, and the measured mean wind at a height above ground zsr : l0 m is U(zrr,
zo) : 11.7 m/s. The mean wind U(z' Zo) at a height above ground z, : 195
m is sought at the Post Office Tower in London, where zo : 2.5 m (2,7 : 0).
zo (m)
0.(n5
0.07
tt
o.83
I .(X)
*l
u a.r
O.07 m and Various Values ar12-421
0.30
r.00
2.50
15
1.33
1.46
I
49
litrrrrr l')t;. 2.2.)t), rt,t O.(Xrti rrr/s. liRilrr 'l'irblt. ).).1. 11 ,711 ,,r 1.4(l; that
is, r.r,,. - l.4l rrr/s. Usltli lit;s. )..2.31, {/(2,,,1,) l-5..1,1 rrr/s. lt is notcd that
this rcsult coincitlcs witlr tlrr. irctuirl rncasLrrctl spccrrl ll .l.ll.
If thc rncan spcc:rl rrcrul rhc l)ost officc'l'owcr ut ;,, r95 rn is calculated
using the powcr law rrroclcl (hq.2.2.28) with thc paranrctcrs cv and 6 suggested
in 12-481, therc rcsults U(2.", z.t) : 13.4 m/s versus the measured 15.3 m/s
speed.
It is of interest to estimate the extent to which the effect of thermal convection
is significant in structural engineering and extreme wind climatological calculatlons. To do this, we use the following expression, based on the work of
Monin and Obukhov f2-2, p. 282;2-51:
U(z):T1,":-r(;))
where
u* : friction velocity, k : von Kdrmdn's constant, zs :
/ :
(2.2.33)
roughness
Monin-Obukhov function, and t : Monin-Obukhov length. If
the stratification is neutral, L: a, tl, : O, and Eq. 2.2.33 becomes the wellknown logarithmic law (Eq. 2.2.18).
The length L is defined by the following expression [I-4, p. 281]:
lcngth,
(2.2.32)
However, subsequent research has shown that the similarity model must be
subjected to empirical adjustments in the case of terrain for which z, ) 0.30
m or so. Table 2.2.3 lists ratios u*/u*1 based on full-scale measurements,
corresponding to zs1 : 0.07 m and various values ze of practical interest
TABLE2.2.3. Ratios u*lu*rf<tr za1,:
,:; lt()W
Effect of Thermal Convection on Mean Speed profiles in
Strong Winds
Then
4O
u4
Nt ()t
2.2.6
luprrrtiorr 2.2..10 tlt:tcrrrrincs tho valr.rc of the friction velocity
{/(;,,,
(2.230)
l,lt{}l ll l:; lNll()lll./()Nlnl lyll{}il/(){il
,
u*l
,-8
(2.2.34)
Qo
T cpp
g : acceleration of gravity (S : 9.81 mls21, T: absolute temperature,
specific heat at constant pressure (co:240 callkg degree [l-4,p. 132]),
air density (p = l.2kglm'|. and Q6 : eddy heat flux (usual orders of
rrragnitude for Qo are 10 to 60 callm2ls [l-4, p.276]).
where
:
:
p
r;,
unstable stratification. In unstable air the following expression will be used
|or tklL):
r (;): I::
, - - 160 t^rT
(2.2.3s)
Equati.n 2.2.35 was pn)pos(:tl irr ltcl. I15. Acc'rding to Rcl,. 2-51 , it
providcs a vory gtxrcl lrt to cxllcrirrrcrrl;rl tllr(ir ovt'r'rrrrilirrrn tcrrain and lor0 >
:ll, > -2. (N<ltc thal /, is by tlt'lirtitiorr nt'|;11'ur,' rt lltc slrlrtilicati<lrr is unstirblc.)
litlrrittiorr 2.2.35 is lcprcscnlcrl irr lirp1. .t..t 5
50
t
Ht
At
M()lit,ilt nto ti()uNt)n ny
tn
yl il
p
3.0
lr
n t M( ll
3.0
2.3
3.5 4.O 4.5
L
FIGURE 2.2.5. Function {(z/L) for unstably stratified flow. From E. Simiu, ..Thermal
convection and Design wind Speeds," Journal of the structural Division, ASCE, l0g
(July 1982), 16l t-1615.
stable stratification. In the case of stable stratification it may be assumed
that
iltc
t
t,nBl,t t
NCL
51
ATMOSPHERIC TURBULENCE
Figure 2.3.1 shows that wind speeds vary randomly with time. This
variation
is due to the turbulence of the wind flow. Information on the features of
atmospheric turbulence is useful in structural engineering applications for
three
main reasons. First, rigid structures and members are suu3e&eo to
time-depen-
dent loads with fluctuations due
in part to atmospheric turbulence. Second,
lThe estimates
were based on the assumption that in unstably stratified flows
290"-
and
7':
(2.2.36)
[2-25,2-51]. The length I is defined by Eq. 2.2.34; however, empirical studies
in [2-58] suggest that under stable stratification conditions it may be
reported
assumed that
L = l. I x
(2.2.37)
l03a3x
a* is expressed in m/s.
Table 2.2.4 lists estimated dcviations from the logarithmic profile (Eq.
2.2.18) for three representativc c.lscs of interest in structural engineering apwhere
TABLE 2.2.4. Deviation of Mcan wirrd speeds from Logarithmic profile [2-59]
(':rsc l "
Elevation
Unstable stratiliculion
Stable stratilicltioll
wirrrl sl)r'r'(l
il
5.O
_z
'llorrlly
'l
Plicitlirlnsl.'l'ltc tcsrrlls ol 'l'irhlc 2.2.4 show llrirl srrt'lr tkrvirrliorrs rnay indeed
bc ncgloctccl wltctr cslirrritlirrg wind prcssuros oll slrucluros (scc Case
l) or
when reducing to it c(tltlttt()ll clcvation largcsl rrr<lltllrly or ycarly wind
speeds
rccorded at a wcathcl' sruti.n (scc case 2). Howcvcr, lbr wind
speeds u(10)
.f the order of 5 m/s thc dcviations from a logarithmic profile are significant
(see case 3). The lattcr conclusion is of interest fbr the
design of structures,
such as smoke stacks, that exhibit a significant across-wind response
at low
wind speeds. This response is usually enhanced if, as in the casl in unstably
stratified flow, the actual mean wind profile is closer to being uniform
than
8q.2.2.18.
2.O 2.5
"Hourly wirrtl s;xt'rl
/'llorrrly wirrtl spct,tl
;t
lrt
:rt
;tt
50
rrr
Case 2t'
200
l'X,
_4%,
l'X,
4'n,
lO nt r'lt'vlrliorr ovt'r ollt'rr lt:r':tilt
lll ilr r'li \:tlI)il rrv(.t rll)(.n l(.il;In
lll ltr r'lIr.rlrrrrr
r'l)(.il l(.il:ilil
'rv(.t
Case 3'
l5m
l-5 m
nr
-4%
-
- LL /O
l5
t2%
*
rrr/s
e
l.)rrr/s
1 rtr/',
Itl(,illl{l,l
1,.1.
l= Wrrr'l
.,lrr'r.rl r('(.()t(l
e
:
50 kcal/m2
s
52
iltt AtMoril'ilt nt(;
tr()t,Nt)At
ty tAyt lt
i,
flexible structures may cxhibit rcsorriurl irrrrlllilicltiorr c:llbcts inclucc:cl by voktcity fluctuations. Third, the aerodynanric bchavi()r ol' structurcs-and, crlrrcspondingly, the results of tests conductcd in thc laboratory-rnay depend
strongly upon the turbulence in the air flow.
The following features of the atmospheric turbulence are of interest in various applications: the turbulence intensity; the integral scales of turbulence; the
spectra of turbulent velocity fluctuations; and the cross-spectra of turbulent
velocity fluctuations. Also of interest to structural designers is the dependence
of the largest wind speeds in a record upon averaging time.
2.3.1 Turbulence lntensity
'l'hc sirrrplcst clcscriptor of atmospheric turbulence is the turbulence intensity.
Lct u(z) dcnotc thc vclocity fluctuations parallel to the direction of the mean
spccd in a t.urbulcnt flow pussing a point with elevation z (Fig. 2.3.1). The
longitudinal turbulence intensity is defined as
ulottz
I(71:
(2.3.1)
u(z)
where U(z) : mean wind speed at elevation z and rl r/2 - root mean square
value of z.J Vertical and lateral turbulence intensity may be similarly defined.
The longitudinal turbulence fluctuations can be written as
u2
:
where z* : friction velocity (see Eq. 2.2.18).It is commonly assumed that B
does not vary with height.+ Values of B suggested for structural design purposes
on the basis of a large number of measurements are listed in Table 2.3.1
12-421.
The averaging time in Eqs. 2.3.1 and 2.3.2 should be equal to the duration
of the strong winds in a storrn. It is commonly assumed that this duration is
between 10 minutes and
t
A3).
TABLE 2.3.1. Values of p Corresponding to Various Roughness Lengths
a
0.005
6.5"
"Based on mcasurcnrcnls rclxrrlctl irt
Scc also 12-l'12l'.
0.07
(r.0
lt ttll
.
tt,ntttJt I
N(
lt
53
2.3.2 lntegral Scales of Turbulence
'lhe velocity fluctuations in a flow passing a point (Fig. 2.3.1) may
be considcred to be caused by a superposition of conceptual eddies transported by the
rnean wind. Each eddy is viewed as causing at that point a periodic fluctuation
with circular frequency <,s : 2rn, where n is the frequency. By analogy with
lhe case of the traveling wave, we define the eddy wave length as )t : (Jln,
where U : wind speed, and the eddy wave number, K : 2rl)t. The wave
lcngth is a measure of eddy size.
Integral scales of turbulence are measures of the average size of the turbulent
cddies of the flow. There are altogether nine integral scales of turbulence,
corresponding to the three dimensions of the eddies associated with the longitudinal, transverse, and vertical components of the fluctuating velocity, u, u,
rrnd w. For example, Ii, Ll,, and L'; are, respectively, measures of the average
longitudinal, transverse, and vertical size of the eddies associated with the
longitudinal velocity fluctuations (-r is the direction of the mean wind u and
of the longitudinal fluctuations z).
Mathematically, 1, is defined as
u: #t:
Ru,ur(x)
dx
(2.3.3)
where R,,rr(x) is the cross-covariance function of the longitudinal velocity comporrents z1 = u(xr, !r, Zr, t) and u2 : u@r -l x, yr, 21, t), defined in a manner
rrnalogous to Eq. A2.29, / : time, and u2t/2 is the root mean square value
of u I (and a2). Note that in horizontally homogeneous flow, tj is independent
olxl and y1 . Similar definitions apply to the other integral turbulence scales.
From their mathematical definition it follows that integral scales are small
il'the cross-covariance functions are rapidly decaying functions of distance,
hour.
tThe altemative notation o, : u2t/2 is also commonly used.
iThis use of the notation should not be confused with its use as the safety index (Appendix
0
zo (m)
n I M( )l;t't
lrrlr cxarrrlllc, il l:
l(l rrr. :1y - 0.07 nr, irrrtl l/( lO) l0 rrr/s, il lolkrws
lirrnt Eqs.2.l.ltJ,2..1.1,1..1.2, and'l'ablc 2..1.1 tlrlr( thc trrlbrrlcrrcc intcnsity
is /(30) : 0.162.
(2.3.2)
Bu'*
ll
0.30
5.25
r.00
4.1t5
2.50
4.00
l2 7t); irrxl rrsctl irr i'orrjrrrrcliorr witlr luls 2.2.23 itntl2.2.25
Velocity fluctuations separated by a distance considerably larger
than the integral scales are uncorrelated, and will therefore act on a structural
c:lcment at cross-purposes. For example, values of r), and Li that are small
compared to the dimensions of a panel normal to the mean wind indicate that
thc effect of the longitudinal velocity fluctuations upon the overall wind loading
is small. However, if D" and Li, uc largc, the eddy will envelop the entire
pancl, and that effect will be signilicrrrrt.
Equation 2.3.3 can be translirrrrrcrtl il it is rrssumed that the flow disturbance
lrirvcls with thc vclocity L/(r) lrrrtl. tlrclt'lirlc. lhir( thc fluctuation u(x1, r -l t)
rrury hc irlcntiliccl with a(,r1 tlll, r). wlrcrt./ tintc (Taylor's hypothesis).
'l'hcn
rrnd conversely.
54
THE AlMOSPI IFRIC R(}I'NI)NIIY IAYI II
Li,
:
:,
#\,:,
Il,,1r)
dr
Cardington
Round Hill
Brookhaven
z(m)
15
t7
16
zo(m) Ii@)
0.01
82
0.04-0.10 55
1.00
36
The following empirical expression was proposed in
range
z:
:
Cz^
It has been
:
0'2Li,
[2-6ll for the height
(2.3.s)
(2.3.6)
suggested that
I
\' =
5
(2.3.1)
0."[.:
(2..1.t|1
1-"rt
(z in rr-rctcrs) l2 l.'l()1. 'l'lrt' cxplcssiorr
/,1,
lM(,:it't il ilt(. il/ntlt,l l N(;r
55
0.01
0.1
1.O
10
zq (meters)
l''l(;tlRE 2.3.2. values of c and m as funcrions of zo [2-611. Reprinted with permis',r.rr l'rom J. counihan, "Adiabatic Atmospheric Boundary Layers: A Review and
;\rr:rlysis of Data from the Period 1880-1972," Atmospheric Environment, 9 (1975),
li/l
905, Pergamon Press.
\virs proposed
where C and m are given in Fig. 2.3.2 and z is the elevation (t) and z in
meters). The application of F,q.2.3.5 to the data just listcd yields, approximately, the values 1:
l5O m (Cardington), 140-120 rrr (lbund Hill), 70 m
(Brookhaven), which are about twice as high as the mcasrrrctl values.
According to 12-611 the integral scales L) and I.i, lrcr, rcspcctively, about
one-third and one-half the integral scale Ij as givcn by lirg. 2.3.-5. However,
according to 12-621, a better estimate of L) is obtuinrrtl lirrrrr lhc cxprcssion
L',
0.001
in
,;rlctl in 12-611.
10-240 m:
Il,
n
(2.3.4)
where R,(r) is the autocovariance function of the fluctuation a(x1, l). Thc lcngth
of the record from which R,(z) is estimated should be the same as that used to
estimate (J and u2 (i.e., about one hour; see Sect. 2.3.1).
Estimates of turbulence scales depend significantly upon the length and the
degree of stationarity of the record being analyzed, and usually vary widely
from experiment to experiment. For example, for open exposure, measured
values of Lj reported in [2-60] (Part 2, pp. 31 and 32) vary between 120 m
and 630 m at 150.8 m elevation (the average value being 400 m); between 110
rn and 690 m at 110.8 elevation (average value: 350 m); between 60 m and
(r50 rn at 80.8 m elevation (average value: 300 m); between 130 m and 450
rn at 50.8 m elevation (average value: 200 m); and between 60 m and 460 m
at 30.U rn elevation (average value: 200 m). Data reviewed in [2-61] suggest
that Li is a decreasing function of terrain roughness. For example, the following
data are listed in [2-61]:
Site
ll
12-641 and confirmed
by subsequent measurements, as indi-
2"3.3 Spectra of Longitudinal Velocity Fluctuations
fhe Energy cascade. It was mentioned in Sect. 2.3.2
that the turbulent
vchrcity fluctuations may be considered to be caused by a superposition of
.rltlics, each characteized by a periodic motion of circular frequency a : 2rn
t.r flt u wave number K :2rl\,,
where x is the wave length). The total kinetic
of the turbulent motion may, correspondingly, be regarded as a sum of
,
'rrtlibutions by each of the eddies of the flow. The function E(K) representing
tlrt' tlcpcndence upon wave number of these energy contributions is defined as
tlrt' cncrgy spectrum of the turbulent motion.
ll'thc equations of motion of the turbulent flow are suitably transformed, it
,;rn bc shown that the inertial terms in these equations are associated with
tr;rrsl'cr of energy from larger eddies to smaller ones, while the viscous terms
,rrtount fbr energy dissipation 12-63]. The latter is effected mostly by the
',rrr:rllcst cddies in which the shear deformations, and therefore the viscous
',lr('srics, arc large. In the absencc of sources of energy, the kinetic energy of
tlrt' ltrr-bulcnt motion will decrclsc thirt is, the turbulence will decay-faster
rl tlrc viscosity eff'ccts are largc, rrr.rc skrwly if'these effects are small.
Molc prcc:iscly, in thc lallsl t'rrst' rlrt' tlt'r'lry tirr-rc is long if compared to the
lrr'riotls ol'thc cclclics irr llrc hi1',lr wlrvr. rrrrrrrlrt:r rangc. Thc energy of these
,,ltlit's rrriry thcrclirrc hc consitlt'tt'tl lo lrt' rrpploxirrrtrlcly stcacly. This can only
lrt'lltt't'trsc il thc: c:rrc:rgy le.tl inlo llrr.trr llrrorrl',lr ilrertiirl lr-utrsl'cr l'nrrrr lhc largcr
ltklit's is lrlrltrrrct:rl by lhc (.ncllly rlr:,r,1|;rlr.tl lltrorrglr visr'ous rlli'cts.'l'ltc srrurll
('n('r'gy
llll
n
lM()i'il'lll lll(; ll()(,Nl)nl lY lnYl ll
,
eddy motion is thcn dctcrrninccl solcly by lhc nrtc ol crrcrgy lrarrslcr'(or'. ctlrrivalently, by the rate of energy dissipation, dcnotcd t (scc [Jq. 2.1.9) antl by thc
viscosity. The assumption that this is the case is known as Kolmog,orov's.first
O
ll
57
N(;t
(2.3.t3)
lr)
lrqs. 2.2.5,2.3.12, and 2.3.13 are used.
e
llu' lrilihcl wrrvc rrrrrrrbcr rlngc t<l which Kolmogorov's first hypothesis applies,
tlrt' rrrllrrt'rrr't'ol (lrc viscosily is srrrirll. ln this subrange, known asthe inertial
tttl,t,rtt,t:r'.llrt't'rltly rnol iort ttury lrt'lrssuntctl to bc independent of viscosity,
;ur,l llrrr:, rlt'lt'rrrrint'tl solcly lry (lrt' r'trtc ol'cnerrjy transfer (which, in turn, is
t't1rr;rl lo llrr'r;rlt'()l ('lr('rl'.y rlrssiplrtiorr). linrrrr this assumption, known as Kolnttt,tltt,(tt".\.\('tttn(l ltvltttlltt',ti,t, il lollows tlrlrt lr rclirtion involving E(r$ and e
Srrlrstituting Eq.2.3.14 into Eq.
:
(2.3.14)
"1*
kz
2.3.1l, if it is assumed that
K:- 2rn
u(z)
Itolrls lot :;ttlltt tt'nlly lrrglr A:
(2.3.ts)
tlrcrc results
l'll',(K ).
(. r |
(2.3.9)
0
nS(z,n): ^^_^_)/,
--- 0.28t--''
- u4
whorc /j(/() is llrc cncrgy pcr ruril wavc nunll)or.
The clirncnsions ol'thc quantitics within brackots in Eq. 2.3.9 are[L3T 2f,
[L-r], and ILtT '1, respectively. From climensional consiclcrations (see Sect.
7.1) it follows immediately that
E(K) :
t:^nzuu
(2.3. r0)
ar62t3Y-s/3
S(rK)
:
(2.3.t1)
aezt3 K-s/3
in which it has been established by measurements that a
:
0.5
12-211.
Spectra in the lnertial Subrange. Measurements carried out in the surface
layer of the atmosphere confirrn the assumption that in horizontally homogeneous, neutrally stratified flow the energy production (see Eq. 2.1.9) is approximately balanced by the energy dissipation [2-3]. The expression of this
balance may be written as
ro dU(z)
p
(2.3.12)
(2.3.16)
I
rvlrcrc the nondimensional quantityt
in which rz1 is a universal constant. On account of the isotropy, the expression
of the longitudinal velocity fluctuation spectrum* [which will be denoted S(K)]
is, to within a constant, similar to Eq. 2.3.10. Thus
lrr
, il,ttltt,il
l{/(:)-.a,1, lrr'
It follows from this assumption that,
'r'A tldrrilctl tlistrrssion 1)l :il)('(
n t Milr;t'ilt ilt{
w I rt'rrr
since small eddy motion is
dependent solely upon internal parameters of the flow, it is independent of
extemal conditions such as boundaries and that, therefore, local isotropy-the
absence of preferred directions of small eddy motion-obtains.
It may further be assumed that the energy dissipation is produced almost in
its crrtircly by (hc vcry smallest eddies of the flow. Thus, at the lower end of
hypothesis.
.r
n known
as the
(2.3.17)
Monin (or similarity) coordinate, and
S(2, n)
dn
:
S(2,
K) dK
(2.3
.
1
8)
'tlrurtion 2.3.15 implies the validity of raylor's hypothesis (see Sect. 2.3.2).
'l'hc left member of Eq.
2.3.16 is called the reducetl spectrum of the lon1'rtrrtlinal velocity fluctuations and is seen to be a function of height. Although
rrrtlividual samples may deviate considerably from the predicted values, Eq.
' ]. 16 is, on the average, a very good representation of spectra in the highI trlrrcncy range
[2-5 1, 2-52, 2-53, 2-&, 2-65, 2-67] and may, for engineering
l)rrl)oscs, be conservatively assumed to be valid for f > O.Z [2_64, p.27,
; (t] , 2-691. As in the case of thc logarithmic law, for high wind speedi such
;r:r iuc irssutlcd in structural dcsigrr (<ll'the order of 2o mls, say or more), it
is
r,':rsorr:rblc t() apply F,q.2.3.1(r tlrRrrrglrout the height range of interest to the
f
r
'.1
nlt'lurll cnginccr.
dz.
is plt'st'rrlt'tl irr Allllcrrtlix A)
Iltls ttst'ol lllt'st:ttttl:tttl ltolrrli()rt / slrrrrtlrl n.t lx tr)rlu.,r'(l rvillr its previous rrsc:rs llrt.(.1yri9lis
l
';il ;[
[('l('r.
ilil
58
n tM()rit,t
I nt(i il()t]Nt)nny tnyt il
:r
Spectra in the LOwer-Frequency Range. 'l'lrc krwcr-li-c(luoncy ltttgcr is tlcfined between n : 0 and thc lowcr cnd ol'thc incrliul subrangc. As rtotccl in
[2-511, 12-52], and [2-65], in the lower-f-requency rangc sirrilarity brcaks down
and the spectra cannot be described by a universal relation. Howevcr, dcscriptions that are useful for engineering purposes may be obtained by noting that:
l.
The value of the spectra for
n : 0 is
s(0)
: +,ft],
U
(2.3.te)
l.
3.
4.
liirrrr lit;. 42.25.)
'l'hc spcctrurn S(n) is lnonotonically decreasing.
'l'hc spcctrum S(n) is continuous at the lower end of the inertial subrange
with the curve S(n) given by 8q.2.3.16.
5. The area under the spectral curve in the lower-frequency range is equal
to the mean square value of the longitudinal velocity fluctuations (Eq.
2.3.2) less the area under the spectral curve S(n) represented by Eq.
2.3.16. (This follows from Eq. A2.15.)
Two comments on lower frequency spectra are in order. First, as in the
case of the mean speed U, the mean square value u2, and the integral scale
Lj, estimates of spectra in the lower-frequency range depend upon the length
of record being used. For consistency, the length of the record from which S(n)
is estimated must be the same as that for (J, u2, and I). As indicated in Sects.
2.3.1 and 2.3.2, for structural engineering purposes this lcngth should be equal
to the duration of the strong winds in a typical storm. Corrmonly this is assumed to be I hour, although record lengths as low as l0 rninutes are used by
some workers. The l-hour period beyond which winds in a typical storm may
be assumed to become relatively weak is sometimes rcf'crrccl to as the "spectral
gap" (or quiescent period) in a conventional reprcscntation of wind activity
corresponding to a continuous range of periods, including daily, monthly, seasonal, yearly, and secular periodicities [2-681. Spcctra ol' longitudinal wind
speed fluctuations for periods longer than about I lrtlttr ctlrrcspond to mesometeorological flow pattems. Thcy wcrc tcntativcly Ittotlclctl by Van der Hoven
rBy virluc ol (he tlelinitiorr ol (lrr's1x't'trirl tlt'rrsily, lit1. 2.1. l() rrrrplics ir vltrisltittgly snrall, rathcr
than it lirritc. t'orrlrilruliorrs ol llrrrlrr:rlirr1l totttlxrnt'nls willt zt'lr ltt'tlucrtty t() lllc tlrolln squatc
virlttc ol lltt' lltttltt;tlrotts
il ttlL ililillt,t I N(:l
59
ll l'l ll'
wlttl ttolr'tl lltt' t'ris(t'rtcc ol's1rt:cllrrl lx.;rks irl pt'r'itxls ol'lrlrout 4 clays.
lilut'lrr:rlions willr;x.r'irxls lorrgcr Lharr tlrtlsc tylricrrl ol'llrc: spcctral gop
ir"
tlrsrt:grrnlcd in slnrclru'irl crrgirrccring rnotlcls. 'l'lris irlkrws thc usc
o1'Eq.2.3.19
:rrrrl ilcttts 2 to 5 trllrvcr as c()n.tponcnts 9l' a rcirs.,rlrflc rnicnrmeteoiological
rrrtxlcl all<lwing thc cslitttittion of longitudinal spcctra fbr periods shorter
than
;rlxrut I hour.
A sccond comment pertains to the relation between the frequency zps11 &t
r'lriclr the curve ns(n) reaches a maximum and the integral scale ri.
As shown
rrr l2-(rll, the assumption has been used in the literature that
rxlu
rr'
is lhc nlcan squilrc valuc of the longitudinal fluctuations, U is
tlrt'rrrt';rrr vckrc'ily, untl /,) is thc longitudinal integral scale.f F;q.2.3.19
Iollows lirrrrr lir;s. 2..1.4 ancl A2.25.
'l'he tlclivltivc ol'S(rr) wilh rcspect to n vanishes atn :0. (This follows
wlrt'r'e
n tM()l ;t,t
"
2t
flpeak
(2.3.20)
it
was pointed out in 12-731 that the estimation of Il based on
of u and npeal can be in error several fold, owing to the
',('rsitivity of Il to the assumptions conceming the spectral
shape between n
o irnd_ n. : ,peak.This shape is in general unknown and, thereiore,
so is the
.
f
lrrr.vsvsl,
rrrt';rsrrrcd values
rll;rtionship between
Expressions for
,roses. The curve
npear and
Il,.
the Spectrum used for structurar Design purt
'l
nS(2, n)
,
u'*
2o0f
(2.3.21)
rrlrrrsc lirrm was proposed in [2-66], approximates very
closely Eq.2.3.16 in
tlr. rrrcftial subrange (zis the height above ground, n isihe trequency inHertz,
rr , ;rrtl.f'are given by Eqs. 2.2.18 and 2.3.r7,
respectively). Ii .un t" verified
rlr;r( lir1. 2.3.21implies that
,/ :
6u2*
(2.3.22)
rrlrtlr, lilr built-up terrain (zo > 0.30 m, see Table 2.3.1), may result in an
,,\('l('stirnation of structural response of the order of 5%.
Requirements pre_
rrrrrr5;fy listed pertaining to the value of .{n) and ds(n)ldn
at-n :0 are not
''.rrr:;lit'tl. However, this is inconsequential
as far as the design of most landlr.r:'t'rl structures is concerned, since their fundamental
frequeicies of vibration
;r(' rriuillly higher than the frequcncy corrcsponding to tire lower end of
the
rrrt rtr:rl subrange. Therefore, pr<lvitlcrl that Eq. 2.3.22
is satisfied, the response
,'l :;ut'll structurcs does not dcpcrrrl sig,rrilit.rrrrtly upon the shape of
the spectrum
rrr llrt' lowor l-rcqucncy rangc (st.t' St.t.l. (). l.-l).
Ilrc tlcvcl.pr'cnt
of'l'\. 2..1.)l ll 701 wrrs rrr.rivutccl by criticisrn of the
l.r'lll;rrrtl rrst,tl irr rhc Natignal Building
lrrlllv*1rt* cxprcssion, pnr;rost.tl irr
('rrrlt' ol ('unutll
l2-721:
60
Tt
tE nTMOSt'Ht-llto tx)(,Nt)ntiy
tnyt il
'
nt
-r'l
''" tl *
ui- :4.0
"'ro"
ns(z..
.1
A l lv'l(
',
il,ill ttl( il,1il
il,1
1 N(.1
6l
(2.3.23)
x : l2OOnlU(l0); n is expressed in Hertz and U(10) is the mean
wind speed, in meters per second, at z : l0 m. Equation2.3.23 was obtained
by averaging results of measurements obtained at various heights above ground
and does not, therefore, reflect the dependence of spectra on height. In the
irbscncc ol' rnodcls capable of describing this dependence-such models were
rrrrf y tlr:vckrpcrl subscqucntly in the 1960s-Eq. 2.3.23 and similar expressions
Prolxrst'tl in llrt' litcr-lrlurc havc pnrviclcd useful first approximations of the
lonpiiiurlirr;rl trllrrrlt'lrct: r.il)(:ctrit in lhc atl'nospheric boundary layer. It is noted
llr:rt llrt' th'Pt'nrlt'rrt'r' ol's;rectllr rln hcight is clcarly suggested by data published
in which
:1 N
-'l 6l-
r5
lJ 7ll (lrip. l.1..lir).
As rrrt'rrliolrr'tl t'trlliu', llrc spccttrl tlistribution in the lower-frequency range
lrlrs littlt' irrllrrcrree on brriltlirrg tcsponsc; however, the magnitude of the turlrrrlcrrl llrrclrrrtiorr r'orrrponcnts at licquencies cqual, or close, to the natural
l'r'cclucncics ol'u tall structurc rnay affect its response very significantly. It is
rrr
thcrolorc ol'intcrcst to comparc thc higher-frequency components inEq.2.3.23
l6 (or, equivalently , Eq. 2 .3 .21). Such a comparison shows
that Eq. 2.3.23 may overestimate the longitudinal spectra of turbulence in the
higher-frequency range by as much as 100-4fi)%, as can be seen in Table
2.3.2 and Fig. 2.3.3b.
It is also noted that Eq. 2.3.23 yields z2 : 6u2*, and that it implies S(0)
: 0, or U:0 (see Eq. 2.3.19), which is physically not possible [2-3].
The von Kiirmrin spectrum [2-1341
0.002
to those of Eq. 2.3.
48q
'U
nS(n)
aU-*
['
_'
Wave number
necessary to have
Ii =
it
cycles/meter
(2.3.24)
,,r(+)l
can easily be shown that
-lLu(r2)
0.01
(a)
Eq.2.3.21
was proposed before the development of Eq. 2.3.16. Equation 2.3.24 satisfies
the conditions S(n) + O and dS(n)ldn : O for n : 0. However, for Eq. 2.3.24
to be consistent with Eq. 2.3.16,
0.005
it
U(10):
30 m/s, eo = 0.08 m
would be
0.303t22, which does not appear to be the case in the
atmosphere. That Eq. 2.3.24 is, in general, not consistent with Eq. 2.3.16 can
be explained physically by the fact, discussed earlier in connection with Kol-
mogorov's hypotheses, that the higher-frequency spectrum is independent of
the large-scale features of the turbulence that determine Lj. Equation 2.3.24
is not used in applications where the magnitude of the higher-frequency components of the longitudinal velocity fluctuations is of interest. However, it can
be used in applications in which the effect of the low-frcqucncy component
could bc irnporlant, suclt irs thc analysis of structurcs with vcry long nalural
n (cycles/s)
(b)
lfl(;UltE 2.3-3- (u) Longitudinal turbulcncc spcctra measured at Sale, Australia (based
20 rccords)12-ill. Frorn A. G. I)lvcrrporl, "'t'hc Spectrum of Horizontal Gustiness
Ncrrr thc Gr<rund in High Winils,'' ettrrt. .l . lilt.1rrl Mcteontl. Soc., g7 (1961):202.
1/r; ('ornp:uison ol'spcctra givcrr by litls. J..1.21 rrrrtl 2.3.23. From E. simiu, ,,wind
rrrr
Sltct'trir rrrxl I)ynurric Akrngwintl ltcsPorrst',"
t9 t0
.l
. ,\rr.rrt.l)ir,., ASCE 100 (1974): lg97_
62
lltE
AtMOS|'l
tLtilo t]()tjND^lty tAyt il
TABLE 2.3.2. Yafues of
nS(n)lu?*
lirr
7.,
:
l).0t1 nr an<l {/(10)
z:100m
n
Cycles per
z:300m
Eq.2.3.16
f
Second
:
or 2.3.21
f
Eq.2.3.16
All
or 2.3.21
of z,
8q.2.3.23
(3)
(4)
(s)
(6)
0.1
0.255
0.70
o.43
0.24
0.15
0.586
0.37
0.23
0.13
0.08
t.4l
0.450
1.125
1.0
2.250
1.172
2.930
5.860
'5r
Values
(2)
0.-5
6lL,-.1',1'
nS(2,
n):\
---u'x
.f <
cz
I
, 2(.1 ,,, .lil i
2!",,(.1', -- .l',,,1
r
.1,( 1.,
* zt,,,,l tnfr.
0.98
0.54
o.34
,
,,,
*2b2f,n
(2.3.259)
o
I*l.f - Pt + b2(f^ -f ,'I
I
--2lo,f^
(2.3.zsh)
b,:t-t.sf.dl
(2.3.2s1)
t'z:0t-azf,-brf?
(2.3.2sj)
llquations 2.3.25 are plotted in Fig. 2.3.4 for k : 0.4, zo : 0.001266 m,
35 m, U(35) :45 m/s (u*: 1.76 m/s), B : 6.0, f,:0.22, U: l8O
rn, andJ, : 0.07. Also plotted in Fig. 2.3.4 isF;q.2.3.23 (intemrpted lines).
I:
Eqs.2.3.25
.f^
(2.3.25a)
*
o.ze.s
azf
''''
t brf,
f^<f<f,
(2.3.25b)
I
I
f--f,
(2.3.25c)
'..r*/-ur
where a* and/are given by Eqs.2.2.18 and 2.3.17, n is expressed in Hertz,
z is the height above the surface (in the case of flow over the ocean, the height
above the mean water level), f, is the lower limit of the inerlial subrange (f,
- 0.2),f-is a parameter allowing changes in the shape of the spectral curve
At:
''
I
j
I
forf < f,,
!,,,,)
lMr,!.t'ilt ilt(; il,lilitit tN(;t
(2.3.2st)
tt2 :
pclirxls ol' vibnrlion (c.g., corrrpliant off.shore platforms, which have motions
with lrcrirxls or irlrotrt 50 io 120 s). A modified form of the von Kdrmiin
ril)r:ctnur, bascrl orr lirs( principlcs and reflecting the variation of the spectrum
witlr lrc:ight irbovc gnrun(|, wils rcccntly proposed by Harris l2-l4}].
ljor thc purposc ol' studying thc sensitivity of tall building response to changes
in the valuc ol'various parameters determining spectral shape, an alternative
expression fbr the spectrum, consistent with Eq. 2.3.16, was proposed in
t2-lo). This expression depends upon the parameter 0 and an additional parameter allowing the modification of the shape of the lower-frequency part of
the spectrum, and is subject to the constraint imposed by Eq. 2.3.2. A similar
expression was developed in [2-74] to study the sensitivity of compliant structures to changes in the values of the parameters B and Ij, and to changes in
the shape of the lower-frequency portion of the spectrum consistent with Eq.
2.3.2. The expression of [2-74] is
( o,f + b,72 + d,13
n
30 rn/s l2-7(ll
(l)
0.2
23
rtrt
and
ariQ)0
(2.3.zsd)
z
8r : 0.26.f ,2tl
o
0 00
0.05
0.15
r1llz)
(2.3.25c)
l,'l(;tJl{lt 2.3.4. Spcctrl ol' hrrrgitrrrlirrrrl vt'locil.y
lluctr.ralions (Eqs. 2.3.25).
ilil
64
n
tM():;t'ilt nt(i lr()t,Nt)nl ty lAyt
I
:|
Il :
nS(n\
.-:4.0u'* -'"
x
(2
+
Clrrh(r,
:
i:
12-1371:
IJ(I0)l/0 (in
(2.3.26)
r
in which
i:
Jl.
lf,,,r(r, n)
+
iSf,,r(r, n)
u1 and u2 indicate that the two records are taken
distance between which is denoted by r.
The coherence function is defined as [2-751
[Coh(r, n)]2
)
nt
-
zr)2
+ C1Jy,
u(t0)
-
yr)tlt,,
(2.3.2e)
I
lil
(2.3.30)
rvirrcl, u(10) is the wind velocity at 7 : 10 m, and the exponential
decay
otrllicients C, C, (or C1r, Cy) are determined experimentally.
ln homogeneous turbulence the quadrature spectrum vanishls
[2-64]. rnthe
;rlrrursphere it appears that the ratio of quadrature spectrum to co-spectrum
is
:.rrurll and that the square root of the coherence function may
therefore be
:rss.rned, for engineering purposes, to be approximately equal to
the reduced
t rr spectrum cu,ur. on the basis of wind
tunnel measurements, it has been
:;rrggcsted in 12-771 that it is reasonable to assume in engineering
calculations
t
(2.3.21a)
:
cf,,ur{r,
a.t
S'u,ur(r,
points M1 and M2, the
n) + ql,,,r(r, n)
:
15f,,,,{r,
n\12
se:;)se,
(2.3.27b)
(2.3.27c)
n)
)
1sf,,1r. r)12
Qituttr.r) : ik:;)Nru n)
lA tlctailctl rlistrrssion ol tlrss spt'tlt:r is ptcst:ttlttl irr Allpt:rulix
A,?.
'A
(2.3.27d)
In Eqs. 2.3.27c and d, S(21, n) and S(22, n) are the spectra of the longitudinal
velocity fluctuations at points M1 and M2.
n) :
Stt2(zr, n)Stt22r,
n1"-i
(2.3.31)
rvlrc,rcf is defined by Eq. 2.3.30 and Cr: 10, C, : 16.* It appears,
however,
tlr:rl the exponential decay coefficients C, C, (or Cv, Cr), iatherthan
being
rrrtlcpendent of roughness, are generally larger for iougir surface
conditions
.rrt'lr as urban areas than for smooth surfaces
l2-el. Moieover full-scale mea_
',rrrr:rnents indicate that the exponential decay coefficients
depend on height
;rlxrvc ground and, quite strongly, on wind speed, as shown in f,igs.
2.3.5 ind
; \.6 12-60, 2-781. The dependence of the exponential decay coeftcients
upon
rvirrtl speed is illustrated in Figs. 2.3.7aand2.3.7b, which
represent Eq. z.i.zs
where
c;tu2tr.
nlc1,k,
Irr llqs. 2.3.29 and 2.3.3O, !r, !2, and 21, Z2 are the coordinates
of points M,,
/ . the line M 1, M2 is assumed to be perpendicular
to the direction of the mean
'
The real and imaginary pafts in Eq. 2.3.21a are known
n):
(2.3.28)
r'. altemativ ely [2-7 6],
as the co-spectrum and the quadrature spectrum, respectively. The subscripts
Q(r,
t
A
The cross-spectruml' of two continuous records is a measure of the degree to
which the two records are correlated and is defined as
:
65
ll
2.3.4 Cross-Spectra of Longitudinal Velocity Fluctuations
n)
n) : (,
.f:
rneters).
S'j,,r(r,
t,ililt,t I N(:t
x2)s'6
r
10001 U(r.)l
I
wlrcrc
I.lt(X)rrl{/(10). Likc F.q. 2.3.23, Eq. 2.3.26 does not reflect the
vlrrilrtion ol'thcr spcrclnurr with height above ground. However, it has over Eq.
2.3.23 tlte: rrtlvirrrtagc that it irnplies a nonzero integral scale of turbulence lf
wlrt'rt
til(
kltowtt its tlltn'()w lt:urtl t.trrss t'orr-crlirliorr) wiri
l)ll)l)()ri(,rl irr lJ-7(rl:
g.
Finally, we mention the spectrum proposed by Hanis in 1968
ti
'l'lrc lirlklwirlg ('xl)t('\\t('rr lot llrt's(luiur
txrl ol llrr't'olrt,n'nt'c lirrrcli6rr (also
Unlikc l:q.2.3.24, Uqs. 2.3.25 atc cotrsis(ctrt with Ilt1. 2.3.1(r. lkrwt'vo', tlrc:y
do not satisfy the requirement dS(n)ldn : 0 lirr rr : 0. 'l'his rcquilotrtr:llt could
be satisfied by modifying Eq. 2.3.25a in thc imrncdiatc vicinity ol'a : 0.
However, such a modification is not necessary in practice since its efl'ect on
results of engineering calculations would be negligible. Finally, it is seen in
Fig. 2.3.4 that Eq. 2.3.23 significantly underestimates the spectral ordinates
at very low frequencies. This is due to the fact, noted earlier, that Eq. 2.2.23
implies that
n I M{ }t;t't
rrrtrtlified modcl of the spatial structure of turbulence proposed
in
t2-147) eliminates the
l'lhrwirrg two drawbacks of Eq. 2.3.31. First, Eq. 2.3.31 does not allow
for negative values of
tlrt'lttl'lrttloncc co-spcctrum, regardless of spatial separation. For homogeneous
turbulence this
rIr|lics lltat' contrary to its definition, thc mcrn ol'lhc lluctuating longitudinal
velocity component
'hrs ttttl vitnish. Sccond' 8q.2.3.31 inrplics lurgc colrclalionsirlthe low-frequency components
' r'r'tr il lltc scpitlitlion is largc. Thc rrtort rt':rlrstit rrlrxlcl pnrposcd in 12-All may result in a
tr'(lrr( li()f) ol lltc calcttlirtctl rcsoniutl ltslxrrs(' ol slt'lrrlt'r slntclrtros
by as rnuch 1ts 25%. Sce also
|
.' t,lr{, 2
t491.
66
THE ATMOSPHFIIIC f]Ot]NI)AIIY IAYI II
r
;,:t
n
lM(lt;t,t il lll(. lUl ilIlt tN(:t
67
0.0
8.0
o
6.0
(:ry
C,,
t/(10)
4.0
-*
|,
o.o
I
0
E
o
-.
ao
I
2.O
3.5
20.8 m/s
0.6
5
o.a
0.4
€
O
O
0102030405060
9oo
u(10) (m/s)
Irl(;llltl,l 2.J.-5. Virliation
Irllrrirr)
ol-cxponential decay coefficient C,,, with wind speed (open
l.l 781.
0.1
0.2
nll
and nlcasurctl valucs ol'thc square root of the coherence function for records
(takcn at points of cqual clevation) with U(10) : 20.8 m/s (Cr:u : 3.5) and
U(10) : 35.2 mls (Cr, : 8.8) t2-601. The dependence of the exponential
decay coeflicients upon terrain roughness, height above ground, and wind speed
is insufficiently documented and therefore represents a source of uncerlainty in
structural engineering calculations.
It was pointed out in Sect. 2.3.2 that relatively large uncertainties remain
concerning the integral scales of turbulence. In view of the close physical
0.3
0.4
yr-!ztl/ul'tol
Run
1
0.1
o.2
nlly*y2ll/u11o)
18
Run
'l
1
7l
lv,I v,]
l(meters) |
I o I o I rz I
l"lolssl
l:l;l::
I
(b)
FIGURE 2.3.7. Measured values of Coh(l
lr _ !zl, n) t2_6O1.
rclationship between turbulence cross-spectra and integral scales,
similar uncertainties can be expected concerning the exponential decay coefficients.
Nevertheless, results of recent research quoted in
lz-sol ,ugg".i that the value
L. = t0 is acceptable or even conservative from a structural design viewpoint.
A similar conclusion regarding the value Cy = 16 follows froil
1Z_St1, ac_
C, is a funcion of the raiio ly, - yrltz, as shown in Fig.
:":d:g^,g,ylich.
2.3.8. Additional research into the vertical and lateral coherence of the lon_
gitudinal velocity fluctuations is reported in
[2-62] and, [2-g2,2-g3,2-g4,
2-85, 2-86, 2-871.
In some applications the longitudinal (along-wind) coherence of the longitudinal velocity fluctuations is of interest. According to
[2-gg], the longitudinal
coherence between the fluctuations at two points M,1x, y,
zj una M2(x2, y, z)
,
can be expressed by Eq. 2.3.28, where
Ii --.'C,lt,
(/t:\ ',1
F'I(;URI,l 2.3.6. Virrirrtiorr ol
(',
with wincl spcctl anrl hciglrl (opcrr lclr:rirr) l2 Tlll
with
cr :
3.0 ovcr warcr.rrtl
tcptlrlccl in l2-t'tt)l suggosts thirt
(',
(2.3.32)
(r.0 rvr:r r.nd. A thcorctical approach
tlrr krrrliitrrrlirrirl cohcrcrrcc clcpcntls up1;n thc
68
lHf n tMosit,ilt llt(; tlotJNt)ntly tAyr lt
:)
rr,\,,,(l:,
rr)
u)r,
50
_
ll
A
lM(l:;l,t il iltl . il,t il tut t
N(
.1..1(r/'
il
69
(2..j.-1.j1
I I l(ll"'
40
cy
30
According to n)casurcnrcnts rcportccl in [2-60, 2 ttOl, thc cn)ss-spectrum of
vcrlical fluctuations ilt lwo points M1 and M2, ol'clcvution z may be expressed
20
ils
S.,-(Ay, n) : S.(2, 11\s-8navtu171
10
o
0.t2345
in which Ay is the horizontal distance between the points M1 and. M2.
The spectrum of the lateral velocity fluctuations may be written as
lv1-vrltz
l"l(,lllltf,l 2..1.t1. l)clrcntlcncc 1rl'(i,. upon ly, - yrllz according to [2-g01. copyright
) l()t{ l by l). ltcirlcl l,rrlrlislring ('ornpany.
nS,(n)
l5f
-;T:(r+riJ-s'
r,
Irrrlrrrlcrrcc ilrtcnsity /(1), thc distance
slrowrr in lrig" 2.3.9.
lr, - rrl,
e3.34)
and integral scale
He),
as
2.3-5 spectra and cross-spectra of vertical and Lateral velocity
Fluctuations
It is shown in [2-31
that the spectra of vertical fluctuations up to about 50 m
may be estimated by the formula
(2.3.3s)
'l'he form of Eq. 2.3 .35 was proposed
in [2-661 . Equations 2.3 .33 and 2.3 .35 ,
irr which the parameter/is given by Eq. 2.3.17, are consistent with the retluirement that, in the higher frequency range, the ratio of the vertical and
lrrteral to the longitudinal spectra is equal to 413 [2-651.
Cross-spectra of lateral velocity fluctuations can tentatively be assumed to
bc given by an expression similar toEq.2.3.3l , with exponential decay coeflicients lower by about 33% rhan those used in Eq. 2.3.31
[z-go, 2-go]. Allcrnative expressions for the spectra of vertical and lateral velocity fluctuaiions,
based on a modified von Kdrmdn formulation which takes into account the
variation of spectra with height, were proposed in [2-140].
2.3.6 Dependence of Wind
Speeds on Averaging Time
It fbllows from the definition of the mean value that mean wind speeds depend
upon the averaging time. As the length of the averaging interval decreases, the
c
J
rnaximum mean speed corresponding
N
to that length increases. The relation
bctween the wind speed averaged over / seconds, u,(z), and the hourly speed,
I
X
L/3u1p(z),
may be written
as
c
o
()
U,(z)
:
Uzr,cnk)
+ cG)71n
(2.3.36)
where c(r) is a coeflicient that depends on t and z, is the longitudinal turbulent
lluctuation. If Eqs. 2.2.18 and 2.3.2 are substituted into nq. Z.Z.Ze .
o
12
nllru
FIGURE 2.3.9. t,ongiruclinal cohcrcncc as a lunction o| nl.i,l IJ f ur thrcc valucs o1.
a : l(2.)lx, - .r.,11.;(r) 12 8el. ('opyrigh( () 1979 hy I). ltcitlcl l)rrhlishing (-orrrpirny.
U,(2,)
'l'hc
:
U,,,,n,(.)
0t '' r(tl \
(l tt 2.5 tnetzu)
/
(2.3.37)
cocllii:icnt ('(1) is clclcrtttittetl on llrc lrlrsis <ll's(atistical stuclics of wind
sPcctl rccorcls. llcsrrlts tll'sur'lt slu(licH w('t(' rclx)tlcrtl l-ly I)urst
l2-9 ll anrl arc
70
ilil
n
tMofit't il til(: n()t,Nt)nily tAyt il
:,,1 ll()1il/()NtAt lY N()Nll{}M()(it
Nt
()U:i
lt{)w:;
71
glirllhic lcirttttcs ol llrc tt'rr;rirr) or'1o tlrc nrclcorrlogit'lrl rurlrrr.t'ol tlrc llow (as
irr tltc cuso ol'ttrrpit'irl tyt'krrrcs rlr ol'thurtrlr-:rstorrrrs). Wlrilc thc structure of
llrrizonlally honrogcrrr:orrs llows is basically wcll rrrrtlclsltxrcl, rcsults obtained
in thc study ol'horizorrtully rxrrrhonrogcncous llows urc lo a large extent still
irrcornplctc or tcntalivc. Sornc of these rcsults arc, ncvortheless, of interest to
lhc designer and will thcrcfbre be discussed hcrcin.
1.5
81.4
5
] t'.
2.4.1 Flow
1.2
ln the case dealt with in the preceding sections, of a horizontally homogeneous
lkrw, it is assumed that the surface roughness is uniform over an infinite plane.
In reality, a site is limited in size; the flow near its boundaries is therefore
rrll'ccted by the surface roughness of adjoining sites.
Useful information on the flow structure in the transition zones may be
l_1
1.0
100
r
1
0,000
(s)
l"l(illl{l'l
2.-1,10. It:rrio o| pnrbablc maximum speed averaged over period r to
that
luvcragctl ovct' ()llc lxrur l2-921.
plotted in Fig. 2.3.10, which corresponds to open terrain conditions (zo
= 0.05
m) and an elevation z : l0 m. values of c(t) consistent with Fig. 2.3.10
are
listed in Table 2.3.3.
Experimental results presented in 12-931suggest that Eq. 2.3.36 is applicable, with the values of the coefficient c(t) of rable 2.3.3, to wind
,p""0,
over terrains with roughness lengths of up to Zs : 2.50 m.
Mean speeds used in the design of tall buildings are hourly averages, while
information on wind intensities is currently provided in terms of fastest
mile
wind speeds at about l0 m above ground in open terrain. Fastest mile wind
speeds are averaged over the time required for thi passage over
the anemometer
of a volume of air with a horizontal length of one mile. From this definition
it
follows that for the fastest mile u|the averaging time in seconds is r : 3600/
UJ, w^here Uyis given in miles per hour. For eiample, if UJ.: 90 mph,
then
t :40 s and the corresponding hourly mean is, from Fig. i.z.to,9ofl.2g
70 mph (31 m/s). A recent study 12-1441 essentially.o-nfi.-, the validity =
of
Fig. 2.3.10. For hurricane winds see Sect. 2.4.3.
2.4
near a Change in Surface Roughness
ohtained by considering the simple case of an abrupt roughness change along
:r line perpendicular to the direction of the mean flow [2-13, 2-94,2-95, 2-96,
) 9l ,2-981 (Fig. 2.4.1). Upwind of the discontinuity, the flow is horizontally
lromogeneous and, near the ground, governed by the parameters Zs1 &\d u.a1.
l)ownwind of the discontinuity, the flow will be disturbed over a height h(x).
'f 'his
height, known as the depth of the internal boundary layer, increases with
(ho distance x until the entire flow adjusts to the roughness length zs2 of the
lcrrain downwind of the discontinuity.
If the investigation is limited to the lower portion of the boundary layer, it
rrray be assumed that the flow is two-dimensional. For steady flow, and neglccting the pressure gradient force-the effect of which was shown to be
insignificant [2-98]-the equations of continuity and of balance of momenta
nray be written as
ua+w9!:taJ
0x 0z pAz
(2.4.1)
AU AW
0x 0z
-+-:0
(2.4.2)
Since Eqs. 2.4.1 and2.4.2 contain three unknowns, a third equation is required
Io close the system. In the solution of [2-96] the mean turbulent field closure
was used (F,q.2.1 .9), which, for two-dimensional flow and with phenomerrolrrgical relations similar to those proposed in 12-91 and [2-10] (see Eqs.
HORIZONTALLY NONHOMOGENEOUS FLOWS
Horizontal nonhomogeneities of atmospheric flows may be ascribed either
to
conditions at the Earth's surface (e.g., changes in surface roughness,
topo-
l.
I.
l0-2. l. 13) takes the form
---J1
TABLE 2.3.3. Coefficient c(t)
ll
t:(r)
l0
3.00
2.32
20
2 (X)
30
r .71
-50
r
.35
100 200
1.02 0.70
3(x)
0.54
600 1000
0.3(r 0. l(r
II
--{
3600
0.(X)
zo1,
u
l,'l(Jtll{lt 2.4.1. l;low
125
" fii,{c
11
zorrt's rlowrrwrrrl
ol
rr
t'lurrrgc irr loLrghncss ol'lcr-rlin.
llll
n
tM()lit'l il til(: tl()t,Nt)nny tn yl n
;,
U A(rlpt w dtrlpl _ritll , I (rtp il(rtptfitlt\
0.t6 0x -o.to a,
p oz az \o.ro az I az )
(rlol3/2
-n
- ',L :0
([
o
l)
Ii,ll.wirrg M.nin
: kz
established at distanccs
(2'4'3)
:
N( )Nt t( )M( )(
,t Nt
(
)llt;
I I ( )wl;
73
rrrorc llrrrn -5 krn downward from the roughness
A more "exact" model of the internal boundary layer growth
: o.28zo,(*)'-
I
I
'l
is
(2.4.10)
12-991,
2.5u*t ln
L
(2.4.s)
w:0
puz*t
U:O
1,"-,
, : olo o,. (#),,,1'
l.
)0;z:zozi)
and rough-to-smooth transition. It is approximately valid for values h(x) <
0.26 where 6 is the boundary layer depth. For additional references on flows
near a change in surface roughness, see [2-100] and [2-138].
2.4.2 Wind Flow over Hills
(2.4.7)
wind tunnel investigations of simulated flows over ramps and escarpments are
reported in [2-101, 2-102,2-1031. For open terrain conditions, ratios (u2lu)2
at various stations given in t2-1011 are represented in Figs. 2.4.2 and,2.4.3.
(uz and u1 denote wind speeds at height z above ground downwind and upwind
of the ramp, respectively.) Measurements of l2-lo2l tend to corroborate these
results. The results of [2-101] and 12-1021also suggest that for ramps with
slopes of about 2O% to 35%, the ratios (J2l()1 are, for practical purposes,
independent of slope. However, for a ramp with a l0% slope, the ratios (U2
- U)lUl are only about one-half as large as in the case of a 2O% slope
[2-101]. More detailed wind tunnel measurements of ratios u2lu1 for escarp-
(2.4.9)
-2.1.8).
Equations 2.4.1-2.4.3 with the boundary conditions, Eqs.2.4.5_2.4.9, werc
solved numerically in [2-96) for various values of the parameter m : ln(261/
zoz). In the case of the smooth-to-rough transition, the calculations indicate that
three regions may be distinguished downwind of the discontinuity (Fig. 2.a.D.
In region I (above line AB, approximately defined by a slope or t : iz.5;, the
velocity is essentially equal to the velocity upwind of the discontinuity. This
result is consistent with conclusions reached independently by other authors
12-941 and 12-95). In region III (below line AC, defined by a slope of about
l: 100) it may be assumed, at least very roughly, that the flow is adjusted to
the new roughness conditions, that is, is determined by the same parameters
Zoz, u*z that would control the flow if the roughness length were everywhere
Zs2. rn region II, as the distance downwind from the discontinuity increases,
the velocity profiles deviate increasingly from the profile given by Eq.2.4.5
and the turbulcnc:c cncrgy varics graclually l'rom linc AB, whtre it is prcsunrably
ncarly lhc sillllc:ts tlpwitttl ol'lhc tliscontirrrriiy, lo lirrc,4(1, wlrcrc it lrrtry bc
i
where zs, is the larger of zs, and zo, [2-100]. Equation z.4.lo was based on the
analysis of a considerable number of data and holds for both smooth-to-rough
(2.4.6)
(2.4.8)
w:o
(see Eqs. 2.1.7
ol
h(x)
Zot
r:
tAl I y
change, (2) for a distancc dowrrwirrtl ol' thc roughness change of less than 500
m the profile is the same as upwind ol'the discontinuity, and (3) in the interval
500 m < "r < 5 km the profile is logarithmic below line AB, with zero speed
at the ground surface, and a speed at elevation x/12.5 equal to the speed at
that elevation upwind of the roughness change t2-421.
e.4.4)
ir is assumed in [2-96] that in Eq. 2.4.3
rlrt' srunt' cxPressiorr lirr /. holils ncar the ground throughout the flow, inituding
(lrc rlistrrrlrt'tl lkrw tlowrrwind ol' thc discontinuity.
'l'lrc lrorrntliuy corrtliliorrs lilr Eqs. 2.4.1-2.4.3
are
IJ
)N
describccl irr lclrrrs ol llrt'I;u;rnr(:l(:r's io2, u*2. For practical purposes it may be
assumed that ( l) tlrc Plrlilt' t'ollt'sporrcling to these parameters is completely
in which l, is the mixing length.
In horizontally homogeneous flow, the validity in the surface layer of the
Iogarithmic law implies the following expression for the mixing length
[2-l]:
L
,t ll( )lit,/(
i
ii
ii
l
i
L
I
I
I
I
I
l
L
l
ri
l'l(;tll{l,l 2.4.2.
Wirrrl lrtoltlt:, r)\'( r ;ul
(':i(
iul)nr('nl
ll lOll
74
iltf ArM(xil,ilf nto t(tt,Nt)Any tAyt
tt
'4 ll()|il,/oNlnl lY
N()Nl t()Ml
,{it Nt (}(,li |()wl;
t f" .l 't.ttt,\lt.\lt ,l
,, J ,, elt,l
(2.4.t2)
lnd
I /L\""
//.r : g
\;/
(The quantity / is the thickness of the intemal boundary layer created by the
change in surface shear stress as the air flows over the hill. This internal
boundary layer is similar to that caused by changes in terrain roughness.) For
any hill symmetric about x : Q, ;(0) can be expressed in terms of Kelvin
lunctions as shown in [2-104]. In the particular case
FIGURE 2.4.3. Wind profiles over an escarpment t2_l0ll.
/x\ :
r\z)
T +1ld
ments with 25%, 50%, and loo% slopes and for a cliff, as well as measurements of the root mean square of the longitudinal turbulence fluctuations, are
reported in [2-103]. The ratios urlu, of t2-1031 are similar to those of Fig.
2.4.2, except at low elevations (about 5 m above ground) where they are larger
by about20%.
Results of theoretical and numerical studies of wind flows over hills have
been reported in [2-lM, 2-105,2-106,2-lo1 ,2-108]. For a hill with maximum
height ft, a longitudinal scale L(L >> /r) and a profile hf(xlL), wheref(xlL)
< 1 (Fig. 2.4.4), the following resulr was obtained in [2-1041:
U,
.l
_-_::
*
ul
(2.4.13)
ho ln2(Llz$tr(o)(", z)
Lln(llz) rLn(zll) + ln(//zo)l
I
(2.4.t4)
in which L is the horizontal distance from the top of the hill to the point at
which the height is half the maximum height ft, the quantity o : l. Values of
l?(0) corresponding to the profile 2.4.14 are represented in Fig. 2.4.5 atxlL:
0 (top of the hill), xlL: -0.5 and xlL:0.5, for llzo: 1gz, Llzo:2.1 t
lOa (curves A), llz(): 10a, Llzo : 3.2 x 105 (curves B), and llzo : 1gs,
Llzo:3.6 x 106 (curves C). Values of t(U2 * U)lUl (L/ft) calculated in
l2-l04l are listed in Table 2.4.1. The analysis and results of [2-104] are valid
(2.4.11)
:
in which
U2 is the wind speed at (x, a), U1 is the wind speed at (x
-@, z),
ze is the roughness length, x is the horizontal distance (see Fig. 2.4.4), z is the
height above surface of the hill at the point considered, il(0) is the approximate
value of a dimensionless quantity representing the perturbatlon to the upwind
velocity due to the presence of the hill,
0
L
li'l(Jllltl,l 2,4.4. l,rolilc: ol l low lrill.
-
01l
l,'l(;tll{U 2.4.5. Vulucs ol
f
;low Ovcr a Low
llill."
r?("'
Qrtrtrt
o4
08;(.))_>. -0.2-0.1 6 iio)e
, "
)=ou
l;rortr l' S .l;rt'ksotr:rtttl . ('. ll. lltltl. "'l'urbulcrrrt
.lt,ut llrtvtl ll,'1,'r,trtl . ,lttt.. ll)l ( 197.5). 929 ().55,
.1
76
llt
n
tM()sl,lFilto n()(,Nt)Ally tAyt
n
\l IYN()Nl l()M()(il Nt t)t,i; lt()Wl;
TAIII-|t 2.4.1. Values of l(U, - Il)lIltl(l,lltl il 'l'op ol'lliil
zll
zull : 10 '
,.t/l - ft) 4
0.0
0.6
2.09
2.46
2.33
2.20
1.0
2.O8
t.5
1.9'7
2.1
l .88
0.1
0.3
;,,// - l0
1.87
t.72
2.13
2.07
1.92
l .85
1.78
1.97
1.87
1.79
1.73
5
t.72
1.66
t.62
lirl hills in rural tcrrain (zo : 0.03 m) with 0.1 < Z < l0 km and with ratios
l(1,,/1,)0 '. For cxample, if a6 : 0.025 rr7, L :500 m, and h : 25 m,
tlrcrr, lirrrrr |tq.2.4.13,2.111 l: 1.0 x l0 3, to which there corresponds, from
'l'ablc2.4.l, UzlUt: l.l2atzll:0.1 (or z 2.5 m). Thetheorybecomes
=
lcss accuratc in rough terrain (zo : 0.5 m), the actual speeds U2 being lower
hll, >
than thosc givcn by Table 2.4.1.
For flow over escarpments (Fig. 2.4.6), the following relation is derived in
[2- lOe]:
!:=r+!!1n(Ltzdh"' (zlLi)2+U+(x/L)12
Ut
L
4r ln(zlzi
klL\2+.1-(xtL)12
(2.4.rs)
in which notations similar to those of Eq. 2.4.1r are used. It is suggested in
12-1091that Eq. 2-4.15 may be applied to flow over escarpments with t <<
5 km and with slopes as large as 202" or so. For example, if L : 250 m,
h:50 m, and zo:0.025 m, for x: L andz: l0mthe ratio U2lU,:
1.19. According to [2-1031, Eq. 2.4.15 provides useful indications of the trends
x and z, rather than dependable quantitative
of the variation of u2lu1 with
results.
Full-scale and wind tunnel measurements of flows over two- and threedimensional hills and over embankments are reported in [2-l0g] (which extends
the analytical approach of [2-1041ro three-dimensional hills), and
[2-tlo,2-lll,
2-112, 2-113,2-114, 2-115,2-l4zl. As noted in [2-ll0], estimates obtained
independently in l2-lo4l, [2-105], and [2-106] agree well with each other and
with the full-scale measurements of [2-110].
A sirtrplc: ttterlltrxl lot r'itlt'ulrrlirrg wirrtl spr:ctl irrt.r't.:rscs ( "spectl-ups") lirr
buildings l<lcalctl ()ll lw() tlitttt'ttsiotral ridgcs or cscarl)nrcnls or orr
axysinrrnctric
hills is includccl irr llrc AS('lj 7-9-5 Stancl:rrcl
12-l'3gl iintl, in corrrputcrized fbrm,
of thc cliskcilc
cornpu(cr-llascrl Vcision of ASCE
7-95 Standard Provisi'ns krr wind Loads"
[17--51 appcnded to this book.
as parl
"l)cvckrpmcntal
2.4.3 The Hurricane Boundary Layer
The horizontal inhomogeneity of a hurricane wind flow over a uniform,
horizontal surface is associated with the variation of the pressure gradient
with
distance from the centerof the storm (see Eq. 1.3.1). tn aerivinfthe
logarithmic description of the mean velocity profilei near the ground
ftq. Z.Z.fS; it
was assumed that the flow in the free atmosphere is geostrophic
lSect. 2.2).
This assumption does not hold in the region of highest winds of the mature
hurricane; the question therefore arises as to wheiher or not Eq. 2.2.1g
is
applicable in this region.
Several analytical solutions of the hurricane boundary{ayer problem
have
been atempted so far [2-116,2-117,2-llg,2-llg, z-l2iJ], att
oi wnicn appty
to steady, axisymmetric mean flows. The solutions of
[2-116] through p lirjl
are based on the assumption that the eddy viscosity is constant,
and they .unnoi
therefore provide a reliable detailed description of the flow near the ground.
A
considerably more realistic modeling of the turbulence effects is
used in
l2-l2ol, in which the equations of motion and continuity are supplemented by
the turbulence closure relations discussed in Sect. 2.1 (Eqs. 2.1.g-2.1.
13). Th;
system of equations thus obtained-in which the expre.ssion for the pressure
gradient field given by Eq. 1.3.1 was used-was solved numerically
assuming
values of the surface roughness of 0.002 m to 0.90 m, differences
between the
high pressure in the far field and the low pressure at the storm center
of 60 mb
to 140 mb, and radii at which the gradient wind has a maximum value of 30
km to 50 km. According to 12-1201, in the lowest 400 m of the boundary
layer
the mean wind profiles differ only insignificantly from the logarithmic profiles
described by Eq. 2.2.18.
As Table 2.4.2 shows, for decaying hurricanes the increase of mean wind
TABLE 2.4.2. Yariation of wind speeds with Height in Hurricanes caror
and
Edna
Height
Carol
above Ground
(m)
l,'l(Jlllll,) 2.4.(r. lrlow
ovL:t'
cscitrpnlcllts noltrlions
l2
1091.
77
Edna
Mean
Max. l-rnin
I 1.3
t4.s
22.8
22.9
18.1
l().
45.7
24.1
t58
r08.2
t25.(l
29.1
l.t tl
Mean
Max. 1-min
t.8
17.0
20.3
25.9
259
30
r
I
rJ
78
IltF ATMospHEntc frot,ND^ny iAyt
y
n
HoFtzoNTAt
spceds with hcight in appnrxintalc ircconlirrrcc wilh tho logarithnric l1w w1s
documented in 1954 following thc passagc ovor lJnxrkhaven National Laboratory of hurricanes Carol and Edna 12-121, p. 461.
More recently [2-122] reported extensive observations of mean wind speeds
recorded at elevations from 9. I m to 390 m during the passage of four decaying
tropical cyclones over northwestern Australia. The mean wind profiles were in
most cases irregular, and as noted in [2-1221, a wind speed maximum was
often observed at 60-200 m. Nevertheless, the profiles corresponding to the
largest l0-min wind speed observed during each storm at 9.1 m were by and
large consistent with the logarithmic law and a roughness length of 1 to 4 cm,
as can be seen in Table 2.4.3, in which the only significant anomaly is the
speed observed during cyclone Karen at 59.7 m elevation.
whether the logarithmic profile holds in the case of mature hurricanes remains an open question. Implicit in the provisions of the 1975 Southern Building code 12-1231is the assumption that hurricane wind profiles are considerably
flatter than would be indicated by the logarithmic law. To date there is no
conclusive evidence that this is the case. Since design wind speeds specified
in building codes correspond to an elevation of l0 m or so, the use of this
assumption in the design of tall structures might be imprudent.
According to a study of tropical storm and hurricane records peak gust
factors for hurricane speeds are about lo% higher than indicated in Fig. z.3.lo
for extratropical storms t2-1351. The conclusions of 12-1351(see also Iz-1391,
p. 155) were based on the analysis of about 12 records.
Reference !2-l4ll contains information on gust factors, longitudinal turbulence intensities, scales and spectra in typhoon Mireille, that traveled over
omura bay (Nagasaki) and passed directly over the anemometer placed at
100-m elevation on a tall building at the shoreline. The gust factors were found
to decrease as the mean speed increased. The turbulence intensity during the
period of the strongest 10-min wind (about 25 mls) was over 25%. Theturbulence scale was estimated to be 780 m during that period and 2g0 m during
the 10-min period preceding it. The von Kdrm6n spectrum (Eq.2.3.24), with
thc lurbulonco scitlc:s.ittsl irrtlicatotl, rnatchctl thc rncirsul'ctl spcctra wcll, cxcept
lirr thc rangc ol'abrlut 0.o25 to 0. l5 Hz, whcrc it undcrcstimate<l the measured
spectra by as much as l0]ol, fbr certain frequencies. Finally, according to
l2-l4ll, surface wincl spccds in the eye can be comparable with or higher than
the estimated speeds at the gradient height level; see also comments following
Eq. 3.3.7 and Ref.
I
N( )Nt t( )M( x
lt Nt ()t,s I I ow$
13-791.
Two more notes on hurricane winds are in order. First, in the immediate
proximity of the eye, flow separation occurs and the boundaryJayer assumptions break down (see Sect. 1.3). The implications of this phenomenon to the
designer are not yet well understood. Second, as the hurricane moves inland,
filling occurs (see Sect. 1.3) and the maximum winds tend to decrease. Empirical descriptions of the wind intensity reduction as a function of distance
from the coastline were proposed in 12-1241, [2-1251, and [2-l5l]. According
to [2-1251, the ratios of peak gusts at 50 km, 100 km, and 150 km inland to
peak gusts at the coastline are, approximately, 0.90, 0.80, and 0.70, respectively. See also [3-57] to t3-601.
A hurricane wind speed record, which clearly indicates the passage of the
eye, is shown in Fig. 2.4.7. The nonstationary character of the record of Fig.
2.4.7 is noteworthy, as is the contrast to Fig. 2.3.1. For techniques to characteize turbulent fluctuations for nonstationary records, which are typical of
hurricanes but characterize other storms as well, see [A2-14] to [AZ-241.
2.4.4 Thunderstorm Winds
The cold air flow which, in a thunderstorm, spreads
horizontally over the
ground was compared in Sect. 1.3 to a wall jet. Just as in the case of the wall
jet, the surface friction retards the spreading flow, which may thus be expected
to be similar, near the ground, to an ordinary boundary layer [2-126, 2-127,
2-1281.
of particular interest to the designer is the so-calledfrst gust (or gustfront),
that is, the wind occurring in a thunderstorm that exhibits a considerable and
relatively rapid change of speed and direction (Fig. 2.a.8). Following l}-l}9l
and [2-130], the wind speed increase and the time interval during which this
TABLE 2.4.3. l0-min Speeds at Various Elevations Corresponding to
Maximum 10-min Speed at 9.1 m during Four Tropical Cyclones
Height
above Ground
(m)
Wind Speed (m/s)
Beryl
(12173 12:00)
9.1
59.7
21
19t.4
32
Trixie
Beverly
Karen
18:30) (3175:2r:O0) (3177; t90O)
(2175;
22
28
31
32.5
39.5
41
279.2
390.
r
30.5
5l
43.5
48.s
36
34
.57..5
Noto: Nurrrbcrs ilr pirrcrrlltcscs irulit'irtc tltc rrronth, ycar, trrrrl lrorrr ((iM'l')
4ti..5
79
l0
MDIIT
f,'l(;tlRl,l
2.4.7
,
I
Plrl
lrrrricrrrrr. wirul sllcctl reconl.
I
lll
n I M(
): it
,l
ll lll( I ll( )l,Nl
)/\t
ty
I n.,
I
I
n lMr
I
c. l77m
f. 444m
'"t ilI
t{tl
.
tt()t ,Nt)nl
t:;{)il ()|t nlJ lt()w
ty lnyl it tllt(
Bl
itttc:rvitl A/: (l) rr;t lo l(X) rrr :lltovt' gtrrrrrrtl wirrtls slrt't'tls v:rry wrllr lrt:iglrt irr
accortlancc witlr llrt'1o1,:rrrlltrrric lrrw, urrtl (l)) irlxrvr' l(X) lrr llrt'vtrr-ilrti1ll gl
wincl spccds witlr hc:iF,lrt is Ircgligiblc. ('t'his is rerrsolrirlrl-y t'orrrplrliblc with the
rccords of Fig. 2.4.8.) Ntl rolation betwccn wirrtl spcctls in tlillorcnt rclughness
regimes, based on a rational model of the thunclcrstorrrr wind lklw, has been
derived so far. To convert thunderstorm wind spccds rccordcd over open terrain
into wind speeds over built-up terrain, the samc procedure is used in current
practice that is applied to extratropical cyclone winds (Eq. 2.2.26), even though
the notions of gradient height and gradient speed have no meaning in the case
of thunderstorms. whether or not this practice is acceptable for structural engineering purposes is a question that merits investigation, particularly if it is
recalled that, according to [2-131], about one-third of the extreme wind speeds
recorded in the United States are associated with thunderstoms.
2.5
ATMOSPHERIC BOUNDARY LAYER EFFECTS ON OCEAN
FLOW
If
suflicient data on wind flows over the ocean are available, it is possible to
model the ocean waves induced by those flows. Such modeling is referred to
as hindcasting. A vast literature on this topic is available (e.g., see [14-34,p.
5201).
b. 90m
e. 355m
Mechanisms by which kinetic energy is transmitted from the atmospheric
boundary layer to the ocean water are exceedingly complex. In some applications one may assume, however, the existence of uniform surface shears at
a hypothetical horizontal ocean/atmosphere interface (see Sects. 2.2.2 and
2.2.3). one among many instances were this assumption is used is the recent
modeling of wind-induced along-shore ocean currents over bottom topography
characterized by comrgations normal to the shore. It was shown in [2-145] that
the equations of motion of the wind-induced ocean motions can be represented
approximately by the equations
* :
a. 45m
a
AyH(x,
y, z) -t eg{x, t)
d
i : - u* H(x. y.;) *
2 : eg(x, z, t)
d.266m
FIGURE 2.4.8. Thunderstom wind speeds recorded simultaneously at six elevations
from 45 m to 444 m above ground near Oklahoma City (courtesy of National Severe
cg2())
(2.s.D
Storms Laboratory, National Oceanic and Atmospheric Administration).
increase takes place will be referred to as the gust size A,V and the gust length
At, respcctively. Depending upon thunclerstorm intcnsity, thc gr-rst sizc rnuy
vary appnrxirtr:t{cly l'roltt S ltt/s to 30 rtr/s, whilc lhc gtrst lengtlr nuly rangc
ll'otn ir l'cw rrrirrulcs (o ?0 rrrirrrrlt's ()r-s().
'l'lrc llrtlllrlt'r'slotttt wiltrl tt'trtllls n';xrr'lt'rl irr
l.) | lOl r;rr1'1'r':;t ilrrrl rlrlrirrl'tltt'
where e is small, x is a basic along-shorc specd, y is proportional to the outol-phase component of a strcarn lirnction lilr rnoli<tn due to the topography, z
is thc cnergy-cnstropy, ancl
,(r
,tj,
= ,:r t
r1y
,.\'. ,(r
I t\tl.
,.
1,, I {r
l)1r,, I tltll
(l-5 l)
82
lltt AIM(xitlll tito tlot,Nt)nny tAvt
H(x, y, z)
:
j.y" + zx
t
,l(r,lil
n
r:).*' 1t'r1l;xn,
nt I I nt N(it
,1;
=I
tD'l
2ll
2-12
and where 6 is the amplitude of the bottom topography comrgations, er is a
friction coefficient related to the eddy viscosity of the ocean flow, and €ze and
er(t) arc, respectively, the steady and fluctuating wind stress at the ocean
surface. The wind stress fluctuations of interest in this problem correspond to
the very low frequencies studied in [2-1431. Reflecting the effect of the bottom
comrgations, Eqs. 2.5.1 form a bistable system capable of chaotic behavior
even if the fluctuating wind stress is assumed to be harmonic, as was done in
2-13
12-t451.
The model used in [2-145] becomes more realistic if it is assumed that the
wind speed fluctuations are random, rather than harmonic. Using chaotic dynamics techniques developed in [6-101], and van der Hoven's results on the
spectra of low-frequency wind speed fluctuations [2-143], the case of random
wind excitation was studied in [2-146]. Among other results, [2-146] provides
estimates of lower bounds for the probability that during a specified time interval, the amplitudes of the wind-induced fluctuating currents do not exceed
a safe threshold associated with the barrier of the unforced system's double
potential well.
REFERENCES
2-l
2-2
2-3
2-4
2-5
2-6
2-1
2-8
H. Schlichting, Boundary Layer Theory, McGraw-Hill, New york, 1960.
L. T. Matveev, Physics of the Atmosphere, TT 67-5 1380, National rechnical
Information Service, Springfield, VA, 1967.
J. L. Lumley and H. A. Panofsky, The stucture oJ'Atmospheric Turbulence,
Wiley, New York, 1964.
H. A. Panofsky and J. A. Dutton, Atmospheric Turbulencc: Models and Methods for Engineering Applications, Wiley, New York, 1984.
A. s. Monin and A. M. Yaglom, statistical Fluid Mechanics: Mechanics of
Turbulence, Vol. 1, MIT Press, Cambridge, 1971.
J. F. Nash and V. C. Patel, Three-Dimensional Turbulent Boundary lnyers,
S.B.C. Technical Books, Atlanta, 1972.
Computation of rurbulent Boundary rrtyers, proceedings of the AFOSR-IFp
Stanford Conference, Vols. I and 2, Stanford Univ., 1968.
A. A. Townsend, The Structure of Turbulent Shear Flow, Cambridge Univ.
Press. Cambridge, 1955.
2-9
2-10
P. Bradshaw, D. H. Ferris and N. P. Atwell, "Calculation of Boundary Layer
Developmcnt Using thc Turbulent Energy Equation," J. Ftuid Mech., 28
11967;.59.1 6l(r.
J. li. N:rslr, "'l'ltc ('rrlctrl:rliott ol''l'hrce-l)irncnsional 'l'urbulcnl Bourxlluy l,lycrs itt lnt'ontplt'ssilrlt. lilow," .1. l;lttitl Mtclt.,.l7 (l()69), 625 M2.
2-14
2-15
2-16
83
(1. dul). l)orurltlsorr, "('lrlculatiort ol"l'ullrrrk:rrl Slrclrr'likrws lirr Atrnosphcric
ancl
(2.s.3)
l;
Vollcx Moliorrs," AlAA.l
., 10, I (Jan. 191')1,4
12.
J. L. l,urrrlcy rrrrrl li. Khajch-Nouri, "(lrnr1'rrrt:rlionll Mrxlcling of Turbulent
Transport," Adv. Oct4thys., l8A, Acadotnic, Ncw York (1914), 169-192.
K. S. Rao, J. C. Wyngaard and O. R. Cotd, "The Structure of the TwoDimensional lnternal Boundary Layer over a Sudden Change of Surface Roughness," "/. Atmos. Sci., 31, 3 (April 1974),738-746.
G. T. Csanady, "On the Resistance Law of a Turbulent Ekman Layer," J.
Atmos. Sci., 24 (Sept. 1967),467-471.
S. R. Hanna, Characteristics of Winds and Turbulence in the Planetary Boundary Inyer, Technical Memorandum No. ERLTM-ARL 8, ESSA Research Laboratories, Oak Ridge, TN, 1969.
G. B. Schubauer and C. M. Tchen, Turbulent Flow, Pinceton Univ. Press,
Princeton, NJ, 1961.
2-17
C. B. Millikan, "A Critical Discussion of the Turbulent Flows in Channels
and Circular Tubes," in Proceedings of the Fifth International Congress of
Applied Mechanics, Cambridge, MA, 1938.
"Similarity Theory and Geostrophic Adjustment," J. Royal Meteorol. Soc., 94 (1968), 586-588.
R. H. Clarke, "Observational Studies in the Atmospheric Boundary Layer,"
J. Royal Meteorol. Soc., 96 (1970), 9l-114.
E. J. Plate, Aerodynamic Characteristics of Atmospheric Boundary ktyers,
U.S. Atomic Energy Commission Critical Review Series, 1972.
2-18 A. E. Gill,
2-19
2-20
2-21 H. Tennekes and J. L. Lumley, A First Course in Turbulence, MIT Press,
Cambridge, 1972.
2-22
2-23
2-24
2-25
2-26
2-27
E. Simiu, "Logarithmic Profiles and Design Wind Speeds," J. Eng. Mech.
Div., ASCE,99, No. EM5, Proc, Paper 10100 (Oct. 1973), 1073-1083.
E. L. Deacon, "Geostrophic Drag Coefficients," Bound Inyer Meteorol., 5,
3 (July 1913),321-340.
A. K. Blackadar (penonal communication, Feb. 1976).
J. A. Businger et al. "Flux Profile Relationships in the Atmospheric Surface
Layer," J. Atmos. Sci., 28 (1971), 788-794.
H. Tennekes, "The Logarithmic Wind Profile," J. Atmos. Scl.,30 (1973),
234-238.
J. Wyngaard, "Notes on Surlace Layer Turbulence," in Proceedings of the
American Meteorological Society Workshop on Micrometeorology, Boston,
1972.
2-28
D. R. Caldwell, C. W. Van Atta, and K. N. Helland, "A Laboratory Study
ofthe Turbulent Ekman Layer," Geophys. Fluid Dyn.,3 (1912),125-160.
2-29
G. C. Howroyd and P. R. Slawson, The Characteristics of a Laboratory Pro-
l,aycr," Iilnnd- Ityer Meteorol., 8 (1975), 201-219.
R. H. Thuillier ancl V. O. L:rp1x'. "Wirll :rntl 'fcmperature Profile Characteristics l'rorn C)bservations orr rr l,lO0 li'lowcr," .1. Appl. Meteorol..,3 (June
t9(A),299 306.
duced Turbulent Ekman
2-30
2-31
A. K. lllackatlaruntl Il.'l'crutt'Lcs.'Asyrrrplolit Sirrrilulity in Ncutnrl Banrtnrpic
llotrttllrt'y l.lryct's," .l . tllrrt,t,r. ,\', r 15 (Nov l(X)li), lOl5 lO2O.
ilil
l .l-l
n tM(): :t't
il lll( . lr()t,Ntr/\ny tn itll
lil I I ilt N(:t
l). M. ('ltr'1, 1. ('. 1';rllrt'll lrrrrl ll A l':rtt,rlr;1.y. 'l'rrrlilcs ol Wrrul :rrul lt.rrr
'lowcls tlvt'r lloilrol'tnt'orrs lt'rr:tr'r,.1 . tlltttl,s.,\,i., -\0 l.lrrly
f)L:ritltllL: l.nrltt
1912t.188 194.
2-33 N. Cl. Halliwcll, "Wintl ()vcr l.otttlotr. it I'nx'rulings ol thc 'l'hird Intcrnrt
tional Confcrcnrc oLt Wind lil.li'r'r,s ott lhtiltlirt,qs und Strudurc.r, Tokyo,
2-34
2-35
197
l,
Saikon, 1972, pp.23 32.
E. L. Deacon, "Gust Variation with Hcight up to 150 m," J. Royal Meteorol.
Soc., 91 (1955), 562-513.
R. I. Harris, "Measurements of Wind Structure at Heights up to 598 ft Above
G round Level , ' ' in Proceedings of the Symposium on Wind ElTects on Buildings
and Structures, Loughborough University of Technology, Leicestershire, U.K.,
2 50
W. W. l':t1',,rtt, Wrrrrl Vclotily ilr llt'llrliorr to llt.ry'lrt Alxrvt'(ilrrlttl." /tir1.3.
/lr'r',. I l.l (M;rv l() 15)" /.ll 7,1.5.
2-51 li. l)lrstltrill, "Wirrtl Slrttt'ltttc irr thc: Alrttosplrt'rit' l}rrtrulruy Laycr." in A Dis
t'ussi.ott rttr At<ltitr'ttuntl A<nxl_vtrutrtics, l'ltil.'l'nut.s. lilty. Stx.. London, A269
Nry',s
York.
2-52
2-53
2-54
2
l1
2
3u A. C. Chamberlain,
2-39
and Wiley, New
1960.
P. S. Jackson, "On the displacement height in the logarithmic velocity profile,"
.t. F.luid Mcch., lrr (1981), 15-25.
"Roughness Length
of Sea, Sand, and Snow,"
Use in Space VehicLe Development, l97l Revision, NASA Technical Memorandum TM X-64589, George C. Marshall Space Flight Center, Ala. (1993
Revision edited by D. L. Johnson.)
2-56
2^57
fttr
2-58
2-40 H. R. Oliver, "Wind
2-41
2-42
Profiles in and Above a Forest Canopy," J. Royal Meteorol Soc.,97 (1971),548 553.
P. Duch6ne-Marullaz, "Full-Scale Measurements of Atmospheric Turbulence
in a Suburban Area," in Proceedings of the Fourth International Conference
on Wind Effects on Buildings and Other Structures, London, 1975, Cambridge
Univ. Press. Cambridge, 1976, pp.23 31 .
J. Bidtry, C. Sacr6, and E. Simiu, "Mean Wind Pr<rfiles and Changes of Terrain
Roughness," J. Struct. Diy., ASCE, 104 (Oct. 1978), 1585-1593.
2-43
S. D. Smith and E. G. Banke, "Variation of the Sea Surfacc Drag Coefficient
with Wind Speed," J. Royal Meteorol. Soc., l0l (1975), 665 673.
2-44
J. Wu, "Wind Stress and Surlace Roughness at Air-Watcr lnterflace,"
ph1,s. Res., 74 (1969) 444-445.
2-45
J. Amorocho and J. J. deVries, "A New Evaluation ol'thc Wind Stress Coefficient over Water Surfaces," J. Geophys. Res., 85 ( l9U0), 433-442.
J. R. Garratt, "Review of Drag Coellicients over Oceans and Continents,"
Mrnthly Wearher Rev., 105 (1977),915-929.
2-46
J.
Proceedings ol the Symposium on Wind Ellects on Buiklings and Structures,
Vol. l. National Physical l-ahoratory. Tcddington, U.K., Her Majesty's Sta-
2-49
tioncry Ollict'. l.ortkrn. l9(r5, pyr.53 102.
Arru'ritttrt Nttliotrrtl ,\lttrtrhtnl A.5ll. I l9lt2, Mitrirrrtrtrr I)asigrt ltxul.s .litr llrtild
itr,q.s trtttl ()llrr't ,\ltrtr'lrn.r'.r, Atttt'tit':rn N:rliornl Sllrnrlrutls llrslilrrlt'. lrrt., Ncw
Yotk.
l()i'i.)
don, 1965, pp. 30-51.
P. Tattleman, D. D. Grantham and I. I. Gringorten, Extreme
Wind Speeds, Gustiness and Variations with Height for MIL-STD210B. Technical Report No. AFCRL-TR-0560, Air Force Cambridge Research Laboratories, 1973.
J. Bi6try, personal communication, July 1976.
S. Schotz and H- A. Panof.sky, "Wind Characteristics at the Boulder Atmospheric Observatory," Bound. Inyer Meteorol., 19, (1980), 155-164.
A. Venkatram, "Estimating the Monin-Oubkhov Length in the Stable Boundary
Layer for Dispersion Calculations," Bound. lnyer Meteorol., 19 (1980), 481
485.
2-59 E. Simiu, "Thermal Convection
2-60
2-61
and Design Wind Speeds," J. Struct. Div.,
ASCE, 108 (July 1982), t67t-t675.
M. Shiotani, Structure of Gusts in High Winds, Parts l-4, The physical Sciences
Laboratory, Nikon University, Furabashi, Chiba, Japan, 1967-1971.
J. Counihan, "Adiabatic Atmospheric Boundary Layers: A Review and Analysis of Data from the Period 1880-1972," Atmos. Environ., g (19j5),8jl905.
2-62 P. Duch6ne-Marullaz, "Effect of High
Roughness on the Characteristics of
Turbulence in the Case of Strong Winds, Proceedings Filih Inlernational Conference on Wind Engineering, Vol. l, Pergamon Press, 1980.
Geo-
2-4'7 G. Hellman, "Uber die Bewegung der Luft in den unterstcn Schichten der
Atmosphire, " Meteorol. Z. , 34 (1916) 273.
2-48 A. G. Davenport, "The Relationship of Wind Structure to Wind Loading,' in
orol. Soc.,98 (1912),469 494.
P. R. Owen, "Buildings in the Wind," J. Royal Meteorol. Soc.,97 (1974),
396 413.
H. C. Shellard, "The Estimation of Design Wind Speeds, in Prot:eedings of
the Symposium on Wind Effects on Buildings and Structures, Vol. l, National
Physical Laboratory, Teddington, U.K., HerMajesty's Stationery Office, Lon-
2-55 N. Sisserwine,
Bound.
Inyer Meteorol., 25 (1983), 405-409.
G. E. Daniels, (ed.), Tenestrial Environment (Climatic) Criteria Guidelines
). 4 1() zl5(r
F. Pasquill, "Sonrc Aspccts of Boundary l,aycr'[)cscription," J. Royal Mete( 197 l
l96rJ.
2.16 O. G. Sutton, Atmospheric TurbuLence, Methuen, London
:
2-63 J. O. Hinze, Turbulence,2d ed., McGraw-Hill, New York, 1975.
2-64 H. W. Teunissen, Characteristics of the Mectn Wind and Turbulence in the
Planetary Boundary Inyer, Review No. 32, Institute for Aerospace Studies,
University of Toronto, 1970.
2-65 N. E. Busch and H. A. Panofsky, "Recent Spectra of Atmospheric Turbu2-66
lence," J. Royal Metartrol. Stx'., 94 (1968), 132 148.
J. C. Kaimal et al., "Spcctritl ('llrnrctcristics of Surlace-Layer Turbulence,"
J. Royal Metertrol. &x.. 9lt ( l()7.1). 56] .5U9.
2-67 I. A. Singcr. N. E. llLlstlr :rrrtl .l A lirittttl't. "'l'hc
Micnrrnctconrkrgy ol the
'l.urbulcnt Flow lricltl ilt lltt'Alttt,':;plu'rr' llrtrrlllrry Surlircc I-aycr." in Pn>
t'ccdings tt lltt lrtltrrrttlirtttrtl l('sr'ttr,lt .\,'rrtittttr ttt Wirrt! l,),llt,t't.s ort l)ttiltlin,q.s
ultl ,t;lrtt(tttr(,r, ()llltwlr. Vol |. Ilrrrlr'r:,rly ol I'olrrrlo Itlt'ss, lirrrrrlo, l()(rll,
pp. .557 5()4.
tl6
ttlt
AIM():;t't
iltrt{.lourJr)nnyrnyr l
I (rtt li. liit'tllt'r rrrrtl ll. A. l':rtt,rlsl'y. t\lrrror,lrlrt'rrt Stlrles :rtul SPt'tlr:rl ( ilrPs."
llull. Arrr. Mcltttntl. Jor'.. 51, Ll (lrtt l()/O). lll.l lll().
2-69 G. H. Ficlttl alt(l (;. li. McVr'lril. "l.ottlirltttlirt:rl rrrrtl Litlclal Spt:clnr ol 'l'ttr
2-70
bulencc in thc Atrnosphcric llorrrul;u.y Liry('r'rrl llrc Kcnncrly Spacc (lcntcr, ./.
AppL. Metcor. (Scpt. 1970), -5 I ().]
E. Simiu, "Wind Spectra and I)ynarrric Akrngwirrd Rcsponsc," J. Struct. Dir.,
ASCE, 100, No. ST9, Proc. Papcr lOltl5 (Scpt. 1974), 1897-1910.
2-11 A. G. Davenport, "The
2-72
2-13
214
2-15
2-76
Spectrum of' Horizontal Gustincss Near thc Ground in
High Winds," J. Royal Meteorol. Soc., 87 (1961), 194-211.
Canadian Structural Design Manual, Supplement No. 4 to the National Building Code of Canada, Associate Committee on the National Building Code and
National Research Council of Canada, Ottawa, 1971.
F. Pasquill and H. E. Butler, "A Note on Determining the Scale of Turbulence," J. Royal Meteorol. Soc., 90 (1964),19-84.
E. Simiu and S. D. Leigh, Turbulent Wind Effects on Tension Leg Platform
Surge, Building Science Series BSS 151, National Bureau of Standards, Washington, DC, March 1983.
H. A. Panofsky and I. A. Singer, "Vertical Structure of Turbulence," J. Royal
Meteorol. Soc., 9l (1965), 339-344.
A. G. Davenport, "The Dependence of Wind Load upon Meteorological Parameters," in Proceedings of the International Research Seminar on Wind
EIJ'eos on Buildings and Structures, University of Toronto Press, Toronto,
pp. 19-82.
B. J. Vickery, "On the Reliability of Gust Loading Factors," in Proceeding,s
1968,
2-71
of the Technical Meeting Concerning, Wind Loads on Buildings and Structures,
National Bureau of Standards, Building Science Series 30, Washington, DC,
1970, pp. 93-104.
2-'78 M.
Shiotani and Y. Iwatani. "Conelations of Wind Velocities in Relation to
the Gust Loadings," in Proceedings of the Third International Conf'erence on
Wind Elfects on Buildings and Stuctures, Tokyo, 1971, Saikon, Tokyo, 1972,
pp.57-67.
2-79
S. SethuRaman, "Structure of Turbulence overWaterduring High Winds," -/.
Appl. Meteorol., L8, (1979), 324-328.
2-8O L. Kristensen
and N. O. Jensen, Lateral Coherence in Isotropic Turbulcncc and
in the Natural Wind," Bound. Layer Meterol., 17 (1979),353-373.
2-81 L. Kristensen,
H. A. Panofsky, and S. D. Smith, "Lateral Coherence of Longitudinal Wind Components in Strong Winds," Bound. lnyer Meteorol., 2L
(le8l). le9-205.
2-82
2-83
C. F. Ropelewski, H. Tennekes, and H. A. Panofsky, "Horizontal Coherence
of Wind Fluctuations,' Bound. Inyer Meteorol., 5 (1975),353-363.
S. Berman and C. R. Steams, "Near-Earth Turbulence and Coherence Measurements at Aberdeen Proving Ground, Md.," Bound. lnyer Meteorol., ll
(1977),485 506.
2-81
J. Kunrlu antl Il, Iloylc:s. "Fuflhcr Consitlcrrtion ol'lhc llcighl l)cpcntk:ncc ol
tlrc l{rrol ('oltt'r't'trt't' irr llrc Nllrrrrrl Wirrtl." Ilttiltl. l'.ttvit,,rr l.l (l()711). 175
I tt'1
nt
IIIu
t{(.1
r,
a7
I t'i-5 It. l'1 . llllrl , A N'lt 'tt
2-tJ6
Vr'tlicrtl ('oltt'r'r'ttct ol Wurrl Mt';rsrlr'tl irr lrn (Jr-ban
Ilorrrttl:ry l.;r-yt'r. llttuurl. lrtt,tr Mtlttttttl ., t) 1l()/-\). 1.17.
G. Il.. Sle:gerr rrrxl ll . l., l'lurrpc, "Vcrliclrl ('ohcrcncc in thc Atmospheric
l,uycl," I'nx'ttlittgs F-ifih ltilcntutiorutl Conlcrence on
Vul. I, l)clgarrxrrr, New York, 1980.
B<runtlary
nccri.ng,
Wind Engi-
2-111 H. A.
2-88
2-89
2-9O
Panof.sky and'1. Mizuno, "Horizontal Coherence and Pasquill's Beta,"
Bound. lnyer Meteorol., 9 (1915),247-256.
H. A. Panofsky et al., "Two-Point Velocity Statistics over Lake Ontario,"
Bound. ktyer Meteorol., 7 (1974), 309-321.
L. Kristensen, "On Longitudinal Spectral Coherence," Bound. Inyer Meteorol., 16 (1979), 145-153.
A. K. Blackadar, H. A. Panofsky and F. Fiedler, Investigation of the Turbulent
Wind Field Below 500 Feet Altitude at the Eastern Te.st Range, F/orida, NASA
CR-2438, National Aeronautics and Space Administration, Washington, DC,
19'74.
2-91 C. S. Durst, "Wind
2-92
Speeds Over Short Periods of Time," Meteorol. Mag.,
89, (1960), l8l-186.
I. Yellozzi and E. Cohen, "Dynamic Response of Tall Flexible Structures to
Wind Loading," Proceedings, Technical Meeting Conceming Wind Loads on
Buildings and Structures, Building Science Series BSS 30, National Bureau of
Standards, Washington,
DC, 1970, pp. 115-128.
2-93
2-94
P. Sachs, Wind Forces in Engineering 2d ed., Pergamon, New York, 1978.
H. A. Panofsky and A. A. Townsend, "Change of Terrain Roughness and the
Wind Profile," J. Royal Meteorol. Soc., 90 (1964), 147 155.
2-95 P. A. Taylor, "On Wind and Shear Stress Profiles above a Change in Surface
Roughness," J. Royal Meteorol. Soc., 95 (1969), 77-91.
2-96 E. W. Peterson, "Modification of Mean Flow and Turbulent Energy by a
Change in Surface Roughness under Conditions of Neutral Stability," J. Royal
Meteorol. Soc., 95 (1969), 561-515.
2-97 R. A. Antonia and R. E. Luxton, "The Response of a Turbulent Boundary
Layer to a Step Change in Surface Roughness, Part I, Smooth to Rough," -/.
Fluid Mech. , 48, (1971), 121-161.
2-98 C. C. Shir, "A Numerical Computation of Air Flow over a Sudden Change of
Surface Roughness," J. Atmos. Sci., 29 (March 1972), 304-310.
2-99 A. S. Monin, "Smoke Propagation in the Surface Layer of the Atmosphere,
in Adv. Geophys., 6, Academic, New York (1959), 331-343.
2 100 D. H. Wood, "Intemal Boundary-Layer Growth Following a Step Change in
Surface Roughness," Bound. ktyer Meteorol., 22 (1982), 241-244.
2-l0l B. G. De Bray, "Atmospheric Shear Flows over Ramps and Escarpments,"
Ind. Aerodyn. Abstr.,4, 5 (Scpt. Oct. 1973) I 4.
2-102 C. Sacri, Influcncc d'unt ctllirtc .sur lu vitcsse du vent dan,s la couche limite
tlc sutfut'e, Centrc Scicr)tilit;rrc t't lt'clurirluc du Bitiment, Nantes, France,
t913.
I 103
A..l . llowcn lntl l). l.irttllt'v. "A Wrrrrl lrrrrrrt'l lrrvt'sligrrtion ol'lhc Winrl Spccd
itlttl 'l'ttrltttlcttcc ('ltltntt lt'rslrr :, ( 'lo:.r' lo llrt ( irrrrrrtl ovt'r' Virriotrs lisr'trrprrrclrl
,Slr:r1rcs," liltruttl. ltt\\'t i'lt'tt't'rr,l l.t, tlt, l/1. .t"t1 .t71,
BB
ilil
ntMo: ;t,t
lll(:l()uNt)Altyl/\\ttl
r.r(
'2-104 I'. S..lltt'ksott ltrltl .l ('. ll. llrrrrt. "'lurlrrilcrrl l;low ovcr
it l.()w llil. .l lirttttl
Mclrontl. Jar'., l0l (1975), ().)t) ()\'l
2-105 W. Frost, J. IL. Matrs, rttttl (i. ll. l'it'lrl. "A lllrntlary l,aycl Anulysis ol'
Atmospheric Motion Ovcra Scrrri lilliplit'rrl Srrllrrcc Obstruction." l*tuntl. 2tvcr
Meteorol., 7 (1974), 165-184.
2-106 P' A. Taylor, "Numerical Studics ol'Ncutrally Stratitied Planetary
Bounclary
Layer Flow Above Gentle Topography," Bountl. Layer Meteorol., 12 (1977),
37-60.
H. Ngrstrud, "wind Flow over Low Arbitrary Hills, " Bound. Inyer Meteorol. ,
23 (1982) rt5-124.
2-lo7
2-
108 P. J. Mason and R. I. sykes, "Flow over an Isolated Hill of Moderate
J. Royal Meteorol. Soc., 105 (1979), 383-395.
2
lo9
2-ll0
2-lll
Slope,
"
P. S. Jackson, "A Theory for Flow over Escarpments," in proceerlings oJ the
Fourth International Conference on Wind Effects on Buildings aml Structures,
London, 1975, Cambridge Univ. press, Cambridge, 1976, pp. 33 40.
N. o. Jensen and E. V. Peterson, "on the Escarpment wind profile, " J. Royal
Meteorol. Soc., lO4 (1978), jl9-j28.
F. Bradley, "An Experimental Study of the profiles of wind Speed, Shearing
Hil1," "/. Royal Meteorol. soc.,
Stress and rurbulence at the crest of a Large
106 (1980), 101-123.
2-ll2
2-ll3
R. E. Britter, J. C. R. Hunt, and K. J. Richard, ..Air Flow Over a Two_
Dimensional Hill. studies of Velocity Speed-up, Roughness Ell'ects and rurbulence," J. Royal Meteorol. Soc., l07, (1981), 9l-110.
G. J. Jenkins et al., "Measurements of the Flow Structure arouncl Ailsa Craig,
a Steep, Three-Dimensional Isolated Hill,"
"/. Royal Meteorol. soc., r07 (l9gl),
833 851.
2-ll4 c. Sacr6, "An Experimental
study of the Air Flow over a Hill in the Atmospheric Boundary Layer," Bound. Loyer Meteorot., 17 (lgjg),3g1_401.
2-ll5 T. Hauf and G. Neumann-Hauf,
"Turbulent wind Fkrw over an Embankment," Bound. ktyer Meteorol., 24 (1982),351 369.
2-116 S. L. Rosenthal, A Theoretical Analysis of the Fie kl ol'Motiptt in the Hurricune
Boundary zrzyer, National Hurricane Research pro.iccr, Ilcp.rl No. 56, u.s.
Department of Commerce, Washington, DC, 1962.
2-ll7 R. K. smith, "The surface Boundary Layer of a Hurricanc,' 'l't,llus,20 (196g),
413-484.
2-ll8
G. F. carrier, A. L. Hammoncl and o. D. George, "A M.dcl of the Mature
Hurricane," J. Fluid Mech., 47 (1971), 145-1j0.
2-ll9 E. Simiu, "variation of Mean winds in Hurricanes," J. Eng. Mech. Div.,
ASCE, 100, No. EM4, Proc. paper 10692 (Aug. 1974) 833_837.
2-120 E. Simiu, V. c. Patel, and J. F. Nash, "Mean wind proliles in Hurricanes.,'
J. Eng. Mech. Div., ASCE, 102, No. EM2, proc. papcr 12044 (April 1976),
265-213.
2-l2l
Survey of Meleorologicttl Frtttors Partincnt to Rt'tlut,tirn of Lpss sf'Lifc und
Propcrty in Hurricutrt' Siluttlitn.r, Nutionul Flurricanc Rcscarch prgjcct Rcp6r1
No' 5, tJ.S. l)cP:trltttt'ttl ol ('onttttcrcc. Wclrlhcr Ilurcur.r, WasIirrgl1trr, IX',
March l()57
.t
r,
Bg
l(..1. Wilsorr. "( lr;rr:rtlt.risl i(.s ()l lll(.srrlrr.lorrtl l.:ryt.l Wrrrrl Slrut.lrue irr'l'r.op
lclrl ('.yt lorr,'s,' llrrrt'lrrr ol Mclt'onrhrl.ly" l)r.lrt . ol St.it.lrt.t. irlrtl 'l'cchnology,
Mcllrottlttc, I'lt'Jr:tli'tl lirr Ittlcrttationirl ('orrli'rr'rrr't'on'llo;lrcul (lyclones, Perth,
Austlllilr. Nov- l()7().
2-123 Srnthcn Stt,tthtrd Buildittg Ci,rlr, llirrrrirrgltrrrr, Ala., 1965, p. l2_5.
2 124 w. Malkin, Filling ond Intensity Changt:; irr Hurricanes over Lanj, National
Hurricane Research Pnrject, Report No. 34, u.s. Department of Commerce,
I lll
Washington, DC, 1959.
2-125 J. L. Goldman and r. Ushiyima, "Decrease in Maximum Hurricane winds
afterLandfall ,".1. Struct. Div., ASCE, 100, No. STI, proc. paper 10295 (Jan.
t974), t29 t4l.
2-126 M. B. Glauert, "The Wall Jet," J. Fluid Mech., f (1956) 625.
2-127 P. Bakke, "An Experimenral Investigarion of a wall |et, J. Fluid Mech.,2
(.t957) 467.
2-128 J. Bumham and M. J. Colmer, On lnrge Rapid Wintl Ftuctuations Wich Occur
when the wind Had Previously Been Light , Technical Report No. 69261 Royal
,
Aircraft Establishment, Famborough, U.K., 1969.
2-129 M. J. colmer, "on the Character of rhunclerstorm Gust-Fronts," Technical
Report Aero 1316, Royal Aircraft Establishment, Famborough, U.K., 1971.
2-l3o R. w. Sinclair, R. A. Anthes, andH. A. panofsky, variationof the LowLevel
winds During the Passing of a Thunderstorm Gust Franr, NASA Contractor
Report No. CR-2289, 1913.
2-l3l H. c. s. Thom, "New
States, .r. Struct.
1801
Distributions of Extreme wind speeds in the United
Div., ASCE, No. ST7, proc paper 603g (July 196g), l7g7_
.
2-132 S. D. Smith, "wind Stress and Heat Flux over the ocean in Gale Force
Winds," J. Phys. Oceanography, l0 (May l98O),'709-726.
2-133 L. Knigermeyer, M. Gninewald, and M. Dunckel, "The Influence of Sea
Waves on the Wind Profile," Bound. lnyer Meteorot.,14(lg7g),403 414.
2-134 T. von Kiirmein, "Progress in the Statistical rheory of rurbulence ," proc. Nat.
Acad. Sci., Washington, DC (1948), 530-539.
2-135 w. R. Krayer and R. D. Marshall, "Gust Factors Applied to Hurricane winds,"
Bull. Am. Meteorol. Soc., 73 (1992), 613-6lj.
2-136 J. Bidtry, Personal communication. 1981.
2-137 R. I. Harris, "The Nature of wind, in The Modern Design of wirul-sensitive
Structures, Construction Industry Research ancl Information Association. Lon-
don, U.K.,
197 l.
2-138 D. M. Deaves, "computations of wind Flow over changes in surface Roughness," J. Wind Eng. Ind. Aerdyn., 7 (1981), 65-94.
2-139 ASCE 7-95 standard, Minimum Desil4rt Loads for Buildings arul other struc,ure.s, American Socicty o1'Civil E,nginccrs, New york, 1995.
2-140 R. L Harris, "Sorttc litrflhcr 'l'lrorrglrls on thc Spcctrum of Gustiness in Strong
Winds," J. Wind l,)n,q. ltttl. .'lt,trnlltr. ll. (1990). 461 461
2-l4l Y. Tamura, K. Shilrrlulrr, :rrrrl lr.. llilri, "Wirrtl l{csponsc o{'a 'lirwcr ('l'yphtxrn
Obscrvation al thc N:tglrs:rki llrrr:, li'rr llrsr'lr l)orrrlorcrr)," .1. Witrtl lin.4. ltrt!.
.
Acnrl. .
-50 ( l9().1). .1O() l
lti
90
lltf
n tMOSI,I
l nt(; nouNl)nny tAyt
n
2-142 A. I). Pcrcira, M. ('. (i. Silvrr, l). X. Vtr'1,',irs, ruttl A. (i. I-opcs, "Wilul 'l'Lrnncl
Sirnulatiorr ol'thc likrw arourttl 'l'wo I)irrcrrsionul l-lills," .1. Wirul lhtg. lnd.
CHAPTER 3
Aerod., 38 (1991), lO9-122.
2-143
I.
Van de Hoven, "Powcr Spcc:(rurrr ol' Wintl Vclocity Fluctuations in the
Frequency Range fiom 0.0007 to 9(X) (lyclcs pcr Hour, " J . Meteor. , 14 (1957),
1254-t255.
2-144 J. Ashcroft, "The Relationship betwccn the Gust Ratio, Terrain
Gust Duration and the Hourly Mean Speed,"
J.
Roughness,
Wind Eng. Ind. Aerod.,53
(r994). 33 l-355.
2-145 J. S. Allen, R. M. Samelson, and P. A. Newberger, "Chaos in an Model of
Forced Quasi-geostrophic Flow over Topography: An Application of Melnikov's Method," J. Fluid Mech., 226 (1991), 5ll*547.
2-146 E. Simiu, "Melnikov Process for Stochastically Perturbed Slowly Varying Oscillators: Application to a Model of Wind-Driven Coastal Currents," J. Applied
Mech., ASME 63 (June 1996),429-435.
2-141 S. Krenk, "Wind Field Coherence and Dynamic Wind Forces," Proceedings,
IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics (A. Naess,
ed.), Trondheim, Norway, July 1995.
2-148 J. Mann, "The Spatial Structure of Neutral Atmospheric Surface-Layer Turbulence," J. Fluid Mech.,273 (1994), l4l-168.
2-149 J. Mann, "Fourier Simulation of a Non-Isotropic Wind Field Model,"' in
Structural Safety and Reliability, (G. Schueller, M. Shinozuka and J. Yao,
eds.), pp. 1669-1614, Bakkema, Rotterdam, 1994.
2-150 J. Blessman, O vento na engenharia estrutural, Editora
da Universidade, Universidade Federal de Rio Grande do Sul, Porto Alegre, Brasil, 1996.
2-151 J. Kaplan and M. DeMaria, J. Appl. Met.,34 (Nov. 1995), 2499*2512.
EXTREME WIND CLIMATOLOGY
Climatology may be defined as a set of probabilistic statements on long-term
weather conditions. The branch of climatology that specializes in the study of
winds is referred to as wind climatology. Wind climatology provides the designer and the code writer with information on the extreme winds that might
affect a structure during its lifetime.* Such information is required for making
rational decisions on the magnitude of the wind loads to be used in design.
This chapter is devoted to a review of problems involved in the description
of the wind climate for structural design purposes and in the development of
criteria for the definition of design wind speeds. Procedures for estimating
extreme winds are presented, and the uncertainties inherent in these procedures
are discussed. Some of the material included herein is heavily dependent upon
probabilistic and statistical notions and tools. These are presented in some detail
in Appendix A1.
The reliability of climatological statements based on the analysis of extreme
wind speed data is clearly dependent upon the quality of the data. This topic
is discussed in Sect. 3.1. The question of the prediction of extreme wind speeds
in well-behaved wind climates and in hurricane-prone regions is dealt with in
Sects. 3.2 and 3.3, respectively. The dependence of extreme wind speeds upon
direction is discussed in Sect. 3.4. Information on the frequency of occurrence
of tornado winds of various intensities in the United States is presented in Sect.
3.5.
In the United States surfacc wirxl spoocls rcpofted by the Weather Service
have traditionally been exprossc(l itt tttilcs pcr hour (l mph : O.447 m/s). In
*Wincls othcr than thoso ol'inlcrcsl
t5.
lirrn
lr
slrr( lurrl sirlr'ly vicwpoittt will bc rlcalt wilh in Chaptor
91
I
X
llil Ml wlNI) ( il tMn
:t I wlNt):it ,l lt) l)n tn
t( )l ( x iY
ltt' lrt'rlrrrrrlly t'xplt'ssul irt rlrrrlrt'rrl rrrlt's (I
nuli - l.l5 rnilc). Iior corrve:rricrrcc, wlrt'r't'lrpPnrplirrtc:, lhcsc trnils will rrlso
Itrtrriclrttc-rollr(ctl wot'k, lt:rrgllts
'l'Alll,lt J.l.l. ('orn.t.liorrs lo
Spt't'tls lrrtliclrlc:tl
be used herein.
By 3-Cup
"S"
1'ypc
Anemometer, mph
3.1
WIND SPEED DATA
To provide useful information on the wind climate at a given location, wind
speed data recorded at that location must be reliable and must constitute a
micrometeorologically homogeneous set.
lrrrlie:rlc<t Wirrrl Sllctrls l-l-21
By 4-Cup Anemometer,
mph [Up to 31 Dec,
[1928-l93l"l
ob-rc
1g2l"l
Whole Miles
per Hour
ob-g
t7-26
+l
9-12
27-35
l3-16
36,44
45-52
t7-20
2t-24
53-61
3.1.1 Reliability of Wind Speed Data
.
The instrumentation used for obtaining the data (i.e. , the sensor and the
recording system) may be assumed to have performed adequately and was
properly calibrated. If it can be determined that the calibration was not adequate, the data must be adjusted-whenever the information needed for that
purpose is available.
I
Example The following information is excerpted from [3-l] regarding the
5-minute winds given in the original U.S. Weather Bureau records taken before
1932 "Up to 31 December 1921 , all recorded wind speeds were the uncorrected readings of 4-cup anemometers. From 1928 through 193 1, all speeds
from the older 4-cup anemometers were corrected to agree with the readings
of the 3-cup instruments, then being introduced, readings from which were not
corrected to true speeds. From I January 1932 onward all readings, whether
from 3- or 4-cup anemometers, were already corrected to true speed in the
original records." Official U.S. Weather Bureau instructions fbr the correction
of 3- and 4-cup anemometer readings are given in Table 3.1.1, which is excerpted from [3-2], and whose use will now be illustrated. At Williston, N.D.,
the original readings of the maximum 5-minute wind in 1922 and 1930 on
record at the National Oceanic and Atmospheric Administration are 56 mph
and 37 mph, respectively. Using the corrections of Table 3 . I . I , the true speeds
(according to U.S. Weather Bureau calibrations) are 56 - l2 - 44 mph and
37 - 2 : 35 mph, respectively.
2. The sensor was exposed in such a way that it was not influenced by local
flow effects due to the proximity of an obstruction (e.g., building top, or
instrument support). For most U.S. weather stations, the existence of such an
obstruction during the period of record is noted, in principle, in Local Climatological Data Summary sheets (LCD Summaries) issued by the Environmental Data Scrvicc ol'thc National C)ccanic and Atmospheric Admlnistration
l3-31.
3.
'l'hc: lrlrrxrsplrt'r'ic
slrlrtilit'irtion lrray
bc:
97-t05
25-28
29-32
33-36
37-39
40-43
44-47
106-1 14
48-5 I
tt5-122
r33-139
52-54
55-58
59-62
140-149
63-65
62-70
Wind speed data may be considered to be reliable if:
assurrretl lo havc hccn rrcrrtr:rl. 'l'his
Corrections in
7
t-79
80-87
88-96
123-132
150-r57
66-69
158-166
70-73
t6t
174
175-184
74-77
78-80
r85-192
193-2(n
8l-84
85-88
89-9r
92-95
96-99
100- 103
104-106
107-t l0
I I l-114
I l5-1 l7
1
l8-l2l
122-125
126-128
129-132
33- 136
t37 -140
"Reference [3-
l].
'Movement of anemometer cups obst.rvt.rl
0
-l
-J
-4
-5
-6
-l
-8
-9
-10
-11
-12
-13
-14
-15
-16
-17
-18
-t9
-20
-21
1a
-23
-24
-25
-26
a1
-28
-29
*30
-31
*32
r
- J-t
l4t-t43
-34
- 3-5
I x tltl Ml wlNt) ( il tM/\ t( )l ( )( iy
94
;llMAll()ll
llsstlllll)ti()ll is itcccp(itblc lirl witrtl spcctls;rt lO rrr;rlrovt'glorrrrtl irr opcll lcll-irirr
in cxccss ol'2-5 rnph rlr so (scc Scct. 2.2.-5).
3.1.2 Micrometeorological Homogeneity of Wind Speed Data
A
set of wind speed data is referred to herein as micrometeorologically homogeneous if all the data belonging to the set may be considered to have been
obtained under identical or equivalent micrometeorological conditions.
These conditions are determined by the following factors, which
brictly cliscussed below:
.
wiil
be
Avcrirgirrg tirnc (i.c., whcther highest gust, fastest mile, one-minute avt'r:rgc, livc nlinulc avcragc, etc., was recorded).
o Ilciglrl irl'rovc gnrurrtl.
.
l{orrglrrrc:ss ol' surnlunding tcrrain (exposure).
.
Avantging 'l'ime.
If various averaging times have been used during the
pcriod o1'rccord, the data must be adjusted to a common averaging time. This
can be done by using Eq. 2.3.31 and Tables 2.2.1 and2.3.3, or Fig. 2.3.10.
(For hurricane wind speeds, see also Sect. 2.4.3.)
Data averaged over short time intervals, such as highest gusts or fastest
miles, may in certain cases be affected by stronger than usual local turbulence
effects, and thus provide a somewhat distorted picture of the intensity of the
mean winds. In principle, it is desirable, therefore, that the data used for the
description of the wind climate be averages over relatively long periods, say
five minutes or so. However, owing to the current data collection policy of the
u.S. National weather Service and the availability of 3-s gust speed data at a
large number of stations in the United States, the ASCE 7-95 Standard lz-1391
uses 3-s gust speeds at lO-m elevation as basic wind speeds.*
2. Height above Ground. If during the period of record thc erevation of the
I
anemometer has been changed, the data must be adjusted to a common elevation
as follows: Let the roughness length and the zero plane displaccment be denoted
by z6 and 27, respectively (zo and za are parameters that clefinc the roughness
of terrain; see Sect. 2.2). For strong winds (i.e., with speecls cxceeding l0
m/s or so), the relation between the mean speeds u(ar) and u(z) over horizontal
terrain of uniform roughness at elevation z1 and Z2 above ground, respectively,
can be written as
xThe National Weather Service and the Federal Aviation Administration are currently implementing the Automated Surface Observing System (ASOS). It is anticipated that by the year 2000
there will be 1700 ASOS units in operation. The ASOS anemomctcr reading is sensed once a
second. Every fivc sccttnds it rlnning ilvcnlgL- is eomputcd, which is rclorrctl kr as a (igsl. A
2-rninute running avcrilgc ol thc 5 s irvcrllgcs is irlso conrprrlctl lrrrtl is rrst'il rrs rr rrrcirsrrrc ol thc
prcvailirrg winrl.'l'ltt'st'tltoiccs rtl:rvt'nrgirrg lirrrc wcle rlct'lrrcrl t() l)('nr()sl rrsr.lrrl lor:rvilrlion
purJxrscs
l.l
(rl{
1 .
rrl r 'ilttl Ml wlNl
I
l(.'.tl
(
/(;'. ,)
r:;t,t tll:; tN wt lt ttt ltnvt t) (,t lMn il:
lnf(;,
lrrl(i,
;',,111:ul
r95
(3. r. r)
:'.,111;'.qyl
Equation 3. l. I lirllows clircctly from Eqs. 2.2.18 ;ttd 2.2.22. For open terrain
: O, and the values ol' the roughness lcngth 2,, can bc taken from Table
2.2.1. The power law (Eq. 2.2.26 and Table 2.2.2) may be used in lieu of
Eq. 3.1.1. As noted in sect.2.2.3, considerable uncertainties subsist with
regard to the values of the roughness parameters in built-up terrain. Good
judgment and experience are required to keep the errors inherent in the subjective estimation of the roughness parameters within reasonable bounds. It is
clearly advisable to investigate in individual cases the effect of such possible
errors upon the predictions of extreme wind speeds.
Za
3. Roughness of surrounding Terrain In many cases anemometer locations
have been changed during the period of record, for example, from a town to
a neighboring airport station. The corresponding records can, in principle, be
adjusted to a common terrain roughness by using the similarity model (Eqs.
2.2.29 and 2.2.31 and rable 2.2.3) described in Secr. 2.2. As indicated in
sect.2.4.1 , this model may be assurned to be applicable in horizontal terrain
at each station the terrain roughness is reasonably uniform over a distance
from the anemometer of about 100 times the anemometer elevation. In terrain
in which sheltering effects by small-scale obstacles are present, the data may
be adjusted by using a procedure presented in [3-4].
A situation commonly encountered in practice is one in which, while the
if
anemometer may not have been moved, the roughness of the terrain surrounding
the anemometer has changed significantly over the years as a result of extensive
land development. In such situations the adjustment of the data to a common
roughness may pose insurmountable problems, unless detailed information on
the phases of the land development is available.
Anemometer elevation and location changes are listed for most u.S. weather
stations in Local Climatological Data Summaries t3-31. wind climatological
information for various locations around the world is available in [3-77 , 3-78].
3.2 ESTIMATION OF EXTREME WIND SPEEDS IN
WELL-BEHAVED CLIMATES
Infrequent winds (e.g., hurricanes) that are meteorologically distinct from and
considerably stronger than the usual annual extremes are referred to herein as
extraordinary winds. climates in which extraordinary winds may not be expected to occur are ref'erred to us wr'l/ ltehuvctl. In such climates it is reasonable
to assume that each of thc tlrrt:r in rr st'r.it's ol'thc largest annual wind speeds
contributcs to the dcscripliott ol'llrc plrb;rbilis( it'l'rr.rhlrvirlrof the extreme winds.
A statistical analysis ol'sttclt rt st'tir's r'lrrr llrt'rt'lir.t' bc cxpcclcd t<l yicld uscful
prctlictions ol' l<lng*lorrrr wirrtl t.xllt.rrrr.:,
'l'htrs, in tt wcll-hchltvtrtl t'lirrrrrlt',:r( :nrv
1'rvcrr st;rliorr ir rirrrtkrn virrilrblt'rrlry
96
I x tilt N/lt wtNl)
(.1 lMn l( )l { )(
l:lllMnllillt
'Y
bc tlclirrr,tl, wlticlr corrsisls ol tlrc l:rrIr':;l yt':rrly wtrrtl sllcctl. ll llrr'sllr(rorr ts
9pc lirr wIich wiltrl rccortls ()vcl lt rrrrtttlrt't ol tttttset'tllit'c yoilrs ittt'lrvltil:rble:,
thcn thc cuntulativc dislributiolr lirrrclion (('ll1'1 ol'tlris ritrlclottt vllriitlllr-: tttay
be estimated to charactcrizc thr: pnrblbilistit' bcltitvi<lr of thc largcst annual
wind speeds. The basic clesign wintl spectl is thcn dclincd as the speed corresponding to a specified value p ol' thc Cl)lr or, cquivalently, to a specified
mean recurTence interval N.* R wind cttrrcsponding to an /V-year mean recurrence interval is commonly referred to as the N-year wind.
This section is devoted to the question of estimating (1) the CDF of the
largest annual speeds and (2) errors inherent in the wind speed predictions.
Such errors include, in addition to those associated with the quality of the data
(sec Sect. 3.1), modeling errors and sampling errors. Modeling errors are due
to an inadequate choice of the probabilistic model itself. Sampling errors are
a consequence of the limited size of the samples from which the distribution
parametcrs are estimated and become, in theory, vanishingly small as the sample size increases indefinitely.
3.2.1 Probabilistic Modeling of Largest Yearly Wind Speeds
Extreme wind speeds inferred from any given sample of wind speed data depend on the type of distribution on which the inferences are based. For large
mean recurrence intervals (".g., N > 50 years) estimates based on the assumption that a Type II distribution is valid are higher than corresponding
estimates obtained by using a Type I distribution, while estimates based on af
reverse Weibull distribution wittL tail length parameter ^Y < 15 , say, are lower.
According to [3-5], extreme winds in well-behaved climates may be assumed
: 9. However,
to be best modeled by a Type II distribution with p : 0 and 7
subsequent research has shown that this assumption is not borne out by analyses
of extreme wind speed data f3-6, 3-1 , 3-91. In [3-6] , 37 year-series of 5-minute
largest yearly speeds measured at stations with well-behaved climates were
ruUi""t"A to the probability plot correlation coefficient test (see Sect. A1.6) to
determine the tail length parameter of the best fitting distribution of the largest
values. Of these series, 72% were best fit by Type I distributions or by Type
distributions with "v : 13 (which differ insignificantly from the Type I
distribution); ll% by Type II distributions with 7 < ^Y < 13; and 17% by
Type II distribution with 2 = I I 7. Virtually the same percentages were
obtained in [3-7] from the analysis of sets of 3'7 data generated by the Monte
Carlo simulation from a population with a Type I distribution. On the other
hand, the analysis of sets generated by Monte Carlo simulation from a Type
II distribution with tail length parameter 7 : 9 led to percentages differing
II
*Rccall that lV: tlf f ' 7r) (scc Appcntlix Al,Irq. Al-).
lDifl'crcnccr hctwccrr spet:tls cslirrrirlctl ott lltc basis ol T'ypc II tlistributions rntl lhc -l'ypc I
tlisll'ibrrlion ilttrt'rrst.lrs.y tlcclclrsr's. l)illi'rcrrt'cs hclwcctt s;rt:ctls lt:tsctl on lhc'l'ypt I tlisllilrtlliott
:uxl rcvt'rse Wt'ilrrrll rlislrilrttliotrs rtlso irtt tt':tse its 1 tlt't'rrl:tst's
ot t"lltt Mt WtNt):;t,t tlrl ;
lNt Wt
lttililnvt
tr(.t tMAilti
97
sigrrilicirnlly ltttltt lltost' r onr'sporrtlirrg to tlrc irclrrirl wirrtl spcc:tl tltrlir. ()rr thc
basis ol'thcsc t'csrllls il r'rrrr lrt'conliclcntly statcd that in woll-bchavccl clirlates
cxtrcmc wintl spr:ctls lrlt' rrrotle lctl rn<lre realistically by thc Type I than by the
Typc II distributiorr with ^y ,., 9. This conclusion was reinforced by studies
reported in [3-9], in which tcchniques similar to those of [3-7] were used in
conjunction with wind speed data at one hundred U.s. weather stations listed
in [3-9].
As indicated earlier, the Type I distribution results in lower estimates of the
extreme wind speeds than the Type II distribution with 7 : 9. An interesting
result obtained in [,{1-36] is that at most stations in the United States even the
Type I distribution appears to be an unduly severe model of the wind speeds
corresponding to large mean recurrence intervals; at these stations a better fit
to the data is obtained by reverse weibull distributions (see also end of Appendix Al). Thus, structural reliability calculations based on the assumption
that the Type I distribution holds are in most cases conservative [Al-36]. For
this reason we will assume in this section that the Type I distribution model
holds. The degree of conservation inherent in this assumption is generally
modest for basic (5O-year) speeds, but it can be very significant for wind speeds
corresponding to nominal ultimate wind loads, i.e., 5O-year wind loads multiplied by a wind load factor (see Sect. A.3.3).
3.2.2 Estimation of and Confidence lntervals for the N-year Wind:
Numerical Example
It is shown in Sect. Al.7 that, given a set of data with a Type I extreme value
underlying distribution, several techniques can be used to estimate the paramof the distribution and, hence, the value of the variate corresponding to
a given mean recurrence interval.* However, inherent in these estimates are
sampling errors. A measure of the magnitude of the latter can be obtained by
calculating confidence intervals for the quantity being estimated, that is, intervals of which it can be stated-with a specified confidence that the statement
is correct-that they contain the true, unknown value of that quantity. Techniques that can be used to estimate the N-year wind, and confidence intervals
for the N-year wind, are discussed in some detail in Sect. A1.7. one of these
techniques is presented and illustrated below.
Using the approximation -ln[-ln(1 - l/N)] = ln N, it follows from Eq.
Al .74 (which is based on the method of moments) that the estimated value
Dp of the N-year wind u1y is
eters
0N
= X I 0.78(lnN -
0.577)s
(3.2.1)
where X and s are, respcctivcly, rlrrr sirrrrplc rncarn and the sample standard
deviation of the largest ycarly wilrtl spt'erls lirr (hc pcriod of recorcl.
*InAppendixAlthisvalucistlcrxrh.rl
lry(i,q711,rllrr.rr./r I
inlcrval, In this chaptcr lhc trottrliolr (i,(
I
l/N
I
r,.
r,, rrr,r.rl
l/N:rrxl Nisllrt, 1rt.ir1 t1.(.rtrroncc
IXIlilMl
I i;llMn ll()t\t
wlNl)(.1 lMnl()l ()(iY
l.& + l.46tln ru -
O.SZtl
the results of lines (l) and (2) ol"rable 3.2.1are small.'rhis is consistent with
the conclusion of Sect. Al.7 that the e{iiciency of the method of moments (Eq.
3.2.1) is generally adequate for structural design purposes.
In Table 3.2.1 the errors in the estimation of the 50-year wind are of the
order of lo% at the 95% confidence level. Since the wind pressures are proportional to the wind speeds (see Chapter 4), the corresponding errors in the
estimation of the pressures are of the order of 2O%. An altemative approach
to accounting for sampling errors, which applies the theorem of total probability, is suggested in [3-51].
To reduce sampling errors, [3-8] resorted to the consolidation of records
from different stations, thereby creating "superstations" with large sample
sizes. This approach, if valid, would be quite attractive: for example, the
information yielded by a "superstation" consolidating 2o-year records taken
at 100 stations would be equivalent to information yielded by a 2000-year
record. However, according to [A1-15], statistical tests did not validate the
"superstation" concept for extreme wind analyses based on peak gust speeds.
For that concept to be valid, the population distributions for the station
records being consolidated would have to be identical, and in addition those
records would have to be mutually uncorrelated. In general, the first of these
conditions cannot be assumed a priori to be true. Second, if the records consist
of peak gust speeds, the observed lack of correlation between records taken at
different stations may be spurious; that is, it would be likely to occur even if
the corresponding mean wind speed records were well correlated. The apparent
lack of correlation would be an artifact due to the strong random variability of
the ratios between gust speeds and mean speeds. For these reasons the "superstation" concept may yield inadequate results. In our opinion this is likely
to be reflected in the quality of the wind speed map specified by the ASCE
7-95 Standard [9-5], which is based largely on the "superstation" concept
applied to peak gust speed records.
I l.l(ln rV - O'S;Zt'l'' ,/n
;
(3.2.2)
wlrt't'c rt is thc samPle size.
Example At Great Falls, Montana, the largest yearly fastest-mile wind speeds
:34) were
rrt l0 rrr above ground during the period 1944-1971 (sample sizen
l3-t)l:
51, 65, 62, 58,
&,
65, 59, 65, 59, 60, 64, 65,'73, 60, 61, 50, 74
60, 66, 55, 51, 60, 55, 60,
51,51,62,51,54,52,59,56,52'
49
(mph). The sample mean and the sample standard deviation for these data are
X:'5g mph and s : 6.41mph. From Eqs. 3.2.1 and3.2.2 it follows that
for N : 50 years and N : 1000 Years'
= 76 mph
|wn = 91 mPh
lso
:
SD(irooo) =
SD(i5o)
3'7 mph
6'4 mph
As shown in Sect. A1.7, the probabilities that u vl is contained in the intervals
68%'
0v1 + SD(01y), 0n t zsD(|il, and 0p + 3SD(0,'v) are approximately
95%,
68%,
the
gi%, ana 99%, respectively. These intervals are referred to as
Great
34-year
the
and 99% confidenci intervals for u7, and are shown fbr
Falls sample in row 1 of Table 3.2.1.
It is also shown in Sect. A1.7 that the width of the confidence intervals can
3-2.3 Methods for Estimating the Extreme speeds at Locations with
TABLE 3.2.1. confidence Intervals for the N-year wind at Great Falls
Mean recunence
s0
interval, N (years)
lnsufficient Largest Yearly Wind Speed Data
95%
687
Confidencc level
50
1000
1000
1000
50
(1) Estimated by
method of
moments
(2) Estinratcd using
C.R. lowcr bounil
76
6.4
16
+ 1.4 9l + 12.8 76 + I l.l
9l I5.0
'76
r
+ 3.1 9l +
76 I .l.l
6.2
gg
bc rctlLtcctl il it lttott't'lltr'rt'ttl t'slitturtor is uscd; lurwcvrrr', llrc intcrvals cannol
bc trarnrwcr lltirtt tlrrst' ohlrrirrr'tl by using the Crarrcr,llacl (C.R.) lower bouncl
(Eq.Al.77). tt<tr tlrr: (irt:;rt Irlrlls sltnple, the confidence intervals based on the
latte r are shown in linc (2) ol 'l'ablc 3.2.1. lt is seen that the differences between
As Prt.vipttsly rurlctl, irtltr.:tcrr( iil lltt't':-lrlrItlt':; ol l'1u 11t'q s:tttlpllttl', t'ltot:.. ll
lirllows l.nrrrr I')t1s' n I 76 lrrrtl Al'/o (wlritlr:tlt'lrltst'tl ott lltc: tttcllrotl ol ttttr
llcllls) llrirt thc staurlanl tlovialiorr ol tltt'srtttplirtg ert'tlt's in thc cslilrr:rtiolt ol
r',y cln hc writtcn as
.\1)0r^) -- 0.781
()t I i: ilil [It wtNt) :;l'l I t)t; lN wt il ilt ilAV' t) (]t lMn il ti
9l I lO.O 7(r I
(). \
9l +
19.2
()l Il5.o
There are about one hundred u.S. weather stations for which reliable and
relatively long wind speed records arc available (e.g., records overperiods of
20 years or more). Some of thcsc sl:rlions c()vcr arcas of tens of thousands of
square miles, over which-lilr rrrc(t:onrkrgical rcasons or owing to topographic
effects-the extreme wincl clirrrrrtt' is rrol rrt't'r,ssirrily uniform. Thcre arises
therefore in practice thc prohlcrrr ol t'slirrrrrlirrl'('xll('tn(: wintl spcccls at various
locations where long-tcntt t'ccottls ol llrc l:rr1'r':rl yt'lrll_y witrtl spcctl tlutl tlo 191
exisf.
lo0
txiltt Mt wtNt)ct
tMnt()t
((iy
r:' Ir;ilMAil()N ()t I . |ltt
Estimates Of Extreme Wind Speerls lrr a Marirrc EnvirOnmenl. l{t'lcrt'rtt't'
(o clt'l'y ottt sttt'lt
1.1-l ll lists ilrrcer rrrcllrorls tlrir( ,iu1', ur lrrrrrt rplt', :rvltilitblc
spcccls
lltt'
cxltctttc
itte itssot'iltlctl
wltt'tt'
cnvirorrrttcrtls
lilr
lrrarinc
cstitnatcs
'l'hc
inlorrnlrkcs
usc
ttl'climattlltlgical
lirst
rrrt'llrrxl
with extratnrpical strlrtrrs.
physical
rclating
ol'
modcls
s(or.rrr
paramctcrs
ol'
thc
iuttl
mation on various
those parameters to the surface wincl spcctls. lt is shown in Sect. 3.3 that such
a method can be applied to estimatc oxlt'cttlc wind speeds in hurricane-prone
regions. However, as noted in [3-lll, owing to the complexity of the surface
wind patterns in extratropical storms, the usefulness of this method appears to
be uncertain in regions where such storms are dominant.
A second method listed in [3-11] is the use of objective analysis schemes.
These consist of (1) an initial guess at the surface wind on a regular grid, (2)
an automated procedure for screening wind reports from ships to eliminate
erroneous readings, and (3) a procedure for correcting the initial guess on the
basis of the usable set of ship reports, which involves relations among the
surface wind speeds, sea-level pressures, and air and sea temperatures. Details
on objective analysis schemes and of errors culrently inherent in such schemes
(which may range from lO% to 30%) are given in [3-11].
The third method listed in [3-11] is referred to as direct kinematic analysis.
The method, which involves subjective judgment by experienced analysts, consists in synthesizing discrete meteorological observations to obtain a continuous
field represented in terms of streamlines and isotachs. Objective or kinematic
analyses applied to a sufficient number of strong storms make it possible to
provide estimates of extreme winds that may occur at any one location. As
indicated in [3-11], one of the major diliculties in conducting such analyses
is that much of the vast store of existing data is currently not accessible in
readily usable form.
Estimation
of Extreme Wind Speeds from Short-Term Records. A prac-
tical procedure for estimating extreme wind speeds at locations where longterm data are not available is described in [3-12]. The method, whose applicability was tested for a large number of U.S. weathcr stations, makes it
possible to infer the probabilistic behavior of extreme winds from data consisting of the largest monthly wind speeds recorded over a period of three years
or longer. Estimates based on the monthly speeds, denoted by 0n.., are obtained by rewriting Eq. A1.74 as follows:
0N.^
= x. + o.zAPn(l2N)
-
0.5771s*
(3.2.3)
X.
and sa are, respectively, the sample mean and the sample standard
deviation of the largest monthly wind speed data, and N : mean recurrence
where
interval in years.
The standard dcviatitln tll'thc sanlpling error in thc cstirlation tll'I'x7.,,,
obtaincd from l'it1s" Al .76 irrrtl A I .70 as
,S/)( /',,r,.,,,
N/,1t
wtNt) l;t 'l It)ti IN wt |t tlt ltAVt t) (.t tMn il:
o /ttl l.(4
)
I
1.4()llrr(
llN)
I r.rlrn(t2N) - o.sztl'l',,
where
n-:
O..5771
(3.2.4)
h:,
sample size.
Example At Great Falls, the sample mean and the sample standard deviation
of the largest monthly fastest-mile wind speeds at l0 m above ground for the
period September 1968 through August l97lx (sample size n*:36) aret^
:42mph, s^:6.96 mph. From Eqs.3.2.3 and3.2.4, the estimates forN
:
50 years and
N:
1000 years are
0s0..
:
74 mph SD(0s0,)
= 6.23 mph
iuxn,^: 90 mph SD(irooo..) : 8.85 mph
i
It is seen that the estimated
speeds based on the set of 36 largest monthly data
are only slightly lower than those obtained from the set of 34 largest yearly
speeds (|so:76 mph and 01es0 : 91 mph; see Sect. 3.2.2); however, the
sampling errors are larger.
Similar calculations carried out for 67 sets of records taken at 36 stations
are reported in [3-12], where it was found that the differences iso,^
- lso,
where 056 is the 50 year wind speed estimated from long-term largest yearly
data, were less than sD(0s0.) in 66% of the cases and less than twice the value
of sD(05s,-) in 95% of the cases. This remarkable result, confirmed by additional calculations reported in [3-13], indicates that the estimates based on
largest monthly wind speeds recorded over three years or more provide a useful
description of the extreme wind speeds in regions with a well-behaved wind
climate.
Inferences concerning the probabilistic model of the extreme wind climate
have also been attempted from data consisting of largest daily wind speeds
13-121, or of wind speeds measured at one-hour intervals 13-141. one problem
that arises in this respect is that data recorded on two successive duys u..
generally strongly correlated. A second and more serious problem is that the
daily (or hourly) data reflect a large number of events (e.g., moming breezes)
that are altogether unrelated meteorologically to the storms associated with the
extreme winds. These events can be viewed as noise that obscures the information relevant to the description ol'thc cxtreme wind climate. Indeed, it was
verified in [3-12] that estimatcs ol' cxtr-cnrc winds based on daily data differ
significantly from estimatcs obllrirrt:tl lor krrrg,lcrrrt rccords of largest yearly
speeds. This conclusion is a.litrtittri ltut'lirl irrlc'r'ctrc:cs based 9n hourly data.
ir
is
*For the actual data. scc thc l,octrl ('liru:rlolo1,r, .rl I ):rt.r
',rrrrrrr;rrit.s li1. tltr. ycitrs l()()ll l.)7.1:
,;'
"'--=\
\
T
i
I
'l
1O2
I x I ill Mt wtNl) (;t tMn t(
)t ( x
iy
l,lstirrlrlcs ()l'L:xirL:nlL: wirttl spct'tls lr;r:;t'rl orr st'ls ol tllrllr itt cxccss ol s1r't'rltcrl
{hrcshokls (scc cncl ol'Sccl . A I.7) lrrt' rt'llrlcrl irr l.t l0l lirt' sltorl r('('()11ls.
It was shown in Scct. 2.4.4 tltal lhrrrttlt'r.slot'rrr wirttls havo I'caiures tlrirt rlillcr
markedly fiom thosc ol'othcr typcs ol'wirrtl. (ic:ncrally, extremc wind spocds
are analyzed without separating thundcrslonr tlata lrcm the other extremes.
Whether it would be useful to extract thcsc tlata l'rorn the mixed sets and analyze
them separately-despite the difficulties this would entail-is still being debated
I
t--Ft--FatsHia-!!txxx,FtsFa__xt
FFrxF!a
I
tF
lF
t0
I
a
I
t0
I
to
I
I
I
I
I
tJq
I
ou
d
rI t<otd
x
I FOJ
rNuI r!
.J
I FIO
I 6F U
t€ra
<'c
tN
a
;
t-n
r>(-J
tua
t=
rd
t!
tN6
lo-:
r+U
Ho
td*
ln
loa
I
I
3.3
ESTIMATION OF EXTREME WIND SPEEDS IN
HURRICANE-PRONE REGIONS
t
/
l\l
"v:-tnl-tnlr-:ll
| \ N/l
I
I
I
I
II
I
I
I
I
I
a
I
!a!
tn
IN
I
I
I
I
I
t
I
-Y
r
Y
.
tNq
I F^
l6l
H No
.. tr
I iO
r
€F
I
Od
I
OU
iu
I
I
€C
r
ou
a
I
I
I
I
I
H
a
o
to
!d
:
trHfil .z
j
=
tr
aF
I
I
I
HoOO
I
I
ld
I
ru+IIF
I
a
I
I
I
I
I
I
t
I
I
I
I
a
I
I
I
I
I
A
lr
I FFF
I Od
ao
u
I (F
I
I
NO
rJ
I
I
I
I
I
l>t
Fo
Hr
Jo
FG
too
t
' :
=
9l
=
-:
O-
lE.>
i6
r cr
ldG
:E-o
r uu
I ox
I >u
lF
U
a
where N is the mean recurrence interval. In virtue of Eqs. Al .43 and A1.45,
a Type I extreme value cumulative distribution function would be represented
in Fig. 3.3.1 by a straight line, whose intercept and slope would be equal to
the distribution parameters p and o, respectively. To the extent that the population of largest yearly speeds would be described by a Type I distribution,
the actual data would then approximately fit a straight line. In Fig. 3.3.1 this
is roughly the case as far as the winds of less than hurricane force are concemed.
However, if-as in Fig. 3.3.l-the hurricane-force winds are included in the
set being analyzed, clearly the fit of a Type I distribution to the data is extremely
poor.
A bettcr fit can bc obtainccl il-a Typc II distribution with a snrall valuc ol'
thc tail lcngth pirrrrrrtclcr is usctl. Howcvcr, As slrowrr in 1.1 (rl. 1ln:1lir'1i1;11r,i,r1'
cxlrclnc wintls irr ltttl-rit'lrltt'plottc tcgiotts b:rscrl ort'l'ypt'll tlisl r.ilrrrliorrs lrlt'irr
Zt
I
t3-s01.
We now consider the prediction of extreme winds in climates characterized by
the occurrence of hurricanes. It was suggested in Sect. 3.2 that in a wellbehaved wind climate each of the data in a series of the largest yearly speeds
contributes to the description of the probabilistic behavior of the extreme winds.
However, in a hurricane-prone region most of the speeds in a series of the
largest yearly winds are considerably lower than the extreme speeds associated
with hurricanes; they may therefore be irrelevant from a structural safety point
of view. This situation is illustrated by the plot of Fig. 3.3.1, which shows the
S-minute largest speeds recorded at Corpus Christi, Texas, between l9l2 and
1948 [3-6]. It may then be argued that in hurricane-prone regions the series of
the largest yearly speeds cannot provide useful statistical information on winds
of interest to the structural designer, much in the same way as the population
of a first-grade classroom-which might include a teacher-is of little use in a
statistical study of the height of adults. That this is the case is suggested below.
The abscissa in Fig. 3.3.1 represents the reduced variate
o
o
a
I
I
I
I
rJL
t o< u
I NFu
I
I
I
I
t
oI
o
Io
o
o
?
n
6
-Frl-FrttsFFtHrHlrF-
oooo
oooo
oooo
COoo
oooo
oooo
oooo
OaFr
EOFF
I
o
I
I
xxx
I FxH
oooo
0o
oooo
oo
oo
I xFx
| -HH
I xFx
-O
=
9
,li
g
I
.
!
I
z
tI
o
J:
=
-H C,
\O
UJ
-5
uo-!
A
=.r
oo
J
ootsoo
UL
l
J
l@
G@
FO
Ud
103
l0,l
Lr l :'llMn ll()N ()l I 1lilt Ailt wtNt,
r,(rIilMt wtNt){]tl\/nt()tI)(iY
:,t
't t lr:, ll! lIrlilili t\|t t,lr{ r,il r ttMAll ..
l(}1r
, .i',( s tlnloalisiic. l'ot cxlttttpl,', ltltrtrl' r'rr{ lr ;l tlistrillrrtiorl trt tllt' l()l'l
lir!r, r,,ol.tl ol'thc llrrgcst yclu-ly slx'('(l:,;rl ('orlrrr:, ('lrlisti wotrltl yicltl, lirr'(lrc
r trr':rtr'rl 1000-ycar wind, a valu('()l l()fr0 rrrplr rr lirlicttlous rcsult l.l ()1.
1t,,.r
',{ uous difficulties als() arisc
il
rrrixr"rl lirt't lrt'1 pnrbirbility distributions arr'
ii ,,1 lt 51. lndeed, sincc hurriciltlcs ill(' l:rl('('v('ll(s, thc number of hurricane
r,,r rr'plcrl cyclone) wind spccd cllttlt irr lt t'e,.'ttltl ol'thc largest yearly winds
,,i,.1 1v1'l at any one stati()n is small (t:.g., irr lrig. 3.3.1 only two of the data
r,l'r(':iL:nt hurricane wincl spccds). 'l'hcrclirrc thc confidence intervals for the
, tr!('nrc wind preclictions arc, in gcncral, unacccptably wide (e.g., for N :
l{x) years, of the ordcr of irgo(l + 0.6) at the 68% confidence level; see
1r l5l). It is for this rcason that the 50-year fastest-mile wind estimated in
1l 5l for Corpus Christi on the basis of a mixed Fr6chet distribution is only
-+-
/(r mph at 30 ft above ground in open terrain. This value appears to be severely
Iow; indeed, in the pcriod 1916-1970 Corpus Christi was hit by three devastating hurricanes 13- l6l with fastest-mile winds of up to 120 mph at 23 ft above
ground in open terrain (see Corpus Christi 1970 Local Climatological Data
Annual Summary).
Because this series of the largest yearly winds does not appear to provide a
0.17 o:11
suitable basis fbr predicting hurricane wind speeds, alternative bases for such
predictions have been proposed in the literature, which are now briefly discussed.
In this procedure, proposed in [3-17], it is assumed that the behavior of the
FIGURE 3.3.2. Probability p, of an annual extreme wind being produced by a tropical
storm. From H. C. S. Thom, "Toward a Universal Climatological Extreme Wind
Distribution," in Proceedings, lnternational Research Seminar on Wind ElTects on
Buildings and Structures, Vol. I, p.682. Copyright, Canada, 1968, University of
extreme winds is described by the cumulative distribution function
Toronto Press.
3.3.1 Procedure
F(u)
Based on the Maximum Average Monthly Speed
:p,"^p[-(;)
"] *
(1
']
- p7)exp[ (;)
(3 3
1)
where zr is the wind speed, p7 is the probability of an annual extreme wind
bcing produced by a tropical storm, and o is a scale parametcr. The parameter
,rr, determined in [3-17] as an empirical function of the mean number of
tropical storm passages per year through a five-degree longitude-latitude square,
is represented in Fig. 3.3.2.The parameter o is given in Fig. 3.3.3 as a function
ol' the maximum of the average monthly wind speeds recorded at the station
concerned over a reasonably long period (e.g., ten years or so).
The application of this procedure is illustrated in three cases: West Palm
llcach (Florida), Boston (Massachusetts), and Columbia (Missouri), for which
pt = 0.43, Pt': 0.72, andPr: O, respectively (Fig. 3'3.2). At West Palm
llcach, thc maximum of the average monthly speeds in the period 1952-1974
(olrtainccl lnrrn thc Local Climatological Data Summaries) was 13.9 mph at 30
It lrhovc gnlrntl. linrnr lrig.3.3.3, o - 5l mph. Thcrcfbre
lt( t'\
on
,.xll (;;)
-'
|
, ,,r, ,.-,,1
(;
'|
)
(r i2)
o 2 4 6 8 t0 12 14 l(i l8 20 22 24 26 2830
32
34
Maxirnrrrrr nr()nlllly,rvcr,r;r. wirrtl speed (mph)
FIGURE 3.3.3. Scalc panunet('r'o. l;rrrr ll (' S.'l'lrolrr, "Toward a Universal
Climablogical Extrcnrc Wirrtl l)islrilrrrlr.rr." nt l'trtt t't'tlirr,q.t. lntcrnational Research
Sctninar on Wintl lillccls otr llriltlirrl':, ;urtl Strrr, trrcs. Vol. l, p. 6112. Copyright,
('rulttllt.
l9(113,
Univcrsily
ol lolonlo
l'r,
,.',
106
l:l
I x I nl Ml wlNl r ( rl lMn l( )l ( )(;Y
r
llll /,ilrv)1" i( lolkrws lrrrrr Ilt1. .1.-1.2 tlrlrl tlrt't'sltrrrtt(t'tl
Recalling that N
50" l(x), irrrrl l(x)o y('iilri itrt: /'5{) , 102 rrrPh, t'1,x,
N-year wincls lirr N
:
l9ti trph, rcsllc:t'livt:ly.
120 mph, and t/l(xx)
At Boston, the highcst of thc avcrirgc rttrtrrlhly spccds rccordcd bctwccn
1950,1914 was 18.8 mph at 30 fi abovc grouncl in opcn terrain. To this valuc
thcrc corresponds o :63.6 mph. With Pr - 0.12, it follows from Eq. 3.3.1
that thc extreme wind estimates are zr5s : 106 mph and z/rur : I 19 mph. It is
norcrl that the estimates presented in [3-5| otc l/5s : 88 mph and ales : 93
rrrph, that is, considerably lower than those based on Eq. 3.3.1.
Al Cblumbia, Missouri, the probability of occurrence of hurricanes is nil
irntl lit1. 3.3.1 becomes
F(u)
:".0[
(;)
']
(3.3.3)
ll:llMn
95 mph and z1ooo : 123 mph. It is noted that the estimated
88 mph,
extremes of [3-51 are lower, that is, u5o : 70 mph and 1]roo : 85 mph. The
extreme speeds at Columbia were also estimated assuming the validity of the
Type I distribution, with parameters inferred from the l95l 1974 series of the
largest yearly speeds at 30 ft above ground in open terrain. The results thus
obtained were t/56 : 66 mph, zroo : 69 mph, and ?1s00 : 8l mph, versus z'so
: 88 mph, urut : 95 mph, and zr1ee,l : 123 mph, as estimated on the basis
of Eq. 3.3.1 with the attendant assumptions of [3-17].
Among these assumptions is the relation implicit in Eq. 3.3.1 and Figs.
3.3.2 and 3.3.3 between maximum average monthly speed and the extreme
wind speeds. No fundamental meteorological grounds are offered in [3-17] or
elsewhere in the literature for this relation which, frorn the evidence available
so far, does not appear to be justified.
ycrrrs)
I
Based on Climatological and Physical Models of
Hurricanes
To illustrate the principle of this procedure, an estimate will be made of the
probability that hurricane winds in excess of 155 mph will occur at any one
specific site on the Texas coast. The following information will be used in the
estimation:
o Average number
per year of hurricanes with-wind speeds in excess
of
155
mph moving inland in the United States, t iss. According to the National
Weather Service, there have been two such hurricancs in thc past 75 ycars
or so, the Labor Day Florida Kcys hurricanc in 1935 ittrtl ltttrricittlc Cltlnillc in 1969 l3-181. A rcasonablc ostirnittc is tlrt'rr ll,tt - 2ltrrlricttrros/
(75 ycirrs) 0.027 htrrr/yc:irr.
ll)i;
lN llllllltl{
nl]l l'lt{rl.ll
r |M/\il:,
llll
.i"i,l lrrrr r'lyeltr.
Avcragc rrrrrrrlrt'r'lx'r'y(:ll
ol'all hun'icattcs trrovittg irrllrrrtl irr 'l-cxas,
a7.
From Fig.'J.J.4. ttr = 2J hurricancs/((r-1 yclr's) = O.43 hurr/year.
o Average width ol'area swept by winds in cxccss ol'155 mph in one
hurricane, I'lz. According to [3-20], thc path of destruction of the Labor
Day Florida Keys hurricane was 35 40 miles wide. It will be assumed
conservatively that winds in excess of 155 mph affected a width W : 30
miles of that path. In the case of huricane Camille, it appears that it may
be assumed conservatively W : 20 miles [3-211. A reasonable value to
be used in the calculations is then W : (30 + 20)12 : 25 miles.
It will be assumed that the average number per year of hurricanes with
in excess of 155 mph moving inland in Texas is
t55
U7
2,100 :
3.3.2 Procedure
Wllil):,1'l
lrr v('irr ol lrutt'it':rttt's nr()vlrl' rttl;rrrrl trr tlrt'I lrrilctl Sllttc:s,
ll(rlrtrtricuttcs/
1r,,,.'l'lris(ilr:ur{r(ytrrrlrt't'slitturtcrlI'nrruliil'.. t.t..[:r,,
'l'hc maximum of the monthly wind speeds recorded between 195 l-1974 was
l-5.7 mph at 30 ft above ground in open terrain so that o : 51 .O mph and zr.e
:
[Il
Avt'r'rrl.:t'rrrrrrlx'r
(63
o
ll()N l)l I "llrl
speeds
(3.3.4)
'-t ,t;tt
U1
(Implicit in Eq. 3.3.4 is the assumption that the probability distribution of the
hurricane intensities, given that a hurricane has occurred, is the same throughout the U.S. Gulf and Atlantic coasts.) The length of the (smoothed) Texas
coast being about 375 miles, the probability sought is
P(u
>
155 mphl
: ,'r"
' :375
:0.00042
(3.3. s)
that is, approximately ll25OO per year.
The estimate just presented has several significant weaknesses. First, the
errors in the estimate of utfs could conceivably be large, the estimate being
based on a 7S-year-long record containing just two relevant data. Second, the
assumption that the rate of arrival of hurricanes is uniformly distributed over
the length of the Texas coastline overestimates the probability of hurricane
strikes over the coastline segment adjacent to the Mexican border (by about
25%), and underestimates that probability (by about25%) nearerthe Louisiana
border (Figs. 3.3.5 and 3.3.6). Third, the reliability of the estimate of er7 is
difficult to ascertain. Indeed, according to Fig. 3.3.6, u7: 1.6/(100 x l0)
entries/yearlnmi of coast x 0.53 huricanes/entry x 330 nmi of coast :0.28
hurr/year, versus 0.43 hurr/ycar, as obtained from the data of Fig. 3.3.4. (This
discrepancy is possibly thrc to lltc courtlirrg ol'ccrtain tropical cyclones as
hurricanes in [3-l9lr';. lirrullr. tlrc esl inurlcs ol'14/:rrc largcly subjccfive, since
l3 l9l w:rs lcvisctl irr l()/li.rrrrl r', ul)(l.rl(,l,rnrrr.rllr lry llrt'N:rtionlrl llrrlritlrrrr'('crrlcr
Arlditiorlrl ittlirttt:rliott otr Norllr r\ll,rrrlr, lrrrrrr,,ur". r', ,r\r;rilirl)lc irr l1 -5 11 lrrtrl. on llrpt',
irr I I 5'1 1. lirr irtlirtrrtrliort tttt Wr':,lcttt Norllr l'.r, rlr, lr{rlr! .rl ( \r lor( s. rt't' I I 551
'r'Rcl'crcncc
l3
-521.
z
4z
U_
Eg
lrrrrl
lrtrtl
lrrrtl
J
'ir
r r r'l'
lrirrlrrrrl
o>
UO
k(J(rU
=o
=or
2
-<
-E
zo&
-rrll
o(/)r
I
i
':tz
d>
ao\
z 64
- trl
o zz
uJ<
9U
>E
KE.
UU
9 -P
FF
zz
UU
L
E!a
dgt
ool E
.c.F !
OO
tlc
o
i
^il
=t
I
:(/o
U(E
>, ,-j
=
dotr
tE
iP
:q
I
.:!d
trE -
c.S
;:
\
--
g
'=o-5".
.: g2\o
cn
, -.
-^O
(t)
TE
.\(r
\90
8'r
o9"i
iin
LF*
dJ9Y
rc
=o
H
5
.r-.i "
O
;h
E
5
_* boU
9()trl
tdE
rIOr
?rH
t%n
<. aJ
.o!
-C
s?e
e4Aa0;
__
llE@
lrrl
nO
rl o-'x
Ao,
--^r=jfA^
t:11\\
108
qm
t'aJ.\
&
\'p
try
/ls
,/e
{| _-*1t1)
-V
^iJ )
.'
u/
't
l!t,l
:- -1,,
!'
'.1
,1,":,
;
'.'i;:'l
l'j
-,
109
tll
Nurrrlr:r
ol
ONN
to
crrtrrr..;/l(X) yrr;rrt/lO ntnt
rr
rt,
of
O
coasl
A
N
il
rrr
dl
^o
_o
',o o
su,4 s.)tlsr.rJl.lr:.il:rl.t.lu0l.)A.l lu.)t(l().tl U() Uotll:tU.t()llil.l() slsll(1 .)(ll u() Jlls ll lll
sJrlrlr(lr:(l().r(l l)Ur/r\.rlt.).t lxJ.l() ifuillolx)rU or.ll ()l r-lrr:().r(l(lu,)All.).).llJ l)lilr JArsuitlr).r(l
-rrr()J v 'spaads pully'r ouea!ilnH 6u//|€tu//ls3 Jo, arnpacotd opec auow
bo
'uorleturlse eql ur pe^Io^ur serlureuecun eql lo cillJrd
InJ8urueew ,{1pcrs,(qd 'Juelo u JeJo feqt 'peeds flqtuoru eSeJa^e rununxuru
eql uo pesEq lepotu eql e{rlun'1eq1 sr s'€'€ puE ?'€'€ 'sbgJo ernlBeJ Injosrr rl
lt
JFts
I {A
_+(,
o5i-At5 tD'O
3 --i
qlnx
!iH='
1a--
Lake Charles,
La.+
=.?
o!1
fii
'oslv 'seleJ esuBJnsur Jo suorluurrxoJdde lsJg JoJ 'eldl'uexe JoJ 'suorSer ouo.l(l
-auBclunq ur seJnlcruls Jo eJnlreJ Jo ,{lrlrquqoJd eqt 1o seleturlse ssoJS Surluilr
JoJ esn etuos Jo eq ,{?lu peurllno lsnl qrBoJddu oql sesselr)lee,$ eseql etrdso( I
'lle^\
se
l4 Jo uorleurlsc crll
lu€JuruSrs eq eJoJeJoql ,{uu ro.ue aql :plou aql ur uo{Bl aJo,r s}uetueJnsuour
ul
oU
3a
o
a
'lZZ-fl
c
3
g
(rtuu)
pelteu sll?^rctul ocu€lslp
IElspoc qlr.iv\ dutu
rotscol .S.t.g .du()i)1.,1
o
St. Marks,
o
3
aa^
o
aD+
Fla.
t
*-/(F-,-
Ft. Myers,
0a.n'3
(-a
9=io
=
v;-Fot!
a5 A;l
Miami,
d
)oz.-\\
Fla.
\,1*
Fla.
l
f
- ?ft u+!r
t6='^
5
=yd
=!
J.9
oxe
ir5
-
Cape Hatteras, N.C
.DD@
-o
3
6 l./
YoI
\o+a;
-J-ur!) 6
-3b
r*^
' a6
ry=-J
:'m
r
9s)99
I
5qo!
Ratio (hurricanes/total srorms)
@
I
@
Alx)t()lvv\It:) (tNtM
tv\t
IIIx
0rr
I
112
I x tttl
Ml wlNl)
I li;llN,4nll()N()l l!llli
(
)'ll' wllerc
tlcvcl6pol in l-3,2-1 l.'l'his ir1-rprxrclr wrrs srrbsctlrrcrr(ly irplllit'tl irr l-l
bltsis ol'
tltc
ott
cstirrratctl
extreme wind speecls associatccl witlr lrtrrricrrncs wLlro
the climatological and physical modcls dcscribcd bclow'
fint WtNt):it't tt)l; tNil( ,tilil(nfJt I.t t.,rJt 1.t tMnil:,
ll3
'l'ltc tttltxilttrrrrr rvrrrtl sl)(.(.(l:tl l0 r1 lrlrpvt'(lrt.6t.t.:rrr
srrr.lltt.c, itvctirgctl
ovcr l0 lrrirrrrlt's, rr p,rv(.n by lhc clnpiriclrl rcllrliorr
')
t/( 10, R)
:
0.865/",(/t)
I 0..5,r
(3.3.7)
Climatological Models
1. The hurricane frequency of occurrence is modeled by a Poisson
with a constant rate.
2.
13-261. This rclation corresponcls to thc avcrage of data observed during
the 1949 hurricane that crossed Lake okecchobee, Florida
[3-27, 3-2g].
whether Eq- 3.3.7 can be assumed to be generally valid is uncertain.
For example, according to [2-l4ll, during typhoon Mireille observed
surface wind speeds over ground (which are lower than over the ocean)
were comparable in the region of the eye with estimated winds at the
gradient height. However, according to [3-79], measurements indicated
that the 10-m level sustained surface winds over water were generally
within 55% to 85% of the winds measured by reconnaissance aircraft at
500 m to 1800 m. Reference 13-791also suggests that the logarithmic
process
The probability distribution of the pressure difference between center and
pcriphcry of the storm, Ap,,"*, is lognormal. To eliminate values of Ap.,*
jrulgctl, in thc light of historical data, to be unrealistically high, the
tlislribrrtion is ccnsorcd stl that AP,."* ( 101.6 mm (4'00 in) of mercury
l.\2.]1.(Ntl(r:tlrirtA/).''.*:l0l.6mmcor.respondstothelowestatmo_
sgrlrcle l)rL:ssurc cvcr rccorclcd worldwide t3-25].) Theoretical studies appcru. (o cottlirttl this bound l3-61].
l. 'l'hc p(rbability distribution of the radius of maximum wind speeds, R,
is krgnorrnal. This clistribution is censored so that 8 km < R < 100 km
to avoid unrealistically "tight" or "broad" storms 13-231'
4. The average correlation coefficient between R and Ap-,^ is about -0'3'
(see 13-221, pp. 68 and 69.) All other climatological characteristics of
law appears to be valid up to about 1g0o m but that at aboul 3000-m
-1_
hurricanes are statistically independent.
The probability distribution of the speed of translation, s, is normal. This
distribution is censored so that 2kmlhr < r < 65 km/hr 13-231'
6. The probability distributions of the distance between any specified point
on the coast and the hurricane crossing point along the coast (or on a
line normal to the coast) are curves matching the historical data. Separate
curves are defined for entering, exiting, upcoast heading, and downcoast
heading storms.
7. For entering storms the probability distribution of the direction of storm
translation is a curve matching the historical data. For exiting upcoast
heading and downcoast heading curyes the distributions are uniform between 130" of the mean directions of storm translation. In all cases the
storm path is assumed to be a straight line.
elevation winds may be less than at the surface.
Let the center of the storm be denoted by o, and consider a line oM
that makes an angle of ll5'clockwise with the direction of motion of
the storm. The l0-minute wind speed at l0 m above the ocean surface
at a distance r from o along line oM is denoted by u(10, r). The ratio
U(10, r)lU(10, R) is assumed to depend on r as shown in Fig. 3.3.7
13-261. Let the angle between a line oN and line oM be denoted by 0.
50
40
30
E
E
d
l0
Physical Models
l.
I
The maximum gradient speed is given by
in which it is assumed that
dp
dn
whr-'l'c rv is ohl:tinctl
Eq' 1'2'8 in which r
:
R' and
1
6
5
4
-
(3.3.6)
dAP,,t,,,
lirr crrrpiricirl tllrtrr l3-2(r.
t,l
.1
2ltl
rrr(;rjrr,r-r..1.7. rr;rrr',,
I
0,,)
lil';li ,
ttrt1111li)
|r
.r(,1
I l4
I
Xi
lll Ml wlNl)
(
,l l[/n l( )l
( )(
ir:r r:iilMn il()u ()r rrrr
() irlolrg lirrc
l'lrc lO tttittrttc wirrtl sllectl l/1 lO, r'. //) trl :t tlis(lttlt't' r' llrttt
(.rN is givcn by Lhc cxPlcssion l'l 2(rl:
U(10,
.
'
r,0)
:
{/(lo'
r') I ,'
cos (/)
center of
witul vckrcity vector has a cornptlncnl. t.lirccted toward thc
lltt.s{tll'llt'().Theanglebetweenthatvcctorandthetangenttothecircle
region O I r
tt'rr(r'tt'rl rrt O varies'iinearly between 0o and 10" in the
' /i :rrrtl l)ctween 10" and 25" in the region R < r < l'zR' and is
r'r1n;rl l,r .15" in the region r > l'2R13-261'
llr,' :,lorttt clccaj results from a decrease with time of the difference
storm
at the center and pressure at the periphery of the
lrt lrl,r'r'il
rr
l)tcssure
()r(llrtlce with the relation
Lp(t)
wll.il,:ir
:
APn,"*
-
0.02[1
+
sin
{lr
I''(U
ll(,r!r
Notc that physical models proposed in [3-28] are in some
rir,rtliliccl wittr respect to the corresponding models o1' l3-26].
dcscribedearlierdefineawindfieldwhichdependsuponthepositionofthe
hurricane.Toeachpositionofthehurricanewithrespecttothesiteofinterest
therecorrespondsawindspeedatthatsite.WindspeedscausedbyahurricaneThe
number of such po-sitions.
at the site are calcurateJ ro, u sufficiently large
caused by the hurricane
largcst among these speeds is the maximum wind speed
atthcsitc.Asctill.mspccdsisthusobtained,whichisusedasthcbasicset
ol'ltttrricrrrtc wind
ol'rllrlu lirr tlrt: cslirrurtion ol'tlrc prtlbilbility ol'()cctrffcllcc
'l'lrt'st' sPr'ctls :tR' tltttkt'tl lry rtrilgtriltttlc 'l'lrc i (lr srrt:rllt'sl spt't'tl ilt lt
lr wiltrl spt't'rl:; is tlt'ttolt'tl lry tt,.
r15
< uln) :
1"',',
(3.3.10)
(3.3.11)
< u, r)
where p(n, r) denotes the probability that n storrns will occur in r years,
Assuming thatp(n, r) is a Poisson process (Eq. A1.34), Eq. 3.3.11 becomes
<
u, r:)
: ! o' (\t)'" ^"
n:O '
nl
(3.3.rZa)
-(4-
- ')"3txt41'
r:0 nt
(3.3.tzb)
_ --)tnl - 1,1
(3.3.t2c)
-(
where \ is the annual rate of occurrence of hurricanes in the area of interest
for the site being considered. For z : r, F(u 1 u, r) is the probability of
occurrence of wind speeds less than u in any one year.
consider now the wind speed, ui.The probability that u I u,in any one
storm is
_t
cases slightly
wind speeds were
lrstirnates of the probabilities of occurrence of hurricane
mileposts (Fig'
56
adjoining
rrtrrrrirrcd inl3-24lby assuming each of the areas
of
characteristics
climatological
r 1.5) to be hit by m : 1000"hurricanes. The
respective
the
from
simulation
llrc hurricanes were cletermined by Monte Carlo
to historical data. For each of the rz hurricanes'
lrKrbabilistic models as fitted
with the physical models
thc climatological characteristics used in conjunction
1:,
: i^ ofu < uln)p(n, r)
n:0
F(U
F(U
Sr'ct.2.3.6.
(1Mn
I u, r). The total
(3.3.e)
(rllrr'rt'tluctionofwindspeedsduetoincreasedsurfacefrictionoverland
t u'irollu'(10) : 0'85' where u/(10) and u'(10)
r" r'rv('n by the
.rtr.lltcl0-minutespeedsatl0-melevationoverlandandoverwater,
rr':,lrcetively.ltcanbeverifiedthatthemodeldevelopedforextratropical
a some'.r,,,,rrs (Flqs.2.2.29 ancl 2'2'31 ' and Table 2'2'3) would yield
rvlurt smaller ratio Ul( l0)/U''(10)'
/ llrt: clependence of wind speeds upon averaging time is modeled as in
s..'t ,,1
rr): ; rN ilr,rrrr{ Ar]r
The probability that U < u in z years is denored by F(U
probability theorem (Eq. Al.5) yields
given in inches of
rrlrr rr. / . travel time in hours, Ap(f) and Ap"'o* are
(0 < d < 180')'
track
rrr'r{ rrry. and @ : angle between coast and storm
in
repotled
12-1241'
llrr', rrttrtlcl is consistent with measurements
sPt't.tls,
,r
l,cl llrt'lllrlxrlrilily llr:tl llrt'wttttl spt't'rl irr rrrry orrt.slorrrr is lcss tlriyr sgl.rrc
vttltlc' tr, bc tlt:ltolt'tl lry /',, l'lrt'pnrblrbility llr:rl llrt. lrililrest wirrtl {/ irr rr st1;rnrs
is lcss lltart ll t':rrr lrt.wttltr.n:rs
(3.3.tt)
,l,lrt.
,r( (
I\,4r
'|Y
A
(3.3.13)
"' -_ m*1
Thus
F(.U
< ui, l) : e ^(t
i/m+t)
(3.3.t4)
For each of the mileposts in Fig. 3.3.5, estimates of hurricane wind speeds
corresponding to various probabilities of occurrence (or mean recurrence intervals) were obtained in t3-24l both at the coastline and at various distances
inland lrom rhe coastlinc.
Results of a rccent stucly inrlicrrlt' llr:rt lrrrrr.it'irrrc wincl speed data obtained
by simulati<ln and ftrrnring lhc brrsis ol llrt't'stirrr:rtr.s ol'13-241 rnay be clcscribecl
by thc rcvcrsc Wcitrull rlistt'ibrrtiorr l l /ll 'llr;rr tlistr.ilrution lurs lirnitcrl uppcr
tail and is consistcnl willr llrt':rsslrnlll rln tlr;rt lrrrrlt.lrrre witttl sPct:tls irrc
hrltltttlltl. Ilcl'crt:ncc 1,1 7l l lllrvi(l(':i 1rl()llrjrtrr)!r {)1 :rt r'r'ssirr1,, llr1;st' tl:rllr (llrt'
rllt(lr is ltlso ltvirilltltlt'irr ll.i ()l). lr:; rr,,.ll tr.. ( r)!nIrttr.r l)r1)t,tiuns lor llrt.(.sllu:tlt()lt
I l6
I
X
llll Ml wlNl)
{
)l lMn l( )l (l(
tY
ol ltttttit'ltttt'witxl
()l tllc t(lvcrst: Wcilrttll tlislt'ilrrrtiorr l)ill;lllrt'l("ls:tttrl
witlr vltt'iotts lllcan rcctlrt-ctlcc irttcl-vltls'
lnrrrr tlrc "tt'rtc" s1.rc:ctls owittg ltt
llsti.urtctl hurricanc win..l ,gr.,,1, tlillr'r
lrrtxlclittg' irrttl sarrtllliug crrors
.5se'vuti.., pnlhabilistil ltx)clolillg, lllysiclrl
(i.t'.'clrrtrscluctothelimitcclsiz,ctll.(Itctllttlrsirttrplc.sbr:ins.u'"..])ljtlrwind
tll'thc ordcr rll'50 ycars' it was shown
spt'etls with lltcan .""u""nt" intervals
()f thc sanrpling errors, duc t<l thc limited
rrr I I ]01 rhirt thc *tunJura deviation
(about 100 years), is about l0% of the
:;t.r .l t.lirrrrt.r.gical Jata availaule
about 15% fot 2000-year speeds'
,1.,1 ,,r':rr,'.1 s1,".,1*' it is
wind speeds-that-are similar conceptually
I'r.t.t.tlrrrt.s ti,'
"*ti,niting-r,u.ri.un"
for the purpose of studying
r. lt .)\l rvt'rt' ,r.,u"r.rf"J-in i: i1]^ana tS-:Zl for
estimating hurricane wind
methoJ
lnil | r, ;ril(' 1't'ttt'trtlccl *uu"'' A simplified
..1,,,,1', \\'it:' l)l()l)()scd in [3-56]'
sPtetls
Boundary'
Shapiro Hurricane'
Mrrrlr.' (--irrlo Simulations Based on the
boundary-laver flow in a translating hurricane
I:ryt.t Mtt<Ial- 'l'hc #;;il;
empirical physical models^based on his\\';,'. .,l,r,r('\trrtrtlocl i"i;-;ii Lt uling
in ittit section. Shapiro [3-57]
r.rr,;rl ,lrrr:rt.l.gicat Jata,-as lndicated earlier
of the hurricane boundary-layer flow based
1|1'r,r.lr11ir'tI ,..' ..pp.o*i*ut" model
equations' complemented by the
orr ,r ,,rrrr1,lrlr,..t ,olution of the Navier-stokes
height 6 and the eddy viscosity are
l()ll{,\\rnr' :rssrtttrptions: the boundary-layer
to the
'), uid th" frictional drag due
(..r.,r,,rr (;, I m; f ='i;1dt;t;
empirical
an
times
velocity
flow
to the square of the
Irrrr',1;rltott vt'ltrcity i,
"qtluf rin"u.iy with flow velocity. Rather than obtaining
rrr"il"J*
trrt t'rr , r,r.llicient
of motion' a simplified approach
tlrr' lrrllv rrottlinear solution of the equations
w;1.'tt''t..lwltcreinthevelocitywaswrittenaSthesumofanaxisymmetrictetm
lttt,lllrt.lllsttwot",m,ofuseriesreflectingtheflowasymmetryduetohurricane
is truncated af'ter the first two terms is
lriilr:,l.rlr()il. The facf that the series
of the fully nonlinear sot':,trrrr,rlt'tl by Shapiro to result in an approximation
p' f q96t' This errorcstirnatc doesnot include
Irrrrrrrr lrr within uAoiZS% t3-57,
Shafully nonlinear solutions wcrc available' the
if
nr,r,l, ltttg crrors;
boundary
the
of
s(ructure
"u""
'f'L
detailed
uut" to iescribe the
1,rr,, rrurtlcl would ";;;;
eye wall [3-57' p' 19951'
the
near
!,rv.r. cspecially
modcl wcrc reported in [3-58]'
liesrrlts of ri-ufution' Uu'"d on ih" Shapiro
approach' The rcsults of the simulations
rvlrrth used Shapiro's truncated series
Owing to a new and'
tlrllt:r from those of 13-241 inone main respect 13-591' conservative than its
less
is
that
ilr ()ur opinion, *"diil. nuing rate model
speeds inland' Otherwise' in
wind
yield.lower
Ref.
t'iOl,the!
toLrnterpart in
to
ln" Sit"pit mode-l'-they yield wind speeds-comparable
spitc of the use
"t
rnual use of data and models from
those of 13-241. Although Ref' t3-581
[3-60])'
;?; basecl tn [3122] (i'e'' an earlier version of obtained
13-601, whereas f3;4f
speeds
wind
E,stimated
this is not a Source of significant dill.erences.
shows cstimatcs fronr [3-24],
are lisrccl;; i;;i;3.3.1, which atso
in
i
&
E
c
tr
QOOO
N
t'
.1
t'i
a
]f
o
r60o
-o
tr
Qr9n
d
d
9'l
r$€r
E
o
o
9N6@
q)
z
tr
6
q)
o
-:
I
O€-O
!
o
al
o.
o.
r
!
I
@
r@ot
n
€660
rr@o
6666
r€r-
o
E
I
ro
€aoh
Eq
9+@N
i
o
o6
@9-$
d
oo
:
>(J
9E
O
.o
N-a$
!
cno$
OO-d
3
.o
o
$tr€
o
liE
|'r
c,
a'
I
r:
+-CO
q,)
!
-:
o
abO
e8
F2
|=
69
:"."
(td
,+
-
,:,,,
&c<rl.l
F6
8r
^r
13_581
t3-29l.arrtl|3-71|.Ntltc(Iur{(lrr:clllllparistlnstll'|3_5ttIllclwcclrtltt:cstitrratcs
-.)
\c
(,
'-:"
!,
i-
'i9'=!
-t
E-'E E!
L
..
!'.
'C
E=
ll7
IIB
I x tltl Ml wllll)
(,1 IMA
l{
<ll'13 5ttlirrrtl 1.1 241irrc irrvirlitl 0wirr1', lo lrrr irrtorrsislt'rtt'y itt lltt'll:tltslirtlttlttiotts
of hourly spccds iukr lirstc:st rrrilc spt'ctls. 'l'ltcse Ir'ltttslitt'tttttlirttts lttt brtsctl itt
(lre e:slirrur(t's ol l.] -5ltl, bLt( ott lht: tlill'crcnt
[3-58] on the moclcl ol'12-1351 lirr
of [2-91] for the cstimatcs ol'1.1-241. lior 50-yc1r spcctls, tlillcronccs
between [3-24] and [3-58] excecd 10"1, lol trrilcposts 2(n. 3(X), ancl l3(X). For
model
2000-year winds the differences excccd lO'/,, l'<tr milcposts 700 and 1700. At
l0 m above open teffain, the hourly spced corrcsponding to the largcst 2000year estimate of [3-71] is about 47 mls at l0 m over water the cstimate of the
largest 2000-year hourly speed would be about 41 x 1.2 : 56.4 mls'
that the wind load factor $,u : 1.3
spccified in the ASCE, 7-95 Standard would in most cases coffespond for windscnsitive structures to nominal ultimate wind loads with mean recurrence intcrvals of, roughly, 500 years or less. For the other sets of estimates of Table
3.3.1, the load factor d' : 1.3 would in many cases coffespond to nominal
ultimate wind loads with even shorter mean recuffence intervals' These results
are reflected in the average estimated ratios of 2000-year speeds to 5o-year
speeds, which are about I .3, 1.4,1.45, and 1.46 for the sets based on t3-7ll'
the squares of these values are
13-241, I3-2g1, and [3-58], respectively, so
uUo"f t.Z, 1.g5,2.1, and 2.15, respectively. The results of [3-71] and Table
3.3.1 therefore indicate that for wind-sensitive structures, the wind load factor
for hurricane wind speeds should be larger than 1.3, even if hurricane design
wind speeds are multiplied by a factor of 1.05, as is done in the ASCE 7-95
Standaid t}-t3gl. For additional details, see [3-80], tA3-3ll and Sect. A3.3'
Load Factors.
lil l:;,Mn
)l { )( iY
It is shown in [3-71]
Estimates of Hurricane/Tropicat Cyclone Wind Speeds for Various Lo'
cations Outside the U.S. Estimates of hurricane/tropical cyclone wind speeds
based on models similar to those of 13-241 were repofted lor French overseas
departments and territories in [3-63] and are summarizcd in Table 3.3.2.
Estimated standard deviations of sampling errors in m/s were 2.2 (3.4),2.5
(4.2),2.0 (4.0), 1.6 (2.5) and 3.9 (12-3) for 50-yr (1000-vr) speeds at Guadaloupe, Martinique, R6union, New Caledonia, and Tahiti, respectively'
Estimates of hurricane speeds are reported in [3-64] for the Eastem Carribean, Jamaica, and Belize and in [3-65] for the Northern Australia Coast.
For information on westem Norlh Pacific tropical cyclones, see
t3-ssl.
Saffir-Simpson Scare. The National Hurricane Center, the National Weather
Service, and emergency management departments use a classification of hurricanes into five categories (Table 3.3.3). The central pressure and wind speed
portion of the classification was proposed by H. Saffir in 1970, while the storm
surge portion was added subsequently by R. Simpson. Thc avcraging timc,
height above grouncl, and surfacc cxposurc (i.c., whcllrcr ()l)cll lcll'llilt or wlttcr)
'l'Alll,l,l
'()N
-1.-].2-
{)t I r.,llA,lt WtNt) :it,t tt)ti lN ,'t.'(
A,t
t,t
tr',
l'lslirrrrlrrl wirrl s;x'r'rrs (r(!r'irr s;x.r.rr irr r0
(.t tMn
'
:,
l1g
rrr ()vcr.irrt.()rcarr)
!.r-6rl
Mcan
rcturn
period
(yrs)
sr.
New
Barthilcrrry (iuatlcloupc
25
50
32
37
Martiniquc
32
Ildunion Caledonia
29
35
100
4t
40
39
500
49
r000
48
52
47
50
51
38
38
40
40
43
48
50
43
48
49
Tahiti
30
34
39
47
52
for the wind speeds are not specified in the classification.
For this reason storrn
effects and evacuation requirements for the various
categories are described for
the use of, among others, emergency management personner
13-661, rather than
structural engineers or building code officiils.
Mixed Distributions. Hurricane-prone regions are also subjected
to winds not
with hurricanes (or tropical cycrones), whose effects can
be accounted for by developing mixed distributions of hurricane
and nonhurricane
wind speeds. Since the occurrence of hurricane winds and
the occurrence of
nonhurricane winds are independent events, it is possible
to write
associated
F(.U
< u) :
F11(J <. u)F7s11(J
I
u)
(3.3.1s)
where F(u ( rz) is the probability that the wind speeds
u associated with any
storm are less than u in any one year, and Fs(U (
z) ancl Fuu(J I u) are
the probabilities that hurricane speeds and nonhurricane
wincl'ri""a, are less
thal a in any one year.-The probability Fs is determined as shown previously
in this section' The probabitity F,vi7 is determined as
shown in sect.
3.2.
TABLE 3.3.3. The Saffir-Simpson Scale
Category
1
2
3
4
5
Mean Wind
Description
Speeds (m/s)
Storm Surge
(m)
33-42
43-49
t.2-1.6
t.7-2.5
50-.5f1
2.6 3.8
#)
3.9- -5.-5
( i r clrlcr
Minimal
Moderate
Extensive
Extreme
Catastrophic
.59
(
ilctrlcr
lltlur (r()
llr:rrr.5.-5
North Atlantic
Examples
Agnes
1972
1964
1965
Camille
1969
Cleo
Betsy
David
1979
l2O
{xllll Ml wlNlr
(.
r
,t wllll, i,il lt I :ilr
)Nn I
il v
r\ vir'
('irlculirliorrs tcptttlt:tl irr 1.1 l':l lsll(rw thlr( lltt'Prrrlrirbility /'lll
'r) yt:1r's'
5O
N
'
lc('llll('ll('r'ilt(t'l'vltls
lulrlly tltc rurn" r,li I,'tr(/ < tt) lirl'lttcltlt
ttl'Itotllttlrricltltc
Fo. N : 20 ycars, cstiptl(ccl wirrtl spce:cls tlr:rt irrcltttlc: tllt: cllcc(
5%' Nrlto that
lly
al-tout
spcctls
wiltcl
wincls cxceed the esl"imatccl hurricailc
whcrc
Hattcras'
tll'Capc
ntlrth
these conclusions are not neccssarily appliclrblc
nonhurricanewindsmaycontroltheclcsigna(coflainlocatitlnsl3_33|.
3.4
WIND DIRECTIONALITY
the
Witrtl cllbcts on various structures and components depend not only on
well'
as
nrilgrritudo of the wind speeds but on the associated wind directions
lr.i this rcason, knowledge of continuous joint probability distributions of exand code develtrcrne wind speeds and directions would be useful for design
0pmentpu,po,",.However,sofarnocrediblemodelsforsuchdistributions
have been proposed in the literature'
20 30 40 50
50
distributions
In the abience of such models, wind effects and their probability
information
of
basis
the
on
may be estimated in well-behaved wind climates
of largest yearly wind speed data recorded for each octant over
data have
periods oT ZO y"uit, tuy, oilong"r (see Sects' 8' l '2 and 8' l '3)' Such
Summary
in
stations
[3-341.
teen published for a number of U.S. weather
consisting
in the
statistics of largest yearly wind data recorded at Sheridan, Wyoming,
in
seen
that
h
is
3.4.1.
Fig.
in
period lg58-1l:i'7 (iee 1'aUte 8.1.2) are shown
norththan
weaker
wincls blowing from the noftheast are considerably
ihi,
or southwest winds.
west"u*"
in which
As shown in Sect. 8.1.3, there are important practical applications
largest
the
of
distributions
information is needed on the univariate probability
directions,
compass
principal
yearly wind speeds associated with each of the
blowing from
una on the correlation coemcients for the largest yearly winds
wind speeds
yearly
largcst
the
any two directions. In well-behaved climates
fitted
always-adequately
not
ioi ony given direction are in most cases-though
corthe
in
indicated
As
values.
[8-141,
UV fVpJI distributions of the largest
comprincipal
eight
of
the
two
in
any
retatlon between wind speeds occurring
correlation
pass directions is in -ort .ur"r weak. For example, the estimated
:
1,2,
i
and
" ' ' 8)
directions
i(i,i
coeflrcients between wind speeds from
fairly
are
values
These
3.4.1.
Table
in
are shown for Sheridan, wyoming,
coefcorrelation
the
of
values
the
where
typical. However, there are stations
values
estimated
28
the
8
of
where
Michigan,
dci"nt, are higher (e.g., Detroit,
are larger than 0.45, although none exceeds about 0'6)'
largest
An important practical p-ut"- faced by the designer is obtaining the
at
directions
compass
principal
eight
of
the
yearly wind ,p""d dutu for each
sourcc
io.ution, not covered in [3-341 . There are two such sources of data- One
Ot:canic
and
Nlr(irlnlrl
thc
hy
storecl
rccorcls
consists of the original LrnpLrblishecl
thc rrcctlctl
Atrn0sphcric Arlrilinisf nrtiorr (NOAA). Ob(aining ittttl e xlrltt'lilrg
A st't'otttl
littlr'('()llsllllllllI
rriltl
irtcortvcrricnl
lrollr
rl:lltr lhltil tlrttst.rct'()ttls is
FIGURE 3.4.1. Summary statistics of largest yearly wind
idan, Wyomine (1958,19j1).
speeds by direction at Sher-
of published Local Climatological Data summaries issued
NoAA. Directionar largest yearly speeds in the published data
differ in a few cases from the corresponding speeds in the original records.
source consists
monthly by
The reason for these differences is that the published data consists
largest daily speed for every day ofthe year and (2) the direction for
of (l)
the
that speed.
TABLE 3.4.1. Estimated correlation Coefficients for Directional wind speeds
in Sheridan, Wyoming
Direction
I
2
3t
4
5
6
1
lt
I
-0.05
1
symmetrrc
0.35
0.01
0.12 0.16
-0.1-5 -0.34
0.04 0.34
|
0 17
I
*0.031
0.03
0.10
0.03
0.01
t
-0.22
0.07
-o.12
-0.16
-0.16
0.20
|
0.07
-0.01
-0.43
0.40
-0.41
0.01
0.32
I
IXiltt Mt
122
WtNt
r,, l,ll()ltnllll llll :; ()t ()(;(:tlilt il t.t(.t o;
)(jtMnt{)l (x;Y
11rt thlnt
)() wtNt):
;
t23
Cgnsidcr, lirr cxartrplc, lIc ctrse wlrr'tt'in lr givr'tt ycltl lltt'lrrr'g.est prrlrlisllt'tl
(r-5 rltPlr,
speeds fbr wincls bckrwing l'nrrrr lhc r)()11lr lrxl lltc c:ltst arc 70 tttplt ittttl
(ha(
occurrccl'
witttl
n<lrtlr
thc
clay
salnc
on
thc
that
respectively. It is conceivablc
the winds blowing from the east were 69 rnph. Thc highcst wind spccd f'rom
the east would not be reflected in the published data'
An exhaustive study of original and published data listed in 13-341 for 24
stations showed conclusively that the extreme wind speed estimates based on
published data differ insignificantly (by about 3% or less) from those based on
ihe original clata. It is, therefore, appropriate to base structural engineering
calculat"ions on the largest yearly directional fastest-mile wind speeds obtained
l'rorn l-ocal Climatological Data summaries t8-14]. A novel probabilistic appnxrch to thc modeling of directional extreme wind speeds, in which the extrcnrc valuc distribution parameters are functions of direction, and which accounts lirr the correlation among extremes across directions, was reported in
I
3-67 l.
In hurricane-prone regions estimates of hurricane wind effects can be carried
out on the basis of hurricane wind speed data generated by Monte Carlo simulation for each of 16 directions, as shown in sects. 3.3 and 8. 1 .3 (Eqs. 8. 1 .218.1.23). Such data-used in 13-241 fot estimating extreme hurricane winds
blowing from any direction-are listed on tape in [8-9] (see also t3-7ll) for 56
mileposts (Fig. 3.3.5).
3.5
PROBABILITIES OF OCCURRENCE OF TORNADO WINDS
consider an area ,40, say, a one-degree longitude-latitude square, and let the
tornado frequency in that area (i.e., the average number of tornado occulTences
per year) be- denoted by D. The probability that a tornado will strike a particular
location during one year is assumed to be
P(S):t-
a
(3.s.1)
Ao
where
vi :
P(viP(s)
P(Z) is the probability that the maximum wind
bc higher than
United States (units are
l0
5
probability per year) [3-351.
which is taken from 13-351. Figure 3.5.1 is based on Eq. 3.5.1 in which D was
estimated from l3-year frequency data, a : 2.82 sq. miles (as estimated in
[3-36] for the state of Iowa), and,46 : 4780 cos S, where @ is the latitude at
the center of the one-degree square considered. Estimated probabilities p(zn)
are shown in Fig. 3.5.2, also taken from 13-351. These estimates are based
upon observations of 1612 tornadoes during l97l and 1972, and the rating
(largely subjective) of these tornadoes according to an intensity scale proposed
in[3-371.* It is noted that in estimating the probabilities of Fig. 3.5.2 it was
assumed that tornado path areas are the same throughout the contiguous United
States.
The maximum speed of the tornado corresponding to a specified probability
where c is the average individual tomado area. In certain applications, for
example, the design of nuclear power plants, rather than the probability P(S)'
it is oi interest to esrimate the probability P(S, Z0) that a tornado with maximum
wind speeds higher than some specified value tr/' will strike a location in any
one year. This probability can be written as
P(5,
FIGURE 3.5.1. Tornado strike probability within 5-degree squares in thc contiguous
(3.s.2)
spccd in a tornaclo will
2,,.
tlsiirlu(cs 6l'p1rl'rlrbililics /'(S) in thc Urtitc:tl Slrrlt's rttt' sllowtt irl
lril',
1.5'
l.
of occurrence can be estimated using Figs. 3.5.1 and 3.5.2. According
to
t3-351, "in order to adequately prorect public health and safety, the determination of the design basis tomado is based on the premise that the probability
of occurence of a tornado that exceeds the Design Basis Tornado (DBT) should
be on the order of 10-7 per year per nuclear power plant." The required
probability P(Ze) is then determined from the relation
P(r/o)P(S)
:
t0-7
(3.s.3)
*According to this scale tomadoes may bc dividcd into the following classes: F0 (maximum wind
speed <72 mph), Fl ('73-112 mph). F2
157 rnph). F3 (l5tt 206 rnph), F4 (207-260 rnph).
F-5 (261-318 mph), and F6 (3 19 3ll0 rnphy
(lll
124
I
X
llll Ml wtNl)
(
;t tMn l( )l ( x iY
1,,
t,t t(rt
rnilll llll :; ()t (xt(;lItl il N(.t ()t t()t tN/\t )() wtNt]:i
r25
FOR ENTIRE CONTIGUOUS
I
t
too
=9
fla
t7
3o
z
FoR ALL STATES wEsT oF 1O5O w LoNGITUDE
;5
4
FIGURE 3.5.3. Calculated tomado wind speed by 5-clegree squares for
bility per year [3-351.
o.l o.2 0.5
I
2
5 lo
20 3040506070 80 90 95 9899
PERCENT PROBABILITY
l'l(;llltl'l
lsl
-1.5.2. Percent probability of exceeding ordinate value ol'the wind speed [3-
wlrt't' lhc value of P(S) for the location considered is taken f'rom Fig. 3.5.1.
'l'lrc wincl speed corresponding to the probability P(lze) so determined is
then
rrlrtrrirrccl l'rom Fig. 3.5.2. The average tornado intensity with a l0-7 probability
P('r ycar for each 5-degree square in the contiguous United States, based on
litl. 3.-5.3 and Figs. 3.5.1 and 3.5.2, is shown in Fig. 3.5.3 13-35].
lior nuclear power plant design purposes, the contiguous United States are
tlivided, in [3-35], into three tornado intensity regions shown in Fig. 3.5.4.
'l'hc corresponding tornado winds are given in Table 3.5.1.
Thc pressure drop due to the passage of tomadoes can be estimated from
tlrc ccluation f<rr thc cyclostrophic wind. Using the relation Vr, : drldt, Eq.
I ..1.2 crrn hc wrillcrr lrs
1.5 -l)
7
proba-
wherep is the pressure, / is the time, 2,, is the translationar speed, p is the air
density, R- is the radius of maximum rotational wind speed, and z, is the
maximum tangential wind speed* t3-351. Assuming R- is typically 150 ft for
intense tornadoes and that Vt = Vrur, Eq.3.5.4, in which the parameters of
Table 3.5.1 are used, yields approximately the values of rable z.s.z
1z-2s1.
Following the development in [3-35] of the estimates summarized in Tables
3.5.1 and 3.5.2, vaious attempts to improve the probabilistic and physical
description of tomado winds have been reported [3-3g, 3-3g, 3-40, 3-41
,
3-42, 3-43,3-44,3-45, 3-46, 3-4i1. Using as a point of departure tornado risk
maps presented in [3-46], a regionalization of tomado risks that divides the
contiguous United states into four areas was proposed in
[3-45] (see also [3-44,
p. 4801. Regional tomado occurrence rate (per mi2 per year) were estimated in
[3-45] from a29-year (1950-1978) data bank maintained by the National Severe
Storms Forecast Center and comprising about 20,000 reported tornadoes. These
regional occurrence rates are corrected in [3-43] and [3-45] to account for:
l.
Failure to record tomado intensity, which affects about lo% of the total
number of reported tornadocs. J'his corrcction is based on the assumption
that unrated tornadocs ntay bc apporlionccl anrong the various intensity
categories according lo lltc tcporlctl tor-rriukr licquencies lor those categories.
(
l0
r'llrt'nrlational spcctl (,,, is tltc tlsttll:rtl ol tlrc tirrrllr'rrtrrl;rrrrl
rirrli:rl vt'ftx,itit.s.
:t,, I'il{ )ilnltil ilil :i
l'Alll,lj J.5.1. llcgirlu:rl'lirrrratkr
()t
(
x;cunt u
N(
.t ( )t
t( )nNn
t ,(
)
wtNt)ti
127
Wirrds
Radius of
Maximum
Maxinturtt
Region
Speed (,,,,*
(mph)
Rotational
Speed tr/,.,
Translational
Rotational
Speed 2,,
Wind Speed R,"
(mph)
(mph)
(f0
70
60
50
150
150
150
il
I
360
300
290
240
m
240
190
in tomado reporting efficiency. The number of reported annual tomado occurrences in the United States has increased from
about 250 in 1950 to 850 rn 1979. The growing trend in the number of
reported tornadoes during this period has been ascribed to a corresponding increase in population density. An explicit relation to this effect has
2. Temporal variations
c.t
c.)
bo
l.)
o
cn
o
o
F
o
$
o
z
la
ar)
iN
Not
\'$rr
N€
F
trr
been proposed in [3-47]. Corrections accounting for tornado reporting
efficiencies were effected in [3-45] by averaging the 1971-1978, 19701918, 1969-1978, and 1950-1978 data and assuming that the true occurrence rates are equal to the largest of these estimates.
1_ Possible errors in the rating of tornado intensities on the basis of observed
damage. The reason for the occunence of such errors is that maximum
tornado winds are in practice not measured, but inferred, largely on the
basis of professional judgment, from observations of damage to buildings, signs, and so forth [3-42].
4. Inhomogeneous distribution along the tornado path of buildings and various other objects susceptible of being damaged. In the possible absence
of such objects over the portions of the tornado path where the winds
are highest-or even over the entire tornado path-the rating of the tornado is bound to be in error. The effect of corrections for such errors is
to increase the estimated probability of occurrence of tomadoes with
higher intensities.
5. Variation of tornado intensity along the tornado path. Accounting to this
factor results in smaller estimated risks of high tornado winds than would
be the case if the maximum tornado winds (by which tomado intensities
TABLE 3.5.2. Regional Pressure Drops and Pressure Drop Rate
Total Pressure Drop
Rate of Pressure Drop
Region
(psi)
(psi/s)
I
3.0
2.25
r.5
2.O
1.2
il
u
0.6
124
I x I tll
Ml wlNl)
(.1 lMn
l( )l ( x iY
arc ftrlc(l) wu'c unili)r'rtt lrlortpl lltc t'rtlttt'p:rllt. ('ot't'ccl itttrs clli'c'lctl in
l3-451, bascd upon lhc irrtalysis ol rhrt'rrrrcn(ctl lorrrarkrcs, lctl to risk
reductions by a llctor ol'abou( livc lirr li4 (orrratlocs antl about tcn lirr
trt
According to [3-72, p. D-ll, the ANSI/ANS-2.3.1983 Standard was not
approved by the Nuclear Regulatory Commission. Efforts to develop an improved standard are under way. Reference [3-73] is an overview of recent
developments conccrning dcsign critcria fbr tomadoes. It notes that ncw Nuclcar Rcgrrlat<lry Corrrrrrission c:ri(cria ckr not clclinc tornaclo clcsign crileri:r on
l;
129
-1.5.J. Slrrrrrlitltl 'l!rt'rtirtlo ('har:rclerislics (llxlraclcrl li-6r1 Arlcrican
Nali'nal slanrl.rrl ANsl/ANs-2.J-l9tt-] with per'rissirrr
publisher, ilre
'l'thc
American Nucletr Socill.y)
Probability ol'
Exceedance, per
ance.
N(;t
'l'Alll,ll
F6 tornadoes.
The corrections for the factors listed involvc subjective judgments that may
be formalized by Bayesian techniques (see Eq. Al.6). In [3-45] the corected
rates of occurrence differ insignificantly from the uncorrected (prior) rates, with
the following exceptions. For the three areas of the regionalization map proposed in [3-451 in which the most intense tomadoes recorded in the period
1950 1978 were F5, it was estimated in [3-45] that rates of occurrence of F6
tornadoes, rather than being zero, are about 1/20 times the rate of occurrence
of F5 tomadoes. For the fourth area of that map, in which the most intense
tornadoes recorded in the same period were F4, it was estimated that the
corrected rate of occurrence of F4 tomadoes is about six times the uncorrected
rate, and that the rate of occuffence of F5 tornadoes, rather than being zero,
is 1l2O times the corrected rate of occurrence of F4 tomadoes.
Reference [3-43] suggests that the velocity ranges associated in [3-37] with
the tornado ratings F 1 through F6 (see p. I I 1) are overconservative by amounts
varying from about 5% for Fl tornadoes to about 2O% or more for F6 tornadoes. The wind speed reductions proposed in [3-43] are used in [3-45] as a
basis for suggesting a reduction of the 360 mph, 300 mph, and 240 mph wind
speeds, specified in [3-35] for regions I, II, and III of Fig. 3.5.4, to 300 mph,
225 mph, and 200 mph, respectively. In the authors' opinion, the arguments
adduced in 13-431 in favor of such reduclions are tentative, in some instances
at least. For example, to support the contention that the maximum wind speeds
in a tornado classified as F3 are lower than the values proposed in [3-37],
[3-43] interprets a 133-167 mph estimate of the velocity causing the collapse
of a chimney during the Xenia, Ohio, tornado of 3 April 1974 13-42, p. 17151
simply as a 133 mph estimate 13-43, p. 16251. On the other hand, it should be
noted that the estimates of [3-35] and [3-37] are also tentative.
A position that is to some extent a compromise between [3-35] and [3-45]
was adopted in the American National Standard ANSI/ANS-2.3-1983 [3-481,
which divides the contiguous United States into three zones, denoted as 1,2,
and 3. Zones I and 2 cover, approximately, region I of Fig. 3.5.4, while zone
3 covers approximately regions II and III. Table 3.5.3 lists the maximum
tomado wind speeds Z-o^, the translational wind speeds Vr,, the radius of the
maximum wind speed R-, and the maximum atmospheric pressure drop po,
given in [3-48] for tomadoes corresponding to various probabilities of exceed-
l lt
Ycar
107
V rrru
V,,
Rnro*
Zone
(mph)
(mph)
(f0
(psi)
I
320
250
70
540
435
t.96
55
180
40
320
0.70
260
200
57
45
453
355
0.85
2
J
106
I
2
l0-5
1.35
1.46
3
140
32
253
o.4l
1
200
45
355
2
150
100
JJ
25
270
0.85
0.47
o.20
3
185
a probabilistic basis, although_the design parameters it accepts for new nuclear
reactordesigns are in the 10 6 range 13-74,3-151. Reference
[3-76] is a study
of tornado climatology in the contiguous United States based on the National
Severe Storms Forcast Center's tomado data base for the period January l,
1954, through December 31, 1983. Strike probabilities were estimated in
[3-76] on the basis of expected tomado areas, conditional probabilities of tornado intensities were based on affected area, rather than on number of occurrences, the intensity distribution was based on a weibull model, and design
wind speeds were based on regional intensity distributions. wind speed io
obtained were 50 to 100 mph lower than the estimates of
[3-35], and tornado
design basis wind speeds suggested in [3-76] are 200 mph and 330 mph,
respectively, for the United States west and east of the Rocky Mountains. For
additional information on tornadoes, see also [3-69] and t3_701.
It was noted in [3-40] that probabilities of a target being hit by a tornado
wind in excess of any specified threshold depend upon the iize olthat target.
This topic is analyzed in detail inf3-44,3-451, where the estimates are based
upon statistics of tomado intensities, path lengths, and path widths on the one
hand, and on the geometric characteristics of the target on the other. It is
suggested in [3-49] that tornado wind loads dominate the design of most transmission lines over 10 miles in length over wide areas of the united States.
REFERENCES
3-l A. Court, "Wind Extrcrrrcs rrs I)esiArr li:rt.t9l.s," .l . ltnttrklirt
1953)
3-2
39
trr.st
.,256
Manualof SutitccOlt.scntttitttt,t,ll.s Wr':rtlrr'r St'r'vir.c. Wirslrirrgt6rr,
p.92.
(.1tly
5s
I)(', l()5 l,
130
3-3
3-4
3-5
txtnt Mt wtNt)(;l
nl
tMnt()t (xiY
Se!tctivt (]uidt ttt ('litttutit'I\tttt,\tutn't',t, Kt'y lo Mclcoxrlogicrtl llt:r'ortls l)oc
umcntation No.4. ll,lirrvinurrrcntll l)lta Sc,r'vicc. [1.S. l)cp:rrtrrtcttl ol ('tltltmerce, Washington, DC, 1969.
J. Wierenga, "An Objectivc B,xposurc (lrrrcction Mcthotl lirr Avcragc Wind
SpeedsMeasuredataShelteredLocation,".l . futyul Mctcttrcl. SttL'.,1112 (1976),
24t-253.
H. C. S. Thom, "New Distributions of Extreme Wind Speeds in the United
States," J. Struct. Div., ASCE, 94, No. ST7, Proc. Paper 6038 (July 1968),
1787- I 801.
3-6
3-7
3-tt
E. Simiu and J. J. Filliben, Statistical Analysis of Extreme Winds, Technical
Note No. 868, National Bureau of Standards, Washington, DC, 1975.
E. Simiu, J. Bi6try, and J. J. Filliben, "Sampling Errors in the Estimation of
Extreme Winds," J. Struct. Div., ASCE 104, No. ST3 (March 1978), 491-501.
J. A. Peterka and S. Shahid, "Extreme Gust Speeds in the U.S.," Proceedings,
7th U.S. National Conference on Wind Engineering, (G. C. Hart, ed.),
Vol'
2,
t993,503-512.
3-9
E. Simiu, M. Changery, and J. J. Filliben, Extreme Wind Speeds at 129 Stations
in the Contiguous (Jnited Stares, NBS Building Science Series 118, U.S. Department of Commerce, National Bureau of Standards, Washington, DC, March
1979.
Winds from Short-Term Records," J. Struct.
(1991), 375-390.
3-11 V. L. Cardone, A. J. Broccoli, C. V. Greenwood, andJ. A. Greenwood, "Error
Characteristics of Extratropical-Storm Wind Fields Specified from Historical
3-10 V. Gusella, "Estimation of Extreme
Eng.
ll7
2511.
Eng.,
ll0,
Struct.
No. 7 (July 1984),1467-1484.
3-14 M. Grigoriu, "Estimates of Design Winds from Short
Records
," J. Struct. Div. '
ASCE, 108, No. ST5 (May 1982), 1034-1048.
3-15 H. L. Crutcher, "Wind Extremes," in Proceedings of the Second U.S. National
Conference on Wind Engineering Research, Colorado State Univ., Fort Collins,
1915.
3-16 A. L. Sugg, L. G. Pardue, and R. L. Carrodus, Memorable Hurricanes of the
United States, National Weather Service, Southem Region, NOAA Technical
Memorandum NWS SR-56, Forth Worth, TX, 1971.
3-17 H. C. S. Thom, "Toward a Universal Climatological Extreme Wind Distribution," in Proceedings of the International Research Seminar on Wind Effects on
Buildings and Structures, Vol. l, Univ. of Toronto Press, Toronto, 1968.
3-18 "The Hurricane Disaster Potential Scale," Weatherwise 27,4 (Aug. 1974), 169'
I 86.
3-19 C. W. Cry, Tnpica! Cyt'loncs of the North Atlnntk' Ocean Trucks urul Freof llurrictmc,s urul 'l'ntpical Stonn.s, l87l l9(t-1 , 'l'cchlricll l)ttpcr No.
.55, tJ.S. l)c:pirrlrrrr:nt ol'('onrrrrcrcc, Wcatltcr llttrcrttt, Wrtslritrg{ott. l)(', l()(r5.
qut:n.cics
l3l
3-20 (i. Ii. l)unn rrrrtl ll. .l . Millcr, Atlantic Hurricanes,
3-21
Louisiana State Univ. Press,
Bakrn Rougc, l9(rO.
H. C. S. 'l'horn ancl R. D. Marshall, "Wind and Surge Damage due to Hurricane
Camille," J. Watcrways, Harbors, and Coastal Eng. Div., ASCE, 97, WW5
(May 1971), 355-363.
3-22 F. P. Ho, R. W. Schwerdt,
and H. V. Goodyear, Some Climatological Characteristics of Hurricanes and Tropical Stonns, Gulf and East Coasts of the
United States, NOAA Technical Reporl No. NWS 15, National Oceanic and
Atmospheric Administration, Washington, DC, May 1975.
3-23 L. R. Russell, "Probability Distributions for Hurricane Effects," J. Watenuays,
Harbours, and Coastal Eng. Div., ASCE, 97,WW2 (Feb. 1971), 139-154.
3-24 M. E. Batts, L. R. Russell, and E. Simiu, "Hurricane Wind Speeds in the
United States," J. Struct. Dlv., ASCE, 100, No. ST10 (Oct. 1980), 2001,2015.
3-25 Worldwide Extremes of Temperature, Precipitation, and Pressures Recorded by
Continental Area, ESSAIPI 680032, Environmental Data Service, U.S. Department of Commerce, October 1968.
3-26 Revised Standard Project Hurricane Criteria for the Atlantic and Gulf Coasts of
the United Sfafes, Memorandum HURT-120, U.S. Department of Commerce,
National Oceanic and Atmospheric ddministration, Washington, DC, June 1972.
3-21 V. A. Myers, Characteristics of United States Hurricanes Pertinent to Levee
Design for lnke Okeechobee, Florida, Hydrometeorological Report No. 32,
U.S. Weather Bureau, Department of Commerce and U.S. Army Corps of Engineers, Washington, DC, 1952.
3-28 R. W. Schwerdt, F. P. Ho, and R. R. Watkins, Meteorological Criteria for
Data," J. Petroleum Technol. (May 1980), 872-880.
3-12 E. Simiu, J. J. Filliben, and J. R. Shaver, "Short-Term Records and Extreme
Wind Speeds," J. Stuct. Div., ASCE, 108, No' STll (Nov. 1982),2511-
3-13 M. Grigoriu, "Estimation of Extreme Winds from Short Records," J.
il ilt Nct t;
3-29
Standard Project Hurricane and Probable Marimum Hurricane Windfields, Gulf
and East Coasts of the Uniled States, NOAA Technical Report NWS23, U.S.
Department of Commerce, National Oceanic and Atmospheric Administration,
Washington, DC, Sept. 1979.
P. N. Georgiou, A. G. Davenport, and B. J. Vickery, "Design Wind Loads in
Regions Dominated by Tropical Cyclones," Proceedings Sixth International
Conference on Wind Engineering, Feb. 1983, Gold Coast, Australia, inJ. Wind
Eng. Ind. Aerod., f3 (1983), 139-152.
3-30 M. E. Batts, M. R.
Cordes, and E. Simiu, "Sampling Errors in Estimation of
Extreme Hurricane Winds," J. Stuct. Div., ASCE, 106, No. ST10 (Oct. 1980),
2tt-21t5.
3-31 V. J. Cardone, W. J. Pierson,
and E. G. Ward, "Hindcasting the Directional
J. Petroleum Technol., (April 1976),
Spectra of Hurricane-Generated Waves,"
385,394.
3-32 E. G. Ward, L. E. Borgman,
and V. J. Cardone, "Statistics of Hurricane Waves
in the Gulf of Mexico," J. Petroleum Technol. (May 1979), 632-646.
3-33 C. S. Gilman and V. A. Myers, "Hurricane Winds for Design Along the New
England Coast," J. Waterways and Harbors Div., ASCE 87, WW5 (May 196l),
45-65.
3-34 M. J. Changery, E. J. Dumitriu-Valcea, and E. Simiu, Directional Extreme
Wind Speeds.for the Design of Buildings and Other Structures, Building Science
Series BSS 160. National Burcau of Standards. March 1984.
132
I x I lll
Ml wlNl)
(
il lMn l( )l ( x iY
nr
] .15 li. ll. Mtrrkt'c, .l . (i. llcckqlcy, ltrtl li lr. Srrrrtlt'ts, l't'r'ltrtitttl lltttit litt Itttt'r'irtt
('rtttt
llt,giotrrtl litrtttttht ('ritt'rirt, Wn Sll I l(X) (ll(' ll), ll.S. Atorrrit littt'tgy
I)(" I974'
3-l(r H. C. S. Thom, "Tornatlo lrnrbabilities," Mrnt. wuttltcr Rcv.,17 (l)cc.
l-5-l ('.
rnission, Ollicc ol' l{cgLrlatittrr, Wirslritrglotr,
1973),
110-736.
'lltnrudt)(s and Hurricancs b,r- Arert
.l-.17 'l'. 't'. Fujita, Proposed Charact.ariz.atiort of
rttrd Inrcn.sity, Satellite and Mesometcorology Rcscarch Project (University of
('lricago), Rcsearch Paper No. 89' 1970.
] ilt y. K. Wcn and S. L. Chu, "Tornado Risks and Design wind speed," J. Struct.
/)ir'., ASCIE (Dec. 1973), 2409-2421.
I l() .l . l{. tiaglcman, V. U. Muirhead, andW. Willems, Thunderstorms, Tornadoes
tttnl lluiaing Damage, Lexington Books, Heath, Lexington, MA' 1975'
\ 40 lt. G. Garson, J. M. Catalan, and C. A. Comell, "Tornado Design Winds Based
on Risk," J. Struct. Dlv., ASCE (Sept. 1975)' 1883-1897'
3-41 R. F. Abbey, Jr., "Risk Probabilities Associated with Tornado wind Speeds,"
Proceedings symposium on Tornadoes, R. E. Peterson (Ed.), Texas Tech. univ',
Lubbock, June 22-24, 19763-42 K. C. Mehta, J. E. Minor, and J. R. McDonald, "wind Speed Analysis of
April 3-4, 1974 Tomadoes," -/. Struct' Div., ASCE (Sept' 1976), 1709-1724'
3-43 L. A. Twisdale, "Tornado characterization and wind speed Risk," /. srruct.
Div., ASCE (Oct. 1978), 1611-1630.
3-44 L. A. Twisdale and W. L. Dunn, "Probabilistic Analysis of Tornado Wind
Risks," J. Struct. Div', ASCE (Feb. 1983)' 468-488'
3-45 L. A. Twisdale et al., Tornado Missile Simulation and Design Methodology,
EPRI NP-2005, Electrical Power Research Institute, Palo Alto, california, Aug.
3-54
Tenth Conference on Severe Local Storms, American Meteorological Society,
Oct. 1977, Omaha, NB.
3-47 R. F. Abbey, Jr., and T. T. Fujita, "The Dapple Method for computing Tomado
Hazard Probabilities: Refinements and Theoretical Considcrations," Eleventh
Conference on Severe Local Storms, American Meteonrlogical Society, Oct.
1979, Kansas CitY'
3-48 American National Standard for Estimating Tornado and Extreme Wind Characteristics at Nuclear Power Sites, ANSI/ANS-2.3-1983, American Nuclear So-
ciety, La Grange Park, IL, 1983'
3-49 L. A. Twisdale, "Wind Loading Underestimates in Transmission Line Design,"
3-50
3-51
Transmission and Distibution (Dec. 1982), 40-46'
L. A. Twisdale and P. J. Vickery, "Research on Thunderstorm Wind Design
Parameters," J. Wind Eng. Ind. Aerodyn-, 4l-44 (1992),545-556'
Y. K. Wen and K. B. Rojiani, Discussion to "sampling Errors in Estimation
of Extreme Winds,,by E'. Simiu eta]l', J. Struct. Div., ASCE 104 (1978)' l815_
1817.
3-52 C. J. Neumann, G. W. Cry, E. L. Caso, and B. R. Jarvinen, Tropical cycktnes
ctf the North Athntic occun, 187 l-1971 , National c)ceanic and Atnrosphcric
Atllninistrirtion. Nutiorral ('lilnatic Ccntcr, Ashcvillc, NC, Junc l97tl. (rrlxlrr(ul
vcrsiott: llislorrclrl ('lirlrllology Scr-ics (r 2' l9(X))
.1. Nerrnlurrr rrntl M. .l . Prsylak, F'rcqutnt:y arul Morion oJ Atlantic Tropical
Cycltnrr.t. N()nn 'l'cchnical Repoft NWS 26, National oceanic and Atmospheric
Administration, Washington, DC, March 1981.
B. R. Jarvinen, and E. L. caso, A Tropical cyclone Data Tape for the North
Atlantic Basin, 1886-1977: Contents, Limitations, and (Jses, NOAA Technical
Memorandum Nws NHC 6, National Hurricane center, National oceanic and
Atmospheric Administration, Coral Gables, FL, 1978.
3-55 z. xue and C. J. Neumann, Frequency and Motion of western North pacific
Tropical Cyclones, National Hurricane center, National oceanic and Atmospheric Administration, Miami, Florida, May 1984.
3-56 J. J. Sanchez-Sesma, J. J. Aguine, and M. Sen, "simple Modeling procedure
for Estimation of Cyclonic Wind Speeds," J. Struc. Eng., ll4 (1938), 352370.
3-57 L. J. shapiro, "The Asymmetric Boundary Layer Flow under a Translating
Hurricane," J. Atm. Sci., 40 (1993), lgg4_lggg.
P. J. vickery and L. A. Twisdale, "prediction of Hurricane windspeeds in the
U.S.," J. Struc. Eng.,l2l (1995), t69t-1699.
3-59 P. J. Vickery and L. A. Twisdale, "windfield and Filling Models for Hurricane
Windspeed Predictions," J. Struc. Eng., l2l (1995), 1700_1709.
3-60 F. Ho, J. Su, K. Hanevich, R. smith, and F. Richards, Hurricane crimatology
for the Atlantic and Gulf Coasts of the rlnited states, Nws 39, National oceanic
and Atmospheric Administration, 1987.
3-61 K. A. Emanuel, "The Maximum Intensity of Hurricanes ," J. Atm. Sci., 45
(1988),1143-1155.
3-62 K. A. Emanuel, "Towards a General Theory of Hurricanes ,,, Am. Scientist, 76
3-58
(1988),371-379.
1981.
3-46 R. F. Abbey, Jr. and T. T. Fujita, "Regionalization of the Tomado Hazard,"
rrlir N(jtis 133
3-63 D. Delaunay, Vents efiremes
dfrs awr cyclones tropicaux dans les DOM-TOM,
Cahier 2078, Centre Scient. et Techn. du BAtiment, 4 av. rect. poincar6, paris
16, France, May 1988.
3-&
A. G. Davenport, P. N. Georgiou, and D. Surry, Hurricane Wind Risk Study
for the Eastern Carribean, Jamaica and Belize with Special Consideration of
the Effects of ropogrctphy, Eng.science Res. Report Blwr-ss3l, University
of Western Ontario, London, Ontario, Canada, 1985.
3-65 L. Gomes and B. J. Vickery, on the Prediction of rropical Cyclone Gust speecls
Along the Nonhern Australia coasr, Res. Report R27g, school of civil Eng.,
University of Sidney, 1976.
3-66 H. Saffir, "Florida's Approach to Hurricane-Resistant Design and Construction, " "/. Wind Eng. Ind. Aerodyn. , 32 (1989), 221-230.
3-67 S. G. coles and D. walshaw, "Directional Modeling of Extreme wind Speeds,"
J. Appl. Stat.,33 (1994),139-158.
3-68
ASOS Tool Box, Surface Observation Modcmization Oflice, 8455 Colesville Rd.,
Silver Spring, MD 20910, Junc
199.5.
3-69 T. P. Grazula, "Significant'lirr.irtl.c:s," .l " Wind llng. Ind. Aerodyn.,36(1990),
l3l-15r.
3-70 L. A. Twisdale antl P..l . Vicke'y, "l,lxr'errc wrntl ltisk Assessnrcnr," pp.46-5
509, Pnbabilistic Strttt trutrl h,llt ltrtrrir..s lltttrrlltntk, ('. Srrrrtl:rrirr:r jln, (ctl.)
Chaprnan lntl lllrll, Nrw York, lt)t)5.
134 txttuMt
.l-7
I li.
3-72
sirrrirr, N.
73
114
A. lltrckcrr,
rrntl
'l'. M.
wlrrrlt'rr,
1,.):;tintttrt',t
ttf'llurrir'ttrtr,witttl
Slttul.s lt.v th<, 'l'tttkl; rn'tr 'l'ltn'sltttltl' 19.1,'11,,r1, NIS'l' 'l'cchnicirl Note l4l(r,
Natitlnal lnstitutc ol'Slarrtlarrl irrrrl 't'ct.llroLrtriy. (i;rithcrsburg, MI), l()()6.
DOB' Stanrlanl l(120-94, LJtritctl Stirlcs l)r:p:rr(rncnt ol' Encrgy, (icnrrankrwn,
MD.
3
wrNr)ct tMAl()t (xly
PART B
1994.
J. D. Stevenson and Y. Zhao, "Mrxlcrrr l)csigrr ol'Nuclear and Other Potentially
Hazardous Facilities," Nuclcar S'zfi,l.y (in prcss).
U.S. Nuclear Regulatory commission, "Final sal'cty Evaluation Report Related
t. the Certification of the Advanced Boiling water Design," NUREG-1503,
V.l. l, July 1994, National Technical Information Service, Springfield, VA
2216t.
i /5
l.s. Nuclear Regulatory commission, "Final safety Evaluation Report Related
rhc ccrtification fo the System 80+ Design Docket No. 52-002," NUREGl;162, Vol. I, August, 1994, National Technical Information Service, Spring_
licltl. VA 22161.
.l . V. Ramsdell and G. L. Andrews, Tornado Climatology of the Contiguous
lltritul states, NUREG/CR-4461 pNL-5697, May 1986, National rechnical
lrrlirruration Service, Springfield, Y A 22161.
WIND LOADS AND THEIR EFFECTS
ON STRUCTURES
I
I
r.
|
](t
|
71
stnt(tural Engineering, Loads-Design Manual 2.2, NAVFAC DM 2.2, Navy
lrar:ilitics Engineering command, 200 Stovall St., Alexandria, virginia 22332,
I7tt
'l'ccltnical Manual-structural
Design Criteria, Loads, Army TN 5-g09-1, Air
Iirrcc AFM 88-3, Chap. l, Departments of the Army and Air Force, 1992.
M. D. Powell and P. G. Black, "The Relationship of Hurricane Reconnaissance
t"light-Level wind Measurements to winds Measured by NoAA's oceanic platlirrms," Int. J. Wind Eng. Ind. Aerod.,36, (1990), 381-392.
E. Simiu, Discussion of "Prediction of Hurricane windspeeds in the u.S." by
P. J. Vickery and L. A. Twisdale, submitted toJ. Struc. Eng. (Apil 1996).
198I
\-l()
l-ttO
.
FUNDAMENTALS
CHAPTER 4
BLUFF.BODY AERODYNAMICS
The subject of aerodynamics covers a very wide range.
of
necessity, therefore,
only a few highlights can be emphasized in the present chapter. The field
received its great initial impulse from the efforts in the early twentieth century
to achieve heavier-than-air flight. Since that time it has continually received
strong contributions from a great variety of aerospace studies, and from the
sustained, intensive development of machines with internal flows, such as jet
engines, pumps, and turbines.
In addition, interesting new advances in applications of aerodynamics to
civil engineering structures have occurred in the last three decades. Dealing as
they do with the natural wind, these applications of aerodynamics are limited
mainly to relatively low-speed, incompressible flow phenomena. In this application, aerodynamics is also closely associated with meteorology and concemed
in particular with turbulent flows in the boundary layer of the earth's atmosphere.
Besides a primary concem with the mean velocity of the wind, two aspects
of these turbulent flows are of interest to the structural engineer: the state of
turbulence of the natural wind approaching a structure and the local or "signature" turbulence provoked in the wind by the strucrure itsell. Since moir
structures in civil engineering present bluff forms to the wind, emphasis is
placed, in wind engineering, upon bluff-body aerodynamics. This fact, characteristic of a new situation not emphasized as strongly in aeronautical and
other previous studies, has occasioned new research on the details of, flclw
effects around bluff forms typical of such structures as buildings, towcrs, ancl
bridges. In this context, interest centers particularly on details ol'thc clcvclopment of body pressures by the givcn flows.
As pointed out by Roshk<l in a rcccnt rcvic:w, "llrc: pnlblcrrr ol' blrrll lxxly
r35
136
nt
lI I
,t I (i{'vt iltltN(, t(Jl,Ail()Nl;
il()l )Y nl ll()l)YNnMl(ll
ll1;w rclrpritrs irlrrrosI errlilcly irr llrt't'trpttit';rl. rlt'st'r'iptive tt'ltltlt ol kttowlt'tlgc"
l4 tttil. Altlxrr.rgh ctx)rnl()us lrtlvlrnt'r:s lurve lrt't'tt lttttl lttc bcittg rrtitrlc itl colttirr tllc sllcltutali6nal lluicl clynarrrics (C-Irl)), so llrt'resrtlts ltitvc boctr Itltltle:sl
(CWts)('onrl)ulirliontrl
wind
cnginccrirrg
ils
known
cializ,cd branch ol-CFD
comlbrt
pcdcstrian
lor
llow
ott
wittcl
rcsrrlts
qualitativc
proviclc
Lllrl)/CWE can
pufposcs (Chapter l5), although cvcn irr this casc no delinitive validations
itppcar to be available [4-89]. Howcvcr, lilr structural engineering purposes,
0wing t() the computational problcms arising in large Reynolds number, turlrulcn(, separated flows (Sect. 4.3), current methods are inadequate and/or
lrnrhihitivcly expensive. For details on the current status of CW-E, see [4-90
ro 4 92, 4-951.
ln this chapter a few basic theoretical principles and experimental facts are
r-r:vicwed that lay a foundation for the study of wind engineering'
4.1
a,,
t.,,*
dxadx.
dxt
FIGURE 4.1.1. Forces on an elementary volume of fluid
It
can similarly be shown that the net force component in the i direction due
to the action of all the stresses o,; is
), uo
o,
0x1
GOVERNING EQUATIONS
Denoting the components of F by F,(i
given by Newton's second law, are
Consider a fixed elemental volume dV in a fluid. The vector velocity* of the
fluid is commonly expressed bY
u:ui+/j+wk
(4.1.1)
j, k are unit vector components
along the usual three fixed rectangular
coordinate axes x, y, 3. For compactness of notation let x, y, Z be replaced
respectively by x1, x2, x3, I't, u, wby u1, tt2, tt3, and the unit vectors i, j, k by
i1, i2, ij so that Eq. 4.1.1 may be rewritten
where i,
t:
3
(4'l'2)
''i"
'?'
i
The force acting on the fluid contained irt the volume dll consists of two
parts. The first part, referred to as the body lforce and caused by some force
held, such as gravity, will be denoted byFp dV, where p is the fluid density.
The second part is due to the net action on the fluid of the internal stresses
oa(i, i : 7,2,3). For example, the contribution to this action of the normal
stress
01 (see Fig.4.1.1) is
-otr dxz ar,
+ (o,, +
#
ff4,,tdxrdx.
(4.1.4)
1:r
4.1.1 Equations of Motion and Continuity
*,) *,
dr, --
:
ai
dxl
oot'
,rv
dr'
cLrt dvl
(4. 1.3)
tlrr :rpltlicirliorrs wlrr:rt' llrclt.t'xisls lt sirtglt' inllx)t'lirtll tttt':tlt llttw vt'lot ily lti lotttltltttir'rl lry vlrri:rblc
c()n)l)on('nls. lltt'nrt':rrr llow is ollctt lirkt'tt:ts ltt'irr1g irt lltt" t rlit't'tliott. tltllt vclotrlv (l(:,il'lllll((l
rrs l/. fltc tcslx'r'livc r. \',.'(()llll)()ll('ttl:i lltt'tt lx'irt11 rlcsrlltt:rlr'rl ltr l/ I tt I' tt
137
Du,
p dV :
A
where the operator
Fip dV
DlDt, known
*
: 1,2,3),
I
the force balance equations,
a",,
(t
,",i,0,
as the substantial
:
1
,2,3)
(4.
r.s)
or the material deivative,
is defined as follows:
Da:a+ Lu-
(4.1.6)
Dt 0t i:r '0*i
Since Eq. 4.1.5 is true for all volume elements, the factor dV may be divided
out of Eq. 4.1.5 and the equations of motion, in component form, of a fluid
particle can be written as
Du,: *,1,#,
.l a"
,i,'
rF,
(i
:
1.2.3)
(4.1.1)
Various forms of this basic equation can be derived depending upon the nature
of the forces d and stresses o4 acting upon the fluid particle.
Before examining these particular cases, it will be useful to recall the principle of mass conservation. This principle states that the rate of increase of the
fluid mass contained within a fixed closed surface must be equal to the difference between the rates of influx to and effiux from the volume enc.loscd by that
suriace. The equations of continuity can then be shown to be [4- | . 4-21:
I
I
rl
i)(Nri\
op
/lr
i)r
(-1.I n)
138
Bl Ut
I
tr()t
)y
*
At n()t)yN^t\,4t(;r;
l'or lttt ittcottt;ltcssiltltr lltritl
whe tt'irr rro r'lr;rrrg,r. irr
tlt'lrsily /) ('(:(:ut.s, llris
rr.rlrrt.t's
Itr
.', arr.
L..',
i r 0.r,
'l I (i()Vl llNlN(i l(Jlln ll()Nri
-l
39
lirrllhcr, by rlivitlirrg tltc wlrolc s(rtss /r'rt,srrl o,, lt( it lltlitl ptlint itt(o prcssttrc
strcss (6r'sirrrply l)ft,,t,\ur(, tltirl is, rtorttutl sltcss) 1t ltttd tltvitrtorll' s(tcss, dclincd
as
0
(4.1.9)
du
:
2r'
(r, _ iu, -i, ,--)
(i,
j : l, 2, 3)
(4.1.1
l)
4.1.2 The Navier-Stokes Equations
tjnlikc a solid, a fluid under static conditions is incapable of suppor-ting any
slcady-state stresses other than normal pressure. In dynamic situations, on the
othcr hand, it may support shear in a time-dependent manner. Most often, in
lltricl-mechanical applications, it has been adequate to assume then that the
strcsscs involved are either normal pressures or ascribable to viscosity only.
lrluicls with internal shear stress proportional to the rate of change of velocity
with distance normal to that velocity are termed viscous or Newtonian. For
cxurnple the shear stress ol2 in the simple two-dimensional flow pictured in
lrig. 4.1.2 is expressed as
otz: lt out
*
(4.1. l0)
where
I ( 6u, 8ri\
eii:i\a*-i,)
(4.1.t2)
" fr. i:j
ur:Lo.
i+j
(4.1.13)
the following breakdown of stress oii can be obtained:
whcrc the proportionality factor is defined as the fluid viscosity.*
:
oij
-pbij + 2r,
("i-
+r,
@.r.4)
-i, "--)
Using this form of stress for a Newtonian fluid results in the equations of
motionx
,
r : \)
Dui _,_op.ig\r*/
:
oF,
0"fi
(",
16,i
.
0x, 7:r dx; (
^4,'^r)\
(4'l'15)
j : 1,2,3)
are the well-known NQvier-Stokes equations. If Flq.4.1 .12 is used, and if the viscosity p may be considered to be
ionstant throughout the fluid, then Eqs. 4.1.15 become
Equations 4.1.15 (i,
3
,o#: oF,- y-,
rl
FiGURE 4.1.2. Linear velocity increase with distance from a wall.
liler.
ilr'('it
lcngth
ily
lorcc timc
vt'kre
lcngth
r
'l'ypitirl v:rlrrcs ol lirl t'x;rnrplt' lirr:rir:uttl wlt(cr irl 20,,
lr
ltuu
0 ()(X)ltt p/t.rrr s,
(irttttttllt ttttil: trtr'/r,rrr'r, r!lrr,lr. I lrrisr. I
111,,,
11lt.rtr
s
-1
a
Z
k:1
@uol\xo)
0*,
(4 116)
)
Further simplification occurs in the case of an incompressible fluid, that is, one
for which Eq. 4.1 .9 holds. Equations 4.1 .16 can then be written in the vector
form
r'l'hc units ol viscosity arc
'
(,; W.
I
.
lcngllr tirrrc
Du
:rrc
PDt
O.Ol 1ilt,rtr
(].()(]lO')
:
pF
P ii -r tt i>I f(txti
- t; I tlX;
s
llrl r/lt
ilior
:r rrrort'tlt'liriletl ir(('()tlrrl s('('.
lirt t'rittttltle,
l'l ll. l'l .ll, l'l ll' or l'l'll
A.t.t7t
140
ilr Ur
I il()t)y
*
nt il()t)yNnMtcl;
;r
:, llowlNA(:llllvl l)l'Alll
v(|l
lllxll()w
t4l
4.1.3 Bernoulli's Equation
lJor a lluirl that, in aclditi<ln to hcirrg irrerrrrprcssiblc, is irryl,rr.irl
is actcd upon by ncgligiblc bocly lirlccs, llt1. 4.1.17 rctluccs to
Du i, al,
e Dr : -,?,;
''
.
(/r
0)
(4.
irrrtl
/I
l. l8)
srnenulrrurs
ll'the coordinate axes are so oriented thatJl corresponds to the direction of
rnotion, and if the flow is steady, it follows immediately from the integration
\
//
Ja
-('
u
dr
\,,/ -\-
ol' [lqs. 4. I . 18 that
jl"l'*4:consr
(4.1.19)
ll
ovory point of a streamline. Equation 4.1.19 is a special form of Bernoulli's
tltt'rtrcm and is most commonly written as
lpu' + p:
wlrcrc
(4.1.20)
const
a is the flow velocity along a streamline. The quantity |puz has the
of pressure and is referred to as lhe dynamic pressure.
'l'his important equation is widely used to interpret
the ie-lation between
rlirncnsions
i)rcssurc and velocity in atmospheric and wind tunnel flows. Detailed comments
on llcrnoulli's equation and its applicability in fluid flows-including flows in
which viscosity is present-are provided in Sect. 3.5 of [4-3].
4..2 FLOW IN A CURVED PATH..VORTEX
FLOW
('orrsicler a two-dimensional flow u"rr")lio locally concentric
streamlines a
tlistance dr apart and having radius of curvature r (Fig. 4.2.1). For the flow
to nraintain its curved path, it must experience an acceleration toward the center
trl'curvature of the streamlines of amountu2lr, where z is here used to designate
tlrc local tangential velocity of the flow. Let the pressure acting on the fluid
e:lcrrrcnt under consideration be denoted by p. The pressure differential from
onc strcamline to the next along r, which is responsible for this acceleration,
is r/7r. The equation of motion for the fluid element is then
FICURE 4.2.1. Flow in a curved
dp
.dr
: pu';
Path
(4.2.1)
Bernoulli's equation (4.1.20) then permits calculation of the pressure along a
curved path of such a streamline flow.
In particular, one may consider the case wherein the flow is completely
circular and the value of p6 in Eq. 4.1.20 is the same on all streamlines. This
is the case of vortex.flow. Differentiation of Eq. 4.1.20 yields
du do
Pui+,rr:o
(4.2.2)
which, when combined with Eq. 4.2.1, yields
du
ur
dr
(4.2.3)
Equation 4.2.3 can then be integrated to yield
rlp dA
:
p d.r dl
ll
r
wltcrc trr is tltc llrritl tlcnsily lu'rd dA is thc arca <lf thc clcntcrrt in ir pllrrrt: rrtlrrrral
to llter plitltc ol'llrt'ligrrlc. 'l'lris rclulion inrlicirlcs lhlrt lltc l)rrsriur'(. t.lurrrge rror.rnirl
l() lllc sll'ctttttlitres ol'lt r'tttvt'tl llrtw irr tlrc: :rbscrnt'e ol irrry ollrt.r' lort.r.s rs
ur:C:const
(4.2.4)
'I'his simplc law sta(cs lirr irrr inconrprcssihle, inviscid fluicl thc thcrllcticlrl
(hypcrb1;lic) rclation bclwct'rr positiorurl t'rrtlius r antl tangcnliltl vclrtcily rr irt lt
.lit't'rrtrtc.r.
142
lil ul
I
il(
)t
)y n t tt( )t )vNn Mtcii
.l
Itt lttt ;tcltt:rl lit't' votlcx, ltowcvt'r. llrt' r'llct l:r ol vrscosily il'(. l)t(',.r(.nl ils
wt'll.'l'llcy llltvc ttol llcctt ittt'ltttlt'rl rtr llrt'srrrrplt tlt'r'iv:rlion irlrovc.'l'lrcst.will
Ititvt', itt l)ilrt, lllc cll'cc( ol'"lockirrg," s()rn('l)()rtr()rr ol lhc lluitl (ltclrr llrt.t'r'lrlg.)
logclltcr alttl citusittg il to ft)lllc lrs lr liliitl lrtxl-y irrslt:lrtl ol'as thc pclli.t.l llrritl
rlcscrilrtxl by l-q. 4.2.4. T'hus locirlly, neru'thc t'crrlcr ol'a I'rcc vgrtcx. llrc
vclocity u intrtu:sc,t with radius, whcrclrs ucconling to Eq.4.2.4 il tlt,crttt,scs
witlr irrcrctrsing r. This latter condition lctually hokls outwarcl fnrrn a tnu1sitiln
tr',t\i.rt irr which rz attains its maximum valuc. 'l'hc value of a in such a region
rs rlt'Perrtlcnt on the values of the fluid viscosity and of the total angular monrcnlunr rrl'thc vortex. Figure 4.2.2 lllustrates qualitatively the pressure ancl
vt'krt'ity rclrrti<lns that hold in a free vortex occurring in a real fluid. It should
lrt'rrrtt'rl tlrirt thc free vortex here described differs from the forced or conttttttttt'rl rrlTr'.r that may develop in a fluid held in a rotating container.
'l'lrr' llcc vortex is of interest in many flows that occur
in engineering aplrlrt';rtirrrs. lior cxample, atmospheric flows along the curved isobars of the
rvt'rrf lrcr nrrl) ilrc described by generalizations of Eq.4.2.1 . These have been
rlt'strilrt'rl irr scct. 1.2, where additional Coriolis forces have been included.
4.3
:l
lt{
)l lNl )n I lY I
AYI
I
l"
n
lll ) :;l l'l\l
tn I l(
'l'lrc lrngc: ol vist'o:;rly v;rlrrt's lo bc lirtrtttl itrrl()rt1l vrtriotts lluids is vory grcat.
'l'ltc viscosity ol lrir lrt rurlrrrirl rrrclorlrolrlgicitl 1tl'cssrtl't:s ancl {ctnpcraturcs howt'vcr has a r-clltively srrlrll virluc. Noncthclcss, in stlrrtc circumstances this small
viscosity plays irrr irrrprlllnt nrlc. An imporlanL rnanif'estation of the viscous
cll'ccts of air occut's in tlro lbrmation of boundary layers.
Consider an air llow over and along a stationary smooth surface. It is an
cxpcrimental fact that the air in contact with the surface adheres to it. This
cituses a retardation of the air motion in a layer near the Surface referred to aS
tlrc boundary layer. Within the boundary layer the velocity of the air increases
l'rom zero at the surface (no slip) to its full value, which corresponds to the
cxtemal (as opposed to boundary layer) flow t2-11. A boundary-layer velocity
profile is depicted in Fig. 4.3.1.
Air, since it has mass, evidences inertial effects according to Newton's
sccond law (or, more specifically, the Navier-Stokes equations). The two most
influential effects in an air flow are then viscous and inertial, and the relation
ol'these to each other becomes an index of the type of flow characteristics or
as a
Phenomena that may be expected to occur. This index can be expressed
nondimensional parameter G", the. Reynolds number, which is a measure of
thc ratio of inertial to viscous forces. For example, consider a volume of fluid
with a typical surface dimension L. Then, by Bernoulli's theorem, the net
pressure p - po caused by fluid flow at velocity U, which is of the order of
p[J2, creates inertial forces on the fluid element enclosed by that volume which
I
p
HEIGHT
u =C/r
VELOCITY
I'r.ssrrrt' :rrrtl vt'krt iry rlisrrilruri.rr rr ;r
'rirrr'\
lr.r'
143
BOUNDARY I AYERS AND SEPARATION
t,
lll(illlll'l 'l.l.f-
)ll
Itl(llllll,l
.1.,!.
L
I\
lr( .rl lrrunrl;trv l;tyt't vt'lot'ily lllirlilt'.
144
ilt
rJr
I il( )t )y l\t il( )t )yNt\Mt(
,l :l
li()t ,Nt)nnY tnYt
ilr; ANt) ::t I'nt tn |()N
145
/){/ /. . ()rr lltt'ollrr't lr;rrr,l. llrt'visr'otts stt't'sst's otr llrr"t'lr'tnt'rtl
Irr.t'ol lltt'onlcl ol lrllll,, sr> visto:;rly rt'l:rlt'rl l()r'('(':j itr(r ol tltt: otrlt'r rtl 1r.llll,
' /,'. 'l'lre lirlio ol'irtctlilrl (o vistorrs lolt t s rs tlrt'rr ol'lho rlttlcr ol'
irr'('()l llr('()r(lt't ()l
(lle
: 'olJ L
pUL ll
p( )l
, I!:
(4.3.1)
lLv
y : pl p is called the kinematic yi^r<'o,ril_y.* (See also Sect. 7.1.) Thus,
wlrcrr 61. is large, inertial effects preilominate; when it is small, the viscous
t'llccts are the stronger ones. It is noted that the concept of Reynolds number
rs. irr rclaticln to the boundaries affecting a flow, a very local thing; that is, the
st'lt'c(ion of the representative length Z for the calculation of G" depends upon
tlrr' irrtcrcst ol'the investigator in local details. Thus a flow over a given object
rrr;ry tlcvckrp a wide variety of Reynolds numbers, depending upon the particrrl;rr rcgion focused on for study. When discussing the whole flow that envelops
rr llivcrr lxily, it is usual to select for the length -L some overall representative
tlirrrrnsion ol that body.
Iirrrrrtlrrry-layer separation occurs if fluid particles in the boundary layer are
srrllicit:rrtly dccelerated by inertial forces that the flow near the surface becomes
n'vt'r'sctl. 'fhcse deceleration effects occur as a result of the presence in the
llow ol'arlvcrse pressure gradients. Such severe adverse pressure gradients as
t'rrrr lrr: llrocluced, for example, by the flow over the comer of a bluff body
I't'nt'r'irlly cause flow separation. Through processes that are not well understrxxl, thc scparation layers generate discrete vofiices, which are shed into the
wrrkc llow bchind the bluffbody (Fig. 4.3.2). Such vortices can cause extremely
lril',lr srrctions ncar separation points such as comers or eaves.
Iilows ol-practical interest have Reynolds numbers ranging from nearly zero
to irs lrigh as 108 or lOe. Steadily increasing the Reynolds number of the flow
()v('r irr) obstacle generally produces a widely varying sequence of flow phen()rncnlr lirr which the Reynolds number provides a convenient index, as is
st't'rr. lirr cxample, in Sect. 4.4.
ll'" as is true in most cases, the flow over a body has separated at some
wlre:rc:
''l vpical valucs of kinenratic viscosity for air and watcr are. respectively:
:
r,".u :
/,i,
0. 150 crn2/s at
20'C
0.01 crn2/s at 20'C
A t orrrnror rrrril lirr kincrrratic viscosity is thc .stote:
I stokc
-
FIGURE 4.3.2. Flow separation at corner of obstacle.
l)oint,* the wake will contain the effects of vortex formation. Depending upon
tlrc magnitude of the Reynolds number, the flow willbe turbulent to a greater
or lesser extent. Many turbulent flows may thus be typically viewed as wake
lkrws in which upstream objects have already "stirred" the flow in some such
rnanner as has been described. Turbulence can bc caused by means other than
thc stirring mechanisms mentioned above (e.g., by thermally induced convection), but for the majority of flows of importance to wind engineering, turbulcnce can be considered to be initiated mechanically, as described. Thus, for
r:xample, trees, buildings, or telrain upstream of a given point play an important
role in developing the turbulence of the wind observed in the atmospheric
lroundary layer at that point. Descriptions of turbulence in the natural wind are
given in Sect. 2.3.
When turbulence is present, one turbulent layer of the fluid tends to produce
Iurbulent motions in adjacent layers, as, for example, in a wake or boundary
ltyer. This takes place through transfer of momentum from one layer to another. A similar phenomenon occurs in the absence of turbulence when a
lirrninar, as opposed to turbulent, boundary layer is created. The difference
bctween a laminar and a turbulent boundary layer is that, in the former, the
transfer of momentum occurs at the molecular rather than the macroscopic
scale. The fluid viscosity p is in fact the result of such molecular transfers of
nx)rnentum. As noted in Sect. 2. I in the context of atmospheric flows, turbulent
lxrundary layers may be viewed as being governed by an equivalent kinematic
viscosity callcd eddy viscosity, whose value reflects the large momentum translcrs induced by turbulcncc.
I crrr/s - 0.001764 ltrls
A rrrclrrl :rplrlrlirrt:rlc lorrrttrl:r lor lltL'l{t:ynoltls rtrrrrrbcr in:rit'lrl lrllrul lO"(':urtl ;rlrrrosplrcrit'
lr( r;sllr( rs (r/ (X)() l// . r|lrt'ri {/ is in Ittt'lr'r's l)r'r s('('on(l :rrrtl /. in nr('l('rs l lrs lrr'r'rrrrrt s (r.) }O l//.
lor l/ rrr lt/:, :lrrl / rr l1-( l
llrt lltt' t'rrsr' ol lrirlirrls (x ( lrrr( r( (' ol llrl,.,r'lrrr.rlrorr r:, rr:rr;rlly tk'sirt'tl lts l:rlc:rs possilrlt':rlorrl',
tlrc lrtxly. in ;rttottlrrtttt rvillr llrr' ,rlnr "l , t'nlr'rllrr,' Irr':,r,lr( rlisttilrttlions 1o iltt li':rsr' lill :rrrrl
tr'tltttr' tlt;t1', lry ttttlttts ol llcrttttt'lttr Iorrr
146
4.4
n
UI
I
Ir( )t
)y n t tr( )t ,yNn Mt( il
.l
'l Wnlll nl.Jl'V()l lllXl0liMnll()Nl;
tl.l tW(llrlMl il:;l{)t!At |()W
WAKE AND VORTEX FORMATIONS IN TWO-DIMENSIONAL
FLOW
lrt thc lilllowing discussion, lho lklw is assrrrrrctl (o bc smooth (laur"inlr) arrd
two-rlirncnsional, that is, indcpcntlcnl ol'lhc c<xrrclinatc normal to tlrc planc ol
vicwing. Consider a two-dimensional llow around the sharp-edgccl flat platc
shrrwn in Fig. 4.4.1 . At a very low Rcynolds number (e.g., ULlu: 0.3,
whcrc L is the dimension of the plate across the flow), the flow turns the sharp
corncr and follows both front and rear contours of the plate (Fig.4.4.la). At
rr slightly higher Reynolds number (&" = l0) obtained by merely increasing
tlrc llow velocity over the same plate, the flow separates at the corners ancl
t'rt:alcs two large, symmetric vortices behind the plate that remain attached t<l
(lrt: back of the plate (Fig. 4.4.1b). At increased Reynolds number (G"
= 250)
thc syrnmetrical vortices are broken and replaced by cyclically altemating vortit'cs that form by tums at the top and bottom edges and are swept downstream
( l;ig. 4.4. I c). A full cycle of this phenomenon is defined as the activity betwecn
llrc occurrence of some instantaneous flow configuration about the body and
llrc ncxt identical configuration. At still higher Reynolds numbers, say Ge 2
It){)t) (Fig. 4.4.1d), the inertia forces predominate; large distinct vortices have
littlc possibility of forming and, instead, a generally turbulent wake is formed
Ilchintl thc plate, its two outer defining edges forming a "shear layer" consisting ol'a long series of smaller vortices that accommodate the wake region
lo thc udjacent smooth flow region. Overall, these results dramatically illustratc
rDe
FIGURE 4.4.1b. Flow past a sharp-edged plate Ge
=
10.
e Q.3
(,
(o)
lll(;llltl,l.l.,l.la.
(b)
lihrw pirsl lr slt:u1r t'tllicrl pl;rtt.(11,. -- O
I
)
l,'l(illllll,l 4..1.lr'. l;kru |1r;l ir "lrirrI
r'rl1'r'r; ,t1,,,. ,t,,
_)
5()
147
t4B
lll l,l
I lt()t)Y At n()t )YNnMt|1
,l
.t w^l,.t
nt.lt
I v(,t illx l()l tMnll()N:; lN tw()trlMl Nt;l{)Nnt ll()w
Qe=l
qe
(o)
149
=20
(b)
@
VON KARMAN VORTEX TRAIL
30 39"
S 5OOO
(c)
WAKE
FIGURE 4.4.1d. Flow past a sharp-edged plate Ge
>
1000.
thc changes in the flow with Reynolds number, proceeding from predominantly
viscous effects to predominantly inertial effects.
Next the renowned case of two-dimensional flow about a circular cylinder
(rig. 4.4.2) is briefly examined. A
number of flow situations can be created
by increasing the flow velocity, each situation being identified by a specific
llcynolds number range. At extremely low values of Reynolds number (G" =
l) thc flow (assumed laminar as it approaches) remains attached to the cylinder
thrrrrgh<rut its complete periphery, as shown in Fig. 4.4.2a. At G" = 20, the
lklw lirnn rcnrains symmctrical but flow separation occurs and large wake
ctklics urc lirnnctl which rcsidc ncar the downstream sudace of thc cylinclcr,
rrs srrggcstt:tl irr Iiig. 4.4.2b. lror 30 < (R" < -5000, al(crnuting vor.ticcs arc
slretl liirrll tltt't'ylilrrlcr rttrtl lirt'nr ir clcar "vorlL:x lr;ril.' rlowrrslreirrrr.'l'his
ltltt'ttolttt'ttott wirs litsl rt'1xrt1ctl by llt<rlrltl l4-.5 1 irtrtl vorr Klrrrrr;ur l.l O; 11r;*.
cuuuS.z{S 2OOOOO
5OOOs.4.3
ZU(JUUU
(d)
4"2ZOOAOO
(e)
l|l(;uRE 4.4.2. (a) Flow past circular cylinder (Re = l. (b) Flow past circular cylinder
ill" = 20. (c) Flow past circular 30 < G" < 5000. (d) Flow past circurar cylinder
.5(XX) < G" < 200,000. (e) Flow past circular cylinder G" > 200,000.
1.1 .2c). The finer details ol this striking occurrence are still not fully underslrxrtl, and the process cor.rlirrrrcs to be the focus of many studies, both experirncrrtrrl and thcoretical 14-21 l. llchintl thc cylinder there is establishcd a stag
gcrctl, stablc arrangctllcltl ol'vorlit'cs lhlrt rrrovcs <tffdownstream at a vcltrity
stttltcwltitt lcss thltrr tlurl ol lltr'sttt.lirrrrtrlirrg llrritl . In this rangc ol'llc:yrroltls
tttuttbcr lltc w:rkc llow is lr.litlrvcly snrtxrllr irnrl rcgrrlirr aplttl I'r-onr lltc vorlit'csr
lllclttsclvt:s. liigrrrt:.tr.,4..1 tlt'1rrr'ts llrt'slrt':rrrrlirrcs ol tlrc wlrkc llow lx'ltiltl ir
150
ilr ur
r
tt()t)Y AI lt()l)YNAMlcl;
't
.l w^t,t Ailll
V(lt
iltx I()llMn il()Ni; tN tw(,t)tMt N:;t()NAt il()w
r5t
r-ffi--r-.-
,
. "eff@f 3 3 ""etrt
o.2
ll
e R"6 STxf
t
^3
I
O Smooth
o.1
o
^o
105
k/D=0.0003
k/D=o.oo12
k/D:0.0101
106
107
REYNOLDS NUMBER
lislablishrncnt, National Research Council of Canada.
l"l(;uRE 4.4.4. Relation between the Strouhal number and Reynolds number for cirt'trlar cylinder. From W. C. L. Shih, C. Wang, D. Coles, and A. Roshko, ,.Experinrcnts on Flow Past Rough circular cylinders at Large Reynolds Numbers," J. wrut
circular cylinder in a water tunnel [4-7] within the above-mentioned Ge range.
'l'hc lkrw in this photograph was made visible by the emission of dye from the
irnd
t'y I irttlcr.
takes on different characteristic constant values depending upon the cross-sec-
l,'l(;tjRli 4.4.3. Vortex trail in water tunnel.
Courtesy of the National Aeronautical
I,)rg. Ind. Aerod., 49 (1993), 351-348.
As ltcynolds number further increases into the range 5000 < G" < 200 000,
(hc irttlchcd flow upstream of the separation point is laminar. In the separated
lLrw (hrcc-dirncnsional patterns are observed, and transition to turbulent flow
(x'('uls irr thc wake-farther downstream from the cylinder for the lower Reyrxrltls rrrrnrbcrs and nearer the cylinder surface as the Reynolds numbers increase
l.l l()l l,or thc largest Reynolds numbers in this range, the cylinder wake
rrrrtlcr-gocs transition to turbulence immediately after separation, and a turbulent
wrrkc is pnrtluced between the separated shear layers (Fig. 4.4.2d).
Ifcyonrl 61" = 2OO 000 (Fig. 4.4.2e) the wake narrows appreciably (giving
lisc lo lcss drag; see p. 158).
( )thcl bluff bodies, notably triangles, squares, rectangles, and other regular
;rrrrl ir.rcgrrlar prisms, give rise to analogous vortex-shedding phenomena.
'l'lrc prorllrnccd regularity of such wake effects was firQt"rgp-pged,by,$trquhal
l.l ttl wlxr pointccl oul that the vortex-shedding phenomenon is describable in
tcrrrrs ol':r rror.trlirncnsional number (the Strouhal number):
ND
s-?
i
wlrerc N, is tlrr. lit't1rrt'rrr'y ol lirll cyclcs ol'votlcx slterrltlilrg. /) is it t'ltltlrtt:lct'islic
tlirrrt'nsiort ol llrc lrtxly lrlojt't'terl ott lr pltrttrr tttll'ttnl ltt lltr'tttr'lttt llow vclocily,
u is the velocity of the oncoming flow,
assumed laminar. The number
S
tional shape of the prism being enveloped by the flow. Figure 4.4.4 t4-g6l
shows the relation of 3 to G" for a circular cylinder in the range 10s < G" <
107. The values of Fig. 4.4.4 were inferred from the unsteady pressure measured in smooth flow at about 90 degrees from the front stagnation point.
('oherent vortex shedding was noted to disappear at Reynolds numbers beyond
and summarized
in l4-9], there was no increase of the Strouhal number to values near 0.5. Table
4-4.1 [4-l0l also lists a number of values of s for different cross-sectional
shapes for Reynolds numbers in the clear vortex-shedding range, the approaching flow being laminar.
A certain amount of debate continues on the question of whether or not
lrcriodic vortex shedding gan still be exhibited at extremely large Reynolds
rrurnbers, say, G" >> 108. If one substitutes an effective eddy viscosity (see
scct. 2.2) for the actual kinematic viscosity of the fluid, it is conceivable that
ir ncw Reynolds number range can be calculated in which altemating vortex
shcdding from extremely largc bluff ob.jects can once more be forecast. In this
wly thc occasionally ohscrvul lrttgc vorlcx trails in ocean currents downstream
ol' islands may possibly lrc tt't'ottt'ilt'tl with srn:rllcr-scalc expcrimcntal <lbscrvlttiotts. Irigttrc 4.4..5, lttl ittslltttt'c, is lr lcpnltlrrcl ion ol'a satcllitc photogrlph
l4 Ill ol'tt voflox tllril irt tltt' itlrrroslrltt'tt' rn;rtlc visiblc by ckrutl Jlrc:sctrctr irr,
tltc votliccrs slttxl lhrrrt lltt' nrrtttttl;rrn l)('irh ()l ( irrrrrltrlrrpc lsltrrrtl ovr'r' 12(X) rrr
4 x l0s, and contrary to results reported by some observers
.l
ill liltlt()t )Y nl ll()l )YNnMl(.1,
152
:
Val0e ol
Prolile drmensions,
in mm
Value 0l
:/
"
r25{l_l___T
0.120
{
rz.s{
0.1 37
P
^"htffi
o.147
l*so--l
r= 1.0
,=0.5
_-_>
,,}
r= 1.0
2.O
*;w
.*
;
t=
tw(rtrtfi,,iltl.,t{rlll\l ttow
.r..\li<{ffCi .,t
*a\\."{f\
q
. ,r*' .,* };1* ,, ,r"
I'Altl,l,l 4.4.1. Slroulral Nurrrlx'r' lirr rr Vrrrit'l.y ol' Slritpr:s
Profile dimensrons,
.t w^t, I /\lillv{)l illxl(}ltMnll()Nt,tt.t
r2s{[-l-
ilH
['r]
0.120
+-t2.5
c<-T
0.150
1..-so-t
0.145
1.0
..,,.s,
'
0.144
0.t42
i$:i.
0.147
i;:-le .
/
0.13
r= 1.5
rzs{f-.1
[-*-]
I
0.145
$-t_
I
[ru!
/
r= 1.0
o 140
L-uo-t
E_l t_
l*rs*zs.l.zs!
0.153
125{l-J
0.134
0.137
0.12 I
0.143
0.168
+
0.156
G
0.160
0.145
,t
1l8OO..*r. tttOO
T-l
Ll
0.200
Q
t
l-.
'/izrrr.r. AS('li, l2(r
\i1,r.r,.. liRrril .'wirrrl lirr.t.t's orr Stlrrt.lrrR's,"
o.1
.'l
(l(x'l), ll.t,l
I''IGURE 4.4.5. Satellite photo of cloud vortices downstream of Guadalupe Island (off
Itlla Califomia) [4-l l]. Courtesy of the National Aeronautics and Space Administralion.
0.135
L_*-J
..,
:.l i!,i,,i.1;*6,'.:i,
:-' .,,-'all$::,"i.'
1
r= l.O
0.145
Cylinder
" q&:
&d'
f.-.-l
I
.-ig
..
l4
0.145
of Mexico. The photograph spans some 250 km.
Assuming, as in [2-1171, an effective value of (kinematic) eddy viscosity an
= -50 m2ls, a full-scale Reynolds number of the order of 1010 for the phenomcnon (based on u
1.5 x l0 5 m'ls; would be reduced to an effective value
=
ol'((R")"s = 3000, which falls well within the laminar vortex-shedding range.
Assuming the island to be about 20 km long, the distance between successive
pcriodic voftex centers is roughly 55 km. Further, assuming a Strouhal number
lirr thc island peak as S : 0.12, a mean wind velocity of U :30 m/s, and
rrrr cfl'cctive island clirnrrrrsion ol'/)
= 6000 m yields the vortex-shedding frchigh off the Pacific coast
(lucncy
0
ll()11
l.)( t())
(il()()
(r - lo IIIZ
I
ill t,t I il( )l)Y n t il( )l )YNn
fi4
.l ',
Ml( ii
lW()l)tMl tl',lol lnl
llll(.1:;()t'l
(-.._--
\-'\)
-
-
-
l;.
lrlll/\l
lrrl lM
t\
I
-_*7
"';ffit
.lllllr
; *:.."s,ffit
I l(.llltl,l 4.4.7. lillcct ol'splittcr plate on flow behind a circular cylinder [4-13'
I r' I I
llow dircction. Thus it becomes possible to inhibit the establishnear wake of the
,.i rir'l;rlurtt l'rocly, as first pointed out in [4-13]. (See Fig. 4.4.7.) The action of
rlrr.. pl:rtt'is to prevent the flow crossover and thus to quiet the entire wake
ll,,s' t.)tr:rlitativcly, the presence of the plate has the same type of effect as
i, rr1'tlrt'rrirrg thc body in the stream direction and causing it to approach, to
.,,irr, ;rppnrxinration, the form of a symmetrical airfoil. Following this type of
,rtri,r();r('lr it can bc seen that elongated bodies, oriented with theirlong dimenr,,rr l';urrllcl to the main flow, tend to elicit relatively narrow wakes, many
r.. rtlr,rrt lrpprcciablc voftex production.
ll llows irbout square and rectangular prisms are compared (Fig. 4.4.8), the
,|il:il(' rs sccn (at reasonably high G") to produce flow separation followed by
,r n'rrlt'. (rrrbulcnt wake, whereas the more elongated rectangular form (dethat
1,, r.lrr11, orr lcngth-to-width ratio) may exhibit separation at leading comers
r. l,rllrrwt'rl rl<lwnstream by flow reattachment and finally, once more, by flow
,p,u;rtr()rr rrt thc trailing edge. Thus it is seen that not only does the bluffface
,,1 rlrr'lrotly prcscnted to the fluid affect the resulting wake, but the streamwise
i, rrlllr rrrrtl gcncral form of the body also play important roles in the wake form.
irr',lr.rr1r tlis(inction to the casc of Fig. 4.4.8b, if the rectangle is placed with
rr. l{)rr)'. tlirrrcnsion normal (o thc llow, the wake exhibits a strong voftexi,,,l,lrrr1, i'lnractcristic, lirlkrwctl at highcr G" by a turbulent wake not unlike
rl,.rr ;rr11f111'gtl by thc sltiu'1'r ctlgctl llirt platc (see Figs. 4.4.|c and4.4.ld).
,r1,rr,r;11 11i11g
lrl(ltll{lt 4.4.6.
Satcllite photo of Jan Mayen Island (Arctic Ocean). From Weather,
1I. l0 (Oct. 1916), 346.
wlrich in turn gives a shedding period of Z : 1/N"
l/7'yiclcls a calculated vortex separation of
S
: 30 x 166l :
50.000 m
:
:
166l s. Employing S
:
50km
with rough fireasurement of the photograph. Anothcr intcrcs(ing photograph ol'largc-scalc vol'tcx shedding is prcscntcd in Fig. 4.4.6
nr
rr ol ir v()flcx trail by placing a "splitter plate" in the
,I
II
rr rlislurrcc consistcnt
l.l t2;.''
lt
lll
WIrcrr contlilions:rlc srrclr thirl ir rlistinct voflcx triril is l)rescn( in lhc wrrkc,
llow ct1)sli()v('r-:rll ol (lrc llrtly occttrs llltl hrts it ('()nllx)n('rrl rrolrnirl lo lltc
lurs
lx'trr lrroul'lrl lo llrr' ,rlllrtlrott ol lltt :tttllllrs llltl lt sitttll;r |rrrlrlr rtr r', ltr';rlr'rl rrr l.l i{)l
!
PRESSURE, LIFT, DRAG, AND MOMENT EFFECTS ON
WO I)IMENSIONAL STRUCTURN
! rlrrrt" ,l.5.l
I
FORMS
srrl',1qesls l s('( l11 )rl ol .r lrlrrll lrotly ltttttr'tst'tl itt lt llrlw ol vclot'ily
it llrt. ll6w will tlt'vt.l91r lpt:rl ;'t,..,.,rrr{.', /, r'\,('l llrr' lrrxly irr ttcr'olltrilt'c willr
!l, r il{}illli':; t't;tIt(iottl
156
tlt t.I
I
il( )t)Y At il(
)t
,t
)YNAM|(:ii
h il |
c
ri oN two DtMl NtiloNAl riffflll:lllll^l
t
.=.t.t^"t
ATTACHMENT
l) t p
pl
r.'orrsl
Ei,
lr'l(;Ljllli 4.4.8. Flow separation
and wake regions of square and rectangular cylinders.
()llMli
157
(4.5.l)
wltetc tlrc cottslirttl lroltls irkrrrg a strcarnline irrrtl l/ rrl)rrsonts thc volocity on
tltc strctttrline itt thc itnrttcrcliatc vicinity ol'tlre hxly (i.c., irrrrrrccliatcly outside
llrc llrundary llyer tltal lirrrrrs on its surlacc). 'l'hcr intcgration of the pressures
ovcr thc body surlitce rcsulls in a nct fbrcc and a lnolnont. The components of
tlrc lirrcc in thc along-llow and across-flow dircctions are referred to as drag
ilnd li.li, respectivcly. 'l'hc drag, lift, and moment are quite obviously affected
lry llrth the shapc ol'thc body and the Reynolds number.
'l'hc body may, lirr cxample, be contoured with the express purpose of
rrrininrizi'ng drag and rnaximizing lift, resulting in an airfoil-like shape. Again,
;rs in rnany civil engineering applications, the shape of the body may not be
;rrucnable to such special adjustment; its form will most likely have been fixed
lry other design objectives than purely aerodynamic ones. Nevertheless, the
lill, clrag, and moment developed by the fluid flows about the structure will
rt'rrrlin of strong interest because these are effects that must be designed against.
ll is usual to refer all pressures measured at a structural surface to the mean
rlynrunic pressure )pU2 of the far upstream wind or the free-stream wind at
rorrrc distance from the structure (e.g., at a point well above it out of the
lrrrrrrrrlary layer). Thus nondimensional pressure cofficients Co are defined by
Lr: P-Po
(b)
|
(4.s.2)
wlrcrc U is the mean value of the reference wind and p - ps represents the
prl'ssure difference between local and far upstream pressure p6. Such nondirrrcrrsional forms enable the transfer of model experimental results to full scale,
irrrtl the establishment of reference values for cataloguing the aerodynamic
;rro;rcflies of given geometric forms.
Arralogously, the net wind-pressure forces (per unit of span) F1 and Fp in
tlrc
lili
and drag direction, respectively, can be rendered dimensionless and
of lift and drag cofficients Cy and Cp as
r'rlrrcssed in terms
L,
rvlte
'
FL
--
iPu'B
t t, :
Fn
t_pu)n
(4.s.3)
(4.s.4)
tc /J is somc typic:al t'cli't't'ttt'c tlittrcrtsiorr ol'lhc s(ructure. For the net flowM lltcr crtrlcsPotttlittg cot'llir'icrrl is
tttrlttt'e:cl lnolncnt
t,'l{;llltl,l 4.5.1. l.ill
irntl tlritg rtn irn itrhil!rt!y ltltt!l hrtly
(
'tt
:
ttt
1,,, ,:
,r:
(4.5..s
)
158
I
tlt Ut
n,, til l(,t:i ()N lw()l)lMt t!!;l(lt,tAt !iililt( illt t^t t()t tMl;
tK)t)Y nt ll()l)YNnMl(
Su br; rit
l59
ical
4r.61
a
U
xlOs
t1
.o
O
+
o
o
O
brr
N'
o
.4c=l.lxlO'
107
106
Reynolds number,4e
FIGURE 4.5.2. Evolution of mean drag coemcient with Reynolds number for a circular cylinder. After L. R. Wooton and C. Scruton, "Aerodynamic Stability," in The
Modern Design of Wind-Sensitive Structures, Construction Industry Research and Infirrmaticrn Association, London, 1971, pp.65-81 and 14-221.*
When the flow is fluctuating as a consequence
of oncoming turbulence,
vortex-associated flow changes, or signature (body-induced) turbulence, the
above quantities become time dependent. In such cases, when time-varying
litrccs ancl moments occur, mean values of force coefficients aS well aS spectral
clcnsity clistributions of these quantities are required for their fuller description.'
(Note that in two-dimensional flow L1 , Fp, ?fid M represent corresponding
valucs per unit of dimension normal to the plane of observation. In threedimensional cases, correct dimensionality is preserved by including an additional factor B in the denominator of each expression.)
Retuming to the prism of circular cross section in smooth flow, the variation
ol' its mean drag coefficient Cpmay be represented as in Fig. 4.5.2, where the
clcpendence on Reynolds number is shown. Note particularly how Cp drops
sharply in the rang6 of about 2 x l}s S Ge < 5 x 10s. This region of sharp
clrop is called the critical region and corresponds to a condition wherein the
layer that forms
f ransition from laminar to turbulent flow occurs in the boundary
on the surface of the cylinder. The turbulent mixing that thus takes place in
tlrc boundary layer helps transport fluid with higher momentum toward thc
surlace of the cylinder. Separation then occurs much farther back and the wake
c()nscquently narrows, finally producing a value of the time-averaged C, that
is only about { of its highest value. As G" increases into the supercritical and
thcn ihe transcritical range (G" = 4 X 101. CD increascs once more but
rcnrains much l<lwcr than its subcritical values.
l,llcccrrl tl:rtu
lrl ll(rl slrow llr:rl tlrt.tlrirg c0cllicicttls itt thc lt'giorr 5
l{)'
(ll, '
lO/ :rtt'stttltllct'
lly ltllrttl l5%' llllrrr lltost'tttrlitltltrl irr lrig 4'5 2 rtlt lltt: hitsis ol t';ttllr': tltlrttttt;tltrttt
Sct' Appr'rrtlir A.)
o"
"o" , ,oaorra'
l,'l(;tlRE 4.5.3. Influence of Reynolds number on
, ylirrtlcr
l2o"
l8o"
pressure distribution over a circular
(after [4-22]).
lrigure 4.5.3 depicts a typical distribution of the mean pressure coefficient
in smooth flow as a function of angular position.
I'lrt' rcsults are evidently sensitive to Reynolds number.*
'l'lrc drag coefficient of an elongated rectangular-section
body in smooth flow
tlrrg. 4.5.4) [4-14,4-231 is also a function of the narrowness of its wake, but
rlrc krwer limit of wake width is approximately the full width of the body. The
n,rrkc width at somewhat lower G." is much greater than the body width, and
tlrrs is accompanied by higher cp;then, when flow reattachment to the body
;rlrorrt the circular cylinder
lx'1iins to occur, the drag cocfficient drops. This is a function mainly of the
r'lrrngirtion blh of the brxly, irs shown in the figure. Flow in the critical region
l:' ilccompanied by turhulcrrcer, irncl thcrcfirrc this region is shown as a shaded
lr;urtl ol'possible valucs in lrig. 4..5.4.
liigrrrc 4.5.5 14-1.5 1 illrrsrrrrrcs tlrt' t'volrrliorr with Rcynolds number of the
rrrt':ttt tlrag cocflicicttt ol'it stlttirtt irt srrroollr llow tlrrrirrg successivc rlodificaItotts ol'ils corncrs. Nolrr llrirt otrly llrt.slttlP t'orrrt'r't'tl stlrrar-o cxhihits practicirlly
rllrt'lrcsstttr:s ('on('sl)on(ling k) /l
ll" irrrrl ll
lltt lltsc l)ti'r{ilr', rt rlx r livrll'
',lrr;',rlrliorr lloirrl :riltl
ll"i{1"
;rrt lt.lt.trt.tl l() itri llt(.plt.ssrrrt.trl
llrt.
160
nt t,t
I lt()t)y At n()l)yNAMt(]ri
4h rIil(lii
il
2.2a--r--
t
'
I8J- r
r
rr
()N twot)tMl Nl;l()Nnt 1;ililt(;ililtnt t()ilMli
r --r--Tt-T-
r --l
----'!-
[-1r,4 f , t, =oozt
, , ,, , l'"_l
r ___r rr
r
---) u--D]n
1.2
161
0.8
r/h = 0.167
0.4
(h)
0L
0
FIGURE 4.5.4. Effect
24
of afterbody
o
6
upon drag
of
a
b/h
rectangular cylinder 14-14),
'F
14-231.
k1h
unchanging drag with change of Reynolds number. This is simply accounted
for by the early separation ofthe flow at the upstream corners and the shortness
of the afterbody that practically precludes the possibility of flow reattachment,
whereas squares with rounded corners tend to possess the same kind of critical
region for the drag coefficient as seen earlier for the circular cylinder. Note
also, in the case of the circular section, the dependence of the drag upon the
roughness of the cylinder surface. This dependence was studied in detail in
14-241. (See also Sect. 11.1.1.)
Because of such effects, certain features of the flow in tests over wind tunnel
models can be expected to be independent of the Reynolds number, while others
may be quite sensitive to it. Thus it can be argued that cerlain Reynoldsnumber-insensitive flow phenomena may be encountered in tests in which the
llow will always break cleanly away at the same identifiable points. certain
types of bodies such as the circular cylinder offer extended regions of possible
llow separation in which the location of the actual separation points depends
rupon Reynolds number. with such bodies the entire structure of the flow will
hc highly Reynolds-number-sensitive (see Secr. 7.3.2).
l'or cxtremely low Reynolds numbers the drag coefficient increases greatly
irs ir rcsult of viscous effects. This is illustrated in Fig. 4.5.6 t4-I41, which
1fL:;)icts Cpfor circularand square flat plates for 10-2 < G., < 107. (Analogous
cllccts on lift and moment do not necessarily follow, though some distortion
is vcry likcly.)
sincc thc prcssurc dil'l'crcnces across a sharp-cornered square vary with time,
llte soctional lili crrcllicicnt will also be a function of time: C1.: C1,Q). Figure
4.5.1 14- l6l illustra(cs tl.rc spcctral density of c7. plottcrl as a lirnction of
rrll/{/, whcrc rr is l'r'c:t;ucrrcy in Hz, B is thc dimcnsion ol'rhc: sitkr ol'tlrc scprarc,
ittttl l/ is ttlcitll (tttr"()trring vt:krcily (irssurnccl to bcconslirrrl llrlorrg.lrorrl tlte lc:gion
I
= 0.007
k/h:
O.OO2
h:0.001
8105 2
r/h:0.5
(circular section)
\--'
8106
8l 07
2
Ee
sanded
---
surface k)
Smooth surface
FIGURE 4.5.5. Influence of Reynolds number, comer radius, and surface roughness
on drag coefficient, square to circular cylinders (r is the corner radius; k is the grain
size of sand). After [4-t51.x
co
o.t
to-2
lo-l
t03
t02
t04
t05
to6
l07
9tt
FIGURE 4.5.6. Typical rlr:rg coc{licicnt as a lirnction o1'Reynolds number [4-141.
*Motc tcccnl tllta lor circtrlitr cylittrlcts
l,l tl(rl rrrt in gcncrirl tprrirliltrlivc itgrcclllcnt wilh tllrsc gl
IriP,.4.5.5 btrl intlicatc lltirl lirl l;tti,ic l{r'ynrltlr rurrrlx'rs thc cylirxk:r rrurglrrrr":ss brings irlxlrt ir
sonrcwhitl slr'ongcr irrcrcitsc irt
tllrg
t62
lil ul I Ir lt)Y n t il( ltlYNAMtcl
50
o
J
t
o
L
llo
J
O
s
lll
tb
lr
N
0.05
./,t
0.02
0.01
rrl
a
h
U
o
O
ii
/'f tr
p
d
!
o
ir
o.2
a
(r
o5"1o"15o20"25"30"35"40"45"
\'\
ANGLE OF ATTACK, o
\:t,
tpj'
.rt"#
,ff
dt
U
]-
/l
N
+
SMOOTH STREAM
F
l
F
O
l
J
L
L
il
0.5
CNr..{r/2pua)u
o
o
2
1
t4
o(
li
f,l
t63
a
I
k;
10
cn
t()t tM:i
E
20
a
N
()N lw()t)lMl N:it{)l.JAl f;lltll(jillilnt
1,, tt|(.11;
0.005
Turbulent
0.002
Smooth -+_o+
stream
lfl(;URE 4.5.8. Variation of the coefficient of fluating normal force, C1y_.. with angle
ol attack for a rectangular prism. From B. J. Vickery, "Fluctuating Lift and Drag on
rr Long Cylinder of Square Cross-Section in a Smooth and in a Turbulent Flow,"
"/.
Iluid Mech.,25 (1966), Cambridge Univ. Press, New York, pp. 481-494.
(
stream
shape and
NB/U
l"lGURE 4.5.7. Spectrum of lift fluctuations on a square-section cylinder for flow
nonnal to a face (G" : l0s). From B. J. Vickery, "Fluctuating Lift and Drag on a
l,.ng Cylinder of Square cross-Section in a Smooth and in a Turbulent Flow," ,/.
lluid Mech.,25 (1966), Cambridge Univ. Press, New York, pp. 481-494.
ol'lkrw under consideration). In both smooth and turbulent flow, a high spectral
1rt'rrk occurs at the Strouhal number nBlU : 0.12.
'l'his is clcar evidence of periodic voftex shedding. For any given bluff body,
tlris shodding is not a purely sinusoidal phenomenon, as seen from the spread
t. rrtlrcr I'rcqucncics ol'thc spcctral peak in Fig. 4.5.7; however, a good first
lrprpIrrxirna(ion lo tho lili lirrcc pcr unit span occurring at the peak Strouhal
rrrrrrrbcr is givcn by
lr,
t,pU)B(-,
sin
<,:t
lift coefficient that depends on the particular
a : 2rn, r? satisfying the Strouhal relation.
where Cl is a mean
0.02 0.05 0.100.20 0.50 1.00 2.00
('1..5.6)
cross section
The root mean square (rms) value of the fluctuating normal force coefficient
on the square section is shown in Fig. 4.5.8 t4-161 as a function of angle
ol'attack a with respect to the mean wind direction. Here the turbulencex is
sccn to lower the highest normal force below, and to raise the lowest normal
lirrce slightly above, the respective laminar values.
Figure 4.5.9 l4-ll presents two photographs of flow over proposed bridge
tlcck sectional forms as visualized in a water tunnel flow containing fine alu(ry,,,,.
rrrinum particles. Figure 4.5.9a shows
a section that produces severe flow
scparation; Fi9.4.5.9b portrays the flow-smoothing effect of a modified section
providing lower lift and drag.
Rcf'erence [4-10] prescnts mcan values of cp and c. obtained under laminar
llow conditions for a largc rrurnbcr ol-scctional shapes common in construction,
irs takcn
lnrm il2-21ntl 14 lltl; scc'l'ablc 4.5.1.l4-171, an<l 14-621.
r'lltc ltttlrttlcttec clt;tt;tt'lt'rislits itt lltt
sr';rlt' l.:l/J. lirlt'r'lrl st:rlt' O.,l/1. lrrrlrrrlt
trlr'rrrrrrrt ol lir1l .l 'r lJ wcrc tlrc lirllowirrli: klrpitrrtlil:rl
! ltl( u.,tl\, lll'i,
nr
164
ilt t,t
I 80t)Y
nt tt()t)YNnMt(;1;
Iit(;t:;
()N tw()l)lMl Nlit()Nnt :;ilIt,(;lUllAl l()ltMt;
1€5
'l'Alll,ltl 4.5.1. 'l'wrl l)irtttttsiottul l)rug urttl l,ilI ('rnllicicrrls lirr Slructrlrul
Slurpes
Prtlth arrrl w[r(l rlilFr lh)il
cD
cL
*M-r.
2.O3
0
r,96
2.01
-
o
-[
-I
2.O4
0
1.81
0
-D#l
FIGURE 4.5.9a. Visualization of water flow over a model bridge deck section. Courtesy of the National Aeronautical Establishment, National Research Council of Canada.
-------
L-
-l
_L
+lJ
-lF
+ll
2.O
0.3
1.83
2.O7
r.99
-0.09
1.62
-
0,48
o
2.O1
nllr
ilhlll
FIGURE 4.5.9h. Visualization ol'watcr flow ovcr a paI1itlly slrcirrrrlirrctl rrrtxlcl lrlitlgc
dcck scction. (-ottrlcsy ol (lrt' Nrtlionul Acntnitulicitl lislitblislrrrrt'ttl , Nitliorrrrl l(cscirrclr
('outrt'il ol ('lrturtLt.
\rrlrrr'. Iilorrr .'Wirrtl lirrtt's ott Sltttr lttrr':
/irrrrr A5('lr. ll(r
(
l()(rl
), I l.l.l
I l()t{ rrrrtl I
ll
Jl
166
Bt Ut
I not)y Al tr()t)yNAMt(;ri
tf, tlt(;l;()N
'l'hc rosults ol"l'ablc4.5.1 are irppliclrblc lo nrcnrbors with luryc lrspr:cl urtio
(ratio of length to width) \, or lo rrtcntbcrs with ond platcs (abutrncnts). For
members with small aspcct ratio (c.g.,
< l0) and no cnd platcs (abutmcnts),
^ drag cocflicients are smaller than in
end flow effects are significant, and thc
Table 4.5.1 (see Sect. 4.6.2). The drag coeflicients are also modified by the
presence of turbulence in the oncoming flow. Experiments have shown that in
most cases of interest in practice these modifications are small Il2-2, 12-51.
For this reason wind tunnel tests aimed at measuring aerodynamic forces or
trussed frameworks with sharp-edged members are to this day conducted in
smooth flow [12-1, 12-6]. Note, however, that in some cases the effect of
turbulence on the drag force can be significant. For members with rectangular
cross section, this effect depends upon (l) the ratio blhbetween the sides of
the cross section and (2) the turbulence in the oncoming flow. If the ratio blh
is small, no flow reattachment occurs following separation at the front corners.
twot)tMt Nlit()NAt l;ltlt t(;lt,ttnt t('ttM!;
167
l)cpcntlirrg ttlxrtt ils tttlt'ttsily, thc) turbulcnccr cirrr crrlrirrrcc llrc llow (,ulti1n1r(.nl
in thc wlrkc ittttl, lltcrcliut. (:itusc stK)n8,cr sucliorrs lrrrtl lirrgcl tlrirtl (1,'ig.
4.5.10a). ll'thc ntlio /r//r is sullicicntly largc, tlrer turhulcrrcrr ciur t'rrrrst. lLrw
rcattachmcnt which wottltl rrot have occurrcrl irr srrrtxrlh llow arrtl llrrrs nsrrll
in reduced drag (lrig. 4.5.lob) 14-25,4-26|. A bcaurilul visualizltion o| r|c
llow around a body with rectangular cross scction (blh :0.4; srn<xrtlr lkrw,
Re : 200) is shown in Fig. 4.5.n [4-87] and may be compared, qualitativcly,
with the smooth flow case depicted in Fig. 4.5.100-see also [4-94]. -the
dependence of the drag coefficient upon turbulence intensity is shown for two
ratios blh in Fig. 4.5.12* 14-261. Additional studies on turbulence effects on
drag and lift of sharp-edged bodies are reported, in [4-271, 14-281, and [4-85].
The effect of turbulence in the case of bodies with rounded shapes is, essentially, to reduce the Reynolds numberat which the critical region (Fig.4.5.2)
sets in. This is shown in [4-291, which includes, in addition, information on
the fluctuating lift and drag forces on a rigid cylinder due to vortex shedding
and to turbulence in the oncoming flow (see also t4-301).
For a recent, wide-ranging review of turbulence effects on bluff-body aerodynamics, see [4-87]. Reference [4-14] is compendium of drag effects that
contains limited data obtained in smooth flow on models of buildings and
structures.
Hrgher
drae
f
!-,/
5
(b)
FIGURE 4.5.10. Separation layers in smooth flow (solid line) and in turbulent flow
(intemrpted line). After A. Laneville, I. S. Gatshore, and G. V. Parkinson, "An
Explanation of Sonrc Ell'ccts ol'Turbulence on Blufl'Bodics," /)rrcclrlirg.r, l,ourth
International Conl'crcttcc, Wirrtl llllccts on Buildings antl Slnrclrrn's, ('irrrrbritlgc t)niv.
Prcss, Carnhritlgc, l()77.
|I'IGURE 4.5.11. Flow around rectangular cylinder (b/h - O.4, G." : 200). From Y.
Nakamura, "Bluff-Body Aerodynamics and Turbulence," ./. Wind Eng. Ind. Aerod.,
49 (1993). 6s-18.
t'Notc llral lir Itlh - l, (), irs ohl:tittcrl in l,l 2{rl lol srrrrxrllr lkrw <lillors by ll-xlrt l0%, lirrrrr tlrc
vitlttc lislcxl in 'l'irhlc 4..5.1. l)illctt'rr('cs ol llri$ ottlcr or l:rlgt'r' irrc c()null()n t:vt:rr lirr rr.strlls ol
sinrplt: wirrtl lunnrl {(:sls.
lrl t,l
168
I lr()l)Y nl lt()l)YNAMI{;l
't
F
nt t,nt 1;t Nln
ltvt I l()w I lil (;t:; ||J ilillt I
t)tMt
N:;t()Nli
169
40
o
(E
F
z
CD
UJ
o
tr
LrUJ
o
o
zo
o.
t- o.4
JUJ
048121620
(E
(E
o
(J
_y,
u.,
enl
FIGURE 4.5.12. Dependence of drag coefficient upon turbulence intensity. After A.
Laneville, I. S. Gartshore, and G. V. Parkinson, "An Explanation of Some Effects of
Turbulence on Bluff Bodies," Proceedings, Fourth Intemational Conference, Wind
Effects on Buildings and Structures, Cambridge Univ. Press, Cambridge, 1977.
4.6
REPRESENTATIVE FLOW EFFECTS IN THREE DIMENSIONS
Most flows have a three-dimensional character, principally as a result of their
contact with boundaries. For example, if a hypothetical laminar flow consisting
of an air mass displaced uniformly as a single unit encounters an object, it will
be diverted in several directions. Also the passage of such a flow along a surface
sets up boundary-layer velocity gradients. Three-dimensionality is clearly inherent in turbulent flows.
Although the general equations for fluid flow remain available for application, few flow problems in three dimensions have been satisfactorily solved in
a purely analytical fashion because of the considerable complexities involved.
As a result, most three-dimensional studies rely partially or wholly upon experiment. Therefore, this section is mainly concerned with broad aspects of
three-dimensional flows, with conditions of testing, and with some representative results obtained by test.
4.6.1
Cases Retaining Two-Dimensional Flow Features
The success of the two-dimensional flow models discussed in the previous
section has in a few cases been considerable because sorlc actual flows retain
certain two-dimcnsional t'catures, at least to a first approxinrllion. Consiclcr,
forexample, lhc casc ol'a long nld ol'squarc cnlss scclirttt itt:rtt lrit llow with
unilorm nroarr vcrkrcily norrrurl (o onc lircc:. lixccgrl rtt'rrr llrr't'rrrls ol llrc nxl,
r1"/D
I,'IGURE 4.6.1. Spanwise correlation of the fluctuating pressure difference across the
t'cnter line of a long square-section cylinder for flow normal to a face (G" : 105;.
lirrm B. J. Vickery, "Fluctuating Lift and Drag on a Long Cylinder of Square CrossScction in a Smooth and in a Turbulent Flow," J. Fluid Mech., 25 (1966), Cambridge
Oniv. Press, New York, pp. 481-494.
llrc mean flow may, in this case, be considered for practical purposes as twotlimensional. However, the effects associated with flow fluctuations are not
itlcntical in different strips, the differences between events that take place at
rrny given time increasing with separation distance. This is shown in Fig.4.6.1
l4-16] for the pressure difference between centerlines of top and bottom faces
ol'the rod under both laminar and turbulent approaching flow.* It is observed
tlrat the three-dimensionality of the flow manifests itself through spanwise loss
ol'correlation R7s between pressure differences (measured respectively between
;xrints,4 and A' at section.4 and points B and B' at section B), this correlation
krss being strongly accentuated when turbulence is present in the oncoming
llow. From this example one may infer that fluctuating phenomena, including
vortcx shedding, cannot nonnally be expected to be altogether uniform along
lhc cntire length of a cylinclrical botly, cvcn if the flow has uniform mean speed
:rntl thc body is gcrlrnctricirlly trrtilirrrn.
ln practicc, rncan llow t'orrtliliorrs rrpwirrtl ol'tall slcndcr structurcs arc usurrlly no( unilorrrr, ls trssrrrrrctl irr (lrt'sirrrplr'r.('ilri('r tliscrrsscrl ahovo; inrlcctl, in
r'l'lrt'lrrrlrrrlt'rttt't'lr;uittlt'lislir's lvr'tr'lltr'',;trrrr';r" ttt llrr'r'r1x'tttttr'rrt ol liil',.'1.5.11.
170
ilt t,t
I il()l)y nt il()t )yNnMr(:l
,l
thc atlttospltcric ltrlLtlttl:rty lltyt:t'tlrt'rrrt'rrrr llow vclocily itrcreirscs witlr lrciglrr.
Also certain tall structurcs (c.g., sllreks):ur lt()t gcorrrctrically unilirrrrr.'l'lrcsc
important features-in addition to thc incitlcnl lurbLrlcncc-furthcr dccrcasc the
coherence of vortices shed in thc wako ol'structurcs.
4.6.2 Structures in Three-Dimensional Flows:
',1
Nlnllvl
ll()Wllll{il:;llJ
Srrrrxrlh l,'low
l.l-lll, l!
2l
Itrrelurtgttllrr l)lirtc
on (inrrrnrl
(Standing on I-ong
l{cctarrgular Platc in Nonnal Wind"
and
differences between drag or pressure
coefficients measured in a uniform and in a boundary layer flow. The existence
of such differences was first pointed out by Flachsbart in 1932 t4-311.
we consider first the case of a rectangular plate normal to the wind in a
lll
llllll I l)l[/l N:;l{)Nl;
'l'Altl,l,l 4.(r.1. l)rrg ('rx'llil'irrrls l'rrr a llccllrrgrrlul l'lrrlt'Nolrrrirl lo Wirrrl irr
Case Studies
The complexities of wind flow introduced by the geometries of typical structures and by the characteristics of the terrain and obstacles upstream emphasize
the need to carry out detailed studies of wind pressures experimentally using
wind tunnel models and simulation. In order to give some idea of the type of
results so obtained and to emphasize the important roles of the boundary layer
velocity profile and of the turbulence in such results, a few examples are cited
bclow.
wind flows about buildings are prime examples of three-dimensional flows
that cannot be described acceptably by two-dimensional models. Ftgure 4.6.2
il5-l ll suggests such a situation. Here a tall model building in a wind flow is
preccded by a lower building. This latter trips off a vortex in the space between
buildings. Air descending close to the windward wall flows through openings
beneath the building at ground level. Regions ofaccelerated flow are produced
around vertical and horizontal corners of the building. In the areas of vortexflow, through-flow, and corner streams, many design problems are presented
by the special characteristics of the locally accelerated flow. (See Sect. 15.3
p. 188.)
A few examples are now shown of
ri lll l'lll
Aspcct
(',,
ratio I .0 2.O -5.0
1.18 1.19 l.2O
10. 20. 40.
1.23 1.48 I .66
Sidc)
oo
I
.98
1.0 10.
l.l0 1.20
oo
1.20
I'l'hc values listed in [4-10] were taken from [2-2]. Some of these values were incorrectly
in [4-10] and therefore differ from those shown in this table.
trrrnscribed
snrooth flow. The drag coefficients depend strongly upon aspect ratio and upon
whether the plate is held in midair, as in the case of a tralfic sign, or stands
on the ground, as in the case of a free-standing wall; see Table 4.6.1. For
rcctangular plates on the ground, the drag coefficients of Table 4.6.1 are reasonably consistent with mean drag coefficients obtained in boundary layer flow
14-931. Reference [4-93] contains additional results on free-standing walls,
rncluding pressures in the presence of a building upwind or downwind from
tlrc wall.
Note that the aerodynamic force normal to the plate is not necessarily largest
when the yaw angle a (Fig. 4.6.3) is zero. For a plate with aspect ratio X :
5, the dependence of the aerodynamic force normal to the face of the plate
upon cv is shown in Fig. 4.6.3. It is seen that for ot : 4Oo the aerodynamic
lirrce is larger by about 15% than in the case cv : 0o. A similar, though
sornewhat smaller, increase was reported in ll2-21for a plate girder with aspect
rrrtio X = 10.
The effect of turbulence on a square plate normal to the flow was studied in
14 251, where drag coefficients were measured for both smooth flow and turbulent flow with 8.3% turbulence intensity and 7 .6 cm longitudinal turbulence
scale; see Table 4.6.2.
Note that the drag coefficients measured in smooth flow differ slightly among
1.2
08
(',,
04
0
o4
l"l(;tJltlt 4.(r.2.
Mrrirr Icit(tttcs ol lhe llow rrnrrrn<l :r lrrll lrrriLlirrll rrrrrlt.l
Il5 III
lr'l(;llltl,l 4.6.J. lX!l)(:nrlt'rrt't'ol tlt:r1'. tocllrr'rctrl lot pl;tlt'willt itspccl ntlio \
tlirccliorr ol lrolizon(rrl wirul I l.) .'l
5 ttpott
172
lll Ul I ll()l)Y nl lr()l)YNAMI(:i;
,l
'l'Alll,lil 4.(r.2. lh'ug ('rx'llicitrrls
ri lll l'lll iil Nln llvl ll()W lllll
lri ll! llllll I lrlMl t..l!;l()l.l!i
113
l'rrr
Sr;uarr: l)latc Nrlrrttrtl lo lhe Mr.un lr'krw
l4-2sl
Wrrrrl
Plate Sizc
(cm)
5.08
10.16
15.24
20.32
x
x
x
x
(',,
Sntrxrlh
Wtrrrl
---d>
'l'urbulcnt
5.08
t.12
10.16
15.24
1.09
l.1l
t.26
L22
t.20
20.32
l. l5
I.t8
themselves and from the value of Table 4.6.1 (CD : l.l8). Note also that as
the ratio between the longitudinal scale of turbulence and the dimension of thc
plate decreases, the influence of the turbulence on the magnitude of the drag
coemcient becomes smaller. These results are further discussed in Sect. 7.3.3.
Figure 4.6.4a shows a model used for measurements reported by Flachsbart
in 1932 t4-311. The measurements were conducted in both smooth and shear
(boundary-layer) flow (Figs. 4.6.4b and c). The measured mean pressure coef'ficients Q, referred to the free-stream velocity, are shown in Fig. 4.6.4d for
smooth flow and Fig. 4.6.4e for boundary-layer flow (interrupted and solid
lines represent pressures and suctions, respectively). It is seen that the differences between the results obtained in the two types of flow are significant.
Similar results were subsequently obtained in 14-321 and [4-33].
Figure 4.6.5a depicts mean flow patterns around a vertical wall of heightto-width ratio I : 1 with uniform approaching flow. Figure 4.6.5b depicts the
same situation in boundary{ayer flow. Figures 4.6.6a and 4.6.6b display the
pressure coefficients developed on the faces of a cube resting on a horizontal
surface (due to flow normal to one face) first in uniform flow, then in a bound-
aryJayer flow. Figures 4.6.7a and 4.6.7b present similar results for a tall
building. It is noted that in Figs. 4.6.5b,4.6.6b and4.6.7b the pressure coefficients are referred to the free stream velocity t4-201.
Loads on structural parts (e.g., cladding) are determined by the algebraic
sum of the extemal and intemal pressures acting on these parts. In the ideal
case of a hermetically sealed building, the internal pressure is not affected by
the external wind flow (Fig. 4.6.8a). If the building has an opening on thc
windward (leeward) side and is otherwise sealed, the wind flow will create a
positive (negative) internal pressure, as shown in Fig. 4.6.8b (Fig. 4.6.8c).
In most cases the opening or porosity distribution over the building envelopc
is not known, and intemal pressures could be either positive or negative (Fig.
4.6.8d). Building standards (e.g., [2-491) specify intemal prcssurc cocflicicnts
generally believed to be conseryative fbr use in design. lrrvcstiglrtiorrs into thc
magnitude of intcrnal prcssurcs and of thcir dcpcntlurt'c orr tirrrt' lrrt' rc;xrr1e:tl
in 14-521 to l4--571, which contuins adclitional rclcrctrccs.
('
lll
o Io
2.O
Srrrnnurry ol rrrrxh'l l(':,1:, rr :;rrlxrllr rrtrl lrottntiary-laycr llow. F'nrtl
\\'rrrlrlrtrt'k lrrrl gcsclrloss('nr' llr(l nllr'rrt ( i('lr;ru(l('." lry ( ). liluchsbafl, in Iirgtltri,t.tt
,1, t lt't!\l\',tttrtti.st'lrttr Vt'r,tttt'lrttit.tt,tlt .it (;tttttut:('tr, lV l,it'li'rrrrrg, 1,. l'rrrrllll, rttttl A.
llt t., (r'rls. ), Vcllirg vott ll . ( )lrlt'trlrotttp, l\ltttttr lt irrr,l ll,'rlirt, l().J2.
l,
llJlllll,,.l.(r.4.
174
Bt
t,t
I
B()l)Y nl lr()l)YNnMl(i:;
4l
z
llll lillAlloll()l
llMl v^l tYlN(it()t
t(:t
t{ r
wt[]D vt
I{
'(
.ltY
175
!-
E
H
r'0
r.0
0
AP
"p-ifr"
FIGURE 4.6.5a. Flow pattern and center line pressure distribution of a wall of heightto-width ratio I : 1 in a constant velocity field. From W. D. Baines, "Effects of Velocity
Distribution on Wind Loads and Flow Patterns on Buildings," Proceedings, Symposium No. 16, Wind Effects on Buildings and structures, held at the National Physical
Laboratory, England, in 1963, published by HMSO London in 1965.
lrlGURE 4.6.5b. Flow pattern and center line pressure distribution over a wall of
hcight-to-width ratio 1 : I in a boundary-layer velocity field. From W. D. Baines, "Eflccts of Velocity Distribution on Wind Loads and Flow Pattems on Buildings," Pro,'rulings, Symposium No. 16, Wind Effects on Buildings and Structures, held at the
Nirtional Physical Laboratory, England, in 1963, published by HMSO London in 1965.
4.7 THE RELATION OF TIME-VARYING
tlurt analytical calculation ol'such rcsults is not possible, it is usual to employ
lirrrnulas l'eaturing unknown cocllicicnls that may be evaluated by experiment.
FORCES TO WIND
VELOCITY IN TURBULENT FLOW
For a given body immcrscd in a wind flow it is of intcrcsl (o crtttvc:tl irrlilrntation
on vclocity lluctualions into inlirnnation on prcssttrcs ovt't lltt'lrotly or on
rcsullanl lirrcc:s iurtl nl()nrc:nls. Sinc:c: tttrtst rr:itl lltlws lttr' sttlltt it'lttly t'oltllllcx
4.7.1 Drag Forces
'l'lrc nct tlt'itg lirrcc c()nsisls ol lltr rt'sttll;url ()v('r'ir givcrrt botly surlircc ol'trll
t'orrr1'roncnls ol'clcrrtcntlrl lirtcr':; llurl ;rrr';rlrlqnr'tl witlr llrt'tlnrg, orirlorrg wintl,
176
tJttJt
I tlot)Y
4
AFllol)yNAMtct
/ lilt ilttAil()N ()t ltMl vnllYlN(i l()t t(;l :; t()
0.2 0
--.5
=-.-.80--_-......-...=-_.70- )
-0 ,0
11y151;
vt t(x;t ly
177
0./0
5
)
-.6 0
-\--.80at--J
-.2
_-.70r'
(_
__--s__--
--.65 -
\--__
-.60
--.,
--/
-
'.(.
rilI,,
99
l/rl
{
f*'no
l*,no
(a)
FIGURE 4.6.6a. Pressure distributions on the faces of a cube in a constant velocity
field. From w. D. Baines, "Effects of velocity Distribution on wind Loads and Flow
Patterns on Buildings," Proceedings, Symposium No. 16, Wind Effects on Buildings
and Structures, held at the National Physical Laboratory, England,
in 1963, published
by HMSO London in 1965.
Fugy
l
lfl(luRE 4.6.6b. Pressure distributions on the faces of a cube in a boundary-layer
't'l.city field. From w. D. Baines, "Effects of velocity Distribution on wind Loads
rrrrtl lrlow
Pattems on Buildings," Proceedings, Symposium No. r6, wind Effects on
lhrildings and structures, held at the National physical Laboratory, England, in 1963,
grrrblished
direction. The time-varying drag FoQ) on a body completely enveloped by
flow is conventionally given by the formula
whcre B is
(b)
:
[pu2gynzc,,
lypicirl lrorly rlirncnsion irnrl (i7, is
lhr.: rrsrrirl
a
(4.7.t)
tlrirll t.rx'llicicrrl
by HMSO London in 1965.
Irr Eq. 4.7.1 a seconcl tcrrrr .l' rlrc lirrrrr pn\au1rlldtlc. is often included,
1r;ulicularly if the lluitl irr tprcstiorr is rclirrivcly rlcnsc, for examplc, as in thc
trtse: ol'walcr; (),,, is an cnrpitit':rl "virlrlrl nritss" erx'llicicnt intcn(lcd lo ucc()111
Ior cll'ccls linkctl trl thc lltrirl itt t t'lt'tirlron. At lrrrrlly llrc cocflicicrrl (1,, trppr:trr.s
Io bc ttsr.:l'trl itt citsc:s wlrt'n'itt lltr'llttirl ruirrr urvolvt'rl is tr;.r1'rrcciirblc rclir(ivc to
178
llr t,t
I
l1()l )Y
4
nl ll()l)YNnMl(;i
/ llll llt tn it()N ()t ItMt v^l tytN{i t()t
t(jt
j; t() wtNt) vt t(x;ilv
l7g
5
.
-0',1 to
-0.49
-0.9
z
f
f *'"0
*'"0
.6
std?
Fro nt
Bock
(a)
FIGURE 4.6.7a. Pressure distributions over the sides and top of a tall building model
in a constant velocity field. From W. D. Baines, "Effects of Velocity Distribution on
Wind Loads and Flow Pattems on Buildings," Proceedings, Symposium No. 16, Wind
Effects on Buildings and Structures, held at the National Physical Laboratory, England,
in 1963, published by HMSO London in
1965.
S
idc
Bock
(b)
lrlGURE 4.6.7b. Pressure distributions over the sides and the top of a tall building
rrurdel in boundary-layer vclocity ficld. From w. D. Baines, "Effects of velocity
l)istributi<rn on Wind Loatls irrrtl lilow P:rltcrns on Buildings," Proceedings, Symposittttt No. 16, Wind E,ll'ccts orr lhriltlings irntl Stnrcturcs, held at the National Physical
l.rrboratory, Iinglancl, in l()(r.1. prrlrlislrt.tl lry IIMS() Lonckrn in 1965.
180
Bl tJt
I
lr(
4
)l)Y n I ll()l)YNn Mloii
/ illl
ilt tAil()N
()t
ltMt VAnytN(i tr)lu F.i trr
6111.11,
vt
t(x.ilv
1Bl
r'':'rrllrt'icttlly stttitll t'ottt;trtretl lo llrc crlntlttliort
rlrslrrrrt'e.s ol llre lltrgllirlrotrs 1,
r'.;ttttl tt', srl lltitl, lirt tlrt'l)ttrl)()scs ol'lhc: lllrrblt'trr rrl lr;rrrrl, llrcsr. lirller rrriry lte
,,rttrltlt'lctl kr bc ptrr'lirt'll-y corlclulcrl. Sirrec irr llrt. lupilr wilrtls rrsulrlly ol'grllrlcsl
rrl{'r('st lo wintl crrgirrcclirrg u(tllU ftrrcly cxr.t't.tls o.l, r' nriry gcnuirlly bc
nt';rl,'r.',r.,,,
(o) HERMETIC
witlr srrritll
c:r'nrr
(b) WINDWARD OPENING
BUILDING
yiclding
I"n(t)
rllrt'rt' thc stoarly
-
F,, 1 plJulrTll(',,
arrcl tlro lluctuating parts
F,,:
WIND
(4.1.3\
of thc clrag lirrce are, respectively,
lpo2cotu,
t
u\Ol
(4.7.4a)
:l!ltl
(c) sucrroN
oPENTNG,r,
Fo: pUu(t)yz'Co
?t_.il't:iro:,X:*.
FIGURE 4.6.8. Mean internal pressures in buildings with various opening distribu
tions. From H. Liu and P. J. Saathoff, "Intemal Pressure and Building Safety," ./.
Struct. Div., ASCE, f08 (1982), 223*2234.
the body mass. One can then visualize it as specifying a hypothetical mass
which, given the acceleration dUldt, accounts for the net force due to all thc
variously accelerated fluid elements in the entire flow around the body' In most
flows of interest in wind engineering. however, the entire term containing C.',,,
contributes only a negligible part to Fp. For this reason it is usually neglectctl
in this context, and it is not retained in what follows.
A three-dimensional flow will have three components, U(t)' V(t), and W(l),
in three mutually perpendicular directions. In the neutrally stratified flows* ol'
strong interest to wind engineering the mean wind velocity -U is horizontal,
and the wind then can be represented as the sum of mean and fluctuating
I'rrrrrr litl. 4.1 .4b it is seen that Fp(t) varies directly as u(t). This is true ro a
lrrr,r ;rpPtrxirnation only, since observation of physical flows reveals that cp
irrirv ilscll'also vary as a function of the frequency components of ,r(/).
Irr orrlcr to cxamine the statistical characteristics of Fb@, it is useful to
,n"r(l('r' ils spcctral density Sro@).one first calculates its autocovariance functirn (s('(' Appcndix A2., Eq. A2.21):
,
111,,(r)
:
W(t) :
V(t)
u(t)
l,
11.(), irtttl
('hitplt'l 2' 1l .l.l
FbG)Fb(t
+ r)
I
prurFilcrpe4t + n
(4.1.s)
rrlllr t'
S6,(n)
r.'\1r1rt'rrrlix
: t
J*
.\'1,;,(rr)
w(t)
Rp,{r;cos 2trnr dr
(4.7.6)
A2, Eq. A2.20), it follows that
(4.7.2t
the means of u, u, and w being zero.
One may then express drag in the horizontal direction by means of Fq.4'7 'l
with U(r) as in Eq. 4.7.2. In general, when time-varying vclocities arc llttts
introduced, the imperfect spatial correlation of thc vckrcily llucttutti(tns Itrtlsl
also be considered. Howcvcr, hcrc it is first assutnctl lhtrt lhc lrtxly in tlttcsliolt
*Sco Clt:rptcr
1fr/2
-' ,l'i f ) ,,rnb{,lnptr + 11 dt :
:
components.
U(t):D + uO)
@.7.4b)
:
p2O2
F
C2rS,1n1
(4.7 .7)
f trr rrlrrrlr rlris through hy (t,p(/; lll)'' yie:ltls rhc spectral density sc, of the fluctn,rtrrl' rlr':tg cocllicicnl :
,!,,,{rr) 't. ;, 't;l'l'
(4.1 .8)
llrr" t'rlttitliott will l.ltr vitlitl ovr't lltl t;rn1:t',r! lrt'rlrrt.rrt'it's ol',\',,(l) prrrvitletl lrll
l\ rr'ntitin ;x'rli.ctl.y cot!rlirlr.rl in .ui!iuuri.{l ;rlrovt'_ llowcver, ltt.r.ttttst. in
r llr'r
182
ttt ut
llll ilt lAil{)| (}t ilMt vnl tylN(i l()l t( t!, lil wtt.Jtt vt t(t{.ilv
4I
I tt()l)Y Al li()l )YNAMlcl
practice this assurrrptiorr tkre:s rrol lroltl, il is rrsturl (o ittcltttlr-: itrt lttlitrsllttcttt
factorto preservc thc validity ol'l')t1. 4.7.t'|.'l'his is tlonc by writirrg (4.7.11) as
56r(r)
- ,\..(rr ) t(r)
4Ci, :';',;' \
-
(4.1.e)
UJ
o
where the newly introduced factor y21r1 is termed the aerodynamic admittance*
of the body in question and represents a modifying adjustment (fbr an actual
body) of the ideal case of a body enveloped by turbulence with full spatial
F
F
correlation. This modification brings the drag coefficient spectrum into alignment with actual conditions.
Thc aerodynamic admittance is a function of body shape and dimensions
ancl of the characteristics of the turbulence. For a given body it is thus a
tiequency-dependent function. Figure 4.7 .l l4-3slsuggests the manner in which
=
z
z
o
=
o
o
o
(E
UJ
12(n) varies for a square flat plate placed normal to a turbulent flow with
uniform mean speed. The decrease of the aerodynamic admittance with increasing frequency corresponds to the fact that the smaller turbulent eddies
have shorter wavelengths; thus those eddies with higher frequencies will suffer
loss of coherence more rapidly than do the large eddies. References [4-36] and
[4-37] appear to be among the earliest to have introduced and used aerodynamic
admittance concepts in buffeting problems.
4.7.2 Relation of Wind Pressures over Slender Buildings to Wind
Velocities
The type of arguments employed in Sect. 4.7.1 in relation to total drag forces
is now applied to the case of a high-rise building of rectangular plan form,
with the horizontal wind blowing normal to one face. In this instance, the
along-wind structural motion is dependent on the windward and leeward pressure distribution in a manner that is conceptually simple.
The pressure acting at a point Q of elevation z on the surface of such a body
in a steady flow of velocity U(z) may be expressed as
p(Q)
:
ip(I2tz'tCo<Qt
- pQ) + p',(Q, r)
(4.1.1t)
.141.
to-r
I
to
^B/U
,'1 ll'irul-Sensitive Stuctures, Construction Industry Research and Information Associ,rrrrrrr, l-ondon, 1971, pp. 42-48. By permission of the Director of the National physical
l;rl)orirtory, U.K., and the Director of the Construction Industry Research and Inforrrr.rtiorr Association,
rvlrt'rc
2
and
p'
U.K.
have the following values:
pe)
p'(Q)
: I
-rrlr+-s!l
I It_(z)l
rpcptetu'k
:)
4! . okfi{n)
oc,Q)u'alz
(4.7.12)
ro,
,r,,
rvlrcre overbars indicate the mean values.
'
A bricf numerical example is in order here. In the atmosphere u21z1t/2 =
rrnd U(z) :2.5u*lnl(z - 2.,1)lz1l (see Eqs. 2.2.18 and.2.3.2 and Table
Forcxample, il'1,, - 0.03 rrr, ?.,1 O, and U1 l0) :40 m/s atz : 50
=
'jrr,r,
,,,' ',
rl'(':)
rt't.t
*Thc ttso ol'tltis letttt irr wirrtl trttgitttrctittg is ltlt cxtcttsitllt ol lls otif itt:tl ttsc itt :rt'trtttltttlicitl
conlcxts l4
lo-2
l.'l(lllRIl 4.7.1. Aerodynamic admittance of a square plate in turbulent flow. After p.
W llcarman, "Wind Loads on Structures in Turbulent Flow," in The Modern Design
(4.7.10)
where p is fluid density and Co is the appropriate pressure coefficient at this
point.
In the case of unsteady flow U(z) : U(<) + u(2, r) the pressure may be
approximated by
p(Q, r)
to-3
',o
()
()llt
lltitl lltc cltrrt ilr rrr-:glct'lirrli lltt' trorrl!rrr':rr lt'rrrr rrr litl .1.7. ll is lcss lhln ),X,.
184
lll ut
I u()t)y
nt n()t)YNnMt(;li
4/
:
)pD'Q)c,,(Q)
(4.1.t4)
p'(Q, t)
:
pOQ)u(Z, t)Cp(Q)
(4.1.1s)
as in the case of many buildings, the horizontal dimensions of the body
are small compared to the scale of turbulence, it is reasonable to assume that
thc fluctuating pressures affecting along-wind response-which consist entirely
ol'thosc on the windward and leeward faces-may be given by
If,
p,(e) :
pU(z)u(2,
t)C,(e*)
(4.7.16)
:
pry1a1u(2,
t)C{e)
(4.',l
p,(e)
where Q", and
.r1)
Q1 are points on the windward and leeward faces, respectively,
and where
,\'j,,,,,t1t,. Q,.
rr)
,tf,,,,tr'. tttN(ttl
(4.1.21t
whcrc S,f,,,r(r, n) is lltr: ('() sl)octrum <lf thc lorrgitutlirrll vckrcity lluctuati()ns .lt
lxrints Q1 and Q2 (Qj bcing thc pro.jection ol'Q, orr u planc. nonnal to the mcan
wind direction, that contains Q), and r is thc clistancc bctween Q1 and Qi.
'l'hc f'unction N(n) is ref'crrcd to as the along-wind cnrss-correlation
coefficient.
11' B1 and Qz are contained in the same vertical plane normal to the mean
wind (i.e., if their along-wind separation is zero), then N(n) = l. For nonzero
rrlong-wind separation, an expression of N(n) is given in Sect. 4.7 .4. In the
t'rtsc Q1 = Qz,
so,(Qr,
d:
c3(Q)p',O2k)s,(2,)
4.7.3 Pressure Fluctuations on the Windward
(4.7.22)
Face of Bluff Bodies
n
theory of turbulent flow around two-dimensional bluff bodies has been dein 14-391, which has subsequently been applied in [4-40] and [4-4ll
to the study of surface pressures generated by turbulent velocity fluctuations.
'l'hc theory is based, essentially, on the following assumptions: (l)
the turbuIcnce intensity is of the same order of magnitude as, or lower than, the turlrrrlence intensity typical of atmospheric flows, (2) the body is sufficiently long
{lrat end effects may be neglected, (3) in the flow region upwind of the body
:rrry velocity fluctuations induced by wake flow are statistically independent of
thc velocity fluctuations caused by the oncoming turbulence and so the latter
t'irn be studied separately. Fundamental to the approach of [4-391 is a generrrlization of "rapid-distortion theory" which allows the linearization of the
er;uations describing the turbulent motion near the upstream face of the body.
'I'his linearization follows from the assumption
that the changes in the mean
llow associated with the presence of the body distort the turbulence sufficiently
r:rpidly so that, during the distortion process, the nonlinear inertial transfer of
cncrgy between eddies of different sizes is negligible.
The equations for the turbulent nondimensionalized vorticity vector a,)i G :
| . 2, 3) are then [4-39]
vclcrped
c,(Q*):
#+
(4.1 .18)
C/Q):ffi
(4.7.1e)
where z is the elevation of point Q, or Q7. As discussed in Chapter 9, it is
usual in current procedures for estimating along-wind building response to
assume that Eqs. 4.7 .16 and 4.7 .17 are valid regardless of the ratio of building
transverse dimensions to the scale of turbulence. This point is brought up again
later in Sects. 4.7.3 and 4.7.4.
In calculating along-wind structural response (see Chapter 5) information is
required on the spatial correlation of pressures applied at any two points Q1
and Q2. Such information is supplied by the co-spectra of fluctuating pressures
(quadrature spectra being assumed negligible). Assuming the validity of Eqs.
4.1 .16 and 4 .7 .17 , the co-spectra take the form
S,',i,(Q,,
tB5
'l'ltc co spt't'tnrrrr,\f,,,,, rrury lrc cxllrcssctl irr llrc lollowilrl' 1,,t"t'
Morc gcncrally ancl anuklgotrsly lo llrr'rllrrg tcsttlls lrlrcatly tliscttssetl irt Scct.
4.'7.1, calc'J,lations reportcd in l4-.ltil itrtlit'irtt: tlrirt (hc lirlkrwing tclations arc
satisfactory, with insignificant erK)r, lirr 7r lttttl 7r':
pQ)
lllr nt tAill )ft()t ilMt vnttytN(it()t ll:t t; t(lwll.lt )vt t()(]ily
Qz, n) :
co(Q)cp(Qr)p'DQ)u(z)sc,,,,(Qr,
Qz, n)
D6,
(4'7 '20)
That is, the co-spectra of the pressures are proportional to the co-spectra of the
fluctuating longitudinal wind components in the undisturbed oncoming flow at
the elevations of the two points. The pressure coefficient C,IQi) rcpresents
windward or lccwarcl vatlucs dcpcnding upon whcthcr lhc poirrl Q, is on the
windwarcl <lr lccwitnl sitlc.
Dt
.i
\fi,
+ cdr^
:t "
dxj
(i:1,2,3)
(4.7.23)
wlrcrc l; (i : 1,2,3) is tlrc norrtlirncnsionalizcd velocity fluctuation vector
'I'hc nondimensionalizccl prcssiur'('7)' is givcn by
lF'
dl;
ittr,
=l
ilt
t"" , ,, t"4\
)i
/ rrI
t)\t
' ilt,/
r r \
(i
r. 2.
3)
(4.1.24\
186
lil t,t
wherc Oi
I tt()t)y
nt n()t )yNnMtcli
4t lilt ilt tnit()N()t ilMt vnnylN(il()l
the high-frequency pressure fluctuations is somewhat greater than the coherence
of the high frequency velocity fluctuations in the undisturbed flow. In structural
engineering computations this "piling-up" effect can be taken into account by
choosing appropriately small values of the exponential decay coefficients in
Eq.2.3.29 or 2.3.30.
4.7.4 Pressure Fluctuations on the Leeward
c
;
*An irrotational flow is one in which the components
,,:#
y,,,,_y,_
.ti;q
_*,
".:y,_y,
-' - ^ -' -/zv-- -
fP-
50.0 7s.0 100.0 125.0
25.o
Time
1s0.0
(s)
lfl(;uRE 4.7.2. Yaiations with time of wind pressure on the windward and on the
k'cward wall of a building. After w. A. Dalgliesh, "statistical rreatment of peak Gust
,rrr Oladding." J. Struct. Div., ASCE, 97 (1971),2173-218j.
forthe function N(n) in Eq.4.7.2l.In
lr.y choosing an appropriate expression
l.l-45| an expression for this function has been proposed of the form
N(n)
:
t
ll
E-
'1(r
15.4nA,x
U
- e-'El
(4.1.2s)
(4.',|.26)
wlrcre u is the mean wind speed at elevation (213)H; Ar is the smallest of the
tlirrrensions B, H, and D; B is the width; F1 is the height, and D is the depth
.l'the prismatic body. Full-scale and wind tunnel measurements reported in
l,t-461, [4-471, t4-48], and [4-49] suggest that this expression is adequate for
pnrctical use.
4.7.5
Peak Local Wind Loads
'I'hc adequate design of roof members, roofing,
cladding, and other elements
srrsccptible to failure due to the local action of wind (e.g., solar collectors
l'1 601) is of foremost impoftance for reasons of both safety and economy. It
is thcrcfbre desirable that wincl-inch.rccil loads on such elements be ascertained
:rs roalistically as possiblc l4-591.
lAccording to
l4-(r71, ltowt:vcr, ul any givon Ircqucncy thc prcssun':rrrrl vchx rlv rpt.r'trr lurve lhc
sanrc skrpc.
--
0.0
Face of Bluff Bodies
however, that the pressure fluctuations on the leeward side are less strong than
indicated by Eq. 4.7 .17 (e.g. , see Fig. 4 .7 .2 taken from t4-501). It is reasonable
to assume, therefore, that the use in design of Eq. 4.7.17 is conservative from
a structural safety viewpoint.
Also of interest for design purposes is the question of the extent to which
pressures on the windward side of a building are correlated to the pressures on
the leeward side. It is intuitively clear that this correlation cannot be perfect.
The correlation will be greater for eddies with large wave lengths-which can
be thought of as enveloping the body in the same manner as the mean flowand will decay as the wave lengths decrease. This dependence can be expressed
zer<r.
t87
o
d
According to Eqs. 4.7.16 through 4.7.19, the ratios of the pressures on the
leeward face to the pressures on the windward face are the same for both
fluctuating and mean pressures. Results of full-scale measurements suggest,
all
:, t()wtNttvt t(x]ty
(i : l,2,
3) is lhc trotttlitttt'rtsionirlizcrl rrrcirn vclocily vc(.[or, wlrosc
field is approximately irrotatiotrirl.* 'l'lrt' lrorrrrtlirry conclitions arc csscntially
the following: (l) at large distanccs lirrrrr (hc b<ily, thc velocitics approach
their values in the undisturbed flow ancl (2) in thc immediate vicinity of the
upwind surface of the body, the velocitics at, cach point are perpendicular to
the outward normal from that surface.
Calculations carried out on the basis of the above equations and boundary
conditions suggest, for example, that whenever alL" < I (where a is the typical
horizontal dimension of body and I is the longitudinal turbulence scale), Eq.
4.7.22 is applicable, on the windward face, up to frequencies r,
= 0.15 Ula,
where U is the mean speed of the undisturbed flow 14-411. For higher frequencies the pressure spectra decay more rapidly than the velocity spectra so
that, for structural design purposes, Eq. 4.7.22 is conservative. That this is the
case appears to be confirmed by experimental results reported in t4-421,
t4-431, and [4-44].r Calculations also suggest that the smallei eddies are "piled
up" against the upward face of the body and therefore that the coherence of
are
t(:l
'l'hc clcrncnts potcntially ittvolvctl irt lirilurcs duc to local wind loads are
trstttrlly rclativcly rigitl so llurl llrt'tlyrr:urric lrrrrplilication ol'thc rcsponsc is
ltcgligiblc. Thc witrcl krlrtl rrclirrl'. ort lrrt clt'rrrt'nl is tlrr:rr cqr.ral to llrc sulrr, ovcr
lltt'crttirc itrcit ol'thc e:lctttcttl. ol llrc inslrrrrllrrrt'orrs l)11:ssutcsi intlrrccrrl by wirrtl.
I)trring cvcly sl()rttt llris loittl tt':tr'ltr's;r pritlr. lltt'clcrrrr:nl coltccrrrrctl, tirrtl its
188
uttl I
connections, must bc closignccl lo srrsllirr lhe llcuk wind load attaincd cluring
the N-year storm, whcrc N is thc lttcan rccut'rotlcc intcrval of the dcsign wind
speed specified for that element.
The total wind force acting on an element such as a roof member or a curtain
wall could, in principle, be measured directly. However, the experimental setups required for such measurements are prohibitively expensive and impractical. For this reason forces acting over an element have recently been measured
by devices that automatically add pressures occurring simultaneously at several
points of the element, weighted by the respective tributary areas. In particular,
such techniques have been used at the University of Westem Ontario to measure
wincl l<rads <rn models of low-rise structures 14-71, 4-72, 4-'13, 4-74, 4-75,
4-16,4-771. These measurements, as well as results of full scale tests, [4-51,
4-1t1,4-79,4-801 have been used to develop new design load provisions for
rnain f-rames and for parts and portions of low-rise buildings that have been
recently incorporated in various standards, including [2-1391; see also Sect.
9.5. Local pressures can have strongly non-Gaussian distributions, especially
at comers and edges; see [,4.2-13].
4.8
Atlt)t Nt){ tM
B()DY nl not)YNAM|(;li
l Bo
rcllcctctl witttl is slroufl('r llrrrrr rlit'ccl wirrtl, irrrrl llrt'rrron'ri() lrs ()lrc is cklscr to
tlrc obslaclc lhrrl rclle't'ts rl. I lravc cxpc:rir-:rrcctl llris ir rrrrrrrhcr-ol'tirncs by
irppnlaching l l()wcl'tlurt is alrrxrst l(X) lbcl lrigh irrrtl is situatccl at lhc nortlr
t'nrl ol'tny ganlcn in Monlb:rrtl. Whcn a stnng wintl l'rkrws lhrrrr tlro s<lutlr, up
Io thirty stcps lirrur llrc towcl onc fbcls stnrngly pushetl, alicr which thcrc is
:rn interval ol' livc ol six stcps where one coascs to bo pushcd and where thc
wind, which is rcllcctcrl by thc tower is, so to spcak, in equilibrium with the
tlircct wind. Aficr this thc closcr one approachcs tho tower, the more the wind
re llccted by it is violcnt. lt pushes you back much more strongly than the direct
wind pushed you forward. The cause of this effect, which is a general one and
t'rrrr be experienced against all large buildings, against sheer cliffs, and so forth,
rs not difficult to find. The air in the direct wind acts only with its ordinary
s1rccd and mass; in the reflected wind, the speed is slightly lower, but the mass
is considerably increased by the compression that the air suffers against the
obstacle that reflects it, and as the momentum of any motion is composed of
thc speed multiplied by the mass, this momentum is considerably larger after
I rc compression than before. It is a mass of ordinary air that pushes you in the
lilst case, and it is a mass of air that is once or twice as dense that pushes you
b;rck in the second case.
I
SECONDARY WIND FLOW EFFECTS
In addition to the wind loads caused by the direct action upon the structure of
the wind flow, it is of interest in certain situations to examine secondary effects
produced by wind, such as the blowing of roofing gravel [4-58,9-63], and the
drifting of snow. Systematic studies of these effects have been reported in
14-63,4-&, 4-65,4-66, 4-81, 4-82, 4-83, 4-841.
Mention is also made of wind action as a factor that influences the energy
consumption of buildings by increasing air infiltration. It is shown in [a-68]
that energy losses due to wind-induced air infiltration can be reduced significantly by the sheltering effect oftrees acting as wind breaks; the energy savings
thus achieved may in certain cases be as high as l5%. The results of [4-68]
were obtained in wind tunnel tests and were subsequently confirmed by fullscale measurements [4-69].
ADDENDUM
For the sake of its historical interest, we reproduce here a note by Count Buffon
describing the flow changes occurring upwind of a tower, for which it offers a
charming (if scientifically no longer tenable) explanation. A translation of the
note follows.
d I'Hitloire
ADDITIONS
A l'Article gui a pour titre: Des
Vents riglis, page zz4.
I.
Sur Ie Vent riflichi, pege 2+2.
T
J r': oors rapporter ici une obfervation
.;,ri rne paroir avoit 6chappC I l'attcncion
Phyficiens , guoique tour le rnonde
'l<'s
li'it
en drat de la viLifer ; c'eft que le
vc'rr riflichi eft plus violent que Ie vent
rlrrcdt, & d'autant plus qu'on eft plus
1,,rt's de I'obftacle qui le rerrvoie. J'en
ri llir nonrbre de fois I'expirience , en
.rl,prochant d'une tour qui a prls de
, cnr pieds de hauteur & qt-ri fe trouve
lirtrdc au nord , ) l'extrCrnirC de rnon
i.rrrlin, I Montbard ', lbrfqrr'il lirrrfflc
rrrr grand venr du nridi, on lc lcrrr lirrr('nrent pouflc iulqu') trcnte p:rs rlt' l:r
r,"'r i ry,ris <1uoi , il y a trn irrrcrv.rllc ,h.
tirrtg t-ru
On Reflected Wind
I must rcporl hcrc lrrr obsr:rvlliorr which it sccnls to Ittt'ltits t'st'rt1tt'tl lltt'ltllcttiion
ol'plrysic'ists, cvt'rr llrorrglr cvt'r'yorrc is in ir posiliott lo vt'ttly rl. ll st't'rrts llutt
Nanrelle. t t
lix
pas
I oi l'on cclli rl'trre
t6
Suppldment
ooufG & or) Ie veur, qtri eft r6flichi par
la tour, fair , pour ainfi dire , iquilibre
avec Ie vent direCt ; aprls cela, plus on
approche de Ia tour & plus Ie vent qui
en eft riflechieft violenr, il vous repou(e
en arriire avec beaucoup plus de force
que le venr direC! ne vous poufibit en
avant, La caufe de cer efter qui eft gin6ral , & dorrt orr peut faire l'ipreuve
contre tous les grands bntinrens, conrre
les collines coupies ) plornb , &c. n'eft
pas difficile I rrouver, L'air dans le vent
dire& n'asir qu€ par fa vireffe & [a
nra(fe ordirraire ; dans Ie vent rdfichi,
la vite(G eft un peu dirninude
,
nrais Ia
eft confiddrablemenr augmerrrie
par la cornprefiion que I'air fo'rftie
rna{Ie
contre l'obftacle qui Ie rdflCchir ; &
comnre Ia guantird de tout rrrouvelrlent
eft compolie de Ia vire(G
nrulripli6e
par Ia nraflb , cette quantird eft bien plus
grande apris Ia courprefiion gu'suparavanr. C'eft unc nra(G d'air ordinaire, qui
vous pouffb dans le prenrier cas, & c'eft
ttrrc rnafG d'air une ou deux fois plus
, U"t vous repcu(G dans le lccond
lt:;:t"
()l n()le {)rl rcllcclcrl wiilrl l,trtttt llt\lt,ilr Nttlutrllt', (ir,ttr,tttlt tt l\trticuliin,, ('(ilttriltutt l(.\
lilrtlttt'.t tlr Itt Nttlurt, Itrrr M. lc lrtttlr rlr llrllltt, Vol I l. lttlcrtrlirrtl rlrr .l:rrrlirt t^l tlrr (':rltitrel rlu l{oi,
rk' l'Atrtrlcrilir liritrrr,'irist', rk' r'clle rli'r :ir t.ilr.r r=lr lo[rr' lrIrrrt'rrrr'. A l'irris, l)r' l,'lrrrPrrrrrrre
I rrr'silrrilc
Itoyrrlc, I
//ll
190
tJt
t,r
I tKltlY
ilt IItil
nt ti()t )YNnMt(;:;
REFERENCES
4
l.|
L. M. Milne-Thompson, 'l'hutrctittl llvlnnlvrutmir'.r', Macrnillan, Ncw York,
r
96-5.
4-2 H. Rouse, Advanced Mechanics tt lluids, Wilcy, 1958.
4-3 G. K. Batchelor, Fluid Dynamics, Cambridge Univ. Press, Cambridge, 1967.
4-4 V. L. Streeter, Fluid Mechanics, McGraw-Hill, New York, 1966.
4-5 H. Bdnard, "Formation de centres de giration ir I'arribre d'un obstacle en mouve4-6
ment," C.r. Acad. Sci.,147 (1908), 839-842.
Th. v. Kiirmiin, "Uber den Mechanismus des Widerstandes den ein bewegter
Kcirper in einer Fltissigkeit erfdhrt,'' in Nachrichten der Kdniglichen Gesellschaft
r Wi s s e ns chaft en, Giitttngen ( 1 9 I 1 ), 509-5 I 7.
de
4-1
4-8
G. A. Dobrodzicki. Flow Visualiz.ation in the National Aeronautical Establishment's Water Tunnel, Aeronautical Report No. LR-557, National Research
Council of Canada, Ottawa, 1912.
V. Strouhal, "Uber eine besondere Art der Tonerregung," Ann. Phys., 5 (1878),
216,250.
L. R. Wooton and C. Seruton, "Aerodynamic Stability," inThe Modern Design
of Wind-Sensitive Structures, Construction Industry Research and Information
Association, London, 197 l, pp. 65-81.
4-10 "Wind Forces on Structures," Trans. ASCE, 126, Part II (1961), 1124-1198.
4-11 NASA Skylab II Photo 73-H-9@, 73-HE-777, released Oct. 2, 1913.
4-9
383.
4-16 B. J. Vickery, "Fluctuating Lift and Drag on a Long Cylinder of Square Crosssection in a Smooth and in a Turbulent Stream," J. Fluid Mech.,25, Part3
. ('ottlt'ltt'sttt' ;ttrrl
and Applied Mathematics, University of Maryland, College Park, June 1972.
4-20 W. D. Baines, "Effects of Velocity Distribution on Wind Loads and Flow
Patterns on Buildings," in Proceedings of the Symposium on Wind Effects on
Buildings and Structures, Vol. 1, National Physical Laboratory, Teddington,
U.K., 1963, pp. 191-223.
4-21 J. H. Gerrard, "Thc Mcchanics of thc Formation l{cgion ol'Vorticcs Bchind
Blull'Boclics," J. l''luid Mach. 25 (1966), 401-41.1.
4-22 A. Iloslrko. "lixpt'rirrrcrrls orr lltc likrw l)lrst lr ('ttt'ttl;rt ('vlttttlt't :rl Vt'r'y lliglt
Itcynoltls Ntttttlrct." .l . I"lrtirl Mt'rlr. lll ( l(X)l), .\'15 lt(r.
lltt
A l ;rrrt'villc, "A
livitlrrrrlrott ol l)rr1i ('ocllicicrtl Mcirsurt'rrrr'rrl:
Cambridge,
4
27
U.K.,
1977.
J. A. Robertson, C. Y. Lin, G. S. Rutheriord, and M. D. Stine, "Turbulence
Effects on Drag of Sharp Edged Bodies," J. Hydr. Div., ASCE, 98 (Jdy 1972),
1187-1201.
4-28 B. E. Lee, "The Effect of Turbulence on the Surface Pressure Field of a Square
Prism," J. Fluid Mech. 69 (1975),263-282.
4-29 R. M. C. So and S. D. Savkar, "Buffeting Forces on Rigid Circular Cylinder
in Cross-Flows," J. Fluid Mech. 105 (1981), 397 425.
4-30 T. M. Mulcahy, Design Guide for Single Circular Cylinder in Turbulent Crossflow, Technical Memorandum ANL-CT-82-7, Argonne National Laboratory,
March 1982.
4
3l
O. Flachsbart, "Winddruck auf geschlossene und offene Gebtiude," in Ergebnisse der Aerodynamischen Versuchanstctlt zu G)ttingen, L. Prandtl and A. Betz
(eds.), Verlag von R. Oldenbourg, Munich and Berlin, 1932.
Jensen, Shelter Effect Investigations, Danish Technical press, Copenhagen,
4-32 M.
1954.
4-33 M.
Jensen and N. Franck, Model-Scale Tests in Turbulent Wind, Danish Technical Press, Copenhagen, 1965.
,4-34 W. R. Sears, "Some Aspects of Nonstationary Airfoil Theory and Its Applications," J. Aero. Sci., 2 (1941), 104.
,1-35 P. W. Bearman, "Wind Loads on Structures in Turbulent Flow," in
The Modern
Design of Wind-Sensitive Structures, Construction Industry Research and Information Association, London, 1971, pp. 42-48.
(1966), 48r-494.
4-17 P. Sachs, Wind Forces in Engineering, 2d ed., Pergamon, Oxford, 1978.
4-18 Swiss Code of Practice, No. 160, Article 20, 1956.
4-19 G. E. Mattingly, An Experimental Study of the Three-Dimensionality of the Flow
Around a Circular Cylinder, Report No. BN295, lnstitute for Fluid Dynamics
lgI
'l'hc Arrrcrit'lut Sot'rt'ly ol Mechaniclrl l,)rrgilrecrs. Nt.rv York.
4-24 O. Giivcn, ('. lrirlcll. lrlul V. C. Patcl, "sulllrt.c llorrl-',ltncss llllccts on thc Mcln
Flow Past ('ircrrlirr ('ylintlcrs," J. Iluid Mtth. gtl (19u0), 61 1-7Ol .
4-25 P. W. Bcarrrrittt, "Alt lnvcstigation ol'tlrc liolcos orr Iilat Platcs Normal to a
Turbulent Flow," .l . lluitl Mach. 46 (191 l). 177,198.
426 A. Lanevillc, I. S. Gartshore, and G. V. Parkinson, "An Explanation of Some
Effects of Turbulcncc on Bluff Bodies," Proceedings Fourth International Conference, Wind ElJects on Buildings and Structures, Cambridge University press,
4-12 Weather,3l, l0 (Oct. 1976), 346.
4-13 A. Roshko, "On the Wake and Drag of BluffBodies," J. Aeronaut. 9ci.,22,
2 (1955), 124-132.
4-14 S. F. Hoemer, Fluid-Dynamic Drag, published by the author, 148 Busteed Drive,
Midland Park, NJ, 1965.
4-15 C. Scruton and E. W. E. Rogers, "Steady and Unsteady Wind Loading of
Buildings and Structures," Phil. Trans. Roy. ,Soc., London L269 (1971),353-
I,;
('otttplrisorr ol ('olt'r'liolr Mcllrrxls Iist.rl
lor I wo l)rrrrt.rrsrorurl ltt.t.l
altgulitr('ylitrtlt'rs," l';rpct No.79-Wn/lrli .1. Arlrrrlrl Mt.t.lirr1i, l)cc. f '/, l()7(),
.1
itt
4-l
tJ(
't-36 H. W. Liepmann, "On
'I'31
,1
lU
the Application of Statistical Conceprs to the Buffeting
Problem," J. Aero. Sci., 19, 12 (1952),793-800, 822.
A. G. Davenport, "Buffeting of a Suspension Bridge by Storm Winds," -/.
Struct. Div., ASCE, 88, No. 5T6 (1962), 233-2&.
R. Vaicaitis and E. Simiu, "Non-Linear Pressure Terms and Along-Wind Response," J. Struct. Div., ASCE, 103, No. ST4, Proc. Paper 12837 (1977),903,
906.
l.l9
J. C. R. Hunt, "A'l'hcoly ol 'l'urbulcttl Iilows Round Two-Dimensional Bodics," ./. Fluid Mcclt., (rl, I'rrr1 .l (l()7 l). (r2.5 7(Xr.
I '10 J. C. R. Htrnl, "A'l'ltcoty ol lilttt'ltt;rlirrli l'r.cssrucs orr Illull'llorlics in'l'rrrbtrlclrt
lil<rws," in Itnx't'ulirt,ti.t rtl lltt' ll l l'..lII l..lllll ,\t'rrtltolitrttr ttrt lilol' lrulrttt,tl Slrtrt'
luntl Viltnttitnt,r, l(lrtlsrttltr'. Wr'r,l ( it'rrrr;rrrv, l()'/1. li. N:rrrrlirsclrct (c(1.). ,Sl)t.ill
gt'r' Vclllrg, lle llirr, l()/,1. pp lrr() -'O l
192
nt
tJil
D()t
lll lllll
)y At n()l )yNnMt(;l
4-41 J. C. lt. Hurrt, "'l'ulhulcrrt Velocities Nr:r'llttl lilttcluirlittg Srrrllrce I'rcssrrrcs
on Structures in'l'ulhulcnl Wintls," irr I'nxvnlitrg.s tl tlrt Iirurtlr lnttnuttiorrttl
Conference on Wind F)ffccts ttrt lluiltlirr,qs tuul Strudur(s, l,undon, 1975, Carnbridge Univ. Press. Cambridgc, 1976, pp. -109 320.
4-42 B. J. Vickery and K. H. Kao, "Drag or Along-Wind Rcsponsc ol' Slcnder
Structures," J. Struct. Div., ASCE,98, No. STl, Proc. Paper 8635 (1912),
21-36.
4-43 P. W. Bearman, "Some Measurements of the Distortion of Turbulence Approaching a Two-Dimensional Body, J. Fluid Mech.,53, Part 3 (19722),451461.
4-44 R. D. Marshall, Surface
Pressure Fluctuations Near an Axisymmetic Stagnation
Pr.,lnl, NBS Technical Note No. 563, National Bureau of Standards, Washington,
DC. r97l
4-45 J. Vcllozzi and E. Cohen, "Gust
.
Response Factors," J. Struct. Div., ASCE,
94, No. 5T6, Proc. Paper 5980 (1968), 1295-1313.
4-46 R. Lam Put, "Dynamic Response of a Tall Building to Random Wind Loads,"
Pnx'ccdings of the Third International Conference on Wind Effects on Buildings
arul Structures, Tokyo, Japan, 1971, Saikon, Tokyo, 1972, pp. 429-440.
4-47 H. van Koten, "The Fluctuating Wind Pressures on the Cladding and Inside a
Building," in Symposium on Full-Scale Measurements of Wind Effects on Tall
Buildings, University of Western Ontario, London, Canada, 1973.
4-48 J. D. Holmes, "Pressure Fluctuations on a Large Building and Along-Wind
Structural Loading," J. Ind. Aerodyn., l, | (1975), 249-278.
4-49 K. H. Kao, Measurements of Pressure-Velocity Correlation on a Rectangular
Prism in Turbulent F/ow, Report No. BLWT-20, University of Westem Ontario,
London, Canada, 1970.
4-50 W. A. Dalgliesh, "Statistical Treatment of Peak Gust on Cladding," J. Struct.
Div., ASCE, 97, No. ST9, Proc. Paper 836 (1911),2173-2187.
4-51 R. D. Marshall, "A Study of Wind Pressures on a Single-Family Dwelling in
Model and Full-Scale," J. Ind. Aerodyn., 1,2 (1975), 177-199.
4-52 G. A. Euteneuer, "Druckansteig im Inneren von Gebduden bei Windeinfall,"
Pressures Induced
4-54 H. Liu and P. J. Saathoff, "Internal Pressure and Building Safety," J. Struct.
Div., ASCE, 108 (19822), 2223-2234.
4-55 P. J. Saathoff and H. Liu, "Intemal Pressure of Multi-Room Buildings," -/.
Eng. Mechs. Dlv., ASCE, f09 (1983), 908-919.
4-56 R. L Harris, "The Propagation of Internal Pressures in Buildings," J. Wind
Eng. Ind. Aerodyn.,34 (1990), 169-184.
4-57 B. J. Vickery, "Intemal Pressures and Interactions with the Building Envelope,"
J. Wind Eng. Ind. Aerodyn., 53 (1994), 125-144.
4-58 J. E. Minor and L. W. Beason, "Window Glass lrtilrrrcs irr Wirrtlstornrs," J.
Struct. Div., ASCE,, 102, No. STl, Proc. Papcr lll3.l4 (l()7()). 1,17 l6O.
4-59 "Hanc<rc:k (illrss l}clrkagc: A ('ornbination ol' littrrs'f "
Rtutnl (Miry l()76), ().
l'.rr.t,ittct't
itt,q
Ncu.,l
103
60 ll. |. Mclk'rur. "Wrrrrl l,orrrl lilli'c(s on l;lll l'irrlr Sol;rt ('ollt't lots." .l ,\trttr'1.
l,.lttg.,lll{lttxry. l,l t l.5l
,1 .61 "Ncw Appnxrtlrt's lo l)t'silr,rr Against Wirxl At'lion." irr A. (i. l)rrvcrrlxrr( (lil.),
Ctnrsc No/r,.r', 'l'lrr' llorrrrtluly Laycr Wirrtl 'l'rrrrrcl l.lrlxrlrrloty. tlnivclsily ol'
Wcstorn Orrtirrio, l,oruLrt, ('anacla, 197L
1-62 J. Blcssllun. Atnnlintinrirt dus Con,slrucor),s, Iililorit tlrr tJnivcrsiditdc, Pontr
Alegre, Brasil, l9tl.l.
,t-63 R. J. Kind arrtl l{. 1,. Wardlaw, Design o.l'Iilxtlir4ts A14,uinst Gravel Bktw-ofJ',
Repoft No. 1.5544, National Research Council ol'Canada, Ottawa, 1976.
4-64 R. J. Kind, "A Critical Examination of the Requirements fbr Model Simulation
of Wind-Induced Erosion/Deposition Phenomena Such as Snow Drifting," ,4rmos. Environ, f0 (1976), 219-227.
4-65 C. Mateescu and H. Popescu, Accumulations de neige sur les constructions,
4-66
Etude explrimentale sur mod?les, Annales de 1'Institut Technique du Bdtiment
et des Travaux Publics, S6rie EM, Pais, 1974.
J. Wianecki, Banc d'essais d'accumulation de la neige due au vent, Annales de
I'Institut Technique du B6timent et des Travaux Publics, Sdrie EM,
Paris,
1976.
4-61 W. Z. Sadeh and J. E. Cermak, 'lTurbulence Effect on Wall Pressure Fluctuations," J. Eng. Mech. Div., ASCE, 98, No. EM6, Proc. Paper 9445 (1912),
t356*1319.
4-69
and E. F. Peten, "Wind and Trees: Air lnfiltration Effects on
Energy and Housing," J. Ind. Aerodyn.,2 (1977), l-9.
G. E. Mattingly (personal communication, April 1977).
4-70
H. P. Pao and T. W. Kao, "On Vortex Trails Over Oceans," Atmos. Sci.,
4-68 G. E. Mattingly
4-7
|
4-72
by Wind," in
Wind Engineering, Proceedings of the Fifth International Conference, Fort Collins, CO, July 1979, J. E. Cermak (ed.), Vol. 1, Pergamon Press, Oxford, 1980.
:,
,l
Der Bauingenieur, 45 (19'70), 214-216.
4-53 J. D. Holmes, "Mean and Fluctuating Internal
l\l(,1
4-73
4-74
4-15
4-76
4-11
tl
18
Meteorological Society of the Republic of China, Taiwan, 3 (1976), 28-38.
D. Surry and T. Stathopoulos, "An Experimental Approach to the Economical
Measurement of Spatially Averaged Wind Loads," J. Ind. Aerodyn., 2, 4 (Jan.
1978), pp. 385-397.
A. G. Davenport, D. Surry, and T. Stathopoulos, 'Wind Loads on Low Rise
Buildings: Final Report of Phases I and II-Parts 1 and 2," BWLT Report SS81977, The Univenity of Westem Ontario, London, Ontario, Canada, Nov.,
t917.
A. G. Davenport, D. Surry, and T. Stathopoulos, "Wind Loads on Low Rise
Buildings: Final Report of Phase III-Parts I and 2," BWLT-SS4-1978. The
University of Western Ontario, London, Ontario, Canada, July, 1978.
L. Apperley, D. Surry, T. Stathopoulos, and A. G. Davenporl, "Comparative
Measurements of Wind Pressure on a Model of a Full-Scale Experimental House
at Aylesbury, England," .1. Ind. Aerodyn., 4 (1979),207-228.
T. Stathopoulos, "PDIi ol'Wind Pressures on Low-Rise Buildings," J. Struct.
Dlv., ASCE, 106 (l9lJ0). ()7.1 990.
T. Stathopoulos, l). Strrr.y,:rtttl A. (i. l)avcnport, "E,fl'ective Wind Loads on
Flat R<xrl.s." .l . Strutl. /)ir', AS('lr..l 107 (l9ltl),281 298.
'f'. Statl-roptrukrs, "Witul l,o;rrls ort lrrtvt's ol l.ow lltriltlings," .l . Strul. l)ir'.
AS('li, 107 (l9t'l l). l1).! I l()t'l
K. .l . liutorr trrrtl .l . l{. M;rVtr,'. Iltr' Mr';r:.rut'rnr'nl ()l Wintl l)ttssutt' ott 'l'wrt
Story llottscs :rt Aylrslrrrty."
.l lrnl
'1'.'tt\lttt
. | (l()75), (r7 l0().
194
ut ut
I l()t )y nt ll()t )yNnMt(it;
Witttl ltttrl.s ()n 1,.,lr'lli,st'Iiltiltlirt.q,t A llryit'w, ('Sll((), l)ivisiorr
of Building Rcscarch, Iliglrctt, Viclorirr, Austlrrliir, I91i3.
4-80 T. Stathopoulos, "Wind Loacls orr l.ow ltisc lluildings: A Rcvicw ol'thc Statc
of the Art," Eng. Struct., 6 (l9tJ4). ll9 l3-5.
4-81 J. T. Templin and W. R. Schricvcr, "l.oarls duc to Drified Snow," J. Struct.
Div., ASCE, r08 (1982), t9t6-t925
4-82 J. D. Iversen, "Small-Scale Modeling of Snow-Drift Phenomena," in Wind
4-19 J. l). llolrncs,
Tunnel Modeling for Civil Engineering Applications,
Cambridge Univ. Press, Cambridge, 1982.
4-83 R. J. Kind and R. L. Wardlaw, "Failure
T. A.
Reinhold (Ed.),
of Loose-Laid RoofWind Eng. Ind. Aerodyn., 9
Mechanisms
Insulation Systems (High-Rise Buildings),"
(1982), 32s-341.
J.
4-84 R. J. Kind and R. L. Wardlaw, "Wind Tunnel Tests on Loose-Laid Roofing
Systems for Flat Roofs, " Proceedings , Second International Symposium on Roofing Technology, National Bureau of Standards, Gaithersburg, MD, Sept. 1985.
4-85 I. S. Gartshore, "Some Effects of Upstream Turbulence on the Unsteady Lift
Forces Imposed on Prismatic Two Dimensional Bodies," J. Fluids Eng.,106
(1984), 418,424.
4-86 W. C. L. Shih, C. Wang, D. Coles, and A. Roshko, "Experiments on Flow
Past Rough Circular Cylinders at Large Reynolds Numbers," J. Wind Eng. Ind.
Aerodyn., 49 (1993), 351-368.
4-87 Y. Nakamura, "Bluff-Body Aerodynamics
and Turbulence," -/. Wind Eng. Ind.
Aerodyn., 49 (1993), 65-78.
"Perspectives on Bluff Body Aerodynamics,"
Aerodyn., 49 (1993), 79-100.
4-88 A. Roshko,
J.
Wind Eng. Ind.
4-89 A. Gadilhe, L. Janvier,
and G. Barnaud, "Numerical and experimental modeling
of the three-dimensional turbulent wind flow through an urban square, ' ' J . Wind
Eng. Ind. Aerodyn., 46-47 (1993),755-763.
4-90 S. Murakami (ed.), "Cunent Status of Computational Wind Engineering," -/.
Wind Eng. Ind. Aerodyn, 35 (1990), l-318.
4-91 S. Murakami et al. (eds.), "Computational Wind Engineering," J. Wind Eng.
Ind. Aerodyn., 46-47 (1993), l-912.
4-92 D.
CHAPTER 5
Laurence and J.-D. Mattei, "Current State of Computational Bluff Body
Aerodynamics," J. Wind Eng. Ind. Aerodyn.,49 (1993),23,44.
4-93 C. W. Letchford and J. D. Holmes, "Wind Loads on Free-standing Walls in
Turbulent Boundary Layers," J. Wind Eng. Ind. Aerodyn., 5l (1994), l-21.
4-94 Y. Nakamura and Y. Ohya, "Vortex Shedding from Square Prisms in Smooth
and Turbulent Flows," J. Fluid Mech., f64 (1986), 77-89.
4-95 A. Baskaran and T. Stathopoulos, "Prediction of Wind Effects on Buildings
Using Computational Methods-Review of the State of the Art," Canadian J.
Civil Eng., 2l (1994),805-822.
STRUCTURAL DYNAMICS
Structural dynamics is the discipline concemed with the study of structural
rcsponse to time-dependent loads. This chapter reviews certain elementary results of structural dynamics theory and derives expressions for the response of
structures subiected to distributed stationary random loads. These results are
then applied in the particular case where the loads are induced by wind to
obtain expressions for the along-wind response, including deflections and accclerations. Several of the results obtained will also be useful in other applications occurring throughout the text.
5.1
THE SINGLE.DEGREE-OF-FREEDOM LINEAR SYSTEM
('onsider the system represented in Fig. 5.1.1 consisting of a single mass ttt
('oncentrated at point B and of the member /8 assumed to have negligible
nrass. The displacement x(r) of the mass nr is opposed by (1) a restoring force
sLrpplied by the member /B and (2) a damping force due to the internal friction
that develops within the system during its motion. It is assumed that the restoring force is linear, that is, proportional to the displacement x(r), and that
rlrc damping is viscous, that is, proportional to the velocity dxldt. It follows
tltcn from Newton's secon(l law lhat the motion of the mass is described by
tlrc cquation
,,ri'I (\ I A.r I'll)
(s. r. r)
/'i/) is tlrc lirtrc tk'gx'ttr['rtl l(]jr(l :r('ltnl', ott (ltt: tttttss, I is thc sllrirrg
t.6rrslllrl (11r lIc slilllrcss) ol llrt'rrrt'rrrlrrr ,'lll, r'ts kttowlt its lltt: cocllit'it'ttl ol
wlrcre:
r95
196
lilN(il l l)t (int t ()t 1ilil t,{)M ltNt Ail r;yl;ltM
!i r llll
-l.trr ,l'
rurt
I
llrc slcatly-stlrle solrrliorr
[-tm
ol
lit1.
r(/)
.5.
1
.2
ItuItQt)
197
is
c<ts(2rnt
(.s.1.6)
$)
wlrcrc
2(t(nln)
d:tan , |
-hk$
H(n)
viscous damping, and the dot denotes differentiation with respect to time.
It
(s.1.8)
(s.1.9)
rrray be written as
-l (2rn1)2x :
F(t)
m
x(t)
(s.1.2)
5.1.2
where
I Ik-
' 2r\lm
.C *
- (nl n,12fiJt1r;7y
F(t) : Fs sin ZTnt
is
common to write Eq. 5.1.1 in the form
Sr
4r2 nlm{lt
'f'lrc quantity F(n) is known as the mechanical magnification
factor or met'lrttnical admittance function of the system with parameters m, n1, and (r.
Similarly the steady-state response to the load
FIGURE 5.1.1. Schematic of a single-degree-of-freedom system.
X -f 2(,(2zrn)*
:
(s.t .7)
,
2,,/km
(s.1.3)
(s.1.4)
are known as the natural frequency and the damping ratio
of the system, respectively.* The quantity zJtcm is known as the critical damping coefficient
:
FoII(n) sin(2rnt
-
g)
(s.1.10)
Response to an Arbitrary Load
Lct the system described by Eq. 5.1.2 be subjected to the action of a load
ct;ual to the unit impulse function 6(r) acting at time / : 0, that is, to a load
rlcfined as follows (see Fig. 5.1.2):
6(o:0
forr + 0)
FLI
t
lllJ, bl)dt:t )
(5.1.1r)
and can be shown to be the value of the damping coefficient beyond which the
free motion of the system is nonoscillatory. The damping ratio is expressed as
a percentage of the critical damping.
5.1.1
Response to a Harmonic Load
It can be easily verified t5-ll that
if
F(/) : Fo cos 2Tnt
xThe quantity
@1.
22"11,
(s. r .s)
is tclcrlctl to as lltc natural circular I'rcquoncy rrrrtl is t.orrrnrrrly rlcrrolctl hy
Il(;lll{1,: 5.1.2. tlnil
irrrprrlsc lillctiorr.
t98
,,
st n[,o I ulln L t)YNAMloti
/1r(rr)
I il[
r"["
crrs'rl
litN(il
t
,t,t1)c.s
J,, .\,,
l"'r"''
H)tr) sinro - ],,
,t, r,\
J,,
sin
I)t
(iltil ()l Iltl Il)r]il/ lll.ll All :;Yr;lltrrt 100
2zutr1(i(
r,) t'.,s )trttt, tlt,
2rnr,G( r.,) sirr 2rttr; tlr,
tlt;
dr,
(.5.1.1(r)
(5.1.l7)
I'ltc addition of Eqs. 5.1.16 and 5.1.17 yields thc firllowing relation between
II(n) and G(r):
H21n1
The response of the system to the load 6(r) is a function of time and is denoted
An arbitrary load F(r) (Fig. 5.1.3) may be described as a sum of elemental
impulses of magnitude F(r') dr' each acting at time z'. By virtue of the linearity
of the system, the response at time / to each such impulse is G(l - r')F(r')
dr' and the total response at time / is
J'
* o,,
-
r')F(r') dr'
(5.t.12)
where the limits of the integral indicate that all the elemental impulses that
have acted before time / have been taken into account. With the change of
variable r: t - r',F,9.5.1.12 becomes
x(t)
LetF(t)
-
F6 cos
2rnt.It
:
*rou -
J-
r) dr
(s. 1. 13)
follows then from Eqs. 5.1.6 and 5.1.13 that
H(n) cost
:
Qz)
f
cos Ztrnr dr
Using now
Iit1s.5.l.l4
I
: I
rrntl 5.1.15,
G(r) sin 2rtt'r tlr
()
itir is possible lo wlilc
Jn
Gr')G(ru) cos 2rn(r1
(s.1. 14)
(s.
,S,(n)
-
12)
dr1dr2
(5.1.18)
: t f) _&(r)cos 2rnr dr
:2[- 1,,, tf'/2
-t r) dt'l
2rnr dr
J - I .* V ) ,'rx(tlx1
lcos
: , J--(r,*T
I f ',- 1- J-r,z
['" ,, IlJoI
x Jo
I tr,,r,, * r -
: t J; *',,
'
cos
2trnr
I.
ls)
er1)F(t
-
11) dr1
rt\ clrtl]ro,
-l) znm a,
[j, *", [J-- ^",' *
11
*
12)
arlarrlar,
: 2-Jo
[ Jo
I tcO)c(rr)cos
and
/y'(rr) sin q5
J;
l)()cess with spectral density Sflrz). The expression for the spectral density of
tlrc response S"(n) can be derived using Eqs. A2.20, A2.21, and 5.1.13:
G(/).
:
f-
5.1.3 Response to a Stationary Random Load
'l'hc case is now examined in which the load ,F(r) is generated by a stationary
FIGURE 5.1.3. Load F(r).
x(ty
:
r-
I
J _ *o,r rt , J,, j,,
2trn(r1
-
12) dr1 dr2
r2lcos2rn(r
I rr - r)
(i(rr)(i(r,)si n 2ntt(r1 - r.) dr1
J ,,r,r,t, ,,
r,)srrrlrrr(r
d(r
*
11
-
12)
dr2
I rr r;)tl(r I 11
r;)
(5.l.lt)l
200
silrt,(;ttlnAl
11
t)YNnMlcli
r. 2" l.
where, in thc last stcp, thc itlcrttity
cos2rnr =
cos
Zrnl(r I
rt
r:)-(r1 -r)l
From Eqs. A2.20, A2.23 and 5.1.18, there follows
Sln)
:
This result is extremely useful in applications. See also
5.2
(s.1.21)
nzg1so1nS
[5-l] to [5-4].
r(2,
Structure
It may be regarded as an experimental fact that a continuously distributed elastic
structure with low damping, when excited by a sinusoidal force, will vibrate
in resonance at certain sharply defined characteristic frequencies. Associated
with each such resonant or natural frequency, there will also be a characteristic,
or modal, form of vibration amplitude distributed throughout the structure. Such
forms are called the normal modes of the structure. For example, Fig. 5.2.1
depicts the first four normal modes of a vertical cantilever beam.
These characteristic deflection modes and associated frequencies are properties ofthe structure, independent ofthe loads, and represent very fundamental
dynamical evidences of its internally distributed inertial and stiffness properties.* In fact, the set of normal modes may be regarded as a fundamental set
of special deflection forms by means of which any general deflection of the
structure may be expressed.
Thus, if z is a running coordinate (e.g., height) of a structure, the modal
deflection forms of lateral ("{ direction) vibration may be written as x; (z), where
/)
:
At
t r:yr;ltM
r(:, /) lllily lrt'exlttt'sst'tl
:ts
2Ol
lltt' sttttt
15.2. I )
)",t.)t,t,l
whcrc the coeflicients t,(r) inclicate what fructiort ol'caclt ntotlc r,(z) ctltcrs (ltc
p,ivcn deflection pattern. The coefficients {r(l) arc callod Lhc gcntruliT.ed ut'
ttttlinat€s of the system.
An important property of the normal modes xi(z) is their mutual orthogonality
with respect to mass weighting, by which is meant that
I,r.,.-
THE MULTI.DEGREE OF FREEDOM LINEAR SYSTEM
5.2.1 Natural Modes and Frequencies of a Continuously Distributed
1 l lll (i1l I ()l llg ltlr)M ttfJt
'l'ltcrr :ttry tlclle:ctiott
(s. r.20)
is used.
Mt
t'(t)"'
(z)m(z) dz
:o
(i + i)
(5.2.2)
wlrore rn(z) is the mass of the structure per unit length.
Since the system is actually continuously distributed but responds at each
rrl its resonant frequencies like a single vibrating entity (or single degree of
ln:cdom), it becomes very useful and convenient to liken continuous systems
to single-degree-of-freedom systems. It is helpful in this context to use the
('rrorgy approach. The kinetic energy of a single mass M is )M*', where i is
rrs vclocity of displacement. We now seek the corresponding energy for the
rlistributed system.
'f'he lateral displacement being x(2, t), the elemental kinetic energy at point
.is
im(z)lx(2,
t)21 dz
'l'lrc kinetic energy (KE) of the whole system is therefore
KE
ll
r
: ; J.r.,.,
.,.,[*k,
t)]2
dz
6-2.3)
thc system is vibrating in the single resonance modex;(z), then
*(2, t)
:
(s.2.4)
xiQ)EiQ)
:.o thaf the kinetic energy becomes
T
KE
- +Mtt?
wlrt'rc
I
FIGUR E 5.2-1. Fint fbur normal modes of a cantilcvcr bcam
M,
\
.l..1 .r, r,,
*Dctails <ln pnlccrlrrtt:s lirrtlclt:r'rrrirring, rtortttirl ntotlcs antl ttitlttt:tl ltt'r1ttr'tttit'r tttrty lrt' lirtttltl. lilr
cx:rtnplc, irr l5 ll ot l5 11.
(s.2.s)
A./,
t
r,(: )l'rrr(.:)
r/r
(5.2.6)
is kttowtr lts lltr.: ,qlrtr'rrr/t ,',1 trnttt ol lltt' rvslt'trr ilt lllt: itlt ttorlttltl lttrxlt'.
ii
l,!
2O2
liltr,otunnr
(illt I ()l lnt I l,()M ltr.ll Alr !;Yr;llM
t)yNAMtoli
ln this scnsc a cttntinttous sys(qnr vibrlrtirrg irr irny orrc ol'ils rlrnrrirl nroclcs
may be viewed as thclugh it worc sintply ir sirrglc{ogrcc<rl-l'rccdr)tl systotn
with a mass M; and velocity {,.
5.2.2 General Expression of the
llrt: gcrrcrirliztrl krrt't' (),(l) will
bcr
I 4,,
Q,u)
Response
lt(:,, l).t,(:,1
I.
o'1"],,
tl:.
xik.) F(t)
=.
Consider a structure for which it may be assumed that the displacement in the
direction x is the same for all points in the structure that have the same coordinate z (Fig. 5.2.2).It can be shown [5-2] that if the damping ratio is small
the generalized coordinates €i(r) satisfy the equations
{,(r)
.11.
203
5.2.3 Response to a Harmonic
ll tr concentrated load
F(t)
t 2(,(2rrt,)fr(/) + 12rni)2tilt):T
(5.2. r0)
Load
:
Fs cos 2rnt
(s.2. 1 1)
of coordinate z1,by virtue of Eq. 5.2.10
tlrt: generalized force in the ith mode will be
rs rrcting on the structure at a point
(,, n,, and M, are the damping ratio, the natural frequency, and the
gcncralizcd mass (Eq. 5.2.6) in the ith mode. The quantity QiG) is known as
Lhc gcncruliz.ed force in the ith mode and has the expression
whcrc
ei(t)
:
\i
,u,t)x{z)
dz
(s.2.8)
is the height of the structure, and pk, t) is the time-dependent load
per unit of length acting on the system. It is seen that each of the equations
5.2.7 is of exactly the same form as the equation of motion of the singlewhere
Q,(t)
t,(t) : Foxik)Hi@)cos (Zrnt - $')
* z)
(s.2.9)
where 6(z - zt) is defined in a manner similar to Eqs. 5.l.ll, that is, if the
structure is subjected to a concentrated force F(r) acting at a point of coordinate
-
2f i@lni)
|-
Fo
(s.2.t4)
(s.2.ts)
r
(nln;)'
-_
t):
+ 4y,2tnl n,\'\"'
(nlni1212
from F,q. 5.2.1 that the response
t txrrdinate z is
of the structure at a point of
I xik)xik)Hi(n)cos(2rnt - $)
(s.2.t6)
is convenient to write Eq. 5.2.16 in the form
x(2,
t) :
FoH(2,, 21,
n)cosf2rnt
-
6(2, zt, n)l
(s.2.11)
rvlrcrc, as fbllows immcrliittcly l'rrrn lrqs. A2.4t and N:2.4b,
tt(t..7.r,
structure.
u,{11
,
It lirllows
lt
FIGURE 5.2.2. Schcrnatic ol'a slcndcr
+rznl
o, : tan
x(2,
H
(5.2.13)
wlrcrc
Hi(n) :
F(t)6(:z
(s.2.12)
ol liq. 5.1.5:
degree-of-freedom system Eq. 5. 1.2.
If the load nQ, t) is such that
:
2rnt
F11xi(z)cos
of Eqs. 5.2.7 wlll be similar to the solution 5.1.6
rrrrtl the steady-state solutions
.F1
nk, t)
:
lr)
: ]
Il
r.,,.,,,,::r)r/,(rr)t.,s,/,,
I
I
I
r,(:)r;(z1rff,f,,l*in,l,
|
|
(.s.2. ItJ)
lt
.
r L;.t;(z).r,(11)//,{rr)silt
o(z' z1' n) : lan
{x1k.'lx1k.1)H,(rr)cos
:
il il Mt,l ll l)l (illl I (ll Iilt I t)()M ItNt
<15,
l5.2.lt);
<},
Similarly, the steady-state response at a point of coordinate z to a concentrated load
F(t): Fssin2rnt
(5.2.20)
l':',:
+
I I";, I J,,
J,"
r;t.,
(i(:.,r. r,\t,1u
?.2.r1)[,2(t
t r\ tlt
At
I f ;Yt;il
M
r
- r,ldr,l
I {'-
I GQ, q, r)F{t -t r - r) dr2
('+ Jo
I G\2. zz, r)F2Q -t r - r)' dr2'l | dr
xI
IJ0
acting on the structure at a point of coordinate Z1 elr' be written
as
|
x(2., t) :
FyII(2, 21, n)sinl2rnt
-
6(2,
\, n)l
6.2.21)
I r: fIJo Gz,21.zlt|lJo\ G{,z.zr,r)Rp,(r * rr -
5.2.4 Response to a Concentrated Stationary Random Load
[,ct thc rcsponse at a point of coordinate z to a concentrated unit impulsive
load 6(t) acting at time / : 0 at a point of coordinate z1 be denoted G{2, 4,
r). Following the same reasoning that led to Eq. 5.1.13, the response x(2, t)
of the structure at a point of coordinate z to an arbitrary load F(r) acting at a
point of coordinate Z1 czn be expressed as
x(:2, t) :
I, ou, zt, t)F(t -
r) dr
(s.2.22)
+
I
r2'sdr2ldr1
l
r- ctz. zz.rzllJo
I f*.. z2'r)Rp,(r t rr
Jo
r
- r27dr-,ldr,
I
f[l+ Jo
I CAz.z1.r'll
'lJo\
G1z.zz.r2)Rp,p,(r
I
rr
-
r2ldr2ldr,
'l
("I f+ Jo
I G(2. 22. ryt | \
Cl,z.21. r2lRprp,(r
*
rr
-
'r2\
LJo
|
|
dr2ldr,
l
(s.2.24)
Note the complete similarity of Eqs. 5 .2.11 , 5 .2.21 , and 5 .2.22 to Eqs. 5. 1 .6,
5.1.10, and 5.1.13, respectively. Therefore, by following the same steps that
led to Eq. 5.1.21, there results
S,(2,
21,
n) : H2(2, 21, n)Sp(n)
(s.2.23)
where S,(2, zr, n) is the spectral density of the displacement x(2, t), the mechanical admittance function H(2, q, n) is given by Eq. 5.2.18, and Sln) is
the spectral density of the force F(r).
5.2.5
Response to Two Concentrated Stationary Random Loads
Let x(2, /) now denote the response of the structure at a point of coordinate z
to the action of two stationary random loads F1(l) and Fr(t) acting at points of
coordinates Z1 and 22, respectively. The autocovariance of the response can be
written
as
R,(2, z)
t l'''
lim .,. \
- t'ql
J
t':
.r(2, r)x(z.
t -t r) dt
wlrcre the definition of the cross-covariance function (Eq. A2.29) was used.
'l'he spectral density of the displacement x(2, t) is
S,(2,
''J- n)
: 2 l"\ &(2, r)cos 2zrnr dr
: 2
[ &(2, r)cos 2trnl(r i rr - rz) * (rt - r)l
- J_' d(r + rr - rz)
(5.2.25)
l.t'r Eq. 5.2.24 be substituted into Eq. 5.2.25. Using the relations
H(2., z,i, n)cos
4,k..:.i. trt
II(2., z.i. rr)sirr y'r(.l, .1,.
rt)
2trnr
dr
(5.2.26)
,r, ,'. .:,, r)sin 2trnr
tlr
(5.2.21\
J,] ",..Z;.7)coS
'
[
.t ,,
206
sillt,ott,nnl
(which arc siurilar to llqs. 5.1.14 irrrtl
5.l.l5)
H(2, zr, n)H(2, 22, n)coslg(2, z.t, n)
:
f
f
Mll il l)l {illl I ()l l lll l l,r )M IllJl All :iY:;ll [/l
nI
r)yNnMtori
G{z' zt'
r)G(z'
-
(5.1..]l
rrrrtl
6Q., zz, n)l
z2' r2)cos Ztrn(rv
-
:-
[-
[-
.1,, J,,
,,.,
/:t-
,
21,
ft) - Qk,
12) dt1 dt2
sj.,,,'{n)
;rrrrl,
zz, n)f
r)G{2, Zz, rz) sin2rn(r1
:
-
sf,,,u(rr)
ilthe
statistical pnrpcrlics of the loads arc thc same, that is'
-
:,2)
S,(2,
dr1 dr2
+
'
ttr.
tl'
zt :
(s.2.34)
n) : 2H2(2, 21, n)Se,(n)
(5.2.3s)
Zz,
S,(2,
5.2.7 Distributed Stationary Random Loads
{S$,r,(nlcosl6k,
(5.2.30)
where S"(2, n) is the spectral density of the displacement at a point of coordinate
z, H(2, Z;, n) are the mechanical admittance functions defined as in Eq. 5.2.18,
Sa(n) is the spectral density of the force F,(r), and Scr,rr(n), Sf,p, are the cospectrum and the quadrature spectrum of the forces F1(r) and Fr(r) defined as
in Eqs. A2.33 and A2.34, respectively.
I lr(' spectral density of the response to a distributed stationary random load
rrury be obtained by generalizing Eq. 5.2.30 to the case where an infinite
rrrrrrrbcr of elemental loads rather than two concentrated loads are acting on the
.,tnrcture. Thus, if the load is distributed overan areaA, and if it is noted that
rrr lhc absence of torsion the mechanical admittance functions are independent
ll
tlro across-wind coordinate
S,(2.
'
Magnitude of the Response
Consider two random stationary loads F1(r) and Fz(r) acting at points of coordinates Zr and 22, respectively, and such that Ft(t) : Ft(t) at all times. By
definition, in this case the cross-correlation equals the autocorrelation, SF,r, :
S",(,e), and Sp,p, : 0 (Eqs. A2.21 and A2.29, A2.20 and A2.33, A2.23 and
A2.34). The loads F1(r) and Fr(t) are said to be perfectly correlated. The
spectral density of the response to the two loads can then be written as (Eq.
s.2.30)
cosld(2, 2.1, tt)
In thc parlicullrr casc wltctt
;'
- $k,
r.,,
z.z,
2H(2.,24, n)H(7.,7.t,
n)llS7,,(rr)
11)
y, the spectral density of the along-wind fluc-
trrrrting deflection may be written as
5.2.6 Effect of the Cross-Correlation of the Loads upon the
'
:
lo;s115.
2H(2, 21, n)H(2, 22, n)
n) : {U'(r, zt, n) + H2(2, zz, n) +
Sp,(n)
I'lr(' spectrum of the structural response to the action of the uncorrelated loads
r:, tlrus seen to be only one half as large as in the case of the perfectly correlated
22, n)Spr(n)
q, n) - 6k, zz, n)l
+ Sf;,o,(n)sinf6(2, zr, n) - 6k, zz, n)l\
S,(2,
if
n) - [H'(2, zt, n) + H2(2, zz, n)]Sp,(n)
wlrich carr bc derived immediately fiom Eqs. 5.2.26 and 5.2.21 , and following
thc stcps that led to Eq. 5.1 .Zl, there results
n) : H2(2, 4, n)Sp,(n) r H'(2,
(s.2.33)
tt
'\1,(rr),
(s.2.29)
S*(2,
)
('tllsitlcr rrpw llre t'rrst'wlrclt'tlrc loatls /"r(l) itrrtl /',(l) irrc strch thlrt llrc:ir
0. 'l'hcn, by Llqs- A2.'1.1 irrrtl A2'14'
, nrss covariipllce: /11,,1,,( r )
(s.2.28)
H(2, zr, n)H(2, 22, n)sinfg(a, Zr,
2O1
n)
: tt
I I Il(2. 21. n)H(z.72. 1)
J,t Jt
x
{Sfio;@)cos[d(2, Zr,
+
Sf;;oi(n)sinI6k, zr, n)
n)
-
-
Q(2, zz, n)f
6Q, zz, n)l\ dAt dAz
(s.2.36)
rVlrt'r'er
/)i anJ pi denotc prL:sstlr(rs
ircting at points of coordinates 11, Z1 and y2,
,, n'sllcctivcly.
It r':ut bc vcriliccl that llirrlr lir1. 5.2.-l(r lltc:rc lirlltlws*
(s .2.3 r )
rll1,rrsitr1,,
', .t l/
li1s.5.2.lllillxl 5.1.1(r.'1 ,' l.l :tllrl 'r'lt.A.r,lrr;rrrtl A].4/r. lrot':rtlclivlrliolrol lil
't. l t)ll
irr l('nns ()l cotttplt'x v;tti:tlrlt'r, r,r'1 1'r
208
l-;lru(;l(,nAt
s,(2.r)
vr\(r
"/
t)yNAMtoti
l,:| lXnMl'l I nl ('lltiWlrllr lltril,r,Nl;l
: -L ) > rr(r)'\/(:)
l6ra 7 7 ,,i ,,i ltt,u,
" {lr "
x
'
ll
r_
(ntn,)zl2 +
4yl@tn;2\{t
-
(ntn)212
//l
t"I
+ +yllntn)2}
,z-
,4'
|
I
- (;)'l l' - (;)'l + 4rci::,JI^ln*,,,,,*/,,,
'
I
ll
sfi,;1n1 dA, dA2
J, J,
*
r,
fr :,1,
- (;)' I -
z
r,
I,[, - (;)' ]]
x1Q)x1Q,)sfi,,{n)dA1dA,)
p(y,z,t) dA
6.2.3i)
I
r-.-
If
the damping is small and the resonant peaks are well separated, the crossterms in Eq. 5.2.37 become negligible and
S,(z,
r)
*?
:I
tO
\
In
*,rr,rr,,rr)
^
I6ranlul {11 -
5.3 EXAMPLE: ALONG-WIND
s'oio;(r) dA I dAz
(ntn)212
+ +y!1zntn,12l
(s.2.38)
/J
RESPONSE
To illustrate the application of the material presented in this chapter, the
of the along-wind response of tall buildings will be dealt with below.
5.3.1
If in Eq. 5.2.8 the load p
per unit of length is independent of time, the corresponding along-wind deflection, which will be denoted by x(z), results immediately from Eqs. 5.2.1 and 5.2.7:
dz
X[Z)
tliig. 5.3.1) may be written
as
p(z):ipG.+
case
Mean Response
XlZr:./t:'l{ Pktx'(zl
i
r'niM;
-
FIGURE 5.3.1. Schematic view of a building.
C)BU2(Z)
(5.3.3)
rvlrt'rc p is the air density, c, and c7 are the values, averaged over the building
rvrtlth, of the mean pressure coefficient on the windward face and suction
, .t'llicient on the leeward face, respectively, and U(z) is the mean speed at
rlt'vir(it>n z in the undisturbed oncoming flow. Equation 5.3.1 then becomes
_
x(z)
:
I
^ _ \- l{ IJ2<zlx,(zt
+ ct)B
,p(c,
l\j4*;
dz
*,1r1
(s.3.4)
(5.3.1)
5"3-2 Fluctuating Response to Wind: Deflections and Accelerations
A:; irrtlicatcd in Chapter4, thc co-spcctrum of the pressures at point Mr,
where
,
, : \: x?k)m(z) az
p denotes thc linro-irrvarian( loacl.
As indicatctl in ('hirlttcl'4, lhc rttcan wincl loatl aclirrg
(s.3.2)
ootlinatcs (yr, zr), (.y:, :.,), rcsl'rcctivcly, may be written
Sl,ii,,t(n) ='S,11'(::,' rrl,f/,1'1;',, rt)('olt(.1',,.v.r, zr . z.z,
orr ir lrrriltlilrg ol'wiclth
n)N(n)
(-5.3.-5)
ol llrr' plt'ssrrrcs irl poinl M,(i - 1,2)
;ttr'lltr':rr'r'oss wintl lrrrtl llrc irlolrg-wintl
. r.rss coltclirliorr cocllit'it'rrl, rcrpr't lrvt'ly lly rlt'lrrrrrorr, il hotlr M, ir,td M, irrt'
,tlr,.'r'c ,S,1/'(l;, rr) is thc slrr't'ltirl rlt'rrrrly
and
M2of
as
:rrrtl ('olt(.)r1,.)r-r, 11, 1.,.
/l) :ttrrl
N(
l)
210
stttu(;tt,liAl
l,
l)YNnMlo:i
on the same-windward or lccwatd--lirco ol' tlrc stl-ucrlutc, N(rr)
expression for Sr(z; , n) is, approximatcly,
so,(zi, n) :
o'czu2{z)s,(zi,
Al I rr.j( i WlNl
) I ll lil 'oNl;l
21
1
Siltrilirlly tlte lirr'1icsl lrcitk ol lhc: itlottg witttl irt'r'clclrrliorr is, lrppnrxinrirtr.ly.
= I . 'l'hc
(.s..r.
t .t )
(s.3.6)
rz)
wlrcrc
where C : C. or C : Cr according as M; is on the windward or on the leeward
face, and Su(zi, n) is the spectral density ofthe longitudinal velocity fluctuations
at elevation z; in the undisturbed oncoming flow
ll I XAMI 'l I
(l : 1,2). Equation
Kr(z)
5.2.38
:
12ln vr(7)Tltt2 +
o.517
12 ln
v1(2.)Tltt2
(5.3.14)
thus becomes
p) r,
..
.\,(:, rr) :
l6rrl
xitzl tc'z, + 2C*C,Nln) +
? ri *i@
.f f f
x
Coh(y1 ,
;trttl
cil
u{z):
xi(z)xiez)u(z)u(z)Stt2(2,)s',''(zr)
J.
!2, Zr, Zz, n) dy, dyz dz1 dz2
(5.3.7)
fS,tz. n) dn
)o
From
Eq. A2.l6b it follows that the mean
square value
lff7e,*-)d,,)
o,(z)
H
u,(z)H
(5.3.8)
u*
Jzk)
:
t4*
where, as indicated in Appendix A2 (Eqs. A2.38 and A2.43), the peak factor
&(z) is, approximately,
0.571
f2ln
v,(z)Tfit2
(s.3.1r)
lff
I
(s.3.18)
(5.3.19)
rvlrt:ro rxg is the mass of the building per unit height at some specified elevation,
L
rr(z):l+1#)' (r: o, t,z,3)
M'
m(Zl
: f'
)'*;tzt '"' dz
z-
and
",(z):
Jzk)
(s.3.r7)
(s.3.10)
K,(z)o,(z)
K,(d : [2ln v^{ilTlt'2 -l
-
tlts
(s.3.16)
is the friction velocity (see Chapter 2)-or any suitably chosen reference
vt'locity-and
in the case of the fluctuating deflection,
r.u*(Z)
4r'
ulz)H _ Jzk)
(s.3.e)
The expected value of the largest peak occurring in the time interval Z is,
fl16
: J(z)
u;
: t./ J; nas,(2, n) dn
pBH Jg(z)
orQ)H: pBH - .
Jt\Z)
)
of the along-wind
acceleration is
olk)
(s.3.1s)
ll is convenient for computational purposes to rewrite the fluctuating re\t)onsc in terms of nondimensional quantities in the form
The mean square value of the fluctuating along-wind deflection is (Eq. A2.15)
oitzl :
I Jff ,os,(2. n1 dn lt'2
n2s,(2., n) dn
J,T S,(z' rr) rln
l"'
(.5.3. l2)
i,'
tt, I
ll+
I
Y
.)/
u
(s.3.20)
(s.3.21)
(s.3.22)
r5 I )lr
212
rit nt,(i
I
t,n^t
Lll IxAMl,l I
t)YNn Mt(;ri
1-.
:
l'"'
J,,
It'
..blt.llY,,(.ll
:
Y,,(f)
:
(,H(iwlr'JtIIilt.t,oNt;t
213
(s.3.24)
'tl
-nH
f:U4
6f,(i)
Ai
(s.3.2s)
(s.3.26)
tr - (ftf,)')z + lzf,(iti)1z
J, J, J: J. [,,
x x,(Z 1tx,(z )) o(z tt
(y) + c?)
otz,t
lt# N+*
+
2c,c,N
l"
(s.3.21)
-
U(Z)
:
U(Z\
--:----:
(s.3.28)
U4
I
5.3.3 Total Fluctuating Response to Wind as a Sum of Background
FIGURE 5.3.2. Spectral densities S(n)lH(n)|2, S(n,)lH(n)12, and S(n).
and Resonant Contributions
Consider a single-degree-of-freedom linearly elastic system with mass m, natural frequency n1, and damping ratio f1 . Let this system be subjected to the
action of a forcing function with a spectrum S(n) such that
S(n)
: So (n = 0)
ll'the damping ratio f1 is small, the bulk of the contributions to the total
ol is due to the "resonant" portion. with reference to Fig. 5.3.2itmay
lrr' observed that if S(n) is not constant, a fair approximation to the integral
v:rlrrc
r-
(5.3.29)
"
I
"i:
Jo
where 56 is a constant. The mean square value of the response can be written
AS
sstlntnll2dn
(5.3.33)
o|, +
(s.3.34)
ronsists of two contributions:
"i
:
cso
Jn
lHtn'112 dn
oj:
(s.3.30)
of,,
rvlrcrc
where
lH(n)l'
:
,l :
o,',
:--
The quantity lH(")l'is an analytic function; therefore, the integral in Eq. 5.3.30
can be evaluated by means of complex integration or integral tables to yield
(see [5-3], p. 501)
,
"' -'
I
ittr
17ni,,1,,,t
Aa
'sl'
(5.3.32\
^@
Jn
S{n,ylHtnllz dn
f- S(nl dn
oi, : .1,,
l'lr:rl is, il' Ar, Az, ancl A1 lrrt. llrt. trlclrs urrtlcr lhc curvcs
l/i(rr)12 ancl S(l), rcsltr,rclivcly (lri,l.5..1 .1). thcrr
lllr)
.'1,:.'1,1.'1,
(s.3.3s)
(s.3.36)
S1n lH1n1l2,
(5..1.17
)
214
rirnt,(:tt,nAt
t)YNnMtcl;
nr
(';, I l(',,,(jN(rr1) I ('i'n[1
((
I c'l )'r
4i
The intcgral ol'tr,q. -5.3.:15 is givtn by lx1. .5..1..12, with
S| =-
(s.3.38)
.5(rrr)
and the integral in Eq. 5.3.36 can be obtaincd if the function S(n) is specified.
In the case of atmospheric turbulence this may be assumed for structural engineering purposes to be a decaying function as suggested in Fig. 5.3.2. Hence
it may be concluded that
f-
I
Jo
strtln(r)12 dn
|
[('-
Tvtr
- --;-r-l
\ s@) dn + +(- r slnl) |
lor ntm- | Jo
I
*- :
/ _\
G:{l-g'l H/
\
di +
J; f"Y,,(i)
#,ii'Ytr(Jr\
(L
: 1,2,3)
I f- v',t|t
4 J,
dI
cl,,+zc,,,c,N(n)t ('i
rlrtry llr-r wt'illcrrt lts
of along-
(s.3.40)
Coh(y', !2, Zr, Zz, n)
:
is
NIC?QI
"^e
[-
+ c10t t
u(
+ U(z)l
Zz
;i
N(n\: lt
i-;f
(5.3.41)
as follows:
Z1
yr)zlt ,J
-)
)
(t
(s.3.48)
- e 2t)
(s.3.4e)
: 15.4nLx
t, _
6.
tt
UGn,
and A"r is the smallest of the dimensions B,
(s.3.s0)
H,
and,
D.
(s.3.42)
REFERENCES
(s.3.43)
": &*^,"'n^'
lll
(s.3.47)
rvlrt:rc C, and C, are known as exponential decay parameters,
It is convenient to define the quantities G and G
is r-tsctl, llrc c;rrirtrlily
G
lrinally, recall that expressions for S,(2, n) in Eq. 5.3.27 are found in Chapter
.' rrnd that, as indicated in Chapters 2 and 4, it is reasonable to assume
acceptable, numerical calculations were carried out for a large number of cases
corresponding to a wide range of typical buildings and terrain roughness conditions. The calculations showed that the approximation is of the order of l%.
It was also verified that for L : 1,2,3 the background term may be neglected,
and therefore
1,,( f ,)
(s.3.46)
rvrrrtl response.
wherefl : nlHlu*.
To verify the extent to which the approximation involved in Eq. 5.3.40
If the notation
Y"(f )l''' odi
l,,tlrurtions 5.3.42 through 5.3.47 may be used for the computation
Irrt: )n OilI)f"Yrl.f ) df
*:
I l-t; J'
rt ('irn be verified that, approximately,
(.-
: ff,fl'r,,tf,l
1.s.1.-15)
I
",,
Lt'l tltc mcan spcctl l/ in litl. -5.3.2li bc rcprcscrrlctl by lhc logirlitlrrrric llrw.
l'ltt'zcro planc clisp-llcclttcttt r,7 will thcn hc a par:irrrrctcr in thc cx1'lrossion lirr
rli Il'thc quantity 63 is dolincd as
The first and the second terms of the sum in Eq. 5.3.39 are usually referred
to as the background part and the resonant part of the response, respectively.
The above relation can similarly be applied to Eq. 5.3.24:
rrrr
7
215
,
(s.3.39)
I
=
l',,t
il ilr l{{.t !,
Y,,(
l,\
(5.3.44)
i I
W. C. Hurty and M. F. Rubinstein, Dynamics of Structures, prentice-Hall, Englcw<xrd Cliffs, NJ, l9(r4.
1.1 .l . I). Robson, An Inlnxlut'tirtrr ttt liltrrtlottr Vihnttion, Elsevier, New York, 1964.
1l 1,. Mcinrvitch, Arutl.vtit'ttl Mt'lltttl,r itt I'iltnttiorr:s, Mac:rnillan, Collicr-Macnrillan
('irilarla, l.l(1.,'li)r'()nl(), l()(r7.
i 'f 'f'.'l'. Srxrng ltlttl M. (iligotrrr, l{tttthutr l'tlrtttlittrt.t rtl Mrtlutttitttl tttttl Slnrt'trtntl
,\'l'.r/.'r,r,r, l)r'crr(ir't' Illrll, lirrlllr'rvlrxl ('lrll',, Nl, ltt().1
I vrrl rllx
CHAPTER 6
AEROELASTIC PHENOMENA
A body immersed in a flow is subjected to surface pressures induced by that
flow. If there is turbulence in the incident flow, this will be the source of timedependent surface pressures. Such stresses are also caused by flow fluctuations
initiated by the body itself.
Further, if the body moves or deforms appreciably under the induced surface
forces, these deflections, changing as they do the boundary conditions of the
flow, will affect the fluid forces which in turn will influence the deflections.
Aeroelasticity is the discipline concemed with the study of phenomena wherein
aerodynamic forces and structural motions interact significantly.
An aerodynamic instability car, be a phenomenon occurring wholly within
the flow alone, as when a trail of vortices or a rapidly diverging wake is shed
from a fixed body. But if a body in a fluid flow deflects under some force and
the initial deflection gives rise to succeeding deflections of oscillatory and/or
divergent character, an aeroelastic instability is said to be produced. A purely
aerodynamic instability such as vortex shedding may occasion structural deflection as well, initiating a phenomenon having aeroelastic character. All
aeroelastic instabilities involve aerodynamic forces that act upon the body as
a consequence of its motion. Such forces are termed self-excited.
The purpose of this chapter is to discuss fundamental aspects of aeroelastic
phenomena that need to be taken into account in the design of certain structural
members, towers, stacks, tall buildings, suspended-span bridges, cable roofs,
piping systems, and power lines. Not all of these phenomena are presently
completely understood. Indeed, only a few theoretical forrnulations from first
principles exist for modcling acrodynamic firrces on oscillirlirrg lrrxlic's. In nrost
investigations, crnpirical ntotlcls arc sct up in which llrc r'sst'rtt't'ol'tlrc: itcr<ldynamics rrrrrs( l-rc cortlribtr(ctl by c:xpcrirttcrtl .'l'lrc t'ottt'slrorttlirrl':rrr:rlylicitl
216
t;l
ll l)l)lN(i nNl, llll
ltx tr [] I'1il t!()Ml N()N
211
rttrxk:ls trstrirlly iltr'ltttlc itsl cn()uglr;latlrtte(crs lo ttrirlt'lr llrt'slnrrrllt'sl olrst'r'vt'rl
It':rlrtros ol'lltt: plrt'rtonr('nit. Srrr'lr rrrotlcrls lrt'llrus nurrlrurlly tlt'sc'r'iptivc. lrrrl
rro( cxplanatory itr llrc serrsc ol' r'cvcaling blrsir' plrysit'itl t'trrrst's; srrlrllr: lrul
rulx)flllnl dctails ol'lltt':tt'ltutl llrritl-structttt'c inlt't'irt'lit)n nrily irr t'cllrrirr t'lrscs
lrr lcli unattcndctl.
lirnpirical modcls rrtay only bc uscd lirr tho prcrliction ol'acnrclastic oll'ccts
rl thc ranges of thc govcrning nondimcnsional paranrctcrs in the modcl arc
, Lrsc: to those of the prototype. Most commonly, it is thc Rcynolds number of
tlrt'llrototype that is not realized in the model. As a result, uncertainties subsist
rrr irrtcrpretation of model test results. (See also Chapter 7.)
Most of the empirical models described in this chapter apply to situations
tlr:r( nray be considered, at least approximately, as two-dimensional. In practice,
tlrn'c dimensional effects are present, owing to any numberof factors such as:
lhrw adjustments near the ends of finite cylinders; spanwise variations, either
,,1 thc body cross section (e.g., for tapered stacks) or of the body deformation;
rr,rrnrnifbrm mean flows; or imperfect spatial coherence of the incident turbuIrrrt:c t)r of the vorticity shed in the wake of the body. Information on three,lrrrronsional effects is in most cases scarce and must be obtained from wind
trrrrncl experiments.
'l'hc topics dealt with in this chapter include vortex shedding and the assor
r;rlcrl lock-in phenomena, across-wind galloping, wake galloping, torsional
,lrvt'rgence, flutter, and buffeting response in the presence ofself-excited forces.
{i.1
VORTEX SHEDDING AND THE LOCK.IN PHENOMENON
lr wrrs seen in Sect. 4.4that under certain conditions a fixed bluffbody sheds
,rltcrrrating vortices whose primary frequency N" is, according to the Strouhal
rt'lltt ion.
" :s
N^D
U
(6.1.
l)
rvlrcrc S depends upon body geometry and the Reynolds number, D is the
rrrss-wind dimension of the body, and U is the mean velocity of the uniform
llow in which the body is immersed. The frequency N" is also that of the net
l,r rnrrtry forces acting transversely to the direction of U while the primary
lrt't;rrcncy of net forces acting in the flow direction will be 2N". Actually the
ru'l lirrcc vector defined by thc intcgral of instantaneous pressures over a given
I'lrrll bocly will vary in magnitutlc lrntl rlircction with time in a fairly complex
rrr;rurcr dcpcnding upon rlc:tirilctl lllrly gc()luotry and Reynolds number of the
llow. Only thc l'rcqucncics ol its prirrt'i1xrl lurnrrlnics urc givcn by N, and 2N,.
ll tltc brxly thll ins(igirtt's llrt'vorlt'r slrt'rhlirrg is cllslicully supp<lrtctl or il'
It rs subjcc( lrl lrtcrtl c()rtl()ur rlclottu;rltrrtt. rl will tlt'llcct wlrolly or locirlly lrnrl,
lry lrtis uc'tirttt, irtllttcttcc llrr' lot;rl llon, Nol rrlrrry ol tlrt' lirll lrrrl.lt' ol' possi
,r,
218
ntllol lnt;ll(;
I \/(rl illx
I'l ll N()MI Nn
bilitics latcnt irr this situation lurvt'lrt't'rr slutlit'tl in tlctiril. l)cliltrttitblt: stccl
shells have givcn risc l.o so-callctl ovrrlling oscillations l6-ll unck:r thcsc ctlnditions. Many examples of cross-wintl rigitl-cttnttlur <lscillalions havc bccn
noted; and in water flows impoftant along-lkrw dcflections have been observed
[6-2,6-3,6-4].
Unless otherwise noted, it will be assumed in this section that the structure
is a cylinder with a rigid surface, the oncoming flow has uniform mean velocity,
the deflections of the body are the same throughout its length, the body is
elastically sprung and possesses mechanical damping in the across-wind direction, and it is rigidly constrained in the along-wind direction. Under the action
of the vortices shed in its wake the cylinder will be driven periodically, but
this driving will elicit only small response unless the Strouhal frequency of
;r ltlrrll cllrsiit'lrrttl.y rrrrtlt'r
at this point that the body mechanical frequency controls the vortex-shedding
phenomenon even when variations in flow velocity displace the nominal Strouhal frequency away from the natural mechanical frequency by a few percent.
This control of the phenomenon by the mechanical forces is commonly known
as lock-in. In dynamical systems theory this phenomenon is referred to as
synchronization. Observations show that during lock-in the amplitude of the
oscillations attains some fraction, rarely exceeding half, of the across-wind
dimension of the body. The effect of lock-in upon vortex shedding is represented in Fig. 6.1. 1, which shows that in the lock-in region the vortex-shedding
frequency is constant rather than being a linear function of wind velocity, as
suggested by Eq. 6.1.1 (and as it in fact is outside the lock-in region).
No completely successful analytical method has yet been developed, starting
from basic flow principles, to represent the full range of response behavior of
trt{
tr [.J I't
1
il u{)Mt Nt)N
ll llrs,
219
irtstelrtl, lrecrt
lorrrtl rcirsontrbly lirrrllrrl lo lrrrrltl errrpirit'irl rtrxlt'ls:rrrtl trr:rlt lr llrr'ir'Pcllirlrrrirrrt't'
l() r(:irlily hy ir jrttlit'iorrs t'lurit'c ol'pitrirtttclers. ltclt'rt'rrt'cs l(r ll lo l6.t.ll.
()5 plovitlc
irrt ovcrvit'w ol'sorrrc ol'lltc rcet'rrt li(cr.rrlrrrt'
l{r tt7l, ancl l(l-9 ll to l(r
1
rrr llris arca.
6.1.1 Analytical Models of Vortex-lnduced
Response
Assurnc first that the circular cylinder dealt with above is fixed not only in the
wind direction but in the across-wind direction as well. In this case a
rt':rsonable first approximation to the across-wind force perunit span acting on
rrhrrrg
tlrt' cylinder is
F : )pUzDCl5
sin
c,r"/
(6.1.2)
rvlrcrc c,r" :2rN, N" satisfies the Strouhal relation (Eq. 6.1.1), and C15is the
lrlt c<refficient. (For a circular cylinder and Reynolds number 4O I G.. I 3
" 105, in a uniform smooth flow C15 = 0.6 16-4, p.721.
An important feature of this across-wind force, however, is that it is imperl'cctly correlated along the cylinder span. When the cylinder is allowed to
oscillate, this simple expression for the forcing function F is inadequate for
rwo rcasons. First, the across-wind force increases with oscillation amplitude
rrrrtil a limiting amplitude is reached. Second, the spanwise correlation of the
:r,'ross-wind force also increases, as indicated in Fig. 6.1 .2. Let y denote the
;r. lrss-wind displacement of a cylinder of unit length for which the effect of
tlrc imperfect spanwise force correlation is not explicitly accounted for.f The
ltluation of motion of the cylinder can be written as
m1)
F
ll l)l )lN(i nNl) llll
llrt';rtlitrt ol vollcx slrctltlirtl,
alternating pressures approaches the natural across-flow mechanical frequency
of the cylinder. Near this frequency greater body movement is elicited, and
the body begins to interact strongly with the flow. It is experimentally observed
:;l
-l cy + l{y : 5(y, i, y, t)
(6.1.3)
rvlrcrc ru is the cylinder mass, c its mechanical damping constant, k its spring
:,tillhcss, and $ its fluid-induced forcing function perunit span, which may be
,lt'pcndent on displacement y and its time derivatives ) and j; as well as on
requency
Iililc.
Much effort has been spent on finding by empirical means a suitable expreslirr $ in Eq. 6.1.3 that fits the experimentally observed facts. The com1'lt'xity of such an expression will depend on the detail and completeness with
rvlrich the experimental facts are observed, on the one hand, and on the needs
to bc rnet by the subsequcnt predictions from the model, on the other.
:,rorr
c
f
o
o
f
f
E
f
o
z
rltctt'rtl slutlios in contputlrlion:rl llttitl rlVrr:rrrrir's lr:rvt't'x:urrinctl a limited numberofsuch
F
low velocity
FIGURIT 6.1.1. I,lvolution ol'vorlcx-shcdcling licqucncy willr wintl vclocity ovcr clas-
tic
slnrcturL:.
|
{r
cases
()l.l
|
.
rllrrs t'lli:tl is irttrrtrrtlctl lirt r'trtpirrtrrlly rr l() .'l (rtr';rlso ('lr;rl)t('r lO, lit1s. 10.2,(r,
10.2.7, and
I
ll).1 l(r).
220
Al
llol lAlill(:
I't lt N()Mt Nn
ri
I
v()t
iltx
:il
<
J
lilt tlx;t(
tN t,ilt N()Mt N()N
221
,orO ..rO
2.8
s(f) .'rr
9
F.
il t)t)tN(i ANI)
.r,
l'r
l0
05
00
.-*--'lti*-.#
ur
t.4
0i
o
.to
a.z
.o5
06
(a)
68
SEPARATION
,/D
o.0B
240
s(0
YID
200
160
0.00
120
80
z
9
40
F
.o.oB
0
J
lrl
c
(b)
E
o
(J
0
004
6
5
YID
SE PA RAT
ION
,/D
S(0
0.000
4
5
2
FIGURE 6.1.2. The effect of increasing the oscillation amplitude al2 of a circular
cylinder of diameter D on the correlation between pressures at points separated by
distance r along a generator: (a) smooth flow; (b) flow with turbulence intensity l1%.
Reynolds number
=2x
t
o
004
t
104. 1After t13-951.)
Among the many empirical analytical models of vortex-induced oscillation
0
(c)
f
lrl(;URE 6.1.3. Across-flow oscillations y/D of elastically supported circular cylinder:
(rr) before lock-in; (b) at lock-in; (c) after lock-in. (After [6-931.)
are a number that recognizethe near-sinusoidal response
ofthe cylinder at each
of two prominent frequencies-the Strouhal and the natural frequency of the
structure. The response in each of these two simultaneously gives rise to a
beating oscillation when the velocity of the cross flow is not precisely at the
lock-in value. Figures 6.1 .3a, b, c depict some illustrative experimental results
for deflection response ofan elastically supported circular cylinder before lockin, at lock-in, and after lock-in, respectively, together with the corresponding
displacement spectra, where f, , f, are the Strouhal and natural structural fre-
quencies, respectively.
A considerable variety of empirical analytical models have been devised to
represent the vortex-lnduced response of bluff cylinders t6-951. one particular
aspect of the phenomcnon itsclf that has been notcd is that thc wakc of the
bluffbody, composctl ol'a "strcct" ol'altcnratcly shcrl vollit'cs, slrows itspccts
ttl'a scparatc "oscillitlot'," couplctl in u lirirly c<lrrrplt.x nr;uur(.1 to llrt'irrilirrting
rrrcchanical body. Another characteristic of vortex-induced oscillation is that,
wlrile self-excited, it never proceeds to divergent amplit,fdes but enters a limit
I
.yclc of relatively modest level.
Numerous qualitative or semi-quantitative attempts have been made to set
rrp associated, descriptive mathcmatical models, in particular several so-called
torrplcd oscillator models govcrnccl by two differential equations, one for the
slrrrcture and another frlr ils wlktr. whilc such efforts have not been unrewardrng, it is oftcn thc casc lhlrt lhc rnos( irrrlxrr(ant cngineering need is to be able
{rr lirrccast the largast suslrtittt'rl r'('sl)()nsc lrrrrplilutlc
ol'the structure alone, that
rs, lhal which <lccr.rrs al lock irr
A tttorc lirrritctl silrgk: rlcp,rn'rtl ltt't'thrttt tnorlt'l is tlrcn <llicn usclirl. ll.cf'-
('11'nct'
l(r-9(lI lurs suggcslctl:
222
nnr()r rnritt(;
iI
t,t tt NoMt NA
mIi + 2r,ry+,?y I )ou'n l
+
Y2(K)
r,,n,
{
(' - #) L
where D is a frontal dimension of the structure,
the Strouhal relation
aD
i
K : DalU,
l;l
ll l)t)lN(i ANI) illl
d)]
and
:2rS
<o
t"
(6.1.4)
satisfies
(6. r.5)
ln this rnodcl, which obviously exhibits aspects of a Van der Pol oscillator,
Y., t, Y2. ancl Cy, arc parameters, functions of K, that are to be fitted to observations. Various cxploitations of this model may occur. In particular, aspects
of nonlincar, scll-lirniting amplitude are inherent in it. in agreement with similar
efl'ects witncssed with vortex-induced oscillation. In effect, the model allows
for linear, fluid-instigated "negative damping" at low amplitudes, an effect
reversed at higher amplitudes. At lock-in o J @r and Y2 = O, C1 : 0, since
at lock-in the last two terms are found to be small compared to the term
reflecting the aerodynamic damping effects. Then Y1 and e remain to be determined from experimental observations.
At steady amplitudes the average energy dissipation per cycle is zero, so
l'ttt
:
dt :
- prrDYl (,
D'/ lr'
,, - ,*)
1"
223
o
(6 r
t(r'l ll)
,,r'
y(/)
=
D
lolD
lr -
((Ai
-
yblA'd
exp (-ayf,Ut/4D3)lt/2
(6.1.12)
rn which
Ot:-(
'fhe value of
cv
pDzYt
(6.1.13)
2m
is determined from the model test as follows: Defining R,
where .4n is the amplitude of y at
lrc cvaluated as
l'lo^y,
Jr
L
t()|l!lr.l ttilt NttMt N()N
'l'lrc rnodel is usclirl irr prctlicting prototypc
aclion lirrrrr thc bchavior ol'laboIirlory tests.
A process by which thc parameters 1, and c may be cvaluated from a model
It'st will be describcd. If, at lock-in velocity, the mechanical model is displaced
Irr irn initial, higher amplitude ! : Ao and then released, it will undergo a
tlccaying response (Fig. 6. 1.4) until this latter levels out at the steady-state
vlluc ye given by Eq. 6.1.10. It can be shown [6-94] that this devolution of
rrrrrplitude is describable by the form
AolA,
that
where c,rZ
rllx
wltt'rc S is tltc Sllrtrlutl rrrrrrrlrt.l=1tr.l.l) irrrtl,f,, lllc,!r',r,/r,/r nrtnrlu't tlr.liur.rl lrs
.
+ c,,1x1sin (c,,/ +
V()l
n
cycles after the release, cv may
6)
2zr. Assuming that y behaves practically sinusoidally,
) : )o cos cdt
(6.1.1)
leads to the results
\',, *:,vtr
:
\inr'at ,vrLo
(6.1.8)
(6.1.9)
Then (6.1.6) yields the steady amplitude solution
{'.lrqdl"
(6. r. r0)
lll(;tJlllt (t.1.4. l)cciryirrg
rrrrlcr vort0x klck-irr.
oscillrtlirrrt lo:lt'ruly ;.lirtt'ol lrlrrll, clrrslic:rlly sprrrrrg rrrrxlcl
224
tnl;llo
nt n()t
I Vrll illx
l'l llN()Ml NA
(Y
-
4ri/)' . 1,,t,i,, lt;vil
l"
u,'l
,,y,1
I ll,,
^,
I
ol
so that Y1 and e are given bY
,,: #1"#+
'
:
r6rrsl
(6. l.
2ma
(6.1. r6)
ody,
ll
t6.
+ 0.43(8rr'sts..)lt tt
l.l7t
6.1.2 An Empirical Model Developed for the Estimation of the
physical significance is discussed subsequently. The term
0. l0
Experimenl
Eq.6.1.17
-\o
'\q,
0.05
o\_
0.00
I .0
\o
l .5
--o_....--o-o
2.O
o
2.5
5.0
5.5
4
.0
Scruton number
FIGURE
6.
/
\1,
1.5. Maxililurn arnplitudcs vcrsus Scnttott llttttthct'(lrlicr
l(rr.)31).
225
r'r l
tr, ) tt
wrillcn itt l6 ttl'll itt thc lirrrrr
I
,J;,'l
,
K'6(U/U,,) is an acrodynamic coefficient, and U,., : alDl(2rS). The
;rlxrvc term is equated to the product -2mlo<'s1, where f, is defined as the
rrt'lrdynamic damping ratio, which may thus be written as
r": -+r.,(*o,)l
-,*il
(6. l
.
l8)
: \D the aerodynamic damping vanishes,
so the structure no longer
t'rpcriences any aeroelastic effects causing the response to increase. The coeflrcicnt \ may thus be interpreted as the ratio between the limiting rms value
ol (hc aeroelastic response and the diameter D.) The total damping ratio of the
:'vslcm is then
(,:(iJ"
A model derived in effect from Eq. 6.1.4 was developed in [6-88] for application to the design of chimneys and towers with circular cross-section. It is
noted in [6-88] that the product pUzYrlXl of Eq. 6.1.4 is considerably less
than mal, so that in practice the term Yz(K)ylD may be ignored. It is also
noted in t6-881 that in the case of alandom motion, the term ,yzlDt of Eq.
6.1.4 may be replaced by the ratio y2l(XD)2, where \ is a coefficient whose
o
/,1/ /))'r(A)
lilr:ti tH t,t il il()Mt u()N
r.vlrcrc
Response of Chimneys and Towers
YJD
ll l)l)lN(i ANI) llll
2a11,r)'K,,"(#)
11'.,r y2l12
r.29
)o:
lrc1. 6. 1.4 is
ls)
liurploying an analytical model of this type for a circular cylinder the maxirrruur arnplitudcs ol Fig. 6.1.5 were obtained. On the same figure (dashed
curvc) an cmpirical fbrmula of Griffin, Skop, and Ramberg [6-33] is plotted.
This lonnula, for circular cylinders, has the form
D
|
(6. r.14)
I
:il
(6.1.19)
f is the structural damping ratio. The aeroelastic effects are, in effect,
rntrrduced in the equation of motion simply by substituting into that equation
tlrc total damping ratio f, for the structural damping ratio f.
'l'he validity of this simple approach was verified in [6-88] by numerical
:trrrlies and by comparisons with experimental results reported in [6-39]. Figure
(r 1.6 shows the dependence of the measured response 4*. : y't''lD upon the
rt'tluced wind speed 2rUlalD for various struclural damping ratios f. Figure
tr. 1.7 shows calculated versus measured ratios y'^^'.lD for various values of the
l)rrameter K, : m(l pd, where yill'" ir the rms response corresponding to the
rrrost unfavorable reduced wind speed. Three regimes are noted in Fig. 6. 1.7,
,,'r'rcsponding, respectively, to (l) vibrations whose character is largely due to
tlrt: random nature of the forces associated with vortex shedding (forced vibratiorr regime), (2) a transition zone, and (3) self-induced vibrations (lock-in
rt'girnc). Vibrations typical of these three regimes are shown in Fig. 6.1.8.
Notc that the ratios of peak to rms response are about 4.0 in the forced vibration
rcginrc, an<I about r.D in th. Iock-in rcgime.
llascd on inferenccs I'nrnr cxpcrirrrcntirl clata available in the literature,
Itr tttll prop<lscd curvcs rcprcscnling ( l) tlrcr tlcpcnclcncc of K,,9,,,,- upon the
Itt'yrr<rlrls ntttnbcr 61": IJI)l t', wltt't't'4,,,,,,,,,, rlt:notcs thc maxitnutn valut: of
K,,r(l.llIJ,,) in snuxrlh lklw (lri1l. (r. l.()). rrrrrl (l) tlrc tlcpcnrlcncc ol'thc ratio
rvhcre
Anlol lAt;ll0 I'lll
NoMl NA
fi r v(lilil x ril ll l)lllN(i ANI) llll
o
E
l()r,h tN t,l I N()Mt N()N
227
Experimental [6-39]
(:itc:600,OOO)
.06
Calculated
F
ul
o
l
F
J(L
" Lock-in
Regime
o
trj
c)
l
o
UJ
E
.O2
89tO
7
2rU
't
"Transition"
D
Regime
I
FIGURE 6.1.6. The response of a model stack of circular section for different values
of structural damping (Ge subcritical). From L. R. Wooton, "The Oscillations of Large
Circular Stacks in Wind," Proc. Inst. Civ. Eng., 43 (1969),573-598.
t--*
"Forced Vibration"
Regime
I
Koo(UlU-)1Ko0',- upon UIU,, for smooth flow and flows with various turbulence intensities
#tt2lU Gig. 6.1.10).
0.1
For a vertical structure experiencing random motions described by the re-
0.2
0.4
0.6 0.8 1.0
2.0
4.0
Ks
lation
Vk):zt?v?rrt
(6.1.20)
I
[6-89] proposes the following expression for the total damping in the ith mode:
(r;:lifl";
(6.1.2t)
u, :-#lrr,, -r,*)
(6.1.22)
Il, K,u(z)
Kri
:
[#l'
1t,i
,112,1 dz.
l,'l(;[JRE 6.1.7. Measured and estimated response in smooth flow. From B. J. Vickery
Ii. L Basu, "Across-Wind Vibrations of Structures of Circular Cross-Section. Part
I l)cvclopment of a Mathematical Model for Two-Dimensional Conditions," J. Wind
I rt,q. lnd. Aerodyn.,12 (1983), 49-73.
;rrrtl
-'
1(r,:-
\tri x,,,,12.1Y!121 dz
\
I',\ .v
itz.l
(6.r.24)
dz.
rvlrr:rc C and L,; are thc structrrnrl irrttl tltc rrcnlclynarnic damping in the ith mode
y?(z) az
(6.1.23)
ol vihralion, rcspcctivoly, /),, is llrt'tlirrttrelet'rrl clcvation z : 0, D(1) is thc
rlilurrclcr at clcvation l, /r is llrr'lrr'ip,lrl ol lhc slnrclul'o, nr,,i is lhc cquivalcnt
nliris por unit lcngilr irt (ltc itlt rrotlc ol vrl)!irlion, tle:lirtcrl as
228
At n( )t l A:i
il(:
t
,t
il
ti r v( )ilil x
N( )Mt Nn
K,/ri,,,
1;1il
I)t)tN(i ANt) ilil
I ( x.h
tN t,lI
N( )Mt N( )N
229
; l'.'
r
lt
Y0
D
K
0.8
d0
{.5
Ks/Kao
=
I
0.1
I
Kr/K"u t
0or
,
fT
L
) ul-
,f
0.8
.n. I
0.9
1.0
1.1
1.2
u
FIGURE 6.1.8. Simulated displacement histories for low, moderate, and high structural damping. From B. J. Vickery and R. L Basu, "Across-Wind Vibrations of Structures of Circular Cross-Section. Part I. Development of a Mathematical Model for
Two-Dimensional Conditions," ,/. Wind Eng. Ind. Aerodyn.,12 (1983), 49-13.
1.3
r.4
1.5
1.6
t.7
u",
I,'l(;URE 6.1,10. Dependence of ratio Kuol Koo-o* upon ratio (JlU.,for various turbulencc intensities. From B. J. vickery and R. I. Basu, "Across-wind vibrations of
slrlrctures of Circular Cross-Section. Part L Development of a Mathematical Model
lrrr Two-Dimensional Conditions,"
.l() 73.
l.
Wind Eng. Ind. Aerodyn., 12 (1983), pp.
(6.1.25)
Ml is the generalized mass in the ith mode. Equations 6. l.2l to 6.1.24 are
lr;tscd on the assumption that aeroelastic effects occurring at various elevations
;nrLl
Mrch No. > 0.15
o
/'
z'
o
Nr.h No <
o
t5
;rrc linearly superposable.
For the relatively small values of the response that are acceptable for chimr('ys and stacks, the estimated response depends weakly upon the assumed
vrrltre of \. It is suggested in [6-89] that the value \ = 0.4 is reasonable for
rrsr: in estimates of the response of concrete chimneys.
6.1.3 Experiments on the Lock-in Phenomenon in Turbulent Flow
l;nrrn tests in turbulent flow with Rcynolds numbers of about 75,000 on a 200cylindrical oscilllr(or with lincar springs, statistics of interest for
l;rligttc studics were obtaincrl irt l(r ()()l on klck-in l'rcqucncy intervals and acrosswirrtl oscillations during irnrl lrlicl krt'k in. l;igrrrc 6. l.ll shows time histories
ol'( l) wind spcctl lluc:tulrliorrs lrrrtl (l) lrr'lrss wirrtl rrulli<lns that cxhibit irrcgrrl;rl' lockotl-in rlscillittiort cpisrxlt'r. l,t'l llrc lowr'r'lrrrtl rrppcr cntl ol'lho lock-in
lrr'tlrrcrrcy irrlcrvlrl bc tlcrrolt'rl 1',t.rrt.rrr'lrlly lry..l trrul /1, r't:sltcclivcly.'l'lre: pr.ob,
rrrrrr diameter
Rt YNOr t)s
Nti!ilil
R
FIGURE 6.1.9. Experimcntal data, and sr.rggestcd dcpcntlcncc ol (,,,,,,,,, rrpon Rcynolds numbcr. Fnrrrr Il..l. Vickcry anrl R. l. Basu, "Ac:nrss Wirrrl Vilrllrtiorrs ol'Stnrcturcs ol' Circular ('trrss Scclirlt. l)iu1 I. I)cvclopnrcnl ol :r Mrllrr'rrurllt lrl Mtrlcl lirr
'f-wtr-l)itttcttsiottirl ('olrrliliorrs." ./. Witttl l,.lrr.t:. ltttl. ,'lt,nult'rt . l: (1,)ttl). .l() 71.
:rlrilislic bclutviot'ol',4 lrrttl /l w;rs lorrrrrl
rrr
ltr t)t); to rlilli'r'lrt't.otliltg lo wlrr.llrt.r.
230
n
t n()t
tn
li
ilo
t't tt N( )Mt NA
67
E
t!
-o
o
t
(D
!
0.40
0.30
0.30
ozs
I
< o.2o
o o.ts -l
o'o -]
0.20
1
0.10
.94
3
E
=10
E8
o.os
0.00
.=4
o_
EC
1.00
1.20
1.25
0.95
0
100 200 300 400 s00
FIGURE 6.1.11. Time histories of longitudinal wind
1.15
Bt
A,
1'10
600
0.90
1.05
speeds and across-wind displace-
0.85
0.95
Time [s]
1.00
ments [6-99].
0.80 0.90
1.00
At
(l)
dVldt is positive when the longitudinal turbulent velocity Z crosses the
lower threshold A and when it crosses the upper threshold B (in this case the
notations A = At andB = 81 are used); (2) dVldtis negativewhen Zcrosses
B end when it crosses ,4 (in this case we denote B = Bt and A = A); (3)
dVldt is positive when Z crosses ,4 (which is again denoted by Ai, changes
sign during lock-in, and is negative as Z again crosses ,4 (which in this case
is denoted by A); and (4) dVldt is negative when Zcrosses B (which is denoted
by Br), changes sign during lock-in, and is positive as Zagain crosses B (which
in this case is denoted by Br). Scatter plots of crossing limits are shown in Fig.
6.1.12, where the abscissa is normalized with respect to the Strouhal number,
which was about 0. 175. Stochastic properties of successive lock-in intervals
were found to be independent. For additional details, see [6-99].
ACROSS-WIND GALLOPING
Galloping is an instability typical of slender structures having special crosssectional shapes such as, for example, rectangular or "D" sections or the
effective sections of some ice-coated power line cables. Under certain conditions that are defined later herein, these structures can exhibit large-amplitude
oscillations in the direction normal to the flow (one to ten or even many more
across-wind dimensions of the section) at frequencies that are much lower than
those of vortex shedding from the same section. A classical example of this
type of instability is the acnrss-wind large-amplitudc galloping ol' powcr line
conductor cablcs that havc rcccivcd a coating ol'icc untk:r contlilions ol'l'rcczing
rain.
1.10
Br
1.20
0
6.2
l.oo
0.85 0.90 0.95 1.00 1.05
Ar
(D^
EO
-{
1.OO 1.10
1.20
Br
FIGURE 6.1.12. Scauer plots of lock-in frequency inrerval limits [6_99].
Early and clarifying analyses of the galloping problem appeared in [6-40],
to [6-50] have dealt with the problem as
lu nonlinear phenomenon. In across-wind galloping the relative angle of attack
of the wind to the structural cross section depends directly on the across-wind
vclocity of the structure. Experience has proved that knowledge of the mean
lift and drag coefficients of the cross section obtained wder static conditions
rrs functions of angle of attack suffices as a basis upon which to build a satislactory analytical description of the galloping phenomenon. Galloping is thus
16-411, and[6-42]. References [6-43]
governed especially by quasi-steady forces.
As in the case of the vortex-induced oscillation, the phenomenon will be
conceived of, and dealt with analytically, as two-dimensional in nature. Further
tluestions related to galloping response are discussed in t6-461 to [6-50]. A
study of a system of two elastically coupled square galloping bars that can
cxhibit chaotic motions, reported in detail in [6-100], is summarized in Sect.
6.2.2.
6.2.1 Analytical Formulation of the Galloping problem
llclirrc presenting thc basic unirlyticirl lirrrnulation, it is of interest to note some
ol'thc rccognizcd litcraturc lhlrl lrr.lrts lhc gall<lping phenomenon. Rcf'crcncc
l6-4131 rcvicws tho siatc ol lltc lttl rrtttl prescrrls a compact analysis 9l'tlrc
pnrhlcrrr. lt irlso poinls orrl tlrt.t.:lly lurtl lrirslt.t'orrll.ibulions ol'(iltrrrul
l(r_401
Ittttl l)t:tt lllrrttlg l6-4 1,6 42 1. l{clt'rt'rrt't's l(r.l llrrntl l(r 4t)lcorrslilrrtt.irrrl.rorlirrrl
232
Atlr()LtAlitt(; t'ltt
ri
NoMl Nn
:, n(:ll( )lil:
wll.Jl
I I iAl I I )l'|lN( i
233
FIGURI.I (t.2.2. Elfcctive angle of attack
FIGURE 6.2.1. Lift and drag on a fixed
on an oscillating bluff object.
bluff object.
contributions particularly toward clarifying the nonlinear questions related to
the aerodynamics. Reference [6-50] offers a critical discussion of existing analytical models of galloping.
Consider a section of a prismatic body in a smooth oncoming flow (Fig.
6.2.1). Assume that the body is fued (i.e., experiences no motion, oscillatory
or otherwise) and that the angle of attack of the flow velocity U. is a. Below
is obtained an expression for the force coefficient in the y direction. First, the
component of the mean drag (mean force in the direction of U,) can be written
rclative velocity of the flow with respect to the moving body is denoted by U,
lrnd can be written as
U,:
l'he angle of attack, denoted by
a,
(U2 +
while the mean
lift
:
(mean force in the direction normal
L(oi
:
(6.2.1)
)pulncrla\
to U,)
: -D(o)sin a -
L(cv)cos
If {(cv) is written in the alternative form
Fn(cv)
:
If the body has mass ru per unit length, is elastically sprung, and has linear
rrrcchanical damping, its equation of motion can be written in the usual form
mly+2lt}1_alyl:F,
whcre
(6.2.3)
cv
(6.2.4)
)pttzBco,1o)
f
is the damping ratio and
c,r1
the natural circular frequency, and where
lrrrdy and for the fixed body are the same so that Fy(cr) is given by Eq. 6.2.4
whcre Cp,(a) is given by F,q. 6.2.6.
Let us first consider the case of incipient (small) motion, that is, the condition
irr the vicinity of i : 0 wherein
(x=
:
(6.2.s)
U,cos a
it follows from Eqs. 6.2.3 and 6.2.4 that
Ce,(cv) : -tcr(a) *
Co(cy)tan alsec
cy
(6.2.6)
The case is now c<lnsiclcrcd in which thc samc hody tt,stillrtlr',r irt lltc ac()ss-
wind direction .y in a lkrw witlr vclocity
l/
(F'ig. 6.2.2|.'l'ltc tnirgnitttrlr-r ol'thc
(6.2.e)
/'',, denotes the aerodynamic force acting on the body. It is assumed that the
rrroan aerodynamic lift and drag coefficients C1(cv) and Cp(cv) for the oscillating
where
U
(6.2.8)
is
The projection of these components on the direction y is then
F,(cY)
lu
arctan
(6.2.2)
|puI,ncr1a1
(6.2.7)
is
AS
D(e)
i\t,t
n
'=0
U
l,or lhis condition
l,'' -.
oF'l
i)rv
1,"
u
(6.2.t0)
,,
wlriclr lcarls (o cxalrrinutiott ol lltt'lttr'tot r/(),,/r/rv lirrrrrtl uptln tlill'crcrtliation ol
lu1. (r.2.(r (o lutvc tltt: vltltle' ltl rv O.
234
nl ltot lnrill(] I't il NoMt
ri
NA
tl('1 ,l
,/t
1,,
("1,,:
,
(6.2. r r)
,',,),,
u[
dl
,,
Thus for small motion the equation o1-Inotion takcs the form
mly+2(rJ+rlyl
: -)pu'n (# . ,"),+
;, n(:ltt):;:;wltnt) (i^t t0t 'tN(i
-'
t-^
235
Y
TJ'
CORNIR RAI]IUS =O O5D
(6.2.12)
Considering the aerodynamic (right-hand) side of the equation as a contribution to overall system damping, the net damping coefficient of the system is
2m(a1
*
)oun
(* . ,,),:
o
(6.2.t3)
5Uo
where, by analogy to the first term of the left-hand side, which is known as
mechanical damping, the second term is referred to as aerodynamic damping.
From the well-known theory of the linear single-degree-of-freedom oscillator
with viscous damping it follows that the system tends toward oscillatory stability if d ) 0 and toward instability if d < O. Since f, the mechanical damping
ratio, is usually positive, instability will occur only if
lrl(;URE 6.2.3. Force coefficients on an octagonal cylinder (G,e: 1.2
(9*.,)
\ dd
/o
<o
symmetry, cannot gallop.
To summarize the problem to this point, the initial tendency of a slender
prismatic structure toward galloping instability can be assessed by evaluating
its time-averaged section lift and drag coefficients and assessing the sign of the
:
the dependence of CTand Cp upon a is known, the coefficients.4, through
, can be evaluated as follows. First, Cp, is plotted against tan cv. Since tan cv
ilu, CF, can then be approximated by the above polynomial using either a
It'ast squares fit or some other technique as desired. Reference [6-48] applies
tlrc method of Kryloff and Bogoliubotr [6-53] to the solution of the resulting
rurnlinear equation, postulating as a first response approximation:
.'1
!: a cos(c,11/ * @)
y : -aor sin(o1r *
*
Cpat a:0.
For many problems of wind engineering this initial assessment suffices to
describe possibilities of incipient instability relative to galloping. For example,
Fig.6.2.3 16-51,6-521depicts the lift and drag coefficients for an octagonal
post structure having a region of wind approach angle (-5' ( cy ( 5") where
the structure is susceptible to galloping according to the Den Hartog criterion.
To pursue the problem further, however, and describe the galloping action in
detail requires full development of Cp, in powers of ylU. Reference [6-48]
suggests an abbreviated power series with several odd powers of ylU and with
an appropriately signed second-power term to smooth the fit:
cr,
^,(L) - ^,(t)' fr -
,,,
106) [6-52].
ll
This is the well-known Glaueft-Den Hartog criterion, a necessary condition
for incipient galloping instability (a sufficient one being d < 0). It is clear
from Eq. 6.2.14 that circular cylinders, for which dCylda = 0 because of their
expression dCl,lda
x
(6.2.14)
(;)' . ,, (.;)' ^,(,r)'
(().2. l-5)
(6.2.16a)
S)
(6.2.16b)
whcre a and S are considered to be slowly varying functions of time. Three
of curv'es Cp" as functions of a and the corresponding galloping
r1'sponse amplitudes a as functions of reduced velocity UID<,:1 are identified
(scc Fig. 6.2.4). The only possible oscillatory motions are those with amplitrrtlcs a traced in full lines in Fig. 6.2.4.If the speed increases fromUoto U2
tl;ig. 6.2.4a), the amplitudc of'thc rcsponse is likely to jump from the lower
t() tllc upper branch of thc solitl curvc. Il thc specd decreases from U2 to Us
tlrc .jump occurs fr<rm (hc ul)l)cr l() llrc: lowcr curvc.
l{clbrcncc [6-491 discusscs llrc loilx)r)sc ol'eklnglrlctl thrcc-climcnsional bodIt's by ttsc ol'thc sccliotltl llrt'oty otttlirtt'rl rrlxrvc lrutl nrcntions tho cll'cct of
lkrw turbulcncc r.rl)on llre g:rllopirrlt. lt is rrolcrl llurl ttrlbulcrrcc clrrr llirrrslirrrn
slt'luly ost'illitliotts ittlo ttttslr'irtly on('s, r('(lu( t'llrt'rrr;rllrrilrrtlc ol'(ltt'irt'txl-yluutrir'
lrrrsic types
236
ntn()LtnsilC l,ilt
{':l WAI\I {inl l()l'lN(i
N()Mt NA
23'7
k1
h1
krz
uo
U1
u2
u/D(4
h2
k2
FIGURE 6.2.5. Schematic of double galloping oscillator.
- hz:6.35 mm and lengths 0.215 m. The spring constants
56 N/m, kz -- 78 N/m, and kn : 145 N/m (Fig. 6.2.5). To prevent
ilr:;pllccments due to drag, the bar ends were attached to fixed points by thin
rvrrcs with lengths r : 400 mm. The bars were observed to gallop in phase,
lrrrt t'xriept for relatively low flow speeds 4 this oscillatory form alternated in
rrrrlrrcrlictable, chaotic fashion with a second oscillatory form wherein the two
l';rrs g,alloped with higher frequency in opposite phases (Figs. 6.2.6a, b). The
urt':rrr cxit time of the system from the region of phase space corresponding to
tlrr in-phase oscillations decreased as the flow speed increased.
( )rrc conclusion of the study concerns basic limitations of empirical fluidi'l;r:rtic rnodels. As is shown in earlier sections and elsewhere in this text, such
rrrotlt'ls can be adequate for some applications. However, it should be rememI'r'rt'tl that the relatively small number of empirical fluid-elastic parameters that
rlt lrnr: the models may not be capable of reflecting in sufficient detail the
, , ,rnplcxities of what is after all an infinitely dimensional fluid-structure system,
,1,'r;r'r'ihcd by a Navier-Stokes equation whose boundary conditions are depenliius with sides ft1
U/D'{U.,
I
FIGURE 6.2.4. Three basic types of lateral force coefficients and the corresponding
galloping response amplitudes tr. From M. Novak, "Galloping Oscillations of Prismatic
Structures," J. Eng. Mech. Div., ASCE, 98 (1972),27-46.
r','rt',1.1
:
damping, and in ceftain cases, depending upon its scale and intensity, destroy
the necessary conditions for galloping. Under certain conditions of an initial
triggering disturbance larger than the steady-state amplitude, certain sections
can experience galloping at much lower velocities than those required in smooth
flow. Finally, it is noted in [6-49] that galloping oscillations also depend upon
the extent to which the mean angle of attack varies as a function of the magnitude of the wind drag.
The closely similar problem of a long flexible beam free to deflect in both
along-wind and across-wind directions is analyzed in [6-54]. Reference [6-55]
discusses the effect of incident wind skewed to the long axis of a galloping
body. For information on galloping tendencies of stranded cables, see [6-901.
rl'rrl rrpon the solution of the system itself. Therefore, unless its range of
6.2.2 Galloping of Two Elastically Coupled Square Bars
I' II WAKE GALLOPING
Reference [6-1001 dcscribcs an cxpcritncnt concluclcrl irr ir wrrlt'r'lrrrrrrt:l on lhc
behavior ol a syslcnt ol'two cl:rstically rcstraincrl arrtl t'orrplt'rl :rlrrrrrinirrrrr :;(lurrrc
,,1
r;rlrtlity is carefully circumscribed, an empirical fluid-elastic model is bound
r,' lrt'inadequate as a predictive tool. We referthe readerto [6-100] fordetails
,rrr tlrt' rnodeling problem for this case study and similar cases.
l'lrc laboratory observations just summarized gave rise to the development
,,1 ;r rrurthcmatical theory of chaotic motions (i.e., motions that are apparently
r;rn(lonr and exhibit sensitivity to initial conditions) applicable to nonlinear
rrrrrltistublc systems subjected to excitation by noise (see [6-10l]). For an apt,lr,;rliorr of the theory to thc problem of wind-induced along-shore currents
rr\r'r ir c()rrugated ocean botltlrrt, scc Scct. 2.5.
llrt't':rst. is rurw coltsitlcrt'tl ol lwo t ylttttlcrs, onr'ol wlrit'lr is locitictl upsllcitln
lltt'otlrct'. Unrkrr ecrlitirr t'olrlrltotu; lltc rlowttsllt'lrttr cylitttlt:t'rrtlty l-rc sttll-
238
nFH()t
lnl;il(: l,l ll N()Ml
li
Nn
:l
Wn
11l
( in I I 1)l 'll..l(
i
?39
FIGURE 6.3.1. Spacer in four-bundle power line.
in a four-cable bundle of a power line.) With the spacers in place, it
r', tlrc cable region between them that is most susceptible to wake galloping
!
'rn(litions since cable freedom of motion is greatest there.
Wake galloping may occur only under conditions where the frequencies of
r,' lx)nse of the downstream cylinder are low compared to its vortex-shedding
lrt'rprcncies and to those of the cylinder located upstream. Just as with the
lrlrt'rrrrmenon treated in Sect. 6.2, wake galloping is governed by parameters
tlr;rl tlcscribe mean (rather than instantaneous) aerodynamic phenomena and can
.r :il)rccr
Time
(s)
lr,' rrrcasured when the body is fixed.
'l'hc wake of the upstream cylinder may be pictured as suggested in Fig.
t, 1 2. Investigating this wake with a "probe" consisting of the downstream
, ylirrder itself reveals a distribution of along- and across-wind forces (Fig.
t' I -1) acting on this cylinder as a consequence of its particular locations in the
urrkc. One important finding is that the across-wind wake forces have a ten,L'rrt'y to center the downstream cylinder, that is, draw it toward the wake
(b)
FIGURE 6.2.6. (a) Observed time history of displacement y'; (b) observed time history of displacements y, (solid line) and y2 (interrupted line). From E. Simiu and G.
R. Cook, "Empirical Fluidelastic Models and Chaotic Galloping: A Case Study," J'
Sound Vibr., 154 (1992\, 45-66.
jected to galloping oscillations induced by the turbulent wake of the upstream
cylinder. This has proved to be the case, for examplc, fbr powcr transmission
line cables grouped in so-callcd buncllcs, that is, lirr grottJrs ol' ctlncluctors
consisting of two, lirtrr, six, cighl, or rrrorc panrlltrl r'irlrles scpltntlctl hy rrrcchanical spucol's irr tlrc rlirccti()rr lliursvL:l'sc to tlrcil splrtt. (l;illrtn'(r.l.l tlc:picts
UPSTREAM
CYLINDER
lll(;Illll,l (r.-1.2. Srr'(ion:rl l',('()nlr'lry. t ylirrtL'r:, irr w:rkt'P,lrlloPirrg
Pltt:rtotttctlott
240
lr:l Wnl,t rinl l{tl,[!{t
Al n()t tnlilt(; l'ilt N()Ml NA
ffiffi=ffiil:t
ft,'n-q{
nffiffi
Wffi
.
LIFT
FIGURE 6.3.3. Qualtitative sketch of the distributions of mean velocity, drag,
lili on a circular cylinder in the wake of another.
and
l'l(;llltlt
f
centerline, contrary to the possible intuitive expectation that, since the outer
flow beyond the wake edges is faster, by Bernoulli's principle it should tend
to pull the downstream cylinder outward, away from the wake center.
An explanation has been sought for this apparent anomaly, which may tentatively be ascribed to numerous criss-crossings of the flow field inside the
wake by time-varying local jets of fluid that have strong components directed
inward toward the center. These jets, or local fluid velocities, would tend to
create repetitive drag forces directed, on the average, toward the wake center.
This view of the phenomenon has been supported to some degree by flow
visualization studies in a water tunnel t6-561. As indicated in Fig. 6.3.3, the
centering lift is strongest at about a quarter of the total wake width outward
from the centerline.
When the downstream cylinder located a few diameters of the upstream
body behind this latter is displaced-for any reason-into approximately the
outer quarter of the wake (see Fig. 6.3.2), it enters a region of galloping
instability. ln this region a galloping motion will begin, growing in amplitude
until an apparent limit cycle is reached. This motion consists of large oscillations in an elliptical orbit with the long ellipse axis oriented approximately
along the main flow direction. The direction of the elliptical orbit is such that
the cylinder moves downstream near the outer edges of the wake and upstream
nearer the center of the wake, or clockwise above the centerline in Fig. 6.3.3
and counterclockwise below it. These directions coincide with the intuitive
assessment that net drag forces will be higher in the outer, faster portion of the
wake and lower in its interior. References [6-56] to [6-65] cover various aspects
of the wake galloping phenomenon. An oscilloscope trace of a developing wake
galloping orbit is shown in Fig. 6 .3 .4 [6-52]. For a useful review of interference
and proximity effects, see [9-ll.
6.3.1 Analysis of the Wake Galloping
ilI
Wul*"**-*
"-*.*rtt"*f#
6.3.4. Amplirude trace of a wake galloping orbit [6-52]. courtesy of
l,rtr.rrrrl Aeronautical Establishment, National Research council of Canada.
the
"rrt' windward, producing a wake, and one leeward, within that wake. The
l.t'wrrfd cylinder will be assumed to be elastically sprung in both horizontal
vcrtical directions about some position (x, y), where X, yare along-wind
'rrr.l
;rrr,l rrcross-wind coordinates conveniently centered on the windward cylinder.
l'lrc cquations of motion tbrthe leeward cylindermay be stated in terms of
tlrt't'xcursions (x, y) of that cylinder away from (X, y):
* lrt'rt:
/r?
mt+d,*IKux*K,ry:F"
(6.3. la)
mli+dry*Kr"x+Knny:f,
(6.3.1b)
is the mass per unit span (normal to the figure) of the leeward cylinder;
,/,. r/,, are respective damping constants; K,,(r, s : x, y) are direct and cross_
,,rrPling spring constants restraining the motion of the leeward cylinder; and
I , , /,',, are the net X- and I-force components.
Ncxl, if c, and c, are defined as the steady average force coefficients referred
r'r lrt't: stream dynamic pressure )p(l')that apply to the cylinder located at point
r \ )'). then it can be shown that the incipient forces in
-r and y directions may
lrr' 1'r111psssd as [6-65.|
U
Y
r@--
EQUILIBRIUM
POStTtON
Phenomenon
Thc phcnonrcnon is lrrrirlyzrtl lrs il'i(s l.xrsic irtgrctliettls wt'tt'lwo tlittrettsiottitl,
:ts wlri tkrrrc irr llrt' pn't'r'rlirrll set'tiotts. ('ortsitlt'l lwo t ylltttlt'ts 1l;i11. (r.1 .5).
241
IYPICAL LIMITING
ORBIT
;';rl
lopirrli lrrlrlysrs.
242
nLn()t
tnlitt(;
t'l ll N()Ml Nn
olllit l\(/), \'(/)l rrriry llc
ctrlt.rrllrlt.rl.
l.l(
.t
24:l
ll tlr.slr,tl. lry rrsirrli
{i I |
: tp'zDt(X . * ft,) - * t-,r, t)
(6.3.2b)
As irr o(hcr trcnlclirsl ic pltc:norrrcrtir, tltrr slr.ut.ltrlrl Plrrirrrrt,lt.rs t.xt,r.l slrorrg
r.rllll)l ovcrlhcchar-nctoristicsol'wakcglrllopirrg. lrrlr:rrl it'ulirr', irrt.irrrylrrgrrrrt
rrr,rtlt'l stuclics thc valucs ol'thc spring consllurls K,,(r., ,r .r,
.1,) r.t.t;uir.e lxrl
Itr'ttl:tt itttcntion. This is cspccially truc irr tlrc rcllrcscrr(irliorr ol'tlrc ucligrr 6l'
xoe)\t
:
Yoert
tox andy in Eqs. 6.3.1
(6.3.3a)
(6.3.3b)
and 6.3.2 and setting the determinant of coefficients
of Eqs. 6.3.1 equal to zero. It follows from Eqs. 6.3.3 thatthe solutions X are
unstable if trr > 0 in the calculated value of form
\:Xr*i\z
(where i : J=), since they then contain a diverging exponential factor. Such
solutions are then sought for the parameters associated with a number of points
Y.
The agreement between the theory and experiment has been found to be
satisfactory, as seen in Fig. 6.3.6 16-56], where the curves indicate points at
which marginally unstable solutions (i.e., where \ : iXz) are found. Forthese
-
;11111
(.1.1.f
.
,;rlrlt's, a sub-jcct that has received rnuch attcrrtiorr
l(r-(r0, 6-(12,
,'rrl:;rrlc thc scope <lf the present discussion.
{i
4
(r-(r-5
1
'l l).
Under the effect of wind, the structure will be subjected to, and will
r l() rcsist, a drag force, a lift force, and a twisting moment. As the wind
i.hrt ity increases, the twisting moment in particular increases also. This in
trrrrr (wists the structure further, but this condition may also, by increasing the
,llctlivc angle of attack of the wind relative to the structure, further increase
tlr. twisting moment, which then demands additional reactive moment from
rlrr' :;lnrcture. Finally, a velocity is reached at which the magnitude of the windrrr,lrrt'r'rl moment, together with the tendency for twist to demand additional
'.rrr( lurlll reaction, creates an unstable condition and the structure
twists to
il':,rrrrc(ion. The problem is one of stability, quite analogous in a structural
to column buckling. Just as column buckling occurs when a critical
",'rr,('
,r
'lrrrrn load is reached, torsional divergence occurs at some critical divergence
r.l't'iry of the wind. The phenomenon depends upon structural flexibility and
tlrr' rrIrnncr in which the aerodynamic moments develop with twist; it does not
ir(
rL 1rt'rtrl upon ultimate structural strength.
lrr tlrc case of thin airfoils, the aerodynamic twisting moment increases with
angle of attack. In other, more complex structures, it may be that
rrr, rt'irsccl
EXPERIMENT
x/D
II
Ali|(i
AX I'J
Y/D
FIGURIt 6.3.6. Mclsurctl :rrrtl prcclictccl stability hottrttlrttit's
561.
lirl
is
TORSIONAL DIVERGENCE
THEORY
-5
but
llrt' phcn<lmenon of torsional divergence was at first most closely associated
rrrrlr :rircraft wings and their susceptibility to twisting offat some excessive
air
',g','t'tl. 'lir fbrm a conceptual picture of what occurs in such
a situation, consider
;r tlrrrr uirfbil, or any other analogous structure, such as a bridge deck (Fig.
(r
x:
Y
l6
litls.
(6.3.2r)
where U is the free upstream velocity and U, is the average wake velocity in
the udirection at (x, Y), and D is the projected across-wind dimension of the
cylinclcr. Expressions similar to Eqs. 6.3.2 were first developed in [6-58] and
16--591. Values of C", C' and their derivatives are obtained by direct measutrnlont ol'timc-avcraged values in wind tunnel model studies. Cases of inlcrcs( havc conccrncd smooth circular cylinders and the rougher surfaces of
s(randccl wire cables.
Analytical solution of the problem, in which the forces given by Eqs. 6.3.2
are clcarly self-excited only, proceeds by assigning values
X,
',r'lrrlions, tltc
,l lr rl l:,1( lll^t lrlvt il( it
, .. "rt'n) , ,', ;, - ,.', ,),, ]
r;" -- )pr./2D[(a#
r,
li
wrrke grrlloping
I"l(illlllt) (r.'1.1. (it'otttt'lty irtt{l l)it!;trttr'l('t', l'r l.r:.i,rr:rl rlrvt.rllt'rrt.t.prrrlrlt.rn
244
At
ti()t
ln:;ilo
t,l
il N()Mt Nn
li4
the acnldynarnic twisting nl()nrcnt rhrtrs ttot lirllow tlris sirrrplc lr.:ttrlettcy. As it
result such structures may not lirllow tlrc pirltcrrr clcscribcd ahovc; in lirct,
depending upon the relation bctwccn acnltlynarrric rnomcnt and anglc of attack,
some structures may be immune to torsional divcrgcnce. Finally, it should be
noted that in most cases of practical interest in civil engineering the critical
divergence velocities are extremely high, well beyond the range of velocities
normally considered in design.
be denoted by
\pU)ll1cr,,
u":
* (t,izrfyl
A,,rv
1(r..t..{
)
rvlrcrc
-t
k" and a, respectively.
245
lrrlrrlrlittg lhc lrcrrttlyrlrrrrit'lo llrc irrlc:r'nirl slnrt.lrrr.lrl rrrorrrt.nl It'irrls lo llrt' t't1ur
LMo -
Assuming that the mean wind velocity is U and that the deck width is B,
the aerodynamic moment per unit span can be written as
N(:l
lt()lt
6.4.1 Analytical Modeling of Torsional Divergence
To analyze the torsional divergence phenomenon, consider, as in Fig. 6.4.1,
the section of a structure that can rotate against a torsional spring about some
pivot point (or elastic center). Let the spring constant and the angle of rotation
l()l t!;lrtNAt l)lvf n( it
acul
,
da lo=o
I
(6.4.s)
llrc divergence problem is summarized (in this two-dimensional description)
lry liq. 6.4.4. We now examine its solution.
l)cfine )t : )pu'zE. Equation 6.4.4 then becomes
(k"-)\C'Mo)o:XCys
(6.4.1)
|pUzB'Cr(o)
where Cy(a) is the aerodynamic moment coefficient about the twist axis. An
example of the dependence of Cy upon cv in the case of an open truss bridge
deck is shown inFig. 6.4.2.
At zero angle of attack the value of this moment is
M,(O) : )pU2BzCro
\Cro
k"
l'lrc solution of Eq. 6.4.6 for
cv
>\Cho
(6.4.6)
approaches infinity (diverges) for the value
6.4.2)
where Cye : CyQ). Fora small change in c away from a :
-
.k"
A:-
O, Mo may be
(6.4.7)
C,,
given to first approximation by
l lris therefore defines the
M,:;pu'Elr^ *
dCrl
do l":n
"]
critical divergence velocity:
(6.4.3)
(6.4.8)
'l'hc problem may readily be generalized to three dimensions, but
this is
rt'scrvcd fora specific application in Chapter 13 (Sect. 13.1.2).It should also
lrt' noted that the problem considered here is that of incipient instability only.
ll rrurre complex structural action with increasing velocity occurs (due to a
o (DEGREES)
rrrrrrc complex curve of Cy vs. a, for example, than that shown in Fig. 6.4.2),
tlrc rlivcrgence problem can be solved by a systematic solution of the relation
t,pl
l)
d {'n,1,r) -
k,,rr
(6.4.e)
FIGURE 6.4.2, Monrcrrt cocflic:icnt lirr a blull'stnrclurc irs ir littttlion ol'irnglc ol
Iol tttty titngc ol'vclocilics rlt'sitt'rl= 'l'lrc prlsrril ol this pr<lblcrrr is l-rcyorrrl lhc
attack.
:rirrr ol' this scction-
246
6.5
n
I I t()t:t Ati
I
l(;
I
'l
ll
N(
r;f, tlilil|t
)Ml Nn
FLUTTER
One of the earliest aeroelastic oscillaliorrs lo bc rcc<lgnizcd was thc lluttcr ol'
airfoils. The term "flutter" has been variously uscd; recently, htlwcvcr, this
use has become more restricted. The most common present uses of the term
employ additional qualifying terms, for example, classical flutter, stall flutter,
single-degree-of-freedom flutter, and panel flutter. All of these terms were
originally employed in aerospace applications, but some have carried over to
wind engineering.
flutter oiginally applied to thin airfoils. The term also finds application today to suspended-span bridge decks. It implies an aeroelastic pheClassical
nomenon in which two degrees of freedom of a structure, rotation and vertical
translation, couple together in a flow-driven, unstable oscillation. Coupling of
the two degrees of freedom-indispensable for thin airfoil flutter under normal
structural circumstances-has come to be the identifying sign for classical flutter.
Stall flutter is a single-degree-of-freedom oscillation of airloils in torsion
driven by the nonlinear characteristics of the lift in the vicinity of the stall, or
loss-of-lift condition. This phenomenon can also occur with structures having
broad surfaces that can stall depending on the angle of approaching wind. Socalled "stop-sign-flutter," the torsional oscillation of traffic stop-signs about
torsionally weak posts, is an example in a nonaeronautical area.
Single-degree-of-freedom may include stall flutter, but may simply be associated with systems undergoing strongly separated flows. Bluff, unstreamlined bodies are typical examples. Prominent among these are the decks of
suspended-span bridges, which can in various instances exhibit single-degree
torsional instability. These are discussed in more detail in Chapter 8.
Panel flutter is a sustained oscillation of panels-typically the sides of large
rockets-caused by the high-speed passage of ait along the panel. The most
prominent cases have been in supersonic flow regimes and so have not appeared
in the usual wind engineering context. Flutter of taut canvas covers and flag
flutter are, however, phenomena related to panel flutter.
It is likely that, in its detail, flutter in practically all cases involves nonlinear
aerodynamics. It has been possible in a number of instances, however, to treat
the problem successfully by linear analytical approaches. The main reasons for
this are two: First, the supporting structure is usually treatable as linearly elastic
and its actions dominate the form of the response, which is usually an exponentially modified sinusoidal oscillation. Second, it is the incipient or starting
condition, which may be treated as having only small amplitude, that separates
the stable and unstable regimes. These two main features enable a flutter analysis to be based on the standard stability considerations of linear elastic systems.
It is characteristic of flutter as a typical self-excited oscillation that a structural system by means of its deflections and their timc clcrivativcs taps <lll'
energy from the wind flow. lf'thc system is givcn an irritiirl tlisltrrbancc, its
motion will cithcr rlccay rlr tlivcrgc (i.r:., its oscillllirlns will lrt'tlrtrrt;rt:tl rtr will
241
1'row itttlcrlittilr.:ly) itct'onlrrrg lo wlrcllrcl llrt'cnctJ'.y ol rrrolton t,xlr;rclt.rl lroln
tlrt'llow is lcss llr:rtt ot'cxccctls tlrc crterrgy tlissipirlt.rl lry llrr'sysl(.nr llrrorrg,lr
rrr,'.'lt:ttticltl dalnping. 'l'hc thcorctical clivitlirrg lirrt. lrt.lwr't.rr llrt' tlct':ryirrg :rrrrl
rlrvt'lgcttt cascs, nalllcly, sustaincd sinusoitlal ost'illirliorr, is tlrcrr r.r.rt.1lgrrizctl :rs
tlr,' t'ritical fluttcr condition.
lrr llro treatment of flutter, in the prcscnt wirrrl cngirrccrirrg corrlcxl, orrly
, l,r:;sicll lluttcr and single-degree-of-fiecdorn llr-rttcr will bo cliscussccl.
|i
5"1
Equation of Motion for an Airfoil or a Bridge Deck
('r,rrsitlcr a section of an airfoil or a bridge deck (Fig.
6.5.1) subjected to the
;rr rr()n ol'a smooth oncoming flow. The section is assumed to have two degrees
r'l lr.trtf<rm: bending displacement and twist denoted by h and cv, respectively.
.\ rrnil span of the system has mass tn, mass moment of inertia 1, static unlr;rlrurcc s (equal to the product of mass m and a distance, ab which separates
tlrt' t'r'rr(or of mass from the elastic center),* vertical and torsional restoring
lirr(('s characteized by spring constant c1"and c", and coefficients of viscous
,l;rrrr1rirrg c1,&ndco.
withthesedefinitionstheequationsofmotioncanbewritten
lt, ()(r. 6-67J
mi+sa+coh+Cph:Lt
S1;+ld]-coarCoa:Mo
(6.5.1a)
(6.s.1b)
0rB4
)*-
l,'l(
jl lltl,l (r.-5.1.
Ntrt:rtions
'Nllt llr;tl willr lr lixt'tl sign (onv(.nliotr, ,\ trr:ry lx. ;rl,rlrvr. {}t n(.},,:tliv(.tlc;.rltlilrg 9rr llrc l6clt(i9rt
rl,r\\,;il(l 0r ttll) 0l lltt'et'ttlt'l ol rrltss $tllr rr..,;r,tI lo llrr.t.l;rsltr.r.t,rtlt.t.
248
nl lt()l
lnlill(;
l'l
I
I r,ilt il
249
where L1, and M,, arc:, rcspcc(ivcly, (lrc st:ll-cxcitul lrcrorlytrirntic lili irntl ttrrlment about the rotation axis pcr uttil sprttt. l)csignating by ru thc radius ol'
gyration of the body about the centcr o1' rotation and using notations sinrilar trl
those of Sect. 5.1, Eqs. 6.5.1 become
mlli + ad
t
2(6a6h
+ af,t4 :
f a..
Il-h
+ & + 2l,a.a + r3rl
lr;
ro
(6.5.2a)
:Mo
(6.s.2b)
whcrc f,,, f,, arc damping ratios-to-critical, and c,.r6, @q are the natural circular
licc;ucncics in h and a degrees of freedom, respectively, defined by
,cn
@n:
-m
,Co
,":
i
(6.5.3a)
(6. s.3b)
In the case of bridge decks that are symmetrical, the center of mass lies in
the vertical plane of the centerline. In this case a : 0. Usually the rotation
axis lies in this plane also, though it may be at some vertical distance from the
center of mass. In the case of bridges with arched decks the effective rotation
axis may lie well below this center. When accounting for the dynamics of the
deck, the mass moment of inertia 1 is calculated above the effective rotation
axis and hence is typically, even for a uniform deck, a quantity that varies
300
*=+
l''l(JURE 6.5.2. Real and imaginary parts of the Theodorsen circulatory function c(K)
tt(K) + iG(K).
'fhe theo,retical expressions for sinusoidally
oscillating lift z and moment M
airfoil are, respectively:
orr :r flat plate
Lr: -pb2((Jra t rti - rbait) -
across the span. Actual determination of the effective rotation axis is a structural
problem outside the scope of the present discussion.
2rpC(k) lUcx + h + n1|
- Ocj
(6.s.4)
Mo:
6.5.2 Aerodynamic Lift and Moment
In the case of thin airfoils in incompressible flow, Theodorsen [6-66] showed
from basic principles of potential flow theory that the expressions for Lp and
Mo are linear in h and a and their first and second derivatives. The coefficients
in these expressions, referred to as aerodynamic cofficients, are defined in
terms of two theoretical functions F(ft) and C,(k) 16-661, where k : balU is
the reduced frequency, b is the half-chord of the airfoil, U is the flow velocity,
and r,r is the circular frequency of oscillation. The complex function C(k) ot
which F(k) and G(k) are the real and imaginary parts, respectively, is known
as Theodorsen's circulation function (Fig. 6.5.2). For aircraft flight regimes
in all velocity ranges, wide research has developed further analytical expressions for all necessary aerodynamic coefficients. There exists a vast literaturc
on the subject, to which [6-67] to [6-701 and [6-951 arc usclirl in(rocluctions.
Attention is confinccl hcrc lo thc low-spccd incomprcssiblc lkrw n'giltrc.
{"(l - euba + rb2([ +
- arbi]
+ 2pUb2r(\ + a)C(k) [Ua + h + U1] - dc1
-pbz
a2)ix
(6. s. s)
wlrcrc
c(k):F(k)+ic(k)
(6.s.6)
I'
balu is the reduced frequency, <,r is the oscillation circular frequency, b
rs crlual to Bl2, B is thc chorcl of the airfoil, p is the air density,
u is the
rr;rlrnrach laminar flow vckrcity , ttlt is thc distance from the midciord to the
r.lirtion point, ancl rr antl /r rrrt.. n's1'lcctivcly, angular rotation and vertical
tlislrlaccrncnt, l6-661, l6-671. 'l'lrt' firrrcri.rrs
(r.5.2.
lior blull'objccls ol'wirrtl
/'Ik), G(k), are shown in
(.nl1ut('(.trrr1, irplrlit':rliotrs,
it
Fig.
has n<ll l<t tlirte ltcerr
*J
25O
AERoflntillo l'lll
N()Ml
|]r, IIiltiltt
NA
possible to develop cxprcssi<lns lilr thc ircrotlytuttttic cocllicicrr(s stltrting I'rtlttt
Lasic fluid-flow principles. Howevcr, it has bccn shown in [6-7 ll that litr srnall
oscillations the self-excited lift and moment on a bluff body may bc trcatcd as
linear in the structural displacement and rotation and their first two derivatives,
and that it is possible to measure the aerodynamic coelficients by means of
specially designed wind tunnel tests. Such tests indicate that just as in the case
oi the airfoil the aerodynamic coefficients of a bluff body are functions of the
reduced velocity.
Various forms for the linear expressions for L1, and Mohave been employed.
Thc classical theoretical (and some experimental) work has used complex numbcr lirrms based on the representation of the flutter oscillation as having the
complcx fbrm ei''. However, in the wind engineering practice developed to
tlatc in the Unitcd States real forms have been employed. Below are stated
commonly uscd lincarized forms of this type [6-71]:
Ln: lpu'nfraftrl Lr+ xuitxrui * xznl(x)a + K'Hf *)
(6.s.1)
L
Lr+ rc$rxrui
* xz,s,t(x)a + K'4*]
Ba
K: U:
B(2rn\
(6.s.e)
../
rvlrcrc the coefficients kL,
quantities a, hlu, and BalU are effective angles of attack and therefore also
nondimensional. The typical term in Eqs. 6.5.7 and 6.5.8 can be viewed as
following the classical pattem of expressions for aerodynamic lift force per
unit span, such as
*The reduccd frcqucncics li, usotl in ucnrnuutical practicc, and K, ttscrl ilt witttl ctlgittt:crirrg' dillct
in that ft is dcfinccl in tcrrrrs 0l tlrr: hull.chortl lt - Ill2, whctcits li)r l(':ls()lls ol toltvt'trit'ttt'c K is
tlolinctl in lontts ol thc lirll cllrll lJ, its irt li1. (r'5.().
rl,v
,,
((r..5.lOl
h
\*I
m'l
1-
U
k';, etc., have
h
,DA -l
d\
+m'i-l
@/
m'6a
(6.s.11)
(6.s.12)
come to be called the "Kiissner coef-
lrt'icnts. "
ln real terms, the following equivalences among the coefficients of the above
rxlrcssions may be verified, when oscillations
U
B is the chord, deck width, or along-wind dimension of the structure, U is the
uniform approach velocity of the wind, and <o is the circular frequency of
oscillatioL(i is the frequency of oscillation). In Eqs. 6.5.7 and 6.5.8 terms in
ii, Ahave been omitted as being of negligible importance in wind engineering.
(ln aeronautical practice terms in ti and it but not h areretained.) The coefficients II,t and Af (l : 1, 2, 3) ate nondimensional functions of K' The
t,ttll'll'l(,'
h
"/
* o:**ki,a+ol::.\
-rpu'b
D@
e/
\0, t
Mu: -rpU'b'
(6. s.8)
where additional terms in h are included and the reduced frequency K is defined
as*
=
lol surall anglc ol'attack a. Formally, (cnns suclr lrs K// jr' or' A'il I lrrc lhrrs
rrturlogtrus to lift coefiicient derivatives tlC1,ldu.'l'lrcsc lcrnrs slrorrltl bc re lcrlctl
Itt as vnrr|loral derivatives, however, and thcy go ovol' into stoacly-stalc rlcriv;rlivos, such as dC1.ldu, only for K - 0 (zero liocluorrcy). Irnrrn an cxpcrintcntal
point of view this means that the aerodynamic cocllicicnts of Eqs. 6.5.7, 6.5.8
, :ur be measured only if the body is in an oscillatory state, whereas dCylda is
ohlrrined under static conditions (i.e., with the body fixed; see Sect. 6.2). The
l:rt'trrrs K or K2 preceding Hf and,4f could just as well be included with these
lrr(lcr in a total coelficient of some other designation if desired, but the evolution
ol lhc theory [6-71] has identified them as nondimensional factors. References
| {r 7 I I through 16-771discuss various experimental techniques used in the United
Stir(cs, Japan, and France for obtaining the nonstationary aerodynamic (flutter)
tlt'r'ivatives. In France, through usage at ONERA (Office National d'Etudes et
rlt' llccherches A6rospatiales), the following alternate forms [i3-45] have been
t rst'tl:
Lr" :
M,: )p(JzP[*fto
t,1t(t'I)('t
251
h:
hoei''
(6.s.13)
Ot :
d6€
ioi
(6.s.14)
.ur'postulated:
K)Hf:*2rk']:-2rKF
(6.s.1s)
K)rr!
4c /t
- -rk'i: -rKl
; lt,o*r\;-')r)
K)H{
: -trki,:
h')rt.l'-,
-*1r,, (j
zr ,l
]rkl.
" I A lll
|
,,) ,,n t "ry:
l(;l
I
^l
\
I
(6516r
(6.-5. r7)
(6.5. lr()
/
'
AEROELASTIC PHENOMENA
KzAf
: -rm'l
-n
'2:
^'l
K2,1,*
:
trKF
(i . ,)
:;1-;t
: t *t
04
00
-04
-,o (". ;) . *o (,, - i)]
In case a
-trm'o
:
+ 2KG(, . ;)]
_B
i
-12
0
-16
-20
I
a 2 4 6 I
101214
:
+lt *f, * ,)
K'H{:-"lro-Tl
(6.s.22)
-0.
-0.
: | rcn
K2AY:;l;-"-T)
rc2t{:llt*"_{91
- 2L32
4)
:I
wo1
D2 4 6 8
U
0
\*
? og
1.6
(6.s.23)
2
0
-2
a 2 4 6 B 10 12 14
0 2 4 6 B t01214
U/nB
lllnB
5
2
0
(6.s.24)
2
-?
XX
w\^
1
(6.s.2s)
(6.s.26)
tD1214
/nB
4
4
0
K,Htr:;"I'-T]
I
x
U/nB
-L.2
K'H7: -2rKF
x''qf
/
3
the above equations reduce to
x,ef
0
4
0 (common for bridges, though not usual for airfoils in aircraft),
KzHt
0
/nB
5
:;lt (* . *) . ,. (" .;) + KG("' - ;)l
-;l+
-:t l,.4 [_
a 2 4 6 B rA1214
l.i
(6.s.2t)
KzA; :
i!
0,8
(6.s.20)
K2A{
tl
L,l
(6.s. re)
4
b
_B
-10
a 2 4 6 I 101214 0 2 4 6 I
U/nB
Il/nB
^
''+_-
NORI\IANDY
CREAT
(6.s.27)
}
TSURUM
ID1214
I
BELT _€_- AIRFOIL
(
EXPERIMENIAI
I
(a)
trf and Af for a thin airfoil (i : 1,2,3)
: l, 2,3,4) shown in Fig. 6.5.3b. After [13-109].
lrlGURE 6.5.3a. Aerodynamic coefficients
(6.5.28)
rrrrcl
three streamlined box decks
(i
.!
fM. rr llr
(6.s.29)
(6.s.30)
Sample experimental values of the coefficients r1f and Af for streamlined
bridge sections are shown in Fig..6.5.3, where forpurposes of comparison the
analogous coefficients Hf and,ef rcr a thin airfoil are also given.
8.5.3 Solution of the Flutter Equations
llccause of $gge_pendgncg
9{ thg aerodynamic terms up,gn K, the analytical
rlrlution .ql-lbg.-fluJte"f problem becomes more involved ihan ttre compai?bte
stubility solutions where quasi-steady aerodynamics holds. under K-dependent
crunditions, a typical solution method is as follows. A value of Kis choien and
thc values of r1f and Af conesponcling to rhut value"iie obtained from plots
ol'these experimental functione, It is then nssumccl that h and cv have soluiions
lrnrportional to €i'r which arc inserted lnt() Eqs, 6.5"2,6.5.7, and 6.5.g, The
,!^\r.."-
{
/
.
254
Al ll()t lAiiilo I'l il N()Mt Nn
ri', IIt,tilil
:]6000
-.-....-
Il.'A.
Ilcrc:ausc
25b
Il.,l,
1{r.5.
of its intsrost in applicatiorrs, u usclirl vlu'itrrrl on thc solrrlion
tl
)
orrl
lrrrctl abovc is skctchcd bclow.* Let
s:-
TSURUMI FAIRIA/AY BRIDGE
Ut
(6.s.32)
B
l,r' rr n<lndimensional time (or distance). Noting that
o:T:#f,:,
31000
GREAT BELT EAST BRIDGE
(6.s.33)
)'vB
lrtlrrrrlions 6.5.2 and 6.5.4 can be reduced to
It" h'
.h
| 2(nKnE
+ Ki;:
oB2
l
h,
Tlorr ; i
tr
KHla,
+ x2H!a + x'nr- L]
Bl
(6.5.34a)
,t" + 2loKoa' +
NORMANDY BRIDGE
K2.u
:
4l
II *nrLB + KAta' + K2Ala + K,4
*f
(6.s.34b)
: BallU, Ko : BaolU.
l'osing now the solution forms
rvlrcrc K6
AIRFOIL
(b)
!:bri't
B B.
FIGURE 6.5.3b. Box decks for three bridges (dimensions in millimeters), and airfoil.
After [13-109].
determinant of coefficients of the amplitudes of h and o is then set equal to
zero as the basic stability condition. This constitutes in fact u .ornpl"" quartic
equation in the unkno*n flutt", frequency c,r, which must then be solved. The
solution obtained will. in general. be of the lorm o : .,r * rc,.r2 with u2 * 0.
and will therefore represent either a decaying (r,lz > 0) or a divergent (co2 (
0) oscillation. A new value of K is then chosen and the procedureis ,"p"ut"d
until the solution_is-purely (or very nearly) imaginary, tirat is, until <,r, Q,
=
so thal Q = @r. To that solution lhere corresponds th! flulter condition
at real
frequency co,. Let l(, be the value of K for which @ : @t Thc critical flutter
velocity is then
a:
-honix'
B
&oei(t'* O) : uoei't
:
(6.5.35a)
(6.s.3sb)
o4eiKt
lrr;rrirtions 6.5.7 take the form
I
| -,r' + 2ifhKhK +
I
f oB2
- l;
oB2 .l
iK'HI +'- K'HT
il lrrlrkt' ltitctitlt wirtgs, britlgc tlt:t'ks
,,1
Ki-4(iK2Hf+x'uttl\
'' m
-'lB
lcvg
: 0
ttt:ry cx;r'ricrrt t' sigrrilic:rrrl
llt('(lriUl lirrcc).'l'lris is txrl litkctt ittlo irtcorrrrt
(6.5.36a)
swly (rnotion akrng thc dircction
rrr tlris st.r'liorr, lrrrl scc
lit;s.
13.1.43.
256
nLnoLIn:;ltC t,ilt N()Mt Nn
l-+
eKzA{ +
(i
FAi'
I
nd :,,q!-4r,n1l*o-u
-i'^
- r -l
Deflning an unknown X
x--
(6.5.36b)
(6.s.37)
{n1
of Eqs. 6.5.36 equal to zer-o results in a complex
degree four. This breaiii down inio-*two real equations,
assuming that X is always real at the flutter condition. These two equations are
' solved successivcly fbr different assumed values of K, and their roots X are
plotted as functions of K. At the point (X,, K,) where the two plots cross,.the
and setting the determinant
rpolyndriiial in
r
I
,
i
Xof
flutter condition is identified [6-66, 6-671.
The flutter problem as treated above is seen to be a semi-inverse oroblem
srqle-the aerodynamic coefficients are functions of the solutioir fiEffil;-unO
a range of frequency parameters K m11st therefore be used to survey the solution
region.
Altemate methods are also available, though they are beyond the scope of
the prbsent discussion. One of the more important of these approaches involves
the use ol aerodynamic indicial funcrions [6-671 ro 16-701 and t6-78. 6-79].
Such functions, derivable from the coefficients H! and,4,I , represent the response of the bluff section to a step change in angle of attack. They also permit
representation of transient response problems under the general hypothesis that
linear superposition of effects remains valid. Reference [6-80] makes use of
individual response functions in predicting bridge response under natural wind
(see also Sect. 6.6). In general, the use of such functions gives rise to more
involved calculations than the stability determinant method sketched above.
Avoidance of the more general indicial function approach is justified in those
cases where structural frequencies and natural modes are not greatly altered by
the aerodynamic forces.
- /
the fluttei equations and the nature of the flutter phenomenon in the case of
bridges as opposed to that of airfoils. In the flutter of airfoils under normal
structural conditions (center of mass not excessively far aft of the rotation point)
it is impossible for single-degree-of-freedom flutter to occur since both degrees
h and u are individuilly positively dampedx (i.e., Hf and A! are negative for
*Because
of the formal similarity between the mechanical damping terms in the left-hancl si6cs
of Eqs. 6.5.2 and the terms containing the coefficients af and;f in eqs. 6.-5.7 and 6.5.g, tho
latterare referred to as aerodynamic damping tcrms. The diflbronccs 2(j,o,,rrr \pUr@)XUl anl
2(.u,,1
l}t.,t
ll llN(i lil :;l'oN:,1 tN illl I'1il l;l NCI ()l nl ll(tl lA!ill(. I'lll Nr}Ml }!n
257
;rll valucs ol'K). 'l'his is tlrc birsic rcirsort wlty t'lirssit'rrl ;rirloil llrtllr'r', il rrrrtl
vrlrorr it occurs, rrrrrst irrvolvc couplcrl llcctkrttts; llrirl ts. ll nnrsl lrr'ir torttltliott
rrr which it is mainly thc coupling (not tltcr tlrrrrrlrirrg) lct'ttts llrrl liovt'nt lltt'
K:,
as
(,
{i
- |oUt{n'1X'l'{{U {,/) arc rclcrrccl to as nct (or total) tlarrrpinll irr tlrr.tlrlrslirtionirl untl tlrt:
rotational mrxlc, rcspcclivcly. (Scc also Scc(. (r.2.l.)
I('sl)OnSC.
On theotherhand, as shown in [6-7 Il, ccrtairt ty;lcs ol'struclurc (c:.g., sotttc:
opcrn-truss s_uspension bridge decks) exhibit,rl-j (torsiorrirl tlarrrping) cocf licicrrls
tlur( change s!gn-from negative to positive with aclvancing values ol'rcduccd
wirrcl velJcity*UlnB (where n : itZn). As a rcsult whether or not coupling
t ocllicients exist, single-degree torsional motion becomes unstable and drives
;r sclf'-excited flutter due to its net negative damping. Thus purely single-degree
lf trttcr, or "single-degree-driven" flutter, can exist for cases where Af evolves
.rs tlcscribed above.
'l'he flutter of three-dimensional structures is essentially based on the twotlrrrrcnsional theory presented above and is discussed in Chapter 13.
6.6
BUFFETING RESPONSE IN THE PRESENCE OF
AEROELASTIC PHENOMENA
Itull'cting is defined as the unsteady loading of a structure by velocity fluctuaIrorrs in the oncoming flow. If these velocity fluctuations are clearly associated
rvith the turbulence shed in the wake of an upstream body, the unsteady loading
rr lclbrred to as wake buffeting. Effective analytical models of the wake bufIt'tirrg phenomenon do not currently exist in the wind engineering field. On the
orlrcr hand, notable contributions [6-82] to [6-85] have been made to the prob-.
k'rn ol the bu.ffe-tjgrg of_linelike stru_qjllres by atmospheric turbulence. Many of
llrt' icleas employed below can be traced to origins in these references.
'l'hc problem dealt with in this section is that of buffeting by incident turlrrrlt:nce that develops in an atmospheric flow over relatively homogeneous
rr'r ririn-open, suburban, or urban. For such turbulence it_is possible, in certain
r :rscs, to
_!hg response to buffeting forces for .bo,th those structures that
"Sp-q_gl
rkr not andihose that do exhibit aeroelastic interaction"with the wind forces.
'i'r'tion 5.3' deals with aerodynamic loadings that are independent of structural
rrrotion. However, structures like slender towers or the decks of suspended,,1r;ur hridges, which exhibit aeroelastic effects, are also of considerable interest
rrr prlctical applications. The present section is concerned principally with the
r{'slx)nse
of such linelike structures.
ti-6-1 Aerodynamic Forces on Linelike Structures
('()nsi(lcra linclikc structurl:. willt sprutwise: r'rxlrrlinatcx, that is being buffeted
lry rrtrrrosphcric turbuloncc. ll tlrc ost'illirliorrs ol'thc stnrcturc in each responding
rrrorlt' itrc srrrall, il rrriry bt: irssrrrrrt'tl tlr;rl tlrt' rrcrotlyttatnic bchavior of thc
',lnr('llrrLr is linciu. 'l'hc: ltcrorlyn;uruc lort't's t'orrsisl ol'rr strpcrposition <ll'(l)
,
1
"'l
258
At n( )t t Al;
nc
I'l il
N( )Mt NA
r;{; l}t,l llllN(i
self'-excited lirrccs ol'llrc tyllc tleralt witlr irr Sct't. (r.-5 irrrtl (2) brrlli:lirrg lirrcc:s
induced by the incidcnt turbulcncr:.
Bufteting Forces. For turbulence intensities typical of winds in thc atnrospheric boundary layer, and for turbulence components with fiequcncics that
are of interest in practice, it
1ryy*.pe .assumed that the squares ancl products ol'
the velocity fluctuations u. u. and, w are negligible with respect to the squarc
of the mean velocity U and that the force coefficients Cp , C1., and, C11o arc
independent of frequency in the'?nge considered. As a result expressions for
the buffeting forces based on quasi-steady theory are acceptable, so that for
scction ,r of the span the buffeting drag, lift, and aerodynimic moment (sec
Fig. 6.6.1) can be written as
6NTE
: c,r(oi f
l+2
-L(r)
: c.r*o
* , u(r.ultlf *lel
D(t)
FNTE
[r
u(x- t\
(6.6.1a)
U
I- dal'd:q0
**",a]ry
(6'6' lb)
Fih lr^*, + cD(ao) #ll' *, ryfl. *1,=,,*f
M(tl
:I
lr l I
tt(v
t\ I
sr |
(6.6.1c)
where B is a typical body dimension such as deck width, ,4 is the
across-wind
area per unit length projected on the plane normal to the mean
wind speed u,
T
lilt;l'ol.l:,r ltJ llll
l,l rl
til N(;l ()l Al ll(,ttA'iil{
I'1ilill)Mt flA
r
2lio
rs lltc tlistrtttcc ol lltc tlt't'k nutss ccrltcr lo lltt't'llt'tlrvr'rol:tltorr;rrt:.. l/
r.{/)iul(l rr(/):rrt: tlrc wintl sPct:tl corttlxrttt:rtls irt llrt';rlorq;' wrrrrl ;rrrtl llrt'vt'r'l rt';rl
rlnt't'(iotts, l-cspoclivcly,'r'iuttl rvly is lltc rrtcirrr ltrrglc ol itllirck rrrrtlcr wrrrtl irtlrorr.
lrr li,t1s. 6.6. lb artcl (r.(r. lc thc dintcnsiottlrrss rtrlio rr'(l)/l/ rr'prr':i('nls :tn ;rr111rrl:rr'
llrrt lrlrtion l'rom thc mcan anglc o1y. ln lic;s. (r.(r. l , lhc tplrrrtity l l I )rt(tll I ll
r', olrtrrined by squaring the sum ll + u(t)ll.l Ilrrrtl negkrc(irrg lltcr st;trirrrr ol ils
,,,'. on(l tctm, as shown in Sect. 4.7.
I
Self-excited Forces. lt was indicated in Scct. 6.-5 that fbr a body oscillating
u,rllr circular frequency o in both the vertical displacernent and the torsional
rrrrrtlcs, the self-excited lift and moment L1, and Mo may be expressed as in
l r1s. 6.5.7 and 6.5.8.
Since the random buffeting load action on a structure may be viewed as a
'.rrptrrposition of elemental harmonic loads (see Appendix A2), the vibrations
nl llrat structure may, conespondingly, be viewed as a superposition of harrrronic responses induced by these loads. Each such oscillation induces, in tum,
.rrr
clcmental self-excited load expressible by Eqs. 6.5.7 and 6.5.8.i
{i.6.2 Buffeting Response of a Suspension Bridget
l'()r nlany types of bridge deck sections the aerodynamic coupling coefficients
6.5.7 and 6.5.8 may be disregarded in first approximation as having
run()r or negligible influence, so the vertical and torsional motions of a straight
lrrtlgc may be taken as uncoupled. The aerodynamic coupling coefficients are
,,1 socondary importance particularly in those cases of common occurrence
rvlrt:rcin single-degree torsional instability is manifest (i.e., where,4f changes
',r1in with increasing UlnB).
lixpressions forthe bridge response will now be sought following a proce,lrrlr closely parallel to that employed in Chapter 5 to study along-wind re',lx)nsc. Here, however, the effect of aerodynamic self-excitation terms will be
t:rkt:n into account in addition to the aerodynamic buffeting forces.
'lirrsion will be dealt with first. Consider a full bridge for which the torsional
rrr l,)qs.
h
rl (luirtions 6.6. lb and 6.6.1c are written assuming that the linelike structure is horizontal (e.g.,
I'ritlgc). In the case of a vertical structure (e.g., a tower), the vertical velocity component w(r)
.r
rrrrrsl bc rcplaced
Lft)
c.m.
=
I
-a-
r.o. =EFFECTIVE ROTATTON
AXIS OF SECTION
I
r.o.
FIGURE 6.6.
l.
Bull'cting r.rccs
MASS CENTER
OF SECTION
'n
sccri.n
.r'. li'clik.
srnr(.rrrc.
by the lateral velocity component z/(t).
'\n crluivalent altemative fomulation is to employ the aerodynamic indicial function approach
l(' /ll. 6 79, 6-80, 6-971 wherein the frequency-dependent information contained in the self, \r'ilrli()n acrodynamic coefficients n! and,4f is first converted into time-dependent indicial
,r,'trxlyruttttic lunctions and the aerodynamic forces are then expressed in terms of an integral over
tlrr' plxluct of an indicial lunction and thc structural motion. This approach, typically employed
!r l,ttsl ttsponsc stuclics lirr aircrrrli, usually lcarls to cxplicit time-history calculations, but these
,rrr'ltvttitlctl in thc prcscrrt con(cx1. llctc lirrtc tlcpcntlcnl lilrrnulations will be transformed into
'.ptt lr:rl, ol lrcrlrrcrrcy tlcpcrrtlcrrl, tlcscriptiorrs ol rt.sporrsc :rrnplituclcs.
Iltts ptrrbk:rtt is llclrtctl rrrolt: y1t'lrt'r:rlly ilt Sr'tt. I i.1.,1.
260
n
f
nol
Asi I t(: I 'l
u
N(
)Mt NA
r;(; ltt,l ll llN(i lll i;l'()rl:;t rN llll I'lrl l;l N(;l ()l At lr()t tA!,il{
response at any spanwisc soclion .r is rv(.r', l). 'l'hc losp()nse ctrrr bc
writtcn in
terms of generalized coordinates as
: \ ai(x)p,(t)
a(tr, t)
*
c,(x)a(x, t)
*
k"(x)a(x, t)
(r.5.tt) will hc irssrrrrrctl lo lirke:
M,,(K)
rc.6.2)
where pr(t) are the corresponding time-dependent generalized coordinates
ol.
the problem and o;(-r) are the torsional vibration modis. The equation
of motion
of the deck section x is
I(x)ir(x, t)
t,r;
:
JfL(x, t)
(6.6.3)
/(r) is thc local mass moment of inertia of the deck about the ell-ective
r.l.ti'n axis ancl r',,(x) and k,,(x) are, respectively, the effective structural damping ancl stillhcss .l'thc sccti'n. To bring the generalized coordinates into
the
Iipi(t)
where
*
2(o,(2rno)bie)
{ is the generalized
+ (2rn,,)2p,(t)l :
ttt
(
(r. (r. ll )
I
will lx: rlrrdorrr
: M.(K) I
Mg6(x
-
x1)cos
2rnt
(6.6.e)
- x1) is the Dirac delta function (see Eq. 5.1.11), so the generalized
I.r('c, Eq. 6.6.7, becomes
rvlrt'rc 6(x
:
pL
)nlU"tXl
+ Mg6(x
-
x,)cos 2rntlu,(x)
dx
(6.6.10)
ol' a(x, t) from Eq. 6.6.2 in Eqs. 6.6.8 and 6.6.10 implies thar calculation
rll bc required of factors having the form
I r:,t'
u
:
I
)ou'n'lnnrror"r", , *',.rfrxr,,
Jlt(x, t)
(6.6.4)
inertia
n
,.1
2Bl
lk:lirrc applying the full random gust ntorncnt, lct a singlc sinusoidal com;xrrrr,:rrl of amplitude Mo and frequency n bc applied at spanwise section x :
r, 'l'lrcn the applied distribution moment is
Mo,
M,,(t)
lirrrt+ (willr
il il{tMt NA
rvlrt'tc K : 2rnBl U while the time-dcpcndont gusl conllillution
tr,cc lrq. 6.6.1c).
whcrc
pnrblcm, Itq. 6.6.2 is used fbr .'(x, r) in Eq. 6.6.3. The result is then
multiplied
through by <r;(x) and integrated over the full span Z, yielding
tltr.:
t,t
J,
I@)a?@) dx
(6.6.s)
f-, and ndi are, respectively, the damping ratio and the natural frequency (Hz)
in the ith torsion mode and Mo, isthe generalized force. Implicit use has been
E 4@ [^t t;t")*,f"1 a,
j
Jo "
rvlrcr.c G;7
: 4
r
Gijpj(t)
(6.6.11)
: I3 oioi dx and
made of the orthogonality relation
ft
)o
l(x)a{x)a1(x) dx
:
O
(i+j)
I, ,U (6.6.6)
:
J,
*o,
:
a;(x1)
(6.6.12)
l'lrt' lirst occurs in M.(K) and the second occurs in the single sinusoidal comrottr'ltt.
Sirrce the modes ai(x) are dimensionless and of arbitrary scale,
rrrt'rrt to normalize them arbitrarily, for example, setting
I
The generalized force M.,(t) has the form
Mo,
xt)oq@) dx
t)a,(x) dx
I f'
:
; Jn aitxl d"x
(6.6.7)
The attention of the reader is drawn at this point to the similarity
between Eqs.
6.6.2-6 -6 -5 and Eqs. 5 -2. r, 5.2.6, 5.2.7, and 5.2. g. Both sets iepict
the usuar
modal approach to a dynamics problem in a continuous structure.
In the present context the distributed moment per unit span will have both
self-excited and active, time-dependent components, the fomer
associated with
the motion and the latter a function of the gust velocity c.rrrprlrcnrs
in thc
atmospheric flow passing <lvcr r.hc structure. Thc scll'-c,xr.it",l ..,,,,,1.,.,,rr:nrs (scc
I
it is conve-
(6.6. r3)
'Wrntl lunncl tests performed by thc wlilcrs havc tcndcd to indicate that the destabilizing effect
''l llrt scll
oxcitcd ftrrces acting on a srtslrt:rrsiorr britlgc dcck is somewhat reduced by the presence
in thc incidcnt llow. 'l'hc rrst' in cirlcrrl:rlions ol'acrodynamic coeflicients H,t and
l' ,rbl:tinql unclcr sm<xrtlr lkrw totttliliorrs is llrt'rclirrr lhorrghl hcrc to bc conscrvative. Model
r \lrliltl('ltls l(r-ll(rl crrtployirtg 1t:r'lrrrirprcs ol rrrrrLrrrr :rrr:rlysis ltavc shctl lurthcr light on thc cli'ect
,'l lrrtl)ttl('tlco ttpott lltc vrtlttcs ol //,+ :url .1,' lirrll t'x;rLllrliorrs ol'lhc cllcct ol'lrpplr)priatcly
'',,r1|rl ltlrlrttlt'ttt't: tttt lltc llrrllertlt'tivitltvt':, ol lr;1111'1' rlt'r'lrs rr'rrr:rilr lo lrc titlrictl orrt.
,,1 lrrtlrttk:ttcc
262
Arnotlnt; n(;
I't
l
N()Mt NA
{;
: L, but llurr. irr gcncral, lilr i *.i, rlrc valucs <ll'
ar1.much less than L.It will bc assurnccl hcro that Gii U +.i ) is nogligihlc,
which is reasonable for bridges in which /(x) is approximately constant acK)ss
the span, as can be seen from Eq. 6.6.6. The net value of the generalized fbrcc
one may then note that G;;
Gi1
;tttt
(;
lJ(,t
II ilN(i ilt :;t,()N:,t tN ilil I'tit l;l NCI ()t nl li()t In:;lt(; t'ilt N()Mt Nn
2ti3
I
"i':1,,,,(,(\)
('ialrvlv( r )l
((r (r.lO)
M*, then is
Mo,=
p[JzB2Ll*ffnff + KzAtK)p,]
+ uoo,1*)cos
2trnt
l'lrrrs, rcf'erring to Eqs. A2.29 and A2.33 (Appontlix A2), tlrc nrorrrcnl cosl)cctnun between sections x1 and x2 may be writtcn
(6.6.14)
s
Equation 6.6.4, which describes the motion of the ith mode, may then be
written with use of Eq. 6.6.14
t
as
Iilpi|) + 2t,",(2Tfi,)b,@ + (2ili.,)2p,(t)l: Msai(x)cos2rnt
wherc new cfl-ective fiequency rio,
and,
damping
that
"r,:
T-,:
n'-,
-
nfrfrj|
I
"?ti
[3
[!
*t@nf
(6.6.17)
- (ntn")'fi4,qUfi
(6.6.18)
where sfa,^a'(n) is the co-spectrum of the buffeting moments M1 and M2 per
unit span which act, respectively, at the coordinates x, and x2.
Equation 6.6.1c describes the applied aerodynamic moment per unit span
due to steady wind and gust components. In this equation, the moment and
drag coefficients Cyand Cp are functions of the mean twist angle as(x) at the
spanwise section x, and the velocity components u and u are also functions of
x and time. For convenience the following notation is introduced:
Cyllus(x)l
=
Clalas(x)
+
C,,1u,,{.01
l' lo
yefas(x
r,,t
*,,(
r I tlC s I .*s(r,rl
1a
)lC'M[ag(x)t
c',a[as(x
)]c
yfc,s(x 1),
.ala.@)l
\!'
8#
fi#
8#)
(6.6.21)
(6.6.16)
a;@)ai@)Sfr,u,@) dxr dxz
t6rafi,t! {ft
2C
r,
-t 2C TaBfas(x)lC
have been introduced such
+
,'l*
*1r,,,.,
i",
u,
(6.6.15)
Equations 6.6.16 and 6.6.17 introduce the effect of the aerodynamic selfexcited forces into the response at frequency n.
Equation 6.6.15 (i : 1,2,3, .. .) is similarin form to8q.5.2.7 forwhich
the generalized force is given by Eq. 5.2.12.In Chapter 5 the system defined
by Eq. 5.2.7 is analyzed under distributed random loading, leading to Eq.
5.2.38. Completely analogous steps hold here, yielding the following resuit
for the spectrum of torsional response:
S,(x, n) =
i
fi,,,{ n ) : | ),
lrt,
(6.6. t9)
liv:rluation of CTap and C'7a at values 11 and -r2 requires knowledge of the mean
rlt'llcction distribution os(x) over the span. This can be obtained by a static
.trrrly of the type discussed in Sect. 6.4.1 or may be described in terms of the
lorsional vibration modes by the expression
os(x)
:
2ltlpu'n'cr4to!"r)lcvi(xr)
i
4T-n;,1,-
dx' - ,..'
o''(x\
(6'6'22\
rvlrich is a result derived from Eq. 6.6.4by neglecting all time-dependent terms.
f lrr: solution of Eq. 6.6.22 for a given wind velocity u requires an iterative
:rppnlach, starting conveniently with ao : 0.
lnEq.6.6.21 the co-spectra Srt*r(n) and lfi.,@) are negative in value and
.rlrlrrcciably smaller in magnitude than lf,,,r(r) and Sfi,,(n); they may conservrrt
ivcly be neglected.
'l'hc root mean square of the fluctuating torsional response
at section x is
o'1x;
:
f-
)n
s*{x. nldn
(6.6.23t
I',"rrk values of the fluctuating torsional response may be obtained by following
',tt'ps sirnilar to those of scct. -5.3. Mcthods of calculation relative to the quanlrlrcs rlrcntioned abovc arc tliscrrssr.:tl in ('haptcr 13.
Il'tlro vcrtical (bcncling) r'eslx)nsc ol tlrc brirlgc is written as
h(.r.r) )J/r,{r)r7,u) (t
l,2....)
16 6.241
264
AEnofl Asltc
t,lt
{;t; ltt,lllllN(illliil'()l'Jl;l
N()Mt NA
where /z;(x) are the verlical bcnding rrroclcs ol'vibration arrtl r7; 1rc lhc gcncralized coordinates for these modes, thcn, by a proccss cor.nplctcly arraklgous
to that described above for torsion, the spcctrum o1'the vertical rosponsc can
be shown to be
Sn(x,
n?@
n) =
i
13
r6ran|,tvt!
[!
h,@,1h,(xr)s?.,r,(n) ctxr ctxz
(ntnp;212 + +yf,1ntrr;t]
{[t -
lt'J
llll I'llllilN(:l
()l nllll)lln:,ll(
l'lllt'l{rlvlltl^
265
Ilrt' rrrcarr, nlciul s(luiu(' irrttl pclk vcrlicirl r'(:slxrrsr'b r'rrtt llrt'tt lrt' t rtlt'til;rlt'rl. irs
lirr lolsirlrtrrl lL:sponsc, by lollownrll rlt';rs srtttillt lo lltrtst'
ol Sc:c:t.5.3.
'lo calculatc |hc ulong-winrl rcsponsc, contpletcly rulrloplorrs pttter'tlttres to
llrosc abovc arc uscd, thc basic lbrcing lirrtcliorr bcirtg tltc tlnrg lrs givcrt by
1,.t1. (r.6. la; a knowledge of along-wind vibra(iou rtttttlcs is irlso rctlrritetl.
rv:rs irrtlica(crl lrlrove
(6.6.2s)
where
6.6.3 Outline of the General Buffeting Response Problem of
Linelike Structures
nL
M:1, m1x1hl1x1 dx
l.r't the across-wind bending and torsional modes of a symmetrical linelike
r,lrrrcturex be representedby hi@) and a;("r) as in Sect. 6.6.2, so that sectional
is the generalized inertia, m(x) being the deck section mass per unit length,
n1r, the natural frequency* in the ith mode, and ;,,, the aerodynamically influ-
tlt'llcctions
h and q (Fig. 6.5.1) under dynamic excitation
enced system damping defined by
:(n : (n,
.
-
pB2L .,.
n
,M UfK) i,
(6.6.26)
where K : 2rBnlU and f1,, is the mechanical damping ratio in the ith mode.
The co-spectrum of the time-dependent lift forces z1 and Lzper unit length of
span, which act respectively at span points xr andx2, is (from Eq. 6.6.lb)
sf,,,(n)
:
l* rr, 4'
l+c,too{,1)l
t
1)lCL6[c.s(x),
2
C
Tfc's(x
c1[cve(x)]
-t
C;s[c.s(x 1)]C Lr1oq(x),
dC,l
El":*0r.,
+
A
U
Cpfuo@)l
a(x,t):lo.{x)p{t)
(6.6.2eb)
Mi[Qi
+ 216(2rnn)ei -t (2rn1,)zqil :
Ii[Fi
+ 2(*,(2rn.,)b, + (Zrn.,)2pil :
I, "O, t)hi@) dx
J.
**,
t).,i@)
(6.6.30a)
dx (6.6.30b)
S(x, r) and 5lt(x, /) are, respectively, the lift and moment per unit span
r of the span.
ln order to obtain the necessary system admittance functions, J(x, l) and
:)lt(.r, /) are alternately specified in the following manners. For lift-associated
r.vlrcre
:rl scction
(6.6.27)
;rr
where
C'yefas(x)l :
(6.6.29a)
Analogously to previous formulations (Sect. 6.6.2) the equations of motion
fi#
fi#)
h(x,t):Vh{x)q{t)
(rrrtrchanically uncoupled about the centerline) become
6#
* 2C yfag(x2)lC Ln[cxor,)t fi#
are
(6.6.28)
*There is no aerodynamic inllucncc in this casc upon the natural Irctlucrrcy,
owirrg to lhc assumcrl
absence in thc basic rrxxlcl ol tcrnts in l.
I
llrrittances,
S,(x,
t) : Lt t
5lL(x'
t) : M"
l"ei2""'67x
-
xrl
ior rnoment-associated adtnillitnccs,
tlror rrnsyrrrrrrctrical slrlrclurcs (,t / 0, lirl rr
lort'c rrrrtl ll)on)cnl rclcrtctl 1o lltt' t'litsltr' ;trtr
r
1), llr(' lr'('irlnl('nl is anakrgous, with aorodynamic
266
At
n()t
lnl;lc I'lr
NoMr Nn
ltl llltl fl{:l !,
-:'1r,,
ril,
26/
REFERENCES
,
M,,,,i,,,,,,61x
-
x,1
I). J..lohns r:t ll., "()n Winrl-lntluccrl lrrslrrlrilrlit's ol ()1x'tr litttk'tl ('itt'ttl:rt
Cylindrical Shclls," in Pnxvulings tl'tltt' (lnt.li't t'ttt'r' ttrt lltlrlt'r' Sltttltttl ,\n rtt'
turcs,The Haguc, 1969. A. M. Haas irrttl ll. virrt Kolerr (etls.). lrrslitrrle 'l'N()
lirr Building Matcrials and Structurcs, I)clli,'l'lre Ncthcllirntls. pp. lt{5 212.
L. R. Wootton, M. H. Warner, R. N. Sainslrrrry, urtrl l). ll. (l<xrpcr', Osrilltttiort
o.f Piles in Marine Structures, Construction lntlustry llcscarch and lnlirrrnation
Association, London, Report No. 4l , 1912.
R. King, M. Prosser, and D. J. Johns, "On Vortex Excitation of Model Piles
in Water," J. Sound Vib.,29 (1973), 169-188.
R. E. D. Bishop and A. Y. Hassan, "The Lift and Drag Forces on a Circular
Cylinder Oscillating in a Flowing Fluid," Proc. Roy. Soc., London, Series A,
rr
I
Modified equations of motion (6.6.30) can then be written that are similar
Eq.
to
6.6.15 though now coupled by the presence of the full set of unsteady
motion derivatives proportional to HI and Af . From these equations, aero-
rr
.)
dynamically modified mechanical admittances can be calculated analogously to
previous results, but now for two coupled equations. The results, representing
( l) the across-wind deflection due to a concentrated harmonic lift at section x1,
(2) the torsional deflection due to a concentrated harmonic lift at x1, (3) the
across-wind deflection due to a concentrated harmonic moment atxy, and (4)
the torsional deflection due to a concentrated harmonic moment at x1, may be
designated, respectively, Hnr(x, xr, /t), Hot(x, xr, n), HnuQ, x1, n), and Ho1a(x,
rr
I
(r
'l
(r
'r
G. V. Parkinson, G. Feng, and N. Ferguson, "Mechanisms of Vortex-Excited
Oscillations of Bluff Cylinders,' ' Proceedings of the Symposium on Wind Effects
on Buildings and Structures, Loughborough University of Technology,
rr
(r
N. Ferguson and G. V. Parkinson, "Surface and Wake Phenomena of VortexExcited Oscillations of BluffCylinders," "/. Eng. Ind., ASME,89 (1967), 831-
']
where L1, and M. are the self-excited aerodynamic
span given by Eqs. 6.5.4.
lifi
and momcnt per unit
277 (1964),5r-74.
xt, n).
Assuming now that the structure is subjected to a distributed buffeting lift
L(x, t) and moment M(x, t) as defined by Eqs. 6.6.lb and 6.6.lc, the spectra
of across-wind bending and torsional response can be calculated by integrating
elemental effects. Designating by 51,p, Sr,u, 5u,4, and S1a,7a, the cross-spectra
corresponding respectively to the lifts and moments at x1 and x2 as suggested
by their subscripts, the following typical expression for vertical response spectrum S6(x, n) is obtained:
s{x' n) :
838.
(t
I
R. T. Hartlen, W. D. Baines, and I. G. Currie, Vortex-Excited Oscillations of
a Circular Cylinder, University of Toronto Technical Report No. 6809, To-
(r
li
rr
1)
G. H. Toebes, "The Unsteady Flow and Wake Near an Oscillating Cylinder,"
Trans. ASME, J. Basic Eng.,9l (1969), 493-505.
ronto, 1968.
fL fL
J. J" lHft6'
x1'
n\Hv(x'
x2' n)s7,1,(n)
+ Hfr@,
x1, n)H61fx, x2, n)Sy,yr(n)
+ HftAx,
x1, n)H1,1(x., xy,
+ Htt{x,
x1, n)H1,1fx, x2 n)S7a,7ar(n)l dxr dxz
n)S11a,yr(n)
where F1* denotes the complex conjugate of F1.
It should be remarked that both the mean speed of the flow and the values
of lift and moment may, in the above expressions, be a function of x. In that
case modal orthogonality relations can no longer be used (e.g., as was done
in Eq. 6.6.14), and the expressions forthe modified admittances become more
elaborate; however, the attendant calculations can be conveniently programmed
for electronic computers.
Possible applications of the expressions for the response of linelike structures
dealt with here include the calculation of the responses of tall prismoidal buildings with strong torsional motions, and those of tall towcrs antl suspcndcd-span
bridges.
Leicestershire, 1966.
)
('
I(
rr
ll
tr
ll
rr I I
7970, Washington, DC, 1970.
R. J. Glass, A Study of the Hydroelastic Vibrations of Spring Supported Cylinders in a Steady Fluid Stream due to Vortex Shedding, ONR Project N001469-C-0148 Final Report, Washington, DC, 1970.
l
R. T. Hartlen and I. G. Currie, "Lift-Oscillator Model of Vortex-Induced
Vibration," J. Eng. Mech. Div., ASCE, 96 (1970), 5'17-591.
R. Sainsbury and D. King, "The Flow-Induced Oscillations of Marine Struc-
l5
turcs," Proc. Inst. Civ. F)t14.,49 (19'71),269 3O2.
L. R. Wooton and C. Scnrtorr, "Acnrtlynanric Stability," inThe Modcrn De-
rr l
(r
V. C. Mei and I. G. Currie, "Flow Separation on a Vibrating Cylinder," Pftys.
Fluids, 12, (1969), 2248-2254.
G. H. Koopm an, Wind-Induced Vibrations of Skewed Circular Cylinders, Civil
and Mechanical Engineering Department Report No. 70-11, Catholic University of America, Washington, DC, 1970.
R. A. Skop, S. E. Ramberg, and K. M. Ferer, Added Mass and Damping
Forces on Circular Cylinders, Naval Research Laboratory Formal Report No.
ol Wirul-Scnsitivr Stutt'ttrrr'.s, ('ortstntcliotr lndustry Rcscarch ancl Inlirrtttitlion Associatiott, l,otttlott, l()7 L
:;ign
{r
l(r (). M. (irillin, R. n. Sko;r. rrtttl (l ll
ortiutl Viblitliotts ol ('ircttl;tt
lirxrprruutn. "'l'ltc Vortcx-lixcitctl ltes
('yltrttlcr:," .l ,\urtul Vilt.,ll (l()71), 2.15 :4().
268
6-11
6-18
6-19
6-20
Alnot
t At;
ilc
t
,r
il
N(
lll l l lll tl(;l l;
)Ml NA
R. A. Skop and O. M. (irillirr, "A Mrxlcl lir tlrc Vorlcx ljxeilt^tl l{cs;xrttsc ol'
Bluff Cylindcrs," J. Stund Vilt., 27 ( 197.\), '225 233.
O. M. Grilfin and S. E. Ranrbcrg, "'l'hc Vortcx-Strcct Wakcs ol'Vibrating
Cylinders," J. Fluid Mech.,66 (1974),553 578.
W. D. Iwan and R. D. Blevins, "A Model for the Vortex-Induced Oscillation
of Structures," J. Appl. Mech., ASME, 4l (1974),581-585.
R. A. Skop, On Modeling Vortex-Excited Oscillations, Naval Research Laboratory, Memorandum Report No. 292'l , Washington, DC, 1974.
F. Angrilli, G. DiSilvio, and A. Zanardo, "Hydroelasticity Study of Circular
(, l5
r, l(r
(r
17
(r
lll
R. King and M. J. Prosser, "Criteria for Flow-Induced Oscillations of a Cantilevered Cylinder in Water," in Proceedings of the IUTAM-IAHR Symposium
on Fktw-lnduced Structural Vibrations, Karlsruhe, West Germany, 1972, E.
(r
l()
6-23
Naudaschcr (cd.), Springer-Verlag, Berlin, 1974, pp. 488-503.
O. M. Griflin and S. E. Ramberg, "On Vortex Strength and Drag in Bluff
rr
.lo
6-24
Body Wakes," J. Fluid Mech., 69 (1975),721-729.
R. King, Vortex Excited Oscillations of Inclined (Yawed) Cylinders in Flowing
(r ,l
6-21
" in Proceedings of the IUTAM-IAHR Symposium
on Flow-lnduced Structural Vibrations, Karlsruhe, West Germany, 1972, E.
Naudascher (ed.), Springer-Verlag, Berlin, 1974, pp. 5U-512.
Cylinders in a Water Stream,
6-22
Water, Bitish Hydro-Mechanics Research Association, Bedford, U.K., Report
6-25
6-25
No. RR 1214,1975.
R. D. Blevins and T. E. Burton, "Fluid Forces Induced by Vortex-Shedding,"
(r.l.l
J. Fluids Eng.,95 (1976), 19-24.
R. A. Skop and O. M. Griffin, "On a Theory for the Vortex-Excited Oscillations of Flexible Cylindrical Structures," J. Sound Vib., 4l (1975),263-274.
rr .l
6-27 W. D. Iwan,
The Vortex Induced Oscillation of Elastic Structural Elements,
ASME paper 75-DET-28, 1915.
6-28
6-29
6-30
6-31
6-32
6-33
Liquids, University of Missouri, Rolla, 1975.
S. E. Ramberg, O. M. Griffin, and R. A. Skop, "Some Resonant Transverse
Vibration Properties of Marine Cables With Application to the Prediction of
Vortex-Induced Structural Vibrations," in Ocean Eng. Mech., N. Monney (ed.),
ASME, New York, 1975,29-42.
R. King, "An Investigation of the Criteria Controlling Sustained Self-Excited
Oscillations of Cylinders in Flowing Water, " in Proceedings of the Symposium
on Turbulence in Liquids, University of Missouri, Rolla, 1975.
S. E. Ramberg and O. M. Griffin, "Velocity Correlation and Vortex Spacing
in the Wake of a Vibrating Cable," J. Fluids Eng., 98 (1976), 10-18.
rr
'l'X,
197-5.
N. Minolsky, Ntutlitrutr O:;cilltttirtrr,s, Vrrn Nos(r;rrll. Nt'rv Yorl. l()(rl.
\
.l,l
rr,l\
tr
.l(r
r,
l
r'
.lli
(r
,l()
(r
\o
(r
\l
(r
i-l
S. E. Ramberg and O. M. Griffin, The Effects of Vortex Coherence, Spacing
and Circulation on the Flow-Induced Forces on Vibrating Cables and BlulJ
Structures, Naval Research Laboratory Formal Report No. 7945, Washington,
DC,1916.
O. M. Griffin, R. A. Skop, and S. E. Ramberg, Thc Rexnunt Vortcx-Excitt:d
Vibrations tf'Struclurcs arul Cablc Syslcms, C)fl,shorc 'l'cr'hrrology ('onl'crcnco
l{. ll. Scirrrlln. "'l'lrcory ol llrc Witttl Arrirlysis ol l,otr;' SPrrrr llritlges llitsctl ott
l)atrr Ol'rtlinlble l.nrrr Se:clion Mrxlcl 'lt:sls," trr I'ttttt'nlirr,qs t2l tltt liturth
Intcntuliottul (lttt.li'n'ttt't'tn Wirul l'),lli'rt:;, l.ttttrlott, l()75, ('artrbriclgc Univ.
l'rcss, Carrrbridgc, 1976, pp. 25()-269.
R. H. Gadc, H. R. Bosch, and W. Ptxlolny,.lr'.. "l{cccnt Acrodynamic Studies
of Long-Span Bridges," J. Stru<:t. Dlr,., AS('li, 102' No. ST7 (July 1976),
1299 13t5.
R. H. Gade and H. R. Bosch, Intcrim Rcport, Wind Tunnel Studies on the
Luling, In, Cable-Stayed Bridge, F. H. W. A., Fairbank Lab, Mclean, VA,
U.S. Department of Transportation, 1975.
G. H. Toebes, "Fluidelastic Features of Flow Around Cylinders," in Pro'
ceedings of the International Research Seminar on Wind Effects on Buildings
and Structures, Ottawa, Canada, 1961 , Yol. 2, Univ. of Toronton Press, Toronto, 1968, pp. 323-334.
L. R. Wooton, "The Oscillations of Large Circular Stacks in Wind," Proc.
Inst. Civ. Eng., 43 (1969), 513-598.
H. Glauert, Rotation of an Airfoil About a Fixed Axis, Aeronautical Research
Committee, R & M 595, Great Britain, 1919.
J. P. Den Hartog, "Transmission Line Vibration Due to Sleet," Trans. AIEE,
sr (1932), tO14-rO76.
J. P. Den Hartog, Mechanical Vibrations,4th ed., McGraw-Hill, New York,
I
W. K. Blake, "Periodic and Random Excitation of Streamline Structures by
Trailing-Edge Flows," in Proceedings of the Symposium on Turbulence in
Papcr O1'(l-23 19, Houston,
6-34
I
l
269
956.
G. V. Parkinson and N. P. H. Brooks, "On the Aeroelastic Instability of Bluff
Cylinders," Trans. ASME, J. Appl. Mech.,83 (1961), 252-258.
G. V. Parkinson and J. D. Smith, "An Aeroelastic Oscillator with Two Stable
Limit Cycles," Trans. ASME, J. Appl. Mech.,84 (1962),444-445.
G. V. Parkinson, "Aeroelastic Galloping in One Degree of Freedom," in
Proceedings of the Symposium on Wind Effects on Building,s and Structures,
Vol. l, National Physical Laboratory, Teddington, U.K., 1963, pp. 581-609.
G. V. Parkinson and J. D. Smith, "The Square Prism as an Aeroelastic Nonlinear Oscillator," Quart. J. Mech. Appl. Math. , l7 , Pt. 2 (1964), 225-239 .
G. V. Parkinson and T. V. Santosham, "Cylinders of Rectangular Section as
Aeroelastic Nonlinear Oscillators," Vibrations Conference, ASME, Boston,
t961.
M. Novak, "Aeroelastic Galloping of Prismatic Bodies," J. Eng. Mech. Div.,
ASCE, 95, No. EMI (Feb. 1969), ll5-142.
M. Novak, "Galloping Oscillations of Prismatic Structures," J. Eng. Mech.
Dlv., ASCE, 98, No. EMI (Feb. 1972),27-46.
G. V. Parkinson, "Mathematical Models of Flow-Induced Vibrations of Bluff
Brdies," in Proceedings of the IUTAM-IAHR Symposium on Flow-Induced
Structural Vibrations, Karlsruhe, West Germany, 1972, E. Naudascher (ed.),
Springcr-Verlag, Berlin, 1974, pp. 8l-127.
Wardlaw, "Wintl 'l'urrncl Invcstigations in Industrial Aerodynamics,"
('rttt. Acnnuut. Spttct'./", ltl, No..l (Mrrrr.'h 1972).
I{. Il. Scanlan anrl ll. 1,. Wirllllw. "ltcrlttt'tion ol Flow-lnducccl Structural
Viblrrlirrrrs," itt l,vtltttitttt ttl lllt'r'ltrttrir'rtl l'iltrttlirttt, Itttltltt, tttul Noi.sc, AMI)
Vol. l. Sccl . 2 ASMli, Nt'rv Yurk. lt,/ l. t,t) \5 (rl.
It. L.
27O
6-53
6-54
6-55
6-56
Ar
nol tnlilt(;
t't
l
N()Mt NA
ltf I l=il| llr t
N. Krylofl'and N. Bogolitrlxtll. lrttnxlil..tit)n tt) Nortlitttttr Mt,<.lttttrit'.t,,lr.lrns.
S.
Lefschetz, Annals of Mathcnratics Sturlics, No. ll, Princcton [Jlriv. l)rcss.
Princeton, 1947.
v. Mukhopadhyay and J. Dugundji, "wind Excited Vibrarion ol' a Squarc
Section Cantilever Beam in Smoorh Flow," J. Sound Vib.,45, No. 3 (1976),
329*339.
R. Skarecky, "Yaw Effects on Galloping Instability," J. Eng. Mech. Div.,
ASCE, 101, No. EM6 (Dec. 1975),739-754.
R. L. Wardlaw, K. R. Cooper, and R. H. Scanlan, "Observations on the
Problem of Subspan oscillation of Bundled power Conductors," DME/NAE
Quarterly Bulletin No. 1973 (1), National Research council, ottawa, canada,
1973 (reprint), pp. l-20.
6-57 K. R. Cooper and R. L. wardlaw,
6-58
6-59
Preliminary wind runnel Investigation ol
Twin Bundlc Sub-Conductor Oscillations, Report No. LTR-LA-4l NAE, NRC,
Ottawa, Canada, 1970.
A. Sinrpson, "stability of Subconductors of Smooth cross-Section," proc.
Ins!. Electr. Eng., ll7, 4 (1970), 741-750.
A. Simpson, "On the Flutter of a Smooth Cylinder in a Wake," Aeronaut. e.
(Feb. l97l),25-41.
tt J)
('/-]
6-61
6-65
6-66
6-67
214 242.
N. Ukcguchi, H. Sakata, and H. Nislritlrni. "Arr lrrvcrlrpiirlrnrr ol Ar'rtx'l;r.trt
Instability ol'Suspcnsion Bridgcs," in I'nx'tt'tlittli,t rtl tltt Lit..,ti(tttttnttl ,\\',tl
Ttsium on Suspcrtsitn Bridges, Laboratorio Nlrt'iorrirl tlc lirrgr.rrlurriir ('rvtl, l.i:,
rr
/5
T. Okubo and K. Yokoyama, "Some Approaches for Improving Wind Stability
of Cable-Stayed Girder Bridges," in Proceedings of the Fourth International
Conference on Wind Effects on Buildings and Structure,r, London, 1975, Cambridge Univ. Press, Cambridge, 1976, pp.241-249.
(r
l(t Y. Otsuki, K. Washizu, H. Tomizawa, and A. Ohya, "A
(r
"/J
(t
ltl
Note on the Aeroelastic Instability of a Prismatic Bar with Square Section," J. Sound Vib.,34,
2 (1914),233-248.
H. Loiseau and E. Szechenyi, "Etude du comportement adro6lastique du tablier
d'un pont ) haubans," T.P. 1975-75, Office National d'Etudes et de Recherches ,46rospatiales, Chdtillon, France.
R. H. scanlan and K. S. Budlong, "Flutter and Aerodynamic Response con-
siderations for Bluff Objects in a Smooth Flow, " in Proceedings of the IUTAMIAHR Symposium on Flow-lnduced Vibratiors, Karlsruhe, West Germany, 1972,
E. Naudascher (ed.), Springer-Verlag, Berlin, 1974, pp. 339-354.
Overhead Transmission Lines," J. Sound Vib.,20,4 (1972), 417-M9.
K. R. cooper, A wind Tunnel Investigation of rwin Bundled power corurucrors, Report No. LTR-LA-96, NAE, NRC, Ottawa, Canada, 1972.
tr
/9 R. H.
J. A. Watts, K. R. Cooper, and R. L. Wardlaw, proposed Wind Tunnel Tests
Programs for Bundled Conductor Subspan Oscillations, Report No. LTR_LA99, NAE, NRC, Ottawa, Canada, 1972.
R. H. Scanlan, A wind runnel Investigation of Bundled power-Line conductors, Part VI. Obsenations on the Problem, Report No. LTR-LA-121, NAE,
NRC, Ottawa, Canada, 1972.
T. Theodonen, General rheory of Aerodynamic Instability and the Mechanism
of Fluter, NACA Report No. 496, 1935.
(r
l{0
tr
R. H. Scanlan and R. Rosenbaum, Aircrafi vibration and Flutter, Macmillan,
New York, l95l (reprint, Dover, 1968).
(r
fl.l A. G. Davenport,
t,
l{;l A. G. Davenport, "The Response of Slender, Linelike
6-68 Y. C. Fung,
The Theory of Aeroelastict4,, Wiley, New
york,
6-69
R. L. Bisplinghoff, H. Ashley, and R. L. Halfman, Aeroelasticity, Addison-
6-70
Wesley, Cambridge, MA, 1955.
R. L. Bisplinghoff and H. Ashley, Principles of Aeroelasti.i/.y, Wiley, New
York,
ntl
1117.
J.-G. B6liveau, R. Vaicaitis, and M. Shinozuka, "Motion of a Suspension
Bridge Subject to Wind Loads," J. Stuct. Div., ASCE, 103, No. 5T6 (1977),
lll
K. R. Cooper and R. L. Wardlaw, "Aeroelastic Instabilities in Wakes," in
Proceedings of the Third International Conference on Wind Effects on Buildings
and Structures, Tokyo, 1971, Saikon, Tokyo, 1972,pp.647-655.
(r
1J2 H. W. Liepmann, "On the Application of Statistical
Concepts to the Buffeting
Problem," J. Aeronaut. Sci., 19, 12 (Dec. 1952),793-800,822.
"The Application of Statistical Concepts to the Wind Loading of Structures," Proc. Inst. Civ. Eng., 19 (1961), 449-472.
Structures to a Gusty
Wind," Proc. Inst. Civ. Eng.. 23 (1962), 389-407.
A. G. Davenport, "Thc Action ol Winrl on Suspension Bridges," in Procccdings ttl thc Inlcnrutionttl S.\,ttrlto.sirtttt tnt Srt,rltorsion Bridges, Lab<lratorio Nu,
cional clc F)ngcnharia ('ivil. l,islxrrr, l(Xr(r, pp. 7() l(X).
(r
i'i.5
Ir
li(r W.-ll. l.in, "l'orcctl
1962.
R. H. Scanlan antl J..1.'fomko, "Airliril and Bridgc l)t.ck lilrrile l)t'rivltivcs,"
J. Eng. Mach. l)iv., AS('li,97, No. IIM(r, I)nx'. l)rryrcr ll(r0t)' (l)ec. l()7 l),
Scanlan, J.-G. B6liveau, and K. S. Budlong, "Indical Aerodynamic
Functions for Bridge Decks," J. Eng. Mech. Div., ASCE, 100, No. EM4
(Aug. 1974), 657-672.
I 189-1205.
1955 (reprint,
Dover, 1969).
6-7 t
119691"
Comparativc S(utly on Acrnrtlynurlic l;orces Act
ing on Cable-Stayed Bridge Girders," in Pnx'tutings ol'tltc Su,rnd I.l.S.-.lu1nn
Research Seminar on Wirul Effects on Slrudurcs, Kyot<1, 1974, Univ. of Tokyo
Press, Tokyo, 1976, pp.27l-283.
6-62 A. Simpson, "Determination of the Natural Frequencies of Multi-conductor
6-64
l{. ll. Scirlrlirrt rtttrl A. Sithzcvltri, "lixpetrttctrlirl Arrrrrlyn;rrrrrr ('nr,llrr tr'irt,i rir
thc Arurlyticirl Slutly ol'Suspcrrsiorr lllitlp.e lrlutlcl ." .l [tlr't li I'trt, ,\, r , l l, t
bon, 1966, pp.273-284.
Proceedings of the Third International Conference on Wind Effects on Buildings
and Structures, Tokyo, 1971, Saikon, Tokyo, 1972, pp. &j-655.
6-63
2l l
(t'14 T. Okubo and N. Narita, "A
6-60 A. simpson,
"wake-Induced Flutter of Circular cylinders: Mechanical Aspects," Aeronaut. Q. (May l97l), l0l-118.
K. R. Cooper and R. L. Wardlaw, "Aeroelastic Instabilities in Wakes," in
!;
:rrtrl St'll lirt'illrl lt('sl)()ns('s ol a lllull s(ntetun. in :r
'l'trl'lttllcltl Wintl," tkrloritl tlnst'tl;tltott. l)r'llrrllrrt'lrl ol'('ivil lingirrcr.,lirrlg,
l'r'incelorr Onivclsity, lt)'/ /.
272
6-87
At I tot
R.
l(l l'l ll
N( )MI NA
l). lllcvirrs.l;lov,-lnrlucrtl l,'iltrttttrtrt,ltl etl., Vrut Noslnttttl ltcitrlroltl, New
York,
6-88
I n fi I
1990.
CHAPTER 7
B. J. Vickery, and R. L Basu. "At'nrss Wirrtl Vibrations ol'Structttrcs ol'(lit'
cular Cross-Section, Part l, Dovckrprrrcnl ol' a 'l'wtl-Dinrcnsional Mtilol lirr
Two-Dimensional Conditions," "/. Wind. Ett11. Ind. Aentdyn, 12 (l9tt3)'
49*73.
6-89
6-90
R. I. Basu, and B. J. Vickery, "Across-Wind Vibrations of Structures of Circular Cross-Section, Part 2, Development of a Mathematical Model for Full
Scale Application," "/. Wind Eng. Ind. Aerodyn., 12 (1983), 15-97.
D. J. B. Richards, "Aerodynamic Properties of the Severn Crossing Conductor," Proceedings of the Symposium on Wind Efects on Buldings and Strut''
tures, Yol.II, National Physical Laboratory, Teddington, U.K., Her Majesty's
Stationery Office, London 1965, pp. 688-765.
6-91
6-92
O. M. Griffin and R. A. Skop, "The Vortex-Induced Oscillations of Structures," J. Sound Vib., 4 (1976),303-305.
K. Y. R. Billah, "A Study of Vortex-Induced Vibration," Doctoral disserta-
6-93
tion, Princeton University, Princeton, (1989).
I. Goswami, R. H. Scanlan, and N. P. Jones, "Vorlex-Induced Vibrations ol'
6-94
6-95
6-96
Circular Cylinders. I: Experimental Data; II: New Model," J. Eng. Mech.,
r19 (1993), 2210-2302.
F. Ehsan and R. H. Scanlan, "Vortex-Induced Vibration of Flexible Bridges,"
J. Eng. Mech., ff6 (1990), 1392-l4ll.
E. H. Dowell (ed.), A Modern Course in Aeroelasticirlr (Chapter 6: "Aeroelastic Problems of Civil Engineering Structures"), Kluwer, Dordrecht, 1995.
E. Simiu and R. H. Scanlan, Wind Effects on Structures, 2d ed., Wiley, New
York.
6-97
1986.
R. H. Scanlan, "Problematics in Formulation of Wind-Force Models for Bridge
Decks," J. Eng. Mech., ff9 (1993), 1353-1375.
6-98 S. Murakami, A. Mochida, and S. Skamoto, "CFD Analysis of Wind-Structure
Interaction for Oscillating Square Cylinder," in Wind Engineering, Proceedings, Ninth International Conference, Eastern New Delhi, Wiley, New York,
pp. 671-682, 1995.
6-99 C. F. Christensen and O. Ditlevsen, "Fatigue Damage from Random Vibration
Pulse Process of Tubular Structural Elements Subjected to Wind," in Proceedings, Third International Conference on Stochastic Structural Dynamics, San
Juan, Puerto Rico, Jan. 15-18, 1995.
6-100 E. Simiu and G. R. Cook, "Empirical Fluidelastic Models and Chaotic Galloping: A Case Study," J. Sound Vibration,154 (1992),45'66.
6-101 M. Frey and E. Simiu, "Noise-Induced Chaos and Phase Space Flux," Physica
D,63 (1993),321-340.
WIND TUNNELS
,\ltlrough the science
of theoretical fluid
mechanics
is well developed
and
orrrputational methods are experiencing rapid growth, it remains necessary to
I't'rlirrrn physical experiments to gain needed insights into many complex effects
,
flow. This is the case in the well-established field of
for which wind tunnels were first developed, and, to an even
;r:sociatcd with fluid
,rt'rorrirutics,
1'rt"rrtcr extent, in the practical study of buildings, structures, and machines that
'.1;rnrl in the earth's near-surface atmospheric layer.
lior the most part such structures have been designed for other purposes than
;,r,rvitling minimal resistance to the air moving about them. They have therelort'. in recent decades, been the focus of what is termed bluff-body aerodyrr,rrrrics. In such aerodynamics there is much emphasis on flows around sharp
, r)nlcrs, on separated flows, and so forth. These situations are among the most
rrr'()n(lite when it comes to both theoretical and computational methods. The
rvrrrl tunnel is thus naturally resorted to as an investigative tool in this context.
I'ypically the full-scale bluff body is immersed in a turbulent atmospheric
llrrw lrlachsbart determined as early as 1932 (see Sect.4.6.2 and Fig. 4.6.4)
tlr;rl sirnulations of the aerodynamic behavior of buildings should be conducted
rrr rl,irrtl tunnel flows with characteristics similar to those of the natural wind.
t 'rrrlcrrtly, the vast majority of tests are carried out in wind tunnels that simulate
,rtrrrosphcric flows. (In some instances tcsts in smooth flow are still accepted,
lrrr 1'x'1111p1s, in the case ol'lrurssccl fiameworks-see Sect. 4.5 and Chapter
| ' ol lirr prcliminary invcstiglrliorrs ol'thc gcrlrnetric shape of bridge deck
',, t liorr rrrorlcls. H<twcvcr, (lrcst' insltrrrt'cs iu'(' tlrc cxccption rather than the
rrrlt').'l'ltr:lc is thcrolillc il iilri)rl', inl('t'sl irr glrinirrg a knowlcclgc-firr latcr
','Ptrttltteliort in lltc wintl luturt'l ol llrt'rlrlrrrc ol wintl llows irr thc ctrrlh's
273
274
/ I lt^til( j l;lMll Alllli nt {rtiutf
wrNl) tt,NNt t:i
(hc witttl tttltttol
boundary laycr; "l"argct" charuc(clistit's to be tltrlllic:alctl in
are acquired from meteorological ilrvcrstiliirliolts ol' tho attnospltcric boundary
layer (see Chapter 2 and l7- ll to l7-41).
Simulation occurs at reduced getltnctric scalc fbr obvious reasons oi cconomy and convenience. The question of scale then opens up the whole area tll'
physical similitude and the necessary underlying theory, which places emphasis
on'u ,"t of dimensionless numbers and/or similarity criteria applicable to both
flow and test models of structures placed in it. With characteristics of the target
flow and scale factors for similitude established, it soon becomes apparent that
certain of the model criteria established for similarity cannot in fact be satisfied
under typical, everyday test conditions. The wind tunnel modeler is thus
launched upon a series of inevitable compromises that render his task complex,
revealing ii as an art of both perfotmance and interpretation rather than an exact
science.
A
basic discussion
of similarity criteria is presented in Sect. 7.1. Wind
tunnels usecl in civil engineering applications are briefly described in Sect' 7'2,
which also includcs comments on some difficulties in achieving similarity between wincl tunnel and atmospheric flows. Section 7.3 is devoted to scaling
problems, insotar as they affect the aerodynamic and aeroelastic behavior of
ihe models to be tested, and to the question of wind tunnel blockage. Section
7.4 reviews some attempts to validate results of wind tunnel tests by comparisons with full-scale *"uru."-.nts. Information on general wind tunnel testing
rechniques is provided in [7-5] to [7-10]. Reference [7-11] is a useful compendium on wind tunnel modeling for civil engineering applications and inin particular, useful information on modern wind tunnel instrumenta-
"lud"r,
tion.
7.1
BASIC SIMILARITY REQUIREMENTS
',r,rttrrl ltlUtlysis ltirsCtl ott lr sct ol'physicirl l)iuilntclt' th it:i:,unt(.(l (!
tlr,' wintl tunncl llow.
tltf ;
215
l,n(ttt lo ;111,.,'1
I I "1 Dimensional Analysis
l'rt t'oltcrctcness, lct it be assumed that thc lilrcc /,'tlcvclo;x'rl sontcwlrr.r't.on
;r lrotly itnmersed in a flowing fluid is a lirrrcliorr only ol'tlrrr lirlkrwirrg six
l';il;ililctcrs: density p, flow velocity U, sonrc typical dirrrcnsion 1), sonrc I'rcrlur'rr('y rr, fluid viscosity p, and gravitational acceleration g. One writes
n
!
p"rfn'rut'g(
(7.1.1)
l'lrt'rt' rv, . . . , f are exponents to be determined. There are three basic quantrtr('ri: nrass M, length L, and time T, to which all of the above parameters are
rlrrirt'rrsionally related. Writing the dimensional equivalent of each of the quanrllrr's in
Eq.7.l. I results in the following dimensional equality:
Y!
t (y)"(n)',.' (;)'(#) (#)'
(7
.1.2)
Ir',rrr which the following three independent equations are obtained by equating
,,,r rt's;xlnding exponents:
M: 1:o*e
L: 1:-3cv+B+-y-e+f
T: _2:_p _6_e_Zf
(7
.1.3)
l lrt'sc cquations may now be solved for any three of the exponents in terms
rrl llrt' r'crnairring three; for example,
In analyzing any problem-more particularly one that is expected to be studied
experimentally-it is usual to identify a set of governing dimensionless parameters. These parameters are in certain cases obtained by first writing the partial
differential equations that describe the physical system at hand. These equations
are then rendered dimensionless by dividing each of the key variables by a
reference value having corresponding dimension. When the process is completed, a number of dimensionless groups emerge as factors goveming the
physical behavior of the system. Maintaining the values of such groups intact
i-- on" situation (prototype) to another (model) will automatically ensure
similarity. In the case of fluid flow, this process involves the conservation
equationi for mass, momentum, and energy, together with the equation of statc
of ttr" fluid. These are written and converted to dimensionless form in thc
manner describccl. In thc prcscnt chapter, however, an ltllcrnitlivc: ltnd simplcr
'l'lris is rt tliltrcnmethod fbr arriving al lho rliutonsionlcss gft)ups will srrllit'c.
trr,l
cv:1-e
0:2-€-6-2f
7:2-e*6+f
(7
.t.4)
irlrr'rrt't' it is seen that
p ! pt eg2-e-b
p' !-- ,l tt tr (
2fD2-E+6+(16rert
'i; )' ( ,,"",,) (?i )'
(7.1.s)
(7
.1.6)
276
wtNf) r UNNII rl
/ t ilntit(] lil[]4il Aililr ilt r,lt ,iltl Ml r!||;
it fbllows that thc dirncnsionlcss lirrcc cocllicicnl* I;lpIJ)l)) is a
function of the dimensionless numbers DnlIl , p"l pIlD, and Dglu).
The dimensionless numbers mentioned are of coursc alrcady wcll known in
fluid mechanics. For example, when n is the frequency n, of vorlex shcdding
from a bluff obiective of cross-sectional dimension D, then
From this
.:s
Dn-
(1
U
tvltclc tt., is lher shcirr, or'll'it'l iorr, vcrlocily, lrrrl .t,, ts
It'rrgllr (scc'l'ablc:
2.2.l). Notc lhirt
U
:GO
(7.1.8)
5t:-
l.t'
'l'hus simple analysis reveals
the several dirnensionless groups that play key
r,lt:s in wind tunnel similitude, particularly in aiding the transfer of results
lrorn experimental model to full-scale prototype.
'l'hough it is not directly pertinent
to the present discussion, it is worth
|.irrting out here that, were thermal effects to be included in the above analysis,
tlrrr:c additional commonly occurring dimensionless numbers would
rurrrtcly:
(7. 1. l0)
I'randtl number:
go :
lpl/)
is tccognizcrl rrs lltc tlynlrtttic l)rcssurc
FCP
K
(7.1.t2)
lickert number:
nu'
Coo
(7.1.13)
F,,
': iil;
lirrrr llrt'llcrrrorrlli
",:+(#)
(7
.1.14)
r'lrcrc Q is specific heat at constant pressure, K is thermal conductivity, and
t/ is absolute temperature. Note that the Richardson number consists of a diilr('nsionless temperature divided by a Froude number; G; plays an impor"tant
r'k: in thermally induced convection in the atmosphere. Because this ihapter
r:r t'oncemed principally with mechanical effects, the last three
numbers are not
,'rrrphasized in what follows.
7.1.2 Basic Scaling Considerations
*Typically coefficients of lift force F. and drag force F, are written
where
"-".g",
v
!6t5 a U4lg
c
.t .t t)
(7.1.9)
viscosity has an order of magnitude near the ground given by
F,
- iirr.
(7
l{ichardson number:
UD
which is sometimes more specifically called the molecular Reynolds number
when z : pl p is the kinematic molecular viscosity of the fluid. In some applications (see Sects. 2.1 .2 and 4.4), a turbulent Reynolds number may bc
employed in which z is replaced by /tu.b, an "eddy" or "turbulence" kinematic
viscosity. It is tentatively suggested, in [7-12] that in the atmosphere such a
c,
u.l
D,,<
is called the Rossby number.
The group y.l pUD is the reciprocal of the well-known Reynolds number
6l€-' oUD
ll7l.
lrinally, thc rocipr<lcal ol'thc group /),q/l/'' is t'rllcrl lft' lirotttlt' ttrtttrltt,t.:
quantity
Dfr
slr.lir(.c rlrrp,lrrrt.ss
lit1. 7. l. lO yit.ltls t.orrslrlcrrtbly lowt.r.virlucs
llr:rrr lh<rsc suggcslctl (irlso lcrrrlativcly) irr 12
.1.7)
is the well-known Strouhal number. When n is n-, a characteristic mechanical
frcqucncy associated with a structure, then Dn^lU is termed the reduced frequ(n(y relative to a steady flow past the structure of velocity U; its reciprocal
Uln,,,D is the associated reduced velocity. The group nzlU-where z is height
abovc ground, n rcprcsents a frequency associated with a component of variablc
wind vcl<rcity, and U is mean wind velocity-is a dimensionless frequency.f'
(called thc Monin coordinate) often used as abscissa in depicting wind velocity
spectra (see Eq. 2.3.17). Further, if n is replaced by the circular frequency./.
: 2<,r sin @, which is the Coriolis parameter (where c,r is the rotational speed
of the earth in radians/second and S is the latitude-see Eq. 1.2.3), then the
lltr
Z7.l
r'r1r:rtiorr (str
lul.
,1. 1.20)
It will bc rccognizcd in lhe: cottsirlt'r';rliorr ol'tlirrrt.nsionlcss numbcrs ab6vc thut
rut tlislincti<ln is lnarlc ils l() s()ut1'(' or.or.i1,.lr ol'lr givcn paritnrclcr: il cirrr ltc
llrritl' s(ructural, <trolltcr. l;ot'cxirttrplt', :r lt'rr1illr, llt.rlucrrt'y, rlcrrsity, or vr'l9t.il-y
ttut,y lrtr itssociltlcltl willr rlrrt'r'lt;tt;tt'lr'u:.l rt ol llrt' llrritl or slrtrclul.r, irrurrt,r.sctl
irr
I
i
278
wlNl) lt,NNl lli
/ t llnl;t(: ritMil Attil r nt rrr fiil
it. This implics IhaI rutirts anr()ng strch rluirrr(i(ir:ri nlr.rsl ltc tttititttltittctl ctltts(itll(
from prototype to model. For cxatttplc, il' p,, antl p/ arc thc ilcrtsity ol' lhc
structure and of the fluid, respectively, thcn
(fi). : (",,),
(:'i,'),,
lcsl
NI:,
2'l.J
(/.1.1())
(''i,')
,r'lr- llrc li'cquency l;t'ulc )r,, l'<tr ull pcrtincrrt
Mt
lr.t't1ut.rrt'it.s
(7. l . ls)
,\u
p refer respectively to model and prototype. Sincc
this holds as well for geometric ratios and geometric shapes in general, it
implies that all model shapes must be geometrically similar to prototype shapes
and that, for example, vibrational modal shapes of prototype structure must bc
maintained in the corresponding model. Likewise frequencies from all sources
must bear the samc ratios to each other in model as in prototype' Further, sincc
oscillatory deflcctions must maintain proper proportionality from prototype to
(1.t.20)
where the subscripts m and
\ilr()s('tcciprocal is the time scale X7..
It rrury be emphasized at this point in this illustrative discussion that \r, \r,,
,rrrrl tr,, lrave been fixed either arbitrarily or in consequence of some unavoidable
i rrt rrrilstilnce. we now inquire as to the consequence of invoking Froude num-
l','r sirrrilitude, requiring
model, dirncnsionlcss damping ratios that affect such deflections must remain
the same in prototype and model.
There now may be examined a typical set of scaling factors together with
the process by which they are set. Three such factors may be arbitrarily chosen.
The first might be an arbitrary length scale:
_D^
(#)^:(#),
r2
'
x.\,
(7.1.16)
'Dp
:l
(7
.t.21)
(7
.1.22)
lrt'tt' L" is the gravitational scale factor. In most instances gravitational effects
rlu.,t lro considered to be the same in model and prototype, so \, : 1, whence
rt
set, for example, by comparison of model size to prototype size. (It will be
seen subsequently what particular considerations enter into the setting of a
length scale when turbulence is involved.) A second choice might be a con-
\r:
J\.
(t .1.23)
venient velocity scale
x,,,
:
U^
-:r
UI'
(1
.t.17)
r'lrt'rr liroude scaling is respected, this may contradict an original choice for
Ar lrr rn<lst cases it is convenient to accede to Froude number scaling, adjusting
,t, ,rlt'orclingly, whence frequency scaling takes the value
\r:l/
set perhaps by available wind tunnel speeds compared to expected natural wind
JI"
(7
.1.24)
speeds, and a third might be a density scale
-P-
(7.r.18)
Pp
\rr'rrtiorr kr gravitational effects may be required for certain structures (e.g.,
or for certain cases where convective air motions are iml,'rrl:ur(. As noted above, the latter are disregarded in the present discussion.
\\'t' rrow rnay examine the effect of invoking Reynolds number scaling:
',rr',Pr'rrsi()rr bridges)
usually forced upon the experimentalist by fixed circumstances (e.g., testing
in airof the same density as that surrounding the prototype, whence Xo : l).
Given the fundamental exigencies of mass, length, and time, the three fixcd
scale choices, once madc, condition all others in conscqucncc ol'the requircment that the dimcnsionlcss groups maintain their conslitrtcy l'trlttt l)K)totypc to
model and vicc vcrs1. 'l'hrrs, lilr cxanrplc, thc rcducctl I't'etlrrcrtt'y rctlttirctrtcn(
("',',,")
ll
Irolo(y1.lr-: ltncl tttotlcl
(+)
(7
.1.2s)
ltlc lrollt itt irit rrrrrlt'r'lrlrrxrspltc:ric conditi<lns, Rcynolds
rrrunlrt'r scllirrg rcrlrrilc:s sirrrlrl-y llrirl
A1tr1 I or'
280
wrNr) rt,NNt
ll
^r,
l/^/
Arcr'prs ,., -,,,,,,,,,,,.
,;:.;"
;,,,:" ,.' ;,, ;,.::, :,':,;,::.',,, ,,.-,
"',:. lcw, il'irrry, lrrlxrr:rloly invr.sligltigrrs
ol lhc lrottntlitly laycI l.ylre,
havc hccrr
t].1.26)
rlt'vtltcd t<l thc sirtttllltliott ol'rlownsl<lpc wintls, lrrrllicirrr.,
crylwalls, trlrnacl'cs,
;rrrtl thundcrstorllls. (Ntltc, lrowcver, thc tcn(ativc sirrrrrllrtion
ol'tornado-inclucctl
which is, in general, in sharp conllict wilh olhcr rcquircr.ncnts sct ahovc, ftrr
example with:
x/: Jt
lorccs in J7-151.)
'l'unnels used fbr civil
engineering purposcs havc cnrss sections that rarcly
t'xcccd 3 m x 3 m. (A notabre exceptirn is thc g
m x g m tunnel of the
Nrrli.nal Research Councir, ottawa, canacra.) Three types
of wind tunnels have
lrt'.rr used for simulating atmospheric flows. They arl referred
to as long tun_
,r'ls, short tunnels, and tunnels with active devices, and are described
in sects.
I )'1., 7 -2.2, and 7.2.3, respectively. Sections 7.2.4 and 7.2.5 comment
on
tlrc possible effects of violating the Reynolds and Rossby number
similarity
rt't;uirements upon the simulation of flow turbulence.
(t .1.21)
in the case of Froude scaling. Thus Reynolds number scaling is seen to be
incompatible with the prior setting of length and velocity scales unless testing
is undertaken at full scale X. : 1.
Another view of the same effect is that, for example, under Froude scaling,
Reynolds number scaling is hugely distorted:
\*":9+:\r,\r-\jrz
(Ge),
To illustrate, if X/.
:
(7
.1.28)
7.2.1 Long Wind Tunnels
1/300, then
/ I \t'' :r,*
I
x'":(:oo)
(7
.1.29)
indicating a tcst Rcynolds number less than one five-thousandth of G" for the
prototype. It is notcd that some aeronautical testing achieves Reynolds numbers
closer to prototype values by using rarefield or compressed fluids, or fluids
with lower kinematic viscosity than air, such as freon. A further recent stage
involves use of gases at cryogenic temperatures t7-131.
Rossby number scaling also proves to be intractable under most circumstances, since an equivalent Coriolis acceleration effect (as represented byl.)
cannot practically be realized to the frequency scale Xn mentioned above. Such
an effect would require some means for imparting lateral acceleration to the
flow, which is not easily achieved, 12-281,12-291, V-141.
Thus normal wind tunnel testing in air under standard gravity and atmospheric conditions typically entails fundamental scale violations of the Reynolds
and of the Rossby number.
7.2
WIND TUNNEL SIMULATIONS OF ATMOSPHERIC FLOWS
To achieve similarity between the model and the prototype, it is desirable to
reproduce at the requisite scale the characteristics of the atmospheric flows
expected to affect the structure of concern (see Sects. 4.6 and 4.7). Thcsc
characteristics have been outlined in Chapter 2.They inclucic (l) thc'variation
of the mean wind spccd with height, (2) the variation ol' lrrrlrrrlt'ncc intcnsitics
and integral scalcs with hcight, and (3) thc spcc(rrr irrttl r'r'oss slx'('tr:r ol-turbulcncc in tho lrlrlng wirttl, rrctrtss-wirttl, ltntl vcrticltl rlitr't ltotts
lrr lrng wind tunnels ([7-16,7-17]) a boundary layer with
a typical depth of
o 5 to I m develops naturally over a rough floor oi the order
oi)o to 30 m in
It'rrgth (Figs. 7 .2.r u-541, 7.2.2, 7.2.3). The depth
of the boundary
layer can
lrt' increased by placing at the test section entrance passive
devices or tn" typ",
rlt'scribed in Sect. 7.2.2. s,ch an artificial increase may
be necessary, particrlrr'ly in simulations of flow over the ocean or over terrain with
low or moderate
r'rrghness. The height
of most tunnels may be adjusted to increase slightly
rvrtlr position downstream. The purpose of such an
adjustment is to achieve a
/('11) pressure gradient streamwise, which would
otherwise not obtain, owing
trr c11c.tt losses associated with flow friction at the
walls and with internal
lrrt'lion due to turbulence.
Atmospheric turbulence simulations in long wind tunnels
are probably the
lr.sl that can be achieved in the present state of the art. However,
even when
l';rssivc devices such as spires are not used, similitude between the turbulence
rrr thc laboratory flow and in the atmosphere is generally
not achieved (see
t'r'rs- 7.2.5 and7.3.1). The rack of similitude becomes
stronger if, for ex_
,rrrrlrlc, spires are employed (see Sect. j.Z.Z).
7
"2.2 Short Wind Tunnels
Irrrrrcls used foraeronautical purposes are usually
designed fortesting in smooth
ll,vv ilnd therefore need not have long test seciions.
tutuny such tJnnels have
lrr't'rr coflvefted for use in civir cngineering
applications by adding, at the test
rirrr cntrance, passivc dcviccs, such as grids, barrierr, r"n."rlund
"r't
spires,
tlr:rr gc:ncrate a thick bounrlirry lrrygr.. 'l'lrr: il<xrr.f
the test section, which is
rr"rr;rlly <ln thc order tll'5 rtr lorrli, is t'ovt'n'tl wilh
nlr.rghncss clcrlcrrts (l;ig.
1 ' "Il Vitritltts typcs. slrrrPes, irrrrl r'rrrrlrirrirliorrs ,rl';lrrssivc
rlcviccs l1rvt, lrct.rr
',rrlilicslctl arrtl corrlrrcnl(:(l ulx)n rrr
|/ l{rl to l7 .}51.
*
1) itt
UC
L)
\
ri,
,\ ={l
\
t |'r:t "r,
;21
;
jtr
L
r:- cblt
a
)4
=.y
OO,
=
4,.
20
o.
{--Dirtance
..1
u:-
11 1)
'- c
z\
.)
cF
,, !' l,' l() ' ilr lil,tl,.
N(,1|
=-J
E
l-J
W)\tt,u
ttl-
ca
'=
\
\-H
l--l
--:
g'E
j
.rril{i,irnr.Ir.
15
I't'
I
IJ
10
from leadinq edqe of
Eell mouth
I f(;uRE 7.2.2. Devclopment of boundary layer in a long wind tunnel.
After A. G.
l):rvcnport and N. Isyumov. "The Application of the Boundary-Layer
wincl runnel to
rlrt' Prcdiction of wind Loading," in proceedings. Intemational
Research Seminar on
wirrtl Effects on Building and Structures, Vor. l, p.221. copyright,
canada, 196g,
I lrriv. of Toronto Press.
<E
a>
.-- ?
-:
a- ,I
--i
.d,
()c
O
a
F
c
F
5- -,
: -.t
*l
v,
f
.d x ':l
!3 r,
I
E
o
cl
\;
.ao
ti
:
=
a
a
c0
.E
0.
)et-t
ii
..
'/,
-o .X n,
-vtt=
0 u-'
trtr
E^-o
'=
=1
Yu',
=ut
,*ui
Ei
iu
!il
a
'a
n
Y
,c
. al
-.
^l
F\e
dc ;,i ijil
I
f i r,lr -l
lrE'i
l'l( jl lltl,l 7.2.-1. 1);rsln';rrl vit'tv ol
I
rrnr'l I .rlrrrl:rloly. I lrt. I Irrtvr.r,.rl\
;r lorr;, ,r.,,,,; trlrrrr.l (r
,,1 \\, ..t, rn ( )ill.rrr,r,
orllr.sy llrrlrrllrry l.trVt.l Wilrrl
2$:l
wlNt) iltNNt
il
I
Wlljl
'
ltll\lNl l:;lMlll
nll()N:i{ll
AlMrr',1'lll
lll(
llrrw',
1'lllr
FIGURE 7.2.5. A proposed spire configuration. From H. P. A. H. Irwin, "The Design
of Spires forWind Simulation," J. Wind Eng.
Ind. Aerodyn., 7 (1981), 361-366.
FIGURE 7.2.4. Spire and roughness arrays in a short wind tunnel (courtesy of thc
National Aeronautical Establishment, National Research Council of Canada).
Reference
[7
-26] proposes the following procedure for the design of spires
'l'he desired mean wind profile occurs at a distance 6ft downstream from the
slrires. According to [7-26], the wind tunnel floor downwind of the spires
sluruld be covered with roughness elements, for example, cubes with height k
srrch rhat (11-261to [7-28]),
:
f *o [(3) '''(?) -
' "''[(a)
0s]
+
'z
]
(7 22)
with the configuration of Fig. 7.2.5:*
1. Select the desired boundaryJayer depth, d.
Select the desired shape of mean velocity profile defined by the power
law exponent, a (Eq. 2.2.26).
3. Obtain the height h of the spires from the relation
2.
h:-
4. obtain
1.396
l*al2
the width of the spire base from Fig. 7.2.6, in which
height of the tunnel test section.
(1.2.t)
rl
is rhc
f,'l(lIJRlt, 7.2.6. Graph fbr obtaining spirc
wirllh. Iinrrn H. P. A. H. Irwin, "'l'lrc
l)r's11'1q 1;l Sllircs lirr Winrl Sirrrtrllliort," ./.
ll irt,l l,.rr,t1 Ittrl. .4rntl.vrr., 7 ( 191{ 1 L l(r I
lrrrsc
xThe base length of thc triangular splittcr plalc in Fig. 7.2.5 is /r/,1. 'l'trc
l;rtt.r;rl sp:rt'irrg bclwccl
the spires is h12. ln praclicc, thc witlth ol lltc lunncl ncctl rrol lrt' trrr intr.1lr:rl rrrrrltiplt' ol /r/2.
It't'
286
il
wrNr) il,NNl r:;
whcrc 1l is lhc sltrrcirrg ol llrc nrtrg.lrrrt'ss t'lt'rrrt'rrls lrrrtl
C,:o116l '|v
'
Equation 7.2.2 is valid in the range 30
ll+rvl
<
(1.2.3')
I
6D2lkt
<
2000.
A study of the dependence of flow features upon the type of passive deviccs
being used was recently presented in U-171. Figure 7.2.1 ll-l1l shows thc
mean velocity, longitudinal turbulence intensity, and vertical turbulence intensity profiles at (l) 6.1 m and (2) 18.3 m downwind of the test section entrancc,
fbr flows obtained by using three different types of spires, the wind tunnel floor
be ing covered by staggered 1.27 cm cubes spaced 5.08 cm apart. In Fig.7 .2.7
the boundary-layer thickncss 6, the mean wind speed U at elevation 6, and thc
power law exponent rr (Eq. 2.2.26) are denoted by deha, Uinf, and EXP,
respectively. It may bc assumed that the mean flow with exponent cy : 0.16
at station x : 6.1 rn, and the mean flow with exponent a : 0.29 at station ,r
: 18.3 m, are approximately representative of open terrain and suburban terrain
4
o
JE
_t)
3
-o
q-o
c\
'-E
f
i'lnfilrrrrirliorr on irrlepnrl st:rlcs lirr llrc wirrtl lrrrrncl llows is rrol rt lrrrlcrl
CDO
qq
:
-_,
-a: L: -: - = cr
jj€:@
-oaF\o)
v_t
(o 6, o)
NOrfi,
o u ,:
!J =\
L
CD
q
oE
@5
4
oo4ir o
q:
I
-s
l
E<l
d)
bO
gd-L
.!
'= E9;
e?
2F\6
--(v)t
Aro\t
o)@r-
:
om
E
.vo.i -.
ii\-r-rr
-t!
cn.
EO
conditions, respectively (see Table 2.2.2).
Some modelers adopt a geometric scale equal to the ratio between the boundary-layer thickness measured in the laboratory and the value 6 of Table 2.2.2,
even though the latter is nominal, rather than physically significant (see Eq.
2.2.15 and Sect. 2.2.4). If this geometric scaling criterion is used for tho
simulations of Fig. 7.2.1 , the geometric scales are found to be 0.751215 :
1136l for the flow with a : 0.16. and l/400 forthe flow with a : 0.29. Thc
respective longitudinal turbulence intensities at 50 rn above ground are about
0.07 and 0.15, versus about 0.15 and 0.225, as obtained from Eqs. 2.2.18,
2.3.1 ,2.3.2, and Table 2.3.1. As expected, the discrepancy between the longitudinal turbulence intensity in the wind tunnel and the "target" value in thc
atmosphere is more severe at the station x : 6.1 m, which would corresponcl
to the fetch available in a short tunnel.
Figure 1.2.8 ll-l1l shows spectra of the longitudinal velocity fluctuations
measured at station x : 18.3 m and elevation z/6 : 0.05 in the three flows
described in Fig. 7.2.7b. Forthe flow with a :0.29, it isseen in Fig. 7.2.1t
that at the nondimensional frequency nzlU(z): 0.8, nS(n)luz = 0.05, versus
0.06, as obtained from Eqs. 2.3.2,2.3.16, and Table 2.3.1. Unlike the turbulence intensity, the higher-frequency spectrum measured in the wind tunncl
is in this instance relatively close to the "target" value.*
The results of []-l7l and of other studies (e.9., U-241, 17-531) indicate that,
regardless of the type of passive devices being used, simulations in short wintl
tunnels generally do not achieve similitude between the turbulence in the laboratory and in the atmospheric flow.
,l
ti
o
-
o
q
q
j,
-e..
F:
yo
:.8€
'oo-E
o7l-P/Z
!
-
^*
*avr\
6 2 >\v
!Ft^
18.h8
o
o
{
-'
{
^t:!
_9
E
E.Ud=
-^ o
[j E ^^o
fo4
{o
d
o_
)<
LrJ
rt(o@
N-E
o ooo
eE
oOO
-(,
o ouro
-(,
Ot-3
(}o 9(oo)
c\ (n @ o)
.-E
)
or@r.(s)
(v
oo<
o
:
! o-i1
.vJ,
G)
-
$
oQ
eat
E.
6
a '. ,-l -'
L
z6)H
(o :f
f,
6t
q
o
I
Eq
ol t-P/z
o
N
<
lGt
oc!
o
:
EFA 3
9/
-_.= \ .:
(Jsdl-:
it'
oo >,=
=
c v >.\+
'r'"bv
sv
<
/.
r'
Yg.i
-'
g5
u
rioou
'. q 6 -S
N^I- o:3
-.
r o'd.\
o\
.-r
E Is={
C(
b +'i
D ".5i:
\JAE
tr€<4,
tt l / I ll
?47
t
C9
q
GI
F
H
o
z
lrJ
F
z
H
uJ
(J
z,
J
lrJ
OE
rO
;
!
o
OG'O
SrS)O
od6;
('ct9
t o
@
d.
d
q-O
-G)!f
O q
c\
$I
.-€
D
6<
o)@l't
t
p
D
F
J
{
Cd
z
6
F
lrj
F
9:
o
e
F
Ed
(J
!
-a-'
f
o
lrj
J
l
.D
d
f
F
J
z.
ts
dg
6
E
I
@
E(D<
O^
q:rQ
{{a
@
E
(o\
o
{
o
E
(.
EJ
d
lC9
o
o
o6roaoe
ino@@scl
oE
p0
o
-tt
(}la
c\
.-e
)
(rt
ooo
(9
qq9
I
q
l-P/z
U
OO(9
OOO
ouro
tt F- O
@
o)crt-
(o^
(9j
o
6<
4
o
J
o
;)
F
H
trr
lr.l
9.
-
H
trJ
2
t^J
.i-Ct
c\
.-E
)
(oo,
o oo)
.o@
F
ri
ID
h
o
o
I
'E
@ae<
,!
Oa
Or:
@.--
.rl
(D5l.
tOv
q;
F.
T
oE o (9o
:0 q qq
l,o
F.!too
F
d
E
.a
\)
o
a
9'-
o
e
d
)
&
;
B<'
I
I
l-P/z
t(\
rd
9o
$JO
d
c,l
F
q
E-1
1
o
oo
at
(9
$(oo) o)
(v)
ao
F-
(rt
Gr@<
F
H
(9
z.
o1
o
E<R
o{E
H
6
o
E
I
o]
ov
FI
c;
(J
F
(Y
L
H
oo
(\,
C9
o
@
o
q
o1 l-P/Z
288
aq<
d,
Gi
trJ
{
H
F
c;
o
(9
z
lrJ
F
z
H
o
o
UJ
o1
u',
a
€
.t
or@1..
I
F
H
!
trO-(',rt
(\, o {
c\
.-e
2
trJ
J
d
g
Esa--
@
o
H
(9
z
o
J
*
I
f d@
e
@
a
f
z
z
H
lrl
d4
H
.n
q
&E
6oo
@<
aD
F
H
oE ocpo
ro
C9OO
dcr;a;
;!
,o99
oc9
l(o
@o
o
I
oc)
:q
o
q
o7l.P/z
oQo
Ic,l
o
I
269
wtNt) tUNflt
t
:
l jllll'
lrllltll
I :;lMl,l n ll()N:i ()l n IM(1: ;l,l ll ltl(. ll()W:,
:'!|
I
D
EXP
o .26
.2O
^o29
gol
soot
csto
r
.
aoo6
t6
.
w6a
t'l(;tJl{lt 7.2.11. Spcctra of longitudinal velocity fluctuations measured at 18.3 rrr
downwind ol spires. Reprinted with permission from J. E. Cermak, "Physical Mocleling of the Atmospheric Boundary Layer (ABL) in Long Boundary-Layer Wind Tunnels (BLWT)," inWindTunnel ModelingJbr Engineering Applications,T. A. Reinhold
(ed.), Cambridge University Press, Cambridge, 1982.
7.2.3 Tunnels with Active Devices
FIGURE 7.2.9. Upstream view of rhe test scction and jets of the 1.20 x 1.70 m
closed-circuit jet tunnel. University of Toronto Institute for Aerospace Studies (courtesy
I)r. H. W.
Teunissen).
In tunnels equipped with jets (Fig. 1 .2.9) it is possible, within certain limits,
to vary the mean velocity profile and the flow turbulence independently of each
other [7-29,7-301. Such tunnels are relatively expensive and do not necessarily
result in superior flow simulations. However, they may be useful for basic
studies in which the effect of varying some flow characteristics independently
of the others can be studied in detail.
Active cascades of moving airfoils (Fig. 7 .2.10) have been recently designed
with a view to creating, and simulating effects of, large-scale turbulence over
bridge deck section models l7-31, 7-321.
7.2.4 Reynolds Number and Turbulent Flow Simulation
It is suggested in [7-33, p.204,7-34, p. 266, andT-35, p.290], that Reynolds
numbers of turbulent flows obtained in the laboratory downwind of square mesh
grids may in some cases be too small to give rise to a turbulence spectrunr
having an inertial subrange. It is further suggested [7-35] that the Reynolds
number based on eddy size should be the order of 105 to ensure existencc ol'
this subrange. Applying analogous rcasoning to a devclopcrl lurbulcnt houndary
layerof depth, say,0.5 rn, in which the intcgral scalc lcrrg,llr /,) (rr rrrcusurc ol'
typical eddy sizc) is :rbotrt 0. 125 rn, u Ilcynolils rrtrrrrlrcr ;rl :r vt'locily ol' lr
m/s may bc culcrrlirlt'tl
Itl( il lltl,l 7.2.111. Mt't'lurrtit;rllv tlr rvcn
sirrrrrl:rliorr
l7
l-?1.
:rrr
l.rl
r ;r:;t :rtlr
lol low l'rctlrrcncy
lurbulencc
292
wrND ruNNr tr
/:'
6le
:
ur
.:.
":
u
l2(o.
r25
x
l0
1..5
)
wtNl) il,NNl t:ilMt,l All()Nr;()l nlM{r'.l,llt lilr
'lrrw',
2q:l
.-10'
Thus typical boundaryJayer simulations of the kind discussed may be expectccl
to develop velocity spectra with satisfactory inertial subranges, though at lower
velocities and turbulence integral scales they may be borderline.
EI
ils
8
.-l
JI
7.2.5 Rossby Numbers and Turbulent Flow Simulation
Failure of Rossby number equivalence in typical test circumstances is due to
coriolis parameter f, above its automatically achieved
full-scale value. Rotating wind tunnels (12-28, 2-29D, or tunnels with porous
walls and acnrss-wind suction imparting lateral acceleration to the flow [7-17]
arc currcntly not used in civil engineering applications. An investigation into
the effect ol'the Rossby number on boundary-layer flow is therefore in order.
It was shown in Sect. 2.2 that the approximate depth of the atmospheric
boundary layer may be expressed as
/////t
Area
of experimental
data
the difficulty of scaling the
0.05
where
0.5
1
t,
l(;t.lRE 7.2.11. Logarithmic plot of velocity distributions in turbulent boundary lay-
F. H. Clauser, "The Turbulent Boundary Layer," Advances
Mech., 4 (1956), Academic Press, New York, p. 9.
r'rs ()vcr plates. After
(7.2.4)
ns(2,
z* : U(h)l{2.5Ln(hlzd\,
U(h) is the mean speed at a reference heighr
h, zois the roughness length, andf, is the Coriolis parameter. Equation7.2.4
can also be written in a form that emphasizes the dependence of the atmospheric
boundary-layer depth 6 upon Rossby number:
6:cGo
0.2
6
l1t1t!.
a = o.zsT
0.1
(1.2.s)
where c = 0.25h1{2.5ln(hlzi} and G.o : U(h)lhf,. The boundary-layer depth
6 is seen to be an increasing function of wind speed. For high wind speeds
such as are of interest to the structural designer, it follows from Eq. 7 .2.5 that
6 is of the order of several kilometers. For example, if z6 : 0.05 m (open
terrain), U(10) : 25 m/sec, andf,: lO-a sec-l(corresponding to an angle
of latitude 6 = 45o, see Table 1.2.1), then 6 = 5 km. The region of interest
to the structural designer, that is, the lowest few hundred meters of the atmospheric boundary layer, is thus seen to amount to about one-tenth or less of
the full atmospheric boundary-layer depth.
As noted in [7-36], both in the atmosphere and in the laboratory the mean
velocity profile is very nearly logarithmic over the region consisting of the
lower one-tenth of the boundary layer or so (see Sect.2.2 and Fig. l.Z.1l).
Moreover, measurements suggest that in this region thc turhLrlcnt cnergy production is approximatcly balanccd by the energy dissipiriiorr (ltig.7.2.l2 ancl
17-371) so that thc lottgitLrclinal vclocity spcctrurn in llrt'int.11 i:rl srrl'rllrngc rnay
bc cxprcssccl in txrrrtlinrr:nsiontrl lilrrrr lrs
Ux
n)
:
0.26
f -r,,
(2.3.t6)
wlrcrc n is the frequency, z is the height, and f : nzlU(z) (see Sect.2.3).
lrtluirtion 2.3.16 is not valid in the upper region of the boundary layer where
tlrt'cnergy production differs significantly from the energy dissipation (see Fig.
t).t2).
('onsider now a long wind tunnel in which the boundary layer develops
rrrrtrrrally over a rough floor and in which the boundary-layer depth is of the
,rrrlcr of I m (Fig. 1.2.2). Assume that the height of the building being tested
rs 200 m and that the model scale is 1/400. Since, as was shown above, the
r,'gion of the atmospheric boundary layer over which the logarithmic law holds
rs (rrnder strong wind conditions) a few hundred meters high, it is reasonable
{() irssume that Eq. 2.3.16 is valid throughout the building height. However,
rrr tlrc laboratory similarity theory suggests that Eq. 2.3.16 can only be applied
lo a height of approximately 0.1 m from the wind tunnel floor, to which
tlrt'rc would correspond a full-scale height of just 40 m above ground.
A schematic representation ol' thc situation just described is given in Fig.
/ I 13, which shows thc bourttllrry llrycr that dcvclops in a long wind tunnel
(lrrll lirrc), and thc atmosphcric lrourrtllrry lrrycr rcrluccd to model scalc (br<lkcn
lrnc). 'l'ho lowcr onc-tcntlr antl llrc orrlt't rtirtc lenllts ol'thc b<luntlitry laycr irrc
tk'rrolctl hy /,,,,, ancl (),,,,, rcsptrt'livcly. lot lltt'witttl lrtttttcl llow, itrrtl by /,,, lttltl
(1,, respr:clivcly, lor llrc lrtrrrrrsplrt'rrt' llow. ll t'lrn lrc sccu itt lrig. 7.2.l.l tlrtrl
rr;r
i
294
WINIJ I UNNFI
/3
Si
WIND IUNNFt lilMt,l
All{,N ()l Al llol)YNnMl(: nNl) Al ll(}l ln:illtr lll llnvl()l I
295
ovcr tttost <ll'thc wirttl lrrrtrtcl lrorrttrlary-llrycr tlcpllr llrc irlrrrosplrt'rrt' llow rrr llrt'
lrtwor laycr L,, is sittrulalctl by tltc llow
in lltc otrtcl
r('p,ron
(),t tt wlrt'lr,
rrccxrrding to similarity thcory, L.,q.2.3.16 woultl rrol hc t'x1rr'r'tctl
lo lroltl.
7.3
WIND TUNNEL SIMULATION OF AERODYNAMIC AND
AEROELASTIC BEHAVIOR OF BLUFF BODIES
'l'his section considers some practical aspects of the dependence of the aerorlynamic and aeroelastic response of wind tunnel models upon the turbulence
t'lraracteristics and the Reynolds number of the flow.
rrc:rodynamic distortions due to blockage effects.
7.3.1
I
v
I
c
.G
_,0
0
0.1 0.2 0.3 o.4 0.52 0.6 0.7
0.8
6
FIGURE 7.2.12. Energy balance in a turbulent boundary layer. After A. A. Townsend., The Structure of Turbulent Shear Flow, Cambridge Univ. Press, New York,
1956, p. 234.
(l
,t
'l'hc details of the dependence of the aerodynamic and aeroelastic behavior of
lrtilies upon the turbulence characteristics of the flow are not fully understood.
llowever, it is clear that for the effects of turbulence on the model to be similar
to those on the prototype (i.e., in order forthe turbulent eddies to envelop or
otherwise affect the body or part thereof in a similar way in the atmosphere
rrnd in the laboratory), it is necessary that the ratio between some typical length
t'haracterizing the turbulence and some characteristic dimension of the body be
thc same in both situations.
lt is convenient to adopt the integral scale Lj (see Sect. 2.3.2) as the charrrcteristic length of turbulence. The geometric scale factor of the simulation,
l), : D^lDo, should then be given by
(7
.3.1)
where (Lj)o and (Il)^ are, respectively, an estimate of the integral scale that
in the atmosphere at some representative elevation (see Sect. 2.3.2),
irrrd the integral scale measured in the wind tunnel flow at the corresponding
clcvation above the tunnel floor. The application of Eq. 7.3.1 is discussed is
rrbtains
Atmospheric boundary layer
(reduced to model scale)
lir--l-
FIGURE 7.2.13. Lowcr and outcr rcgi.ns of the bouncr,ly lirye'i'rrrr: wintl tunncl
and in thc atmosphcrc.
also briefly discusses
Effect of Turbulence Characteristics of the Flow
Dr:w
,l
i,
It
|
7-381.
Equation 7.3.1 is violated in many instances because of the difficulty of
rrchieving sufficiently large integral scales in the laboratory, particularly in short
wind tunnels (see Sect. 7.2.2). However, even if Eq.7.3.1 is nominally satisliod, it should be recalled that integral scales are poorly known and can vary
lhrm measurement to measurcmcnt by a factor of five or even ten (see Sect.
).3.2). thus, the assumed valuc ol'lhc ratio (lj)-lUi), can differ significantly
lnlrn its actual value.
An atlompt to asscss crnrrs rhrt' lo llrt' irrrl'lcll'cct sirnulation <ll'thc intcgrul
scirlc <ll'turhuloncc is rcporlr:tl in l7 lt)l lirr. llrt' l)rcsslrrcs at vlrriorrs lxrinls ol'
296
il
wtNt) iltNNl l:
/:r wtNl) iltNNl I i,lMt,l n iii
)t.t I
I nl lt()l lyNnMt(] Ail! ) nl |{
)t
tl\"il,
nt |/\\/ti )tt
l Alll,l,l 7.-1.1. l{:rlios ol'l't:th, Mt:ut, :tntl l{NlS I'r'rsrurr.r orr l/l(ll) ;rrrrl
Nlrxltls lo ()orrrspolrrlirrg lDrt.ssrrrcs orr l/SlX) Morlel"
ti t(x)
I
I
lAl'
30.5
.',)" 1.34(0.93)
t'/t' 0.90(0.97)
(,t' 1.00( I .02)
I I r' 0.69(0.75)
t/' 0.84(0.83)
(rlli' 1.05(1.07)
t
a
68
111
Pcak
.l
Mcan
nls
l't'ltk
Mr'.rrr
/-l)
nr:,
1.09(0.51)
0.62(0.es)
O.(XXO.
o.'/t{( Lo
I)
L.)(r( .f .,1O1
o.()i,i( l.o/)
1.67( 1.90)
l.l3(1.90)
0 u4(0.tt
I)
L20(0
r.00(0.43)
0.60(0.67)
0.83(0.91)
0.80(0.67)
0.96(0.93)
l.40( 1.40)
l/lll)
I/"rl|
r.r6(r.48)
r.6-5(0.19)
?1lI
LO t(() f{t{)
(
).()l(o.l.iti
)
I)
1.0,1(0.71{)
0.63(0.7rJ)
r.00(0.7-5)
0..53(0.57)
0.90(0.95)
0.83(0.90)
0.81(0.79)
0.83(0.e 1)
0.73(0.81)
n
r.07(0.97)
'Nrrrttbers not between parentheses comespond to open exposure. Numbers between parentheses
rrrrcspofld to built-up exposure.
'Srrc(ions.
,
I
1.0 m
I
|
a
r'l
6
)q
47
a
' 4.9
m
24.4 m
FIGURE 7.3.1. Schematic view of building with pressure taps (After I7-391).
'r c ssu
res.
;rntl lluctuating pressures associated with separated flows can be properly simrrlrrtcd even if only the small scale turbulence is correctly reproduced. This
r'vould require (l) the correct reproduction of the longitudinal and lateral turlrrrlcnce intensity, and (2) the use of sufficiently large models. Thus, as is the
( rsc for the prototype, higher-frequency components of the longitudinal velocrly spectrum that affect the separated flow would be contained within the inertial
rrrbrange; see Sect. 7.2.5, Eq. 2.3.16.
/.3.2
the building represented schematically in Fig. 7.3.1.In the investigation of
U-391 the integral scale was not varied independently of the other flow features.
Rather, the wind tunnel boundary-layer flow was kept unchanged while the
dimensions of the model were increased. It was estimated that the integral scale
Iiinthe wind tunnel was equal to about l/500 times a nominal integral scalc
judged to be typical of atmospheric flows. Measurements were made on l/500,
ll25o, and l/100 models of the same building. Ratios between the peak, mean,
and rms pressures measured at several points on the 1/100 and 1/250 models,
and the corresponding pressures on the l/500 model, are listed in Table 7.3.1.
It is seen that in some instances the influence of the model size upon thc
test results is significant (e.g., for the peak pressures at tap 29, 1/100 scale,
or tap 1 I l, l/100 scale and 1/250 scale). Note also that the pattern of variation
of the ratios of Table 7.3.1 is irregular. This may be due, at least in paft, rtr
the fact that by changing the height of the model by the factors 2.5 and 5, thc
turbulence intensities at the elevation of the points under consideration also
changed.
The effect of turbulcncc features upon the modcling ol'lrcrorlyrr:rrrric hohavirlr
17 4t)l to l7 421, Accorcling ro 17,40. 7 .lll. rlrt. nrle ol' thc
intcgral (urbttlcttc'c: scitlc ilt wirttl (unncl sirrrrrlirliorr is rrrrrror tl rro( 1t'pligi$t:.
is discusscd in
Reynolds Number Effects
Slrarp comers tend to cause immediate flow separation, independently of the
licynolds number of the flow. For this reason it is generally assumed that if
tlrc flow is adequately simulated, pressures on rectangular and other sharpr'or'flered structures are adequately reproduced in the wind tunnel. However,
lrlrrllbodies with long afterbody extensions downstream may exhibit flow reatt:rchment, which does depend on the Reynolds number. Such circumstances
rrny affect the values of the across-wind forces experienced by the body. Few
lrrll scale supporting data on this topic are available to date. Note also that if
tlrr: details of a scale model require extremely small dimensions (as, for ex:rrrrple, in modeling the members of a truss structure at a scale of 1/500 or
lrt'low) it may be that the drag coelicient applicable to such a member can be
rrrrtluly influenced (raised) by Reynolds number effects. Figure 4.5.6 bears out
tlris tcndcncy.
lrt thc case of bodies wilh curvc:cl surfaces, Reynolds number deficiencies
,;rrr llavc significant cll'ccts. 'l'lris is sirrrply illustratcd by the evolution of both
rttr'lttt tlntg crlcflicicnt untl Slrotrlr;rl rtrrrrrlrt'r'lir rr cir-cular cylindrical section as
:r lirrrction ol' llcyrrolrls trrrrrrlrt'r (st'r' liilrs. .1..1.,1 :rrrtl 4.5.2).
As irrtlic:rlctl ilr ('lrirplt'1 ,1 , lltt':r,'torlyn;unr( l)('lritviorol'srrch b<xlic:s rlcpcnrls
,rlt wltt'lltt't (ltt'lrotttttl:tt'y l:ryt'ls olt llrr', rrrvt'tl :,rul:rtt's;u('llrrrrinlrror (lllrrlitrlly
298
wrNt) tuNNt
l
/it wtNt)
or fully) turbulcnt. Sincc bounclary laycrs occurring
ll
tt,NNt
I
l;tMl,t n lt()N ()t Al ll( )l)YNn Mtc
n Nt
, n I il(
,t I n !;t
t(
lr! ItAVtilI I
?!lgl
lrigh llcylroltls rrurrrbcrs
it is logical t() attcmpt tho rcpr<lduction ol'lull-scalc ll<lws ar<lunrl
smooth cylinders by changing laminar boundary layers into turbulcnt oncs.
This can be done by providing the surfacc with roughness elemcnts (scc
14-151, U-431 to [7-46], and Fig. 4.5.5). Ir is suggested in 17-441rhar rhc
thickness e of the roughness elements should satisfy the relations
are turbulent,
lirrroollr rrrorirl
Morlcl:;
willr 0.U nrrr
attrl
wrtlr 1 rnrrr wires
Ue
400
U
e
D
l0-2
-2-10+1
whcro U is thc rncan wind speed, a is the kinematic viscosity (u = 1.5 x
l0 s m2lsec in air), ancl D is the characteristic transverse dimension of the
ep
model.
For exarnplc, in the case of the DMA tower (Fig. 15.3.22), the roughness
was achieved by fixing onto the surface of the 1/200 model thirty-two equidistant vertical wires. Three sets of experiments are reported in[7-44] in which
the surface of the cylinder was (1) smooth, (2) provided with 0.6-mm wires
(elD - 7 x 10-3;, and (3) provided with l-mm wires, respectively. It was
found that the highest mean and peak pressures were more than twice as high
on the smooth model than on the models provided with wires. The differences
between pressures on the model with 0.6 mm and the model with l-mm wires
were small. The influence of the roughness on the magnitude of the mean
pressures at 2O m (full-scale) below the top of the building is shown in Fig.
7.3.2 in which the mean pressure coefficient 4 is defined as follows:
Ln: jPul
p is the measured mean pressure, p, is the static reference pressure, U,
is the mean speed at top of the building , and p is the air density.
Approaches of the type described above were found to yield acceptable
where
l,'l(iURE 7.3.2. Influence of model surface roughness on pressure distribution [7-44].
Corrections for blockage depend upon the body shape, the nature of the
of concern (i.e., whether drag, lift, Strouhal number, and
so fbrth), the characteristics of the wind tunnel flow, and the relative body/
wirrd tunnel dimensions. Basic studies on blockage are summaized in 17-471
rolT-491and in [7-50], which also contains a bibliography on this topic.
It is concluded in [7-50] that, in the case of drag, the following approximate
rclation may be used for the great majority of model configurations in all flows,
rrrcluding boundary-layer flows:
rrt:rodynamic effect
results in cases not involving aeroelastic motions. However, if aeroelastic effects are present, wind tunnel tests in which such approaches are used can
provide an utterly misleading picture
of the behavior of the prototype
CD
I + KSIC
(7.3.2)
(see
Chapter 10).
7.3.3 Wind Tunnel Blockage
wlrcrc Cp, is the corrected drag coefficient, Cp is the drag coelficient measured
rrr thc wind tunnel, S is the refercnce area for the drag coeffrcients Cp, and Cp,
rurtl C is the wind tunnel cnrss-scctional area. The ratio S/C is referred to as
A body placed in a wind tunnel will paftially obstruct thc passagc of air, causing
the flow to accclcratc, This cffect is refbrred to as bkrcklrgr:. ll'thc blockagc is
substantial, thc ll<lw anttttttl lhc nrodcl , and thc rnork:l's rrcrrrtl,yrtrnric.hchuvi11r,
arc no longcr rcprr-:srrrrlirlivc: ol' prrrlolypc c<lntliliorrs.
'l'hc c<lcflicicnt K has bccn rlclcrrttirtc:tl only for a limitcd numbcr ol'situirliotts. lior oxalnplc, in thc clrse ol'ir lrlrt'with a rcc(angular cK)ss sccli()n sl):ut
rrirrg thc crrtirc hcight ol'a wirttl lrrrrrrt'l willr rrornirrirlly snroolh lkrw, A wlrs
tlt'ltrrtttittctl lo tlcpcrttl ttpon lltc rirlto ttll, irs slrowrr in lrig. 7.-1.,1 (rr lrrrrl /r lrlt'
lrkrckagc ratio.
WIND TUNNELS
'eH.9"
l:i
r--o--"-1
ll
r
:,i
" ,-
-
:"e
60
1l:l:,j;
Fr '1, all
=rtilr€tr
Li1
-d- ut*-4
H'P.,=
g eCA
d
0E9.
=
<-z
=
]]: ::']1:j:.*
E
z-
;
6-
5 E H3
!3
x
-\,
:
-=6a
=
=
rntrHov
-ad
X
tsc '
!LAW.
5 Or:5\O
Oy'r{\O
I
oed,:
ts
o
ts-q-c
ch
o q)
4,^'
=
h..oo9il
-a E E
@d
=o.
C 3;F.r
.tzlEx
V
-5-R' S
i 69
E FF
.. E,H O U
lteE
=s
* *c,.U.
p
60?\
ts=s
cr9
FO.o,5
:^
i,;
sd
t_t:
lll03 lSnsslud t^isws
60
l
lll0l
lSnsslSd
=
NVlw
l-iNFll\
a/b
FIGURE 7.3.3. Blockage correction factor K for two-dimensional prism with alongwind dimension a and across-wind dimension b in nominally smooth flow [7-49].
the dimensions of the along-wind and across-wind sides of the rectangular cross
section, respectively).
The effect of turbulence on blockage by flat plates was studied in [7-48] for
flows with uniform mean speed, and was found to be negligible in most situations. On the other hand, it is stated in [7-50] that this effect can be significant.
Thus, according to [7-50], turbulence does not increase the drag on a square
plate, as concluded
in 17-251
(see Table 4.6.2). Rather, the increased drag
reported in 14-251 was only apparent, and it was the blockage effects that were
affected differently by various turbulence levels. For a basic study of blockage
effects on bluff-body aerodynamics, see t7-551.
Despite such ongoing debates and various continuing uncerlainties. it may
be assumed that for blockage ratios of 2% the blockage corrections are likely
to be about 5%, and that the magnitude of the blockage correction is proportional to the blockage ratio [7-50].
EEbg
CF,
>v o:
toro
: 9i.F"&
<-z
I-----1
l.l
-
Oz*
6-
o o XFlr
BE
F-fr
I
e :8e
Bx-,65
E6
o
d) @r
:\d
E E;T"E
oVA
>
]
:..H'o!
-ad
E
!
= 3E<5S
'"
iJ>es
-.-=r\
'8:
=Z
.r I r
F- tr Y=:.
7.4
VALIDATION OF WIND TUNNEL TESTING
Despite the numerous full-scale measurements reportcd in lhc litcraturc, the
number of dependablc comparisons between modcl anrl plototypc rcsults rcmains relativcly srnall. liigurcs 7.4.1 m,J 7.4.2 slrrrw ir cotttllrlisorr hctwccn
rL
5€
,
*
cJ o=
P:,A
tiliNFz-o
E
oi,
i
301
302
wrND rUNNr rs;
/4
wind tunncl and l'ull-sclrlo nrcasurcnrt.rrls ol prt'ssurcs on llter ('olrrrtrcrcc ('orrt't
tower (Fig. 15.3.17). '['hc winil tttttncl vrrlrrui wcrc providccl at lhc rlcsign stagc
and are represented by opcn circlos. 'l'hc solitl lincs join avcftlgc valucs ol'
estimates derived from actual observations ol'prcssurc dillbrences on thc builcling; the shaded areas indicate the standarcl clcviation of the full-scale estirnatcs
l7-5 U. It is seen that the agreement between model and full-scale measurements
of the mean pressures is satisfactory.* However, it appears from Figs. 7.4. I
and 1 .4.2 that local fluctuating pressures attributable to vortex shedding (fluc,
tuating lift) differ at some points significantly in the wind tunnel from thc
pressures on the prototype.
Further data for this building are available in [7-51] and [7-52]. Figure 7.4.3
shows acceleration spectra obtained from full-scale measurements and from
tests on a model of the building with seven lumped mass levels. It is seen that
in this case the model tests tended to underestimate the response in the intermediate-frequency range but appear to be adequate at the low and high ends
of the spectrum.
Model/full-scale comparisons for pressures on low-rise buildings are also
reported in 14-74,7-4Ol to f7-42,7-53,7-561. According to V-561, compari-
VAI
ll)All()N ()l Wllllt ltllltll I ll 'iilf lrr
ilo;l
ril'll+l
rtl,{" A,lr
"
i"* 'tl
e.
Lti
E
1.2
o-u U.al
0,4
L
o
O
2-O
-
O
O
N
N
o 10J
6
E rc2
E
a
z.
ul
o
1
E.
F
1
00
200
J00
400
Azimuth, degrees.
I
-a
o
fuli
Scole
CSU
UWo (rough exp,)
UWO (smooth exp,)
rc0
(a)
(h)
l'l(;Illtlt
7.4.4. wind pressure coefficients on the Texas Tech Experimental Buildlirll-scale and wind tunnel measurements: (a) wall pressures; (b) corner roof pres'.rtrr's. Iinrm W. H. Tieleman "Problems Associated with Flow Modelling Procedures
l,u l,ow-Rise Structures," J. Wind. Eng. Ind. Aerodyn., 4l-44 (1992), 923,934.
_t
a 10'
rrl'
E
fi-2
0.0
o.2
0.6
FREQUENCY (Hz)
FIGURE 7.4.3. Full-scale and model north-south acceleration spectra, Commercc
Court Building. Reprinted with permission from E. A. Dalgliesh, "Comparison ol'
Model and Full-Scale Tests of the Commerce Court Building in Toronto," in Wind
Tunnel Modeling for Civil Engineering Applications, T. A. Reinhold (ed.). Cambridgc
Univ. Press, Cambridgc, U.K.,
I
J00
I
I
IJ
(L
*In Figs. 7.4.
2OO
I Full Sco e
CSU
-o UWo (rough exp.)
o UWO (smooth exp,)
rc1
C)
(L
=
O0
Azimuth, degrees-
J
rnoan l7-5 I l.
1.0
1982.
and'7.4.2 tltc ltlthrcviatiott IIMSM clcrxrtcs r'(x)l nrr'irr \(lllir('virlu('itl)()rl lll('
'.rrrrs bctween full-scale and wind tunnel measurements on low-rise gable-roof
I'rrrltlings suggest that the wind tunnel does not model accurately the flow
'.i'|':illrlion on the windward roof, so roof pressures often differ significantly in
tlrt' rrrrltlcl fiom the prototypc. Sirnilar discrepancies occur between pressures
nr,'rrsurr(l 0n models of diflcrcrr( scirlcs; scc Fig. 7.3.1 and Table 7.3.1. Figurc
/'1.'l shows rncasuremcnts ()n ir lirll sclrlc:'l'cxas Tcch University cxpcrirncnlar
I'rrrltlirtg ittttl Coloratlo Slirlc: []nivcrsity lrrxl I lnivcrsity of Wcslcrn Orrtirrio wintl
Ittttttt'l ttttttlcls rll'tlritl hrrilrlirrg. Wrttrl ltttrrrt'l nre:rsulcnlcn(s lrrcl scrr:rr lo llc
;rtr't'lrlltl'rlc lilr llrc w:tll ltrcsstrtt's ltttl t;tttlt'irr;rrlcrqrrlrtc lirr lhc nrol r'()lnr.t.
304
wtND I(JNNI I ri
ilt
However, accorcling lo l7-41 l, a corrsitlcllrlllt: irrrpnlvcrttcnl ol tlrc wirrtl lunncrl
modeling of roof corncr and ttthcr prcssul'cs can bc achicvcd by placing srnall
spires directly upstream of the modcl lo sitnulatc correctly thc turbulencc intensities, as well as the spectral densities at a I'requcncy l}U(h)lB, lbr botlt
the longitudinal and lateral turbulence (U : mean wind speed, h : building
height, B : characteristic dimension equal to h for low-rise buildings and ttl
the least horizontal building dimension for tall buildings).
REFERENCES
7-l
D. A. Haugen (ed.), Workshop on Micrometeorology, American Meteorological
Society, Boston,
1-2
1-3
MA,
1973.
Charat:lcristics fi'Windspced in the Lower I'ayers of the Atmosphere near the
Ground: Strong Wintl,r (Ncutral Atmosphere), ESDU Data Item No. 72026'
Engincering Scicnccs Data Unit, London, 1972.
Charude ristics 0.f'Almlsphcric Turbulence near the Ground, ESDU Data Items
Nos. 74030, 14031,75001, Engineering Sciences Data Unit, London, 1974.
1975.
7-4
7-5
J. Counihan, "Adiabatic Atmospheric Boundary Layers: A Review and Analysis
of Data from the Period 1880-19722," Atmos. Environ., 9 (1975), 871-905.
A. Pope and J. J. Harper, Low-Speed Wind Tunnel Testing, Wiley, New York,
t966.
7-6
1-7
S. M. Gorlin and I. I. Slezinger, WindTunnels andTheir Instrumentation, Israel
Program for Scientific Translations, Jerusalem, 1966'
R. C. Pankhurst and D. W. Holder, Wind Tunnel Technique, Putnam, London,
1-8
E. Ower and R. C. Pankhurst, The Measurement of Air Flow, 4th
1968.
ed.
, Pergamon,
Oxford. 1969.
1-9 P. Bradshaw, An Introduction to Turbulence and Its Measurement, Pergamon,
Oxford, 1971.
7-10 W. Merzkirch. Flow Visurtlization, Academic, New York, 1974.
7-ll T. A. Reinhold (ed.), Wind Tunnel Modeling for Civil Engineering Applications,
1-12
Proceedings of International Workshop, Gaithersburg, MD, April 1982, Cambridge Univ. Press, Cambridge, 1982.
R. Britter, "Modeling Flow over Complex Terrain and Implications for Detcrmining the Extent of Adjacent Terrain to be Modeled," Wind Tunnel Modeling
for Civil Engineering, Applications, T. A. Reinhold (ed.), Cambridge Univ.
Press, Cambridge, pp. 186-196.
1-13 High Reynolds Number Research, D. D. Baals (ed.), NASA
CP-2O09 (19'17)
Proceedings of Workshop, Langley Research Center, Hampton,
197(r.
VA, Oct'
7-14 D. R. Caldwell and C. W. Van Atta, "Ekman Bountlary l.aycr Instabilitics,"
J. Fluid Math..44, P'arr l (Oct. 1970), 19-957-15 M. C. Jischkc and Il. I). l,ight, "l-aboratory Sirrrulrrtiott ol 'l'olttiulic Wintl l,oirtls
on a Cylirrtlr-iell Slrrrt'lrrlc," in Witul l'.)r,q,ittt'rrirt.ti, I'totr'r'rlittg,s ol- tlrt: lrilill
lllil
N(;t
l;
305
Itllctrtltliottttl ('ottlr'lcltr.'e. lrorl ('ollirts, ('olorirtlr..lrrly lrl/()..1 .li. ('crrrurk (ctl.),
Vol.2, l)l). 104() lO.5(), Pclgrrrnorr ltress, ()xlonl, lt)ll0.
A. (1. I)avcnporl :rrrtl N. lsyrrrrrov, "'l'hc Alrlrlrt'rliorr ol rlrr: I]ourrdary-Laycr
wintl rtrnncl to thc Prcdiction ol'wirrtl l,.rulirrg." rn l'nx.ccrlings of'rhe Inter-
ttl
tttttitrral Rcscur<'h Serninar on wirul l'.,lli'rt,t ttrt llttiltlittg,,; ttnd Structures, univ.
ol"li)()nto Prcss, Toronto, t96lt, pp. 20 1 2.10.
.l . B. Ccrrnak, "Physical Moclcling ol'thc Atrrrosphcric Boundary Layer in Long
llrundary-Laycr wind runncls," in wintl 'lfunnel Modeling for civil Engincering
Applicutitn.s, T. A. Rcinhold (Ed.), Carnbridgc Univ. Press, Cambridge, 1982,
1tp.97-125.
Counihan, "An Improved Method of Simulating an Atmospheric Boundary
l.ayer in a Wind Tunnel," Atmos. Environ., 3 (1969), 197-214.
/ ltl .l . Counihan, "Simulation of an Adiabatic Urban Boundary Layer in a Wind
'l'unnel," Atmos. Environ., 7 (1913), 673-689.
/ .'o N. J. Cook, "On Simulating the Lower Third of the Urban Adiabatic Boundary
Layer in a Wind Tunnel," Atmos. Environ.,7 (1913),691-705.
I .'l N. J. Cook, "A Boundary-Layer Wind Tunnel for Building Aerodynamics," ./.
lnd. Aerodyn., I (1975), 3-12.
N. J. Cook, "Wind-Tunnel Simulation of the Adiabatic Atmospheric Boundary
l.ayerby Roughness, Barrier, and Mixing-Device Methods," J. Ind. Aerodyn.,
3 (1978), 157*t76.
/ .'l N. M. Standen, A Spire Array for Generating Thick Turbulent Shear Lctyers for
Nutural Wind Simulation in Wind Tunnels, Report No. LTR-LA-94, National
Acronautical Establishment, National Research Council, Ottaw a, 197 2.
| .,,1 .l . A. Peterka and J. E. Cermak, Simulation of Atmospheric Flows in Short Wind
'l'unnel Test Sections, Fluid Mechanics Program Research Report,
Colorado State
tJniversity, 1974.
.l . c. R. Hunt and H. Fernholz, "wind runnel Simulation of the Atmospheric
ll.undary Layer: A Report on Euromech 50," "/. Ftuid Mech., 70, part 3 (Aug.
/
l|,1
.1.
I
.'(t
l{.
te7s),543-559.
P. A. H. Irwin, "The Design of Spires for Wind Simulation,,' J.
lnd. Aerodyn. T (1981), 361-366.
Wincl Eng.
l. S. Gartshore, A relationship
slrape .fbr
between roughness geometry and velocity profile
turbulent boundary /ayer.r, National Research council of canada, NAE
l{cp. LTR-LA-140 (Oct. 1973).
l{. A' wooding, E. F. Bradley
and J. K. Marshall, "Drag due to regular arrays
.l'nrughness elements of varying geometry," Bourul. lnyer Meteorot., s (1973),
28.5 308.
I ttl
/
/il
r()
ll. w. Teunissen, "Simulation of the planetary Boundary Layer in a Multiple.lct Wind Tunnel," Atmos. Environ., 9 (1975), 145-1i,4.
ll. M. Nagib, M. V. M.rk.vin..l. T. yung, andJ. Tan-atichat,.,On
Modeling
Atrnospheric Surtircc l,uycrs by lhc countcr-Jet Technique," AIAA Jourruil,
14, No. 2 (1976). ltl-5 l()0
.l
ll.
llicnkicwict,. J.lt,. ('clnr:rk. .l . I'crcr.krr, irntl ll. H. Scanlan, "Activc Mrxlclirrg ol' Lurgc-Scalc 'l'rlrlrrrlt.rrr't'." .l ll'irrtl 1,,'tr,q. Ittl. Acnxlvtt. ll ( lgul),
'I(t5 4'/(t
306
wtNt) ll,NNl
il
l:i
j-32 .l . Ij. ('crrrrak. l]. lJicltkiewit'2,;rrrtl .l . l't'1t'tIrt. .'|lit't'Mrulrlirt,q, rtf lirrlutlr'rtr't'
Jir Wintl'l'uttru,! Stutlit,s ol'tlritl.4r fVItr!cl,s.ltclxrrl No. lrllWn/Rl) 13l/l':llJ lictl
eral Highway Adrninistratiott, Mcl,cart. Vir.. Iicbrrrary I913-1.
J. O. Hinze, Turbulence, McGraw-Hill' Ncw York' l9-59'
1-33
7-34 H. Rouse (Ed.), Advanced MechuniL:s rtl'Fluid:;, Wiley, Ncw York' l9(r-5'
1-35 H. Tennekes and J. L. Lumley, A First coursc in Turbulcnrc, Mll' Prcss.
Cambridge. 1972.
7-36 H. Tennekes, "The Logarithmic Wind Profile," J. Atmos. Sci., 30 (1913),234
238.
1-37 J. L. Lumley and H. A. Panofsky, The sutface rf Atmospheric Turbulencc,
Wiley, New York, 1964.
7-38 N. J. Cook, "Dctermination of the Model Scale Factor in Wind-Tunnel Simufation of the Acliabatic Atnrospheric Boundary Layer," J. Ind' Aerodyn',22
(t911-18),311 321.
1 l9 A. G. Davcnporl. l). Surry. T. Stathopoulos, Wind Loads onLow Rise Buildings,
liinal Rcport ol' l)hrscs I and ll, BLWT-SS8-1977, University of westem ontario. l,otttlolt, Canada,
1977.
'l-40 W. H.'l'iclcrlan,
"Pnrblcms Associated with Flow Modelling Procedures 1or'
Low-Risc Structures," J. Wind Eng. Ind. Aerodyn., 4l-44 (1992),923-934'
l-41 W. H. Tieleman, "Pressures on Surface-Mounted Prisms: The Effects of Incident Turbulence," J. Wind. Eng. Ind. Aerodyn., 49, (1993),289 300'
1-42 D. Surry, "Consequences of Distortions in the Flow Including Mismatching
Scales and Intensities of Turbulence," in Wind Tunnel Modeling for Civil En'
gineering Applications, T. A. Reinhold (ed.), Cambridge Univ' Press, Cambridge, 1982, pp. 137-185.
7-43 E. Szechenyi, "supercritical Reynolds Number Simulation for Two-Dimensional
Flow Over Circular Cylinders," J. Fluid Mech., 70, Part 3 (August 197 5), 529
542.
7-44 J. Gandemer, G. Bamaud, and J. Bi6try, Etude de lct tour D.M.A. Partie l,
7-45
7-46
Etutle des e.fforts dfis au vent sur les faqades, Centre Scientifique et Techniquc
du Bitintcnt, Nantes, France, 1975'
B. J. Vickery, "The Aeroelastic Modeling of chimneys and Towers," in wintl
Tunnel Motleting .for Civil Engineering Applications, T. A. Reinhold (Ed.).
Cambridge Univ. Press, Cambridge, 1982, pp. 408-428.
O. Giivcn, C. Farell, and V. C. Patel, "surface-Roughness Effects on the Mearr
Flow Past Circular Cylinders," J. Fluid Mech',96 (1980)' 673-701'
1-47 P. Sachs, wind Forces in Engineering,2d ed., Pergamon Press, oxford, 1971t.
7-48 V. J. Modi and S. El-Sherbiny, "Wall Confinement Effects on Bluff Bodics irr
Turbulent Flows," Proc. 4th International Conference ctn Wind Effects on Build
ings and Structures, Heathrow, U.K. (1975)' pp' 121-130'
7-4g J. Courchesne and A. Laneville, "A Comparison of Correction Methods Usctl
in the Evaluation of Drag Coelficient Measurements ftrr Two-dimensional Rcct
angular cylinders," ASME Wintcr Meeting, Papcr No. 79 WA/FE3 (19'79).
Blockagc Flll'ccts irrrtl ('otrt'l:tliotts," inWitul
'l'. A lit'irrlroltl (etl.), ('irrtr
Tunnt'l Mrxltlirr,q litr ('ivil l,)rgirttaritr,q Altltlit'tttion.s,
(';trrtbt'itlgc'
l()ll2' pp. l()/ ) l('
britlgc tJnivcrsily l)tt'ss,
j-5O W. H. Melbournc, "Wincl Tunncl
(r:,
30/
/ 5l W. n l)lrlPltt':;lr. '('{rrl):uisorr ol Motlt'l/l,rrll Srz,. lir..rlr. Wrrrrl l,rr.ssrrrt,s
.rr :r
lliglr ltist' llrultlrrrlt," .l Irttl. ..lttt,rlrvr., | ( l{)l.r). .)i (,()
I \)' W- A. l)lrlp,lit'slt, "('ottt;rrttisott ol Mtxlt'l ;rrrrl l,ull Sr:rlt. lt.sls ol llrc (,orrrrrcrc.t:
(lourt IJuiltlirrg irr'lonrlto," in Wirtrl l'rutttt.! ll!,,,lr,ltrr11
fitr.(,ivi! 1,.)tt,qitrtt,rittg
Al4sl.iuttiotts, 'l'. A. l{oinho[l (otl.), (,irrrrlrritll,t. t trriv,.r.sity l)rcss,
Carrrhrirlgc:,
l9tl2. pp. .575-589.
/ 5-l I{. D. Marshall, "A Study .l'wintl I'r1'ssrr1's
:r sirrglc-Family Dwelling in
M<del and Full Scale," .l . lnd. Acnxlrvr., 1,2 'rr
(ocr. l9'5), llj_19g.
l5'+ R. D. Marshall, "wind'r'urrncrs Applictr t. wind Engineering in Japan,"
-/.
Struct. Eng., ll0 (1984), tZ03 122t.
/ -5-5 H. utsunomiya, F- Nagao, y. Ueno, ancl M. Noda, "Basic Study of Blockage
Effecrs on Bluff Bodies," J. wind. Eng. Intr. Aerocryn., 49 (lggi),247-256.
/ 5() G. M. Richardson and D. Surry, "comparisons of wind-Tunnel and Full_Scale
Suriace Pressure Measurements on Low-Rise pitched-Roof Buildings, "
J. wincl.
Eng. Ind. Aerodyn., 38 (1991), 249_256.
;
CHAPTER 8
ltI
l,lt()rItrilltt
,t{)ilt:;llMA|N(,I,il()llntilt
jl,
t,1,.|ililil1t0ll,.
.l(l!)
ust'. A st:t'ontl 1rt.tt'rlrrrt' tlilt ttsst'tl irr sct'1. li. I "r. l:;;rllrr()l)r:rlr. l{rr r1rr.,1r..,11,11
,tl clrttklirtg lurrl olltt.r rrrt.lrrlrr'rs lrol srrlrjt't'lt'tl (o:.r1,1111sq;rrrl rl1,rr.rrlrr, ,rrrrlrlllg
r'lttitltl <lr :totrlclttslit' cllt'c'ls.
ll
rrlilizcs
(l)
t'xlrt'rrrt. wnr(l :,lr('i.(l rl;rl;r rt.r lrr1.rl
oI cslitrtatccl lilI circlr ol tlrc ti (rlr l(r) plirrt'iplrl ( (]rnl]:r:i:. tllt.t lr.rr:;, ;rrrrl 1.,;
;rt'rodynarlic clata, llascd ott wincl tunncl lcsls, ()n llr('(l('lx.rrrlt.rrt.t.rrpon tlrrct lrorr
ol thc wind cflcct being ctlnsidcrcd.'l'hc wirrtl spt't'tl :rrrrl lr.'lxly,r,,,r,,,.tllrlrr ;rrt.
rrsccl to create timc series of cxtrclnc wirrtl crllL't'ls, liirrrr wlrit'lr il rs
lxrssilrlc {rr
t'slitnate a univariate probability distribu(ion ol tlrc: llu-gcsl wirrrl cllcct, irs wgll
;rs lhe requisite design loading (c.g., thc winrl kracl with a
-50-ycar nlcan rc('rrrrcnce interval). The practical application clf this proceclurc-par-ticularly
fbr
t'lirclding design-is simple and straightfirrward. The third p.o""dur", discussed
WIND DIRECTIONALITY EFFECTS
rrr Sect. 8.1.3, utilizes the eight univariate probability distributions
of the largest
ve:arly wind speeds recorded for the principal compass directions
[g-2, a-11,
:rrrtl the fact that the time series of the largest yearly winds blowing from
tlillbrent directions have as a rule weak mutual correlations
lsee Sect. 3.+;.
'l'lris procedure can
be applied to any type of structure or structural member,
rrrcluding structures or members subjected to wind-induced aerodynamic
am,
lrlilication or aeroelastic effects.
Wind effects on structural members depend upon direction for climatological,
aerodynamic, and structural reasons. The extreme wind climate at any one site
is, in general, nonuniform with respect to direction owing to basic atmospherit.
circulation pattems and/or the presence of local obstructions. Aerodynamic
behavior depends upon direction for most structural members; examples rangc
from cladding to bridges. The dependence upon direction of the structural
response of a member subjected to a given aerodynamic load can be simply
8.1.1 Procedure Based on Theory of Random processes
lr this procedure the mean wind velocity is regarded as a stationary
twotlirrrcnsional random vector process, u(r), with speed u(r) and direction d(r).
lirrilure is assumed ro occur tf u(0) > g@) (i.e., the curve g(d) is the failure
lr.undary in the velocity space; see sect. A3.1.2). Forexample, if the relation
lrctween the wind effect Q@) and the wind speed u blowing from direction
0
illustrated in the case of a circular flagpole in horizontally homogeneous terrairr,
anchored to its foundation by four bolts located at the comers of a square basc
plate. For any given wind speed. the uplilt lorce on the anchor bolts is grcalcr
by a factor of V2 when the wind direction is parallel to the diagonal of thc
base plate than when it is parallel to one of the sides.
This chapter describes procedures for estimating probability distributions ol
Iargest yearly wind effects which account for the dependence upon direction ol
the extreme wind climate and of the aerodynamic and structural behavior ol'
the member. Also described in this chapter are procedures for estimating prolr
abiiities of failure and safety indices for members sensitive to wind direction
ality effects.
8.1
PROCEDURES FOR ESTIMATING PROBABILITY
DISTRIBUTIONS OF LARGEST YEARLY WIND EFFECTS
Three such proccdurcs arc currcntly availablc;. Thc lirst pnrt't'tlrrr.c is htsctl orr
the theory o['s(ationirry nrnclorn pn)ccsscs I13-1, lt-2. u ll l( is slrowrr in Scr't.
8.1.1 that in (hc prcserrl st:tlc ol'lhc rrll tlris proct'rlrrn'is rrol :,rrlr'rl lir.pllrcliclrl
308
Q@
:
h(iluz@)
(8.1.1)
thcn the boundary g(0) has the expression
s(0)
wlrcre R is the
prrncl).
:
ttt n
Lnot
lt/z
(8. r.2)
I
limit state (e.g., the wind pressure causing failure of
I
a cladding
'fhc mean rate at which the vector
u(r) crosses the boundary g(0) in the
rrrrtward direction is denoted by u,rand may be referred to
as the mean failure
rrrrc. If the values of u1| arc srrrall, failure is a rare event and
its probability
rrury bc assumed to be .l' lhc l).iss.n typc. The probability
that in the time
rrrlcrvatl 7'no luilurc will oct'tl.(i.t'.. lhc:
llnrblrbility that the velocity vector
will not cnlss llrc liriltrrc lrrttutrl:rry,r;(//) irr llrt'ou(wirnl direction) can be written
t'lll. tl
t
'1't
(li. L.l)
t
wlNt) t)ilil(itl()Nnl llY llll(.1:i
310
(Eqs. A1.34, A2.39) so ihlrt tlrc pnrbtrbilily
4 : t
ol
lrrilrrrr': tltrrirrg tirrrc 7'is
(, ,1!,t
(tt.1.4)
The fact that there is no failure during the time interval Z means that thc largcst
wind effect Q occurring during that interval is less than R. Thus Eq' 8.1..1
yields the cumulative probability distribution of Q corresponding to the valtrc
Q:
R:
Fo(Q<R):g-'nr
(8.
I"
:
,Etr 1it,(')lu(,)
s(0)lfu,ots(01,
,l
It
(8.1.6)
[8-1, 8-2], where U, : derivative with respect to time of the projection of thc
velocity vector U(0) on the normal to the boundary g@), U(0) : wind speed,
fu.o : joint probability density function of wind speed and direction, and Efi'
: average of the positive values of U,, given that U(0) : g(0).
Attempts to evaluate the mean rate vp have been reported in [8-1] and
[8-3], where in addition to the assumption of stationarity of the wind velocity
process, the assumption that U, and U are statistically independent was used'
SO
Etrtu"@)lu(0)
:
s(o)l
:
Efftu,@)l
I'llr)r I lrlllll :, l ()ll l:;llMn
llu(i
l'ltl,lll\ltll
ll,
Irl',iltlill,ilillj
l:rlgt'.'l'ltt'sc rlist'tt'p:trrt'rr's tltrr lrt':rlllilrtrlr'tl lo llrt'rr,,r'r,l rrrrrl,,1rr.r.rl rl.rl;r
lo lr l:rrgt' t'xlcrrt willt ltrt'lt'onrlop,rtlrl Plrt'rrurrrr.u;r {r. l, rronlntt,
lrtcczcs) thltl itrc ttttrcl:rlctl to tltc slt'ott1'. willls ol lrr(t'rt.,,l lt .,lltr lul;rl ,lr.,,r;'rr
As ntllccl in Scc:t.3.2.3, such n:ctlttls crut lrlrvrtlt':r rrrr:,lr':rrlnrl', lr.l,r:, l'r .,1.r
tislical inf'crcnccs ctlnccrning cxlrcnlc: wirrrl spt'r'tls l,irr tlrr:, r(':r,on, rnr1,..,..
rt'liablc cstimates of the tcrrns lifflIl,,lllt0l
,r](//)l rrrrtl .l ,, ,ltt.l,ll :ur. rrurtlt. orr
;rssocitttctl
tlrc basis of data pertaining to stft)ng wirttls, llrt'rrrt'llrtxl n'vit'wt.rl irr llris st.r.liorr
r';ulnot be used with confidcncc lirr stnrc(unrl tlcsigrr l)utl)()srrs.
I .-s l
whcre zp is a function of R. In particular, if Z: I year, Eq. 8.1.5 represents
thc cumulative distribution function of the largest yearly wind effect, that is,
thc probability that thc largcst actual wind effect in any one year is less than l
spcci(icd wind effect R.
Thc mcan outcrossing raLc vp may be obtained by using Eq. A2.47, which
is valid r,rnclcr thc assumption that the random process is stationary. If polar
coordinatcs arc uscd, it fbllows from Eq. A2.47 that
,o:
ll I
(8.1.7)
The quantity EfflU,(:0)l can be estimated fiom spectra of wind velocities on
the basis of Eq. A2.l6a. Under the assumption of stationarity, these spectnr
and the probability density fu.oof Eq. 8.1.6 can be estimated from continuotrs
wind velocity records or from comparable types of records, such as wintl
velocities recorded at one- or three-hour intervals [8-1, 8-3]. Once Eff[U,,([t)l
andfu., are obtained, Eqs. 8.1.5 and 8.1.6 can be used to estimate the ctr
mulative distribution function of the largest yearly wind effect.
In [8-3] largest ycarly wind loads estimated by thc procctlrrrc tlcscribcd irr
this section wcrc conlparcrl in a nurnbcr ol cascs witlt llrrgt'st ycrrr-ly lo:rtls
obtained on (hc bltsis ttl'itc(tutl trtcasttrcttlcnts. 'l'ltc rlistn'Plttrt'it's lrt'lwt:cll tltt'
rcspcctivc cunrrrllrlivt' tlis(t'ilrtrliolt lirnc(iorts wt. tt' lotttttl lo lrt' ttltltt t't'pllrlrly
B-1.2 Procedure Based on the Time Series of the Largest Yearly
Wind Effects
'l'lris procedure is applicable if the wind effect
Q@) can be described by an
('\pression of the form
Q(0)
:
)pC@)C,,(0)U'(h,
(8. r.8)
0)
rvlrcre p : air density, C(0) : coefficient transforming wind load into wind
cll'ccts (if Q(0) is a wind pressure or suction, C(0) : l), Cp(0) : aerodynamic
t'rrcfficient corresponding to wind blowing from direction 0, and U(h, 0) :
rncan wind speed corresponding to the direction d at the reference height lr
:rbove ground. It is assumed that the influence coefficient C(0) is independent
,tl' U(h,0). It is also assumed that the coeflicient Q,(0) is independent of or
,rrrly weakly dependent upon U(h,0). These assumptions exclude from considt'ration members subjected to significant dynamic amplification or aeroelastic
t'll'ccts.
'Ihe details of the procedure discussed in this section differ according to
whether the region being considered has a well-behaved wind climate or can
t'xperience hurricane winds. The two cases are therefore treated separately.
Struclures in Well-Behaved Wind Climates. Let U,(h,0) denote the largest
vrrlue of U(h, 0) during year i. The largest wind effect Q; during that year is
tlrc largest of the values Qi(0) obtained by substituting U,(h,0) for U(h,01in
lrc1. 8.1.8:
e,
:
)p max[C(0)Co(ilU?(h,
(8.1.e)
0)]
Notc that in conventional engineering practice, wind directionality effects are
rrtrl taken into account; that is, the largest yearly wind effect calculated for
,lcsign purposes, Ol""', ir assumed to be given by
Onlrrrr ]/) frurxl('(//)(),((/)lnrt;x1U!{h,
l,1t(',r,,.,r1
l,' {l t I
011
(8.1.10)
(lJ. r . r r)
312
wlNt) t)lll oil()Nnt
ly
I
IIrt lltl :, I()l l l:;lllvln llN(i I'l t()lrnlilt ll, ttr',ililnililr,il',
l tct:i
whercU;(h) - maxp[ U;(/r,0)l dcnotcs llrc lurgcst uttnttal wittcl spcc:cl tcgarcllcss
of direction, and
C*o^
:
(8. r. r2)
max[C(0)C,,(0)J
0
Example Consider a 100-m tall building located in an urban environment, for
which it was estimated in [8-5] that U(h,0)lUJ(0) : 1.39, where Uy(0) :
f-astest-mile wind speed at l0 m above ground in open terrain, and lr : 3ltt
m. The largest yearly fastest-mile wind speeds Uf.(0) recorded during a given
ycar in the region being considered are listed in row I of Table 8.1.1. Thc
r.ncasured peak suction coefficients Cp(O) reported in [8-5] for a cladding panel
located at 94-m elevation near a conrer of the building are listed in row 2 ol'
Tablc 8.1.1. Thc corrcsponding suctions, Q(0), calculated by Eq. 8.1.9 in
which C(g) = I ancl p : 1.25 kg/m3, are listed in row 3 of Table 8.1.1, and
represent thc largest suctions induced in the cladding panel by winds blowing
from the cight directions of the compass during the year being considered.
It is seen in Table 8.1.1 that the largest suction induced during the yearol'
concem by winds blowing from any direction is
Q1
:
maxlQ(0)l
0
:
(8.1.13)
703 Pa
effects were not taken into account, it would follow from
Eq. 8.1.11 that Ol" : ) x 1.25 x 3.33 x (1.39 x 31.3)2 :3939 Pa. It is
seen that in this example the value obtained by ignoring wind directionality
effects is considerably higher than the actual value of the largest yearly wind
suction, Qt : 1O3 Pa. Note, however, that this would not have been the casc
had the directional distribution of the wind speeds and/or of the suction coel'ficients been relatively uniform, or had the directions corresponding to thc
If wind directionality
maximum values of Ct,@) and Uy.1(0) coincided.
If extreme wind speed dara Ui(h,0j) (j : 1,2, ... , 8 or 1,2, ... , 16)
are available for a su{licient number n of consecutive years (:e.9., m - 20),
a set of largest yearly load data, Qi, (i : 1,2, ... , m) can be calculated by
TABLE 8.1.1. Largest Yearly Wind Speeds, Suction Coefficients, and Largest
Yearlv Sucfions
DiTectionNNEESESSWWNW
u1.{:0)
(mis)
2
ct,(0)
.J
Q,@)
(l'a)
t2.5
8.9
0.07 l.(Xr
t3
tot
r0.3
426
22.3 10.3
0.51 0.61
106 ll(r
22.3
22.8
0.(r(r
.)()(r
/o
3 r .3
l.12 0.2,1
i
284
rrsing I,)t1. lJ.l.().
linltr llrt'st'tllrtlr it is lxrssrlrlt- lo r.:,llnt;tlr.llrc
:ll:l
l11.1,l
11111111,
rlrsllibtttiott ol'Q trntl v:rriorrs stirlistics tlrirl lrriry lrt.rr:t'rl lul .1...,,,,,, l)lulx)s(.5
()ttc sttch slalistic is tlre wintl krlrtl Q,y col't1'slxrrrtlrnli lo ilry nr(':ur r('(ulr('n(('
rrrlcrvtrl N.
It is convonicnt lirr ctlntpulatiottal prtrpost's lo tlt'lirrt' llrc rlrr;rrrlily, rt'lt.r'lt'tl
to lrs cquivalcnt wind spccd,
,t
"'' -
\
|
nurx,,l
Q({/)l
]pc),,,,,*
}r
(8.1. l4)
J
rvlrcrc C,-o^ is defined as in Eq. 8.1.12. The largest yearly equivalent wind
:;pccd during year
i
is
u.q,i:l+r,*,*)"
(8.1.
ls)
Stltistical analyses of sets of data, Ueq,i, reported in [8-6] have revealed that
tlrt' probability distributions of the largest annual equivalent wind speeds may
I't' assumed for design purposes to be Extreme Value Type I. This assumption
rs trsually conservative. Note that if the equivalent wind speeds have an Extreme
Vrrlue Type I distribution, the distribution of Q is not Extreme Value Type I.
I-ct U.op denote the equivalent wind speed corresponding to a N-year mean
r('currence interval. Assuming that the distribution of U"q is Extreme Value
l'ype I, it is possible to write (see Eq. 3.2.1)
U"qN
=
a"o
*
b"o ln
N
(8.1.16)
wlrore
a.q:7.q - o.45s"o
b.o : 0.78s.0
(8.1.17)
(8.1.18)
;rntl X.o and s"o : sample mean and sample standard deviation of U.0.,. The
wincl effect Qlr corresponding to the mean recurrence interval N can then be
rvrittcn (Eqs. 8.1.15 and 8.1. i6) as
Qn
=
ipC,-u^(a.q
I
b.o ln
N;2
(8.1.19)
It is clcar that failure to takc wind directionality into account (i.e., the use of
lrrrgcst ycarly wind effects oslirrrirtctl by Eq. 8.1.10, rather than by Eq. 8.1.9)
wottltl rcsult in some citscs itt trrrreirlisticirlly inllatccl cstimates of the wind load
toltcsptlncling to an N-ycrrl rlr(':ur rr'( unt'rtt'c irrlcrvtrl. This is shown in thc
lollowirrg cxltrttplc, prcse:ttletl irt tlr'l;rrl ()n :r('r'()lnrt ol'thc plrcticul ilrrl-lol'lutrcc
ol sttclt crtlculrrliotts irr clirtklirrp', 1,,1;r:rs tlt'srlirr
314
wtNt) t)ilil (:il()Nnt
ily I ll t(
*
t,,
(:l l)l,l ll l; l()l I llillMn llN(, I'il(ltiAilil il r trl..iltilrt lil()N:;
Example'l'hc largcst ycurly Iirs(csl rrrrlt'wrrrtl spt'r'tls lrl l0 rn irlrovt'glorurtl irr
open teffain, U7, clbtaincrl lirrrrr rt't'oltls (rrkt'n rrl Slrcritlirrr, Wyorrrirrg, irr llrt'
period 1958-1911 arc listcd in rnph irr 'l':rlrlt' 13.1.2. (Surrrrrrary sttrtistics lirr
these data are shown in Fig. 3.4.1 .) Wc scr:k thc -50-ycar wincl-inrluccrl suc(ion,
QN:so, on the cladding panel of the prcvious cxample, lor which thc acrody
namic coemcients are given in row 2 ol"lablc 8.1.1, and the estimat.cd ratio
U{h,0)lUJ(0) is approximately equal to 1.39. From Eqs. 8.1.9 (in which C.(//)
: l) and 8.1.15, it follows that the largest yearly equivalent wind spcctls
during the period 1958-1977 have the values shown in Table 8.1.2, where thc
corresponding sample mean {o and sample standard deviation s"u are also
shown. From Eqs. 8.1.16 through 8.1.19, h:so:974 Pa (20.3 psf).
If the load is calculated without taking wind directionality into account, thc
nominal 5O-year load is (Eq. 8.1.11)
(8. 1.20)
TABLE 8.1.2. Largest Yearly Fastest-Mile Wind Speeds, Ur, Loads, Q,, and
Equiralent Speeds.
{,,;.,.t
wltct'c {f lr
-50-yclrr-lirs(cst-ltrilc wirrtl slx'r'tl (rrr rrr/:,) t'strrrurlrrtl l.nrrrr thc sct
ol'largcst ycarly spcctls rcgartllcss ol tlircclrorr. lirorrr llrt: tllrttr ol-'l'ablc 8.1.2,
I/1so:74.25 ntph (33.2 nr/s), irrrtl Qfi"",,, ,l,l,lo I,l (92.5 psf), versus the
:rctual -50 ycar load, 0t s, -,914 l'rr (20..1 lrsl ).
Directional largcst yearly lirstcst-ruilc wirrrl spccd data at a number of weather
stations in the United states arc availablc in [8-7]. Similar data that may be
rrsed for design purposes can also be obtained fiom monthly Local climatoIogical Data summarics publishcd by the National oceanic and Atmospheric
Administration (see Sect. 3.4).
structures in Hurricane-Prone Regions. In hurricane-prone regions the load
rlata used for inferences conceming design loads are not yearly maxima. Rather,
lhey are associated with hurricanes, which occur at irregular intervals. The
rrpproach used in this case is the following. A large number m of hurricanes is
gcnerated by Monte Carlo simulation on the basis of climatological information
trn hurricane storms, as shown in Sect. 3.3. For each hurricane the load e; and
the corresponding equivalent wind speed U"o,; (see Eqs. 8.1.9 and 8.1.15) are
then obtained. Following exactly the same steps used in Sect. 3.3.2, the curnulative distribution functions of the largest load and of the largest equivalent
wind speed occurring in any one year are found to be
Largest Annual Fastest-Mile Wind Speed at 10
m above Ground in Open Terrain (mph)"'"
Year
NE
1958
t959
1960
1961
1962
1963
t9&
1965
t966
28
20
41
25
36
25
2t
l8
22
23
3r
t4
22
33
l5
36
1967 44
1968 36
1969 28
t970
t97t
28
33
23
1972
t9l3
1974
1915
1916
t97l
2l
l4
l9
l6
l3
l5
l9
28
24
22
23
28
22
31
24
20
44
Notc: X..,
"l rrplr /'V:rlrrt:s
31
-
SW
23
19
16
30
22
23
18
20
19
16
19
15
20
22
26
t9
t9
19
28
19
16.11 rn/s;
O,447 nr/s.
,r,.,,
50
29
34
36
36
33
34
33
34
40
3s
36
35
31
36
32
37
27
33
40
2.1
I
23
25
26
21
16
36
19
t7
14
36
2t
22
37
22
37
15
25
28
38
36
50
40
43
47
41
63
54
66
51
51
39
53
61
49
55
46
57
39
47
34
Fo(Q
NW Q,(Pascals)
sl 70
38 60
4s 60
38 60
52 60
48 s'7
54 60
43 55
39 6I
4t 62
40 47
34 66
37 s3
31 41
44 47
39 64
49 56
33 st
33 47
44 s6
nr/s.
irr ilrrlits:rrc l:ul'r'\t y(''lrly rvirrrl spt't'rls lirrrn;rll rlirctlrorr,
U"u.i(m/s)
103
399
18.4
13.9
553
16.3
718
728
643
782
708
419
464
438
459
598
18.6
384
13.5
553
419
653
16.3
289
ll.t3
64t3
17.6
15.1
5
t.\
18.7
t7.6
19.4
18.5
14.2
315
< Qi): Fu"r(U,,t<
_ - Xll-i,ttr{
U.q.i)
t)l
(8.1.21)
where \ is the annual occurrence rate of hurricanes in the area of interest for
the site being considered. Continuous probability distribution curves, F(e <
17) and F(U"q I u"o), that best fit Eq. 8.1 .21 (e.g., reverse Weibull or Extreme
Value Type I distributions) can be estimated by using standard statistical techrriques. Note that the mean recurrence interval of the load e, and of the equiv:rlcnt wind speed U"o , is
15.0
14.5
I _ e-Atl
t4.9
17.0
t4.2
t]
.l
i/(m+ t)l
(8.1.22)
A similar approach is reported in [8-4].
A computer program for estimating hurricane-induced wind loads in accortlirnce with the procedure outlined here is described briefly in [8-8], and is
rrvailable on tape in [8-91 . Stored in the program are hurricane wind speecls
t'orrcsponding to the 16 compass rlircctions at 56 mileposts located atdistanccs
-50 nar-rlical rnilcs along thc: (irrll'rrnrl Atlunlic coasts (see also [3-7 ll). Thcsc
spcctls worc <lbtaincd 1'rolrr t)(X) Irrrlrit'lrrrt'wintl licltls gcncratecl hy Morrlc ('trllo
sirrrrrlir(iorr lrI cach lllils:1.rosl , rrs tlt.sr.rilrt'rl in St.t.l,3.3.2, unrl wcr.e rrst'tl irr
ll"i l0l rrntl irt llrc tlcvt:lo1'rlttt'nl ol llrt'rvrrtl :;;x't'rl rrr;rp irrclu<lc:tl ilr llrt'Arrrt.rit.;rrr
Nlrliortrrl SllrrttLrrtl n.58. I l()li.) lt{ I ll
ol
316
wtNt) I)iltt (;il()NAt
ily I IIIcr
llllllll:.
In cases whcrc it is.iutlgctl tlr:rl tlrt'lltrb:rlrili(y tlistlibrr(iorr ol lhc
lirrgesl
yearly loads, Fy(Q I q), ntay bc irlleclcrl by lxrtlr hurricanc urrtl rllnhrrrricrrnc
winds, the following expression should bc uscd:
Fo(Q
<0:
Fa"(Qn
(
4)Fo*"(QNu
<
(8. r.23)
4)
where Fq^(Qs < 4) : cumulative distribution function of hurricane-inducul
wind loads, Qs, estimated as shown in this section, and Fp*"(Orvu < Q) =
cumulative distribution function of loads induced by nonhurricane winds, cstimated as shown in the previous section (see also Sect. 3.3.2).
8.1.3 Procedure
Based on the Univariati Probability Distributions of
the Largest Yearly Wind Speeds Recorded for Each of the Principal
Compass Directions
A
simplc pnrccdurc is now presented that may be applied to any type ol'
structurc, including structures subjected to aerodynamic amplification of aeroelastic effects.
It was pointed out in Section 3.4 that the correlation between extreme wincl
speeds occurring in any two directions is generally weak. As shown in Appendix Al (Eqs. A1.64), two uncorrelated variables having a joint Extremo
Value Type I distribution are statistically independent. It can be shown thal
statistical independence also holds for any number of uncorrelated variablcs
whose joint distribution is of an extreme value typelAl-241 . In practice it can
therefore be assumed that the largest yearly winds blowing from the eight
principal compass directions are statistically independent. The cumulative probability distributions of the largest yearly wind effect may thus be written as
Fo(Q
<R) :
:
Prob(ut
1
r'r, u2 1
Prob(rrr
<
a{) Prob(t,,
ut2,
...,
us
<
(8.1.24a)
uL)
< ut21... Prob(u3 <
r.,/3)
g.l.24b)
where erj is the wind speed from direction i causing the occurrence of the wintl
effect R [8-14]. Note that if the wind speeds occurring in all directions wcrc:
perfectly correlated, then
Fo(Q
< R) :
Prob(ar
<
utr)
(8.1.2s)
where I < k < 8. Equation 8.1.25 indicates that Eq. 8.1.24b is conservativc
from a structural design viewpoint. Bonferroni techniques applicable to bivar'
iate extreme value distributions were used in [8-15| to cstimatc bounds lirr
probabilities Fo(Q < R). The estimates showcd that in clinrirlcs rx)( pK)nc (o
hurricane occurrenccs, Eq. 8. 1 .24b typically ovcrcsliuurlcs rrnrr;rl l:rilurc prrrl'l
abilities by a lactol lrl'lcss than two. Sirrrilar (cclrrritlrrr's rrury lrr'rrpplicrl to
hurricanc wirrtls" lirl t'xrrrrrplc, rrsing cs(ittt:rlctl tlittt liorr;rl tl:rl;r :rv:rillrlrlc irt
llt 9l or
1.1
7ll
nlllr',r\il
iy it\ll,l{.1:;
,lll
8.2
ESTIMATION OF FAILURE PROBABILITIES AND SAFETY
INDICES FOR MEMBERS SENSITIVE TO WIND DIRECTIONALITY
EFFECTS
'lo detemrine whether a mcmbcr scnsitivc lo winrl tlircctiorlrlily cllccts is rrr'
ccptable from a safbty point o1' vicw, ils rrorrrinirl lailLrrc probability (or its
safety index) is compared to that ol' a rtrcrrrbcr.juclgcd to be acccptablc. 'l'hc
rnember is then redesigned as nccdcd until thc rcsult of this comparison is
satisfactory. An application of this rcliability-based approach to the design of
glass cladding for a tall building is presented in Chapter 9. This section describes procedures for estimating nominal failure probabilities and safety intlices required for the application of this approach.
8.2.1 Estimation of Failure Probabilities
Consider a member whose resistance is R, and denote by Q the largest load
cffect acting on the member during any one year. Failure occurs for any pair
of values R, Q such that R - Q < 0. In most applications R and Q may be
russumed to be independent, so the probability of failure in any one year can
hc written as
P,
:
(.Jn
Fn@tfs@) dq
(8.2. r)
(Eq. ,{1.21), where Fa : cumulative distribution function of R, and fq :
probability density function of Q. The function/q is related by Eq. Al.ll to
lhe cumulative distribution function, Fg, estimated as shown in Sects. 8.1.2
rrnd 8.1.3. The probability of failure during the n-year lifetime of the structure
t'an be obtained from Eqs. 8.2.1 and A1.31. The probability of failure so
obtained is conditional upon a given set of values of the random parameters
(lrat determine the functions R, Q, Fn, and fq. Conditional probabilities of
llilure can be useful in certain applications in which the objective is to assess
tqualitatively the relative reliabilities of various members.
Unconditional failure probabilities can be estimated by using an expression
sirnilar to Eq. A3. I, provided that (1) the probability distributions of the various
rrrrdom parameters that determine R, Q, Fa, and fpare known and (2) such
t'stimates are not computationally prohibitive. In a number of situations of
practical interest it is in principle possible to use reliability-based design methrrrls that employ the safety index as a measure of structural reliability. Nevertlrclcss, difficulties pertaining to thc choice of the target safety index for at least
sorrrc situaticlns remain unsolvr.rtl.
8.2.2 Estimation of Safety lnclices
'l'lrt'ltursl t'orttlttttttly ttst'tl s;rlt'ly ur(l('\ ()rr rvlri,lr rt'li:rlrilily t';rlt'rrl:rliotrs irrt'
lr;rst'rl lr:ts lltt' t'xPn'ssiort
318
lr:'r:;ilMn|ol.|'rt
wlNl) t)ililcil()Nnt ily tiltct
tnilt,ilt
I't
t()iln||illilt :,At.lt ,,.r\t tt,
Ilt rt{t,
itl tt:rltts tll'tttt';tlts. t'ot'lltt'lt'ltls ol vltt'iltlirltt, ,lilrtl t ont'l.rltotr t ot'llrr rr.1l., 6l
ru,,,rr,
lrlttlorrrv:rriirblcsl,(i l,l,..,,a)irlrrl
.\,(/
llrt.
,u I l.rti I .'.
rtl.
lrighcr-tlrclcr tcrtns irr tlre:sc cxprcssions lrt.ilrp rrt.1ilt.r.{t.rl l lior r.r;rrrrlrlr,:, ol :,rit.lr
clrlculations, soc uqs. A3:26 n3.2tt irrtl lll lll)r lrr rno:.r rlt.:,r1,11 :,rru;rtr()n:,
tlrlrt involvc wind action. R arrcl Zn errrt lx't'slinurlt.rl rrrlt'pt.rrtlt.rrtly ol (),,;rrrtl
f
vo,, I
u(h,0)
Sfructures with Specitied Orientation. 'l'lre t'rrrrrrrlirtivc: tlistributiel lirlclion Fy"o,, of the largest lil'ctirnc c:(luivalont wirrd spcccl 4u, ili
F
l1'l(JtJlllt
tt.2.l.
Wincl dircction, 0, and angle of orientation of structure, a.
",))
:
lF u"u(u.,)1"
(8.2. s)
:rnd the standard deviation J(a"q,).
in addition to wind loads, arc
we then treat the case of structures with unknown orientation, which
is of interest for the development of building code provisions on wind loads.
In general,
Q,: A, + Q',
: itB + p') (e,^o^ + c,^^*) 1O"o, + U!r,)2
Q, =
present.
(8.2.3)
X,)
(8.2.4)
where, for example, the random variables XiG : 1,2, . . . , m) may denotc
memberdimensions and material strength, and X1(j : m * l, m I 2, . . .,
n) may represent aerodynamic and micrometeorological parameters. * Equations
8.2.3 and 8.2.4 can be expanded in Taylor series; approximate expressions for
the mean values and coefficients of variation of R and e,, can lhcn bc obtaincrl
tThe cxprcssitlns lilr /? ltlttl
Q, ttt:ry tortliritt lr rrrrrrrbcr ol toltrrrrorr v;rrr;rlrlr':,. llr;rl is.
.y
X, lin
(8.2.6)
whcre the bars and primes indicate mean values and deviations from the mean,
rcspectively. we assume that the values p and c^"" used in calculating u"0,,
(lrc1. 8.1.15) are the mean values of these two variables. The following apprrrximate relations follow from Eq. 8.2.6:
also consider the case where gravity loads,
r -: ltr :rrrtl i - trt I l.
",,,,(u
(lrq. A.3.2), where n : lifetime in years and Fu.,t: cumulative distribution
lirnction of the largest yearly equivalent wind speed u"o obtained as shown in
soct. 8.1.2. From the distribution Fy"u,, it is possible to estimate the mean u"on
(8.2.2)
(see Eq. A3.29), where R and vp: mean value and coelficient of variation
of the limit state, and Q, and vg,, : mean value and coefficient of variation ol'
the largest lifetime load effect.
we consider here only members that do not experience significant dynamic
amplification or aeroelastic motions [8-13]. Load effects for such members can
be described by Eq. 8.1.8. Expressions for Q, and V9, are first developed ftrr
the case of structures with a specified orientation angle, a (Fig. 8.2.1). Wc
sornc vafrrcs
t
Associated with the largest lifetime equivalent wind speed u.q,, is the largest
lil'ctime wind effect Qn (see Eq. 8.1.14):
lnn-nQ,
^ vTii;tr
u:
R:R(Xr,X2,...,X.)
Q, : Q,(X,+t, Xm+2, . . .,
t
vb,,
lpC,,'u^A?r,,(l
= vl + vzr^.^, +
I
Vtr"o,,)
vL",,,
(8.2.7)
(8.2.8)
'l'hc coefficient of variation, 26,,,"*,
reflects the variabilities of the influence
t trclhcients c(0) and of the aerodynamic coefficients Co@). For example, the
inllucnce coefficients transforming pressure on cladding into maximum tensile
slrcss in the panel depend upon the panel thickness, which, forthe same nomrnirl thickness, may actually vary somewhat from panel to panel. Aerodynamic
r'ocllicients for any given structure can-and usually do-vary from experiment
lo cxperiment. It is possible to write
rsirtrPlcr rnanipulations are possiblc in ceftain
instances; see Eq. 8.2.6 and subsequent deriva-
lr()lls-
llris is lnlc r:von in (hc casc ttl gl;tss cl:rrklirrg, which experiences latigue under wind loading
,ttrl wltosc sllctlSllr is inllttcrrtt'tl l)y lll('nirlurc ol thc wincl prcssure fluctuations. As shown in
( ltrrPlt'r (), lhis ittlltrcncc crttt lrr' ittr'orlxrr;rlt'tl irr llrc cxprcssion ol thc load
Q, so values of R an6
ll t;ttt lrt'trsctl ilt lit;. IJ.2.2 tlr:rt (()l('rlrrrrl kr rrrrrvt'rrliorlrl l():l(ling pa(tcrns indcpcndent ofthe
lrtnt ltislory ol lltr':rt'lrr:rl wilrrl lo:rrlr {lol t.rrltrIlt'. l(, (()usl:utl lo:rtls ol (r0 s rlrrnrlion).
320
*
wlNl) l)llil oll()Nnl llY llll(.1:;
vi,_,,
t
,' r
l'r
(
,
t{.2.u;
where V6 and Vg,, are the coeflicicnts ol' vitr-ilrtittlt ol' C:({/) antl Q,(0)-'r'
We now derive expressions for U.u, and Vu",t,, lirr thc casc whcrc it nray bc
assumed that the data (J"r,1(Eq. 8.1.15) are best fittcd by an E,xtrernc Valttc
Type I distribution. As mentioned previously, this assumption is gcncrrtlly
conservative. From Eqs. A3.8-A3.9 it follows that
:
s(a.',) :
U"q,
whcrc
X,.,, ancl s,.,,
X"o
+ 0.78s"q ln n
(8.2. r0)
(8.2. l r)
s.o
arc dcfincd fbllowing Eq. 8. I . 18 and
n
:
lifetime of structurc
in ycurs.
'lir cstirttirlc Vrt,,r,, wc considcr the relation
O
(8.2.t2)
:,,,c2c-,coc"D![f
",r,
where U"u,, and 7!!,," : estimated and true (but unknown) mean value of thc
largest lifetime equivalent wind speed, and c1, c2, c3, c4, c, : coefficients witll
mean equal to unity that reflect, respectively, (l) errors in the measurement ol'
the fastest-mile wind speeds over open terrain, (2) errors in the transformatioll
of the fastest-mile wind speeds over open terrain into mean wind speeds at l0
m above ground in open terrain, U0(10), (3) errors in the transformation ol'
U0(10) into mean wind speeds at 10 m above ground near the building sitc,
U(10), (4) enors in the transformation of U(10) into mean wind speeds at thc
elevation, h, near the building site, u(h), and (5) sampling errors in the estimation of (J.q, due to the limited size m of the sample of data U"r.iQ : 1,2,
. .. , m). The coefficient of variation Vu.r, can therefore be written as
vru.u,: vL, + vl, + vf,. + ,i^
*
*.
lff
(8.2.13)
whete V,, (j : 1,2,3, 4) are the coefiicients of variation of c;, and s, ''
standard deviation of sampling error in the estimation of U.on.
Approximate estimates of the sampling errors s, can be obtained by noting
from Eqs. 8.2.10 and 8.1.16-8.1.18 that, fot
n:50
years, Ueqn:5gyr =
U"qru:soy' so that the respective sampling errors are approximately the sanrc
for these two quantities. From F;q.3.2.2 it then follows that
tt =
where ru
:
size of data sample U"u.,
4
*-
(8.2.14t
t",
(i : l, 2, . . .,
8:' t:;ltM/\il(irl o, tAilUlu I'tt()ilnillt ililr.
r.,/\t tty tNl)t{:t :t
:12 I
Example lisl irrrtrtt. tlrt' srrlt.ty rlrrlt'x, /i (llt1. l{ .r..)). lor /r' /.5 psl'1}.590 l)u),
V4 - O.2), p
1.2.5 k11/rrr\. (,
0.0.5, (),,,,, t.|.1. f(
U, Vcr,:0.i,
X.u,, 16..17 rrr/s.,.r",, l.l I rrr/s, irrrtl rrr 20 (scr. ,l'irblc lJ. 1.2), 1,, : V,,
: V,, : V,., : O.05, rt . -50 ycars.
From Eqs. 8.2. l l, ti.2. 10, 8.2.t4, 8.2. 13, 8.2.9, 8.2.8, anc| 8.2.7, s(u"en)
- 2.ll.,mls, U4t,,:22.61 mls, J., : 1.89 rn/s, Vu,,tn: O.tOt, Zc.^,u* :0. I,
Vo,, : 0.3+0, Q, : lO92 Pa. From Eq. 8.2.2, p :'2.94.
case where Gravity Loads Are present. Equations g.2.7 and g.2.g are
if the gravity loads acting on the member may be neglected (as in
the case of cladding panels subjected to wind roads). However, if"(l)
the effect
applicable
of the gravity load is significant and, (2) the most unfavorable load combination
occurs when the wind load reaches its largest lifetime value while the gravity
load has an "arbitrary-point-in-time," rather than an extreme, value.*
then
Q,:Qao-o + G
v;: ' vba"-o-+ YL
f r + --c )' /, * o'"o\'
\ Qac=o/ \'
C I
(8.2.1s)
(8.2.16)
O"tg: g ?nd.Vo,l.":s : me?n value and coefficient of variarion of largest
l!"1"
lil'etime wind load esrimared by 8q.8.2.7 and g.2.g. respecrively. and
6ind
v6 : mean value and coefficient of variation of "arbitrary-poinrin{me"
gravity load, respectively.
structures with unknown orientation. A procedure is now presented for
r:stimating safety indices for members of structures whose orientation is not
known. such a procedure can be useful for the development of building code
provisions on wind loads.
The unknown orientation of the structure can be considered as one among
scveral uncertain factors that determine member reliability (member resistance,
:rcrodynamic coefficients, influence coemcients, etc.). To the extent that struc-
trrre orientation is included as a random variable in a properly conducted relirrbility analysis, the reliability of members in a structure sampled at random
will be acceptable regardless of structure orientation, just as the reliability of
will be acceptable even though, owing to
:rrry properly designed steel member
tlrc variability of the steel strength, actual yield stresses might be lower in some
t':rscs than the average yield stress.
'l-he mean value and thc variance
of the largest lifetime loads acting on the
rrcrnber undcr consideration, avcragcd over all possible angles of orientation
rv ol'thc structurc to which llrc rrrcrrrbcr hclongs (Fig. g.2.li
are
nt).
+Equations tl.2-7 lrrxl ll.2.ll:rrt: trpploxirrurtc hccitttsc (l) lcrrrrs rtl rtrtllt lttl'lrtr llt;rlt lw(l illt'
ncglcctctl,:rlrtl (2) it is;rssrrrrrul llurt llrt'vlrriirbililit:s ol lltt'llttkrtr, llt,tl rIlcrtrtlttt't,,,,,, lrrrtl l/,.,,,,
,h'1x'rrl lrt l'lif iltlv tllr'lt rlltt r ltoll
nNt
a,,
l"
I
.lo
ql,,,,r)/(,r)
'Sct' tlisr'rrssiorr irr A1r1x'rrrlir A l lqllrrs 1rl, I tl .\ I
,)
r/rv
(8.2.11)
.t
322
wtNt) t)ilil (;il()Nnl ilY I ttlcr:;
(Q.,,
-
Q,,)''
lil Iilrl
- I.:' 10,,(rv)
0,,(,v)l.r
(n.1. llt)
l'(u) tlu
:
a is uniformly distributed, that is,/(cy) : ll2r. Other assurnptituts
can be made, as necessary, if predominant structure orientations are known to
sume that
exist.
In addition it is assumed that wind speed data are available from 8 compass
tlircclions. (ll-data arc available from 16 directions, the number 8 in the equa
tirrrrs that lirlkrw rnust bc changed into 16.) Using Eqs. 8.2.11 and 8.2.lti it
carr bc slrowrr
alicr soutc algcbra that
Q,:
jPcrn,*
i,,?,
U.-.,,,ta, )2 [
(
= ftl
+ stf,
+
v2r,^^.y*
t::,
- L* t:: , u,,,(o,)2 tr
[*t::
, u*,(o,)2 rt +
+
+
I
vL.,,,(d,)t
,t.r.
I
'r,
u"qJ.,Jo
[t + 6 v'r.u,,(o,)l
Vrr"",,@,\l
jI
)
v2u,u,1"'ril
(8.2.20,
j
REFERENCES
A. G. Davenport, "The Prediction of Risk under Wind Loading," Procecdingl
Znd
International Conference on Structural Safety and Reliability, Munich,
Scpt
.
1977, pp. 51 l-538.
8-2
Y. K. Wen, "Wind Dircction and Structural Reliahility."
8-3
lApril 1983r. I028- r04L
Y. K. Wcn, "Winrl l)ircction utrtl Stntclttritl llcli:rlrilitv ll
ll0 (l9tt4).
125.1 1264,
.l
.
Srrrtct.
.l
ling.,lllt)
.,\trtrt't .
l,.rr,q.,
llitul Iirtttrtl
Strt,lt',,1 .ltl.ntttt ()111,'t'lilttlrltrt,tl.
Fluid Mcchanics and Wincl lingirrccriltg l'trr1r,r':rrrr. ('ollt'1it' ol llrrl'rr11'1'1i111i. 1',,1
orado Statc Univcrsity, Ft. (lollirrs, Nov. l()7t3.
E. Simiu and J. J. Fillibcn, "Winrl l)ircclion lrllccls on ('lutltling lrrl .Slluc(ulul
Loads, " Eng. Struct., 3 (July l9t3 I ). ltt I ltt(r.
lJ6
M. E. Changery, E. Dumitriu-Valcca, and E. Simiu, Directional Extreme
rJ-7
Wind
Datafor the Design ofBuildings and Other Structures, Building Sciences
Series BSS 160, National Bureau of Standards, Washington, DC, March 1984.
E. Simiu and M. E. Batts, "Wind-lnduced Cladding Loads in Hurricane-Prone
Regions," J. Struct. Eng.,109 (Jan. 1983), 262-266.
Hurricane-Induced Wind Loads, Computer Program, Accession No. PB
Speed
r.t-8
9
82132259, National Technical Information Service, Springfield, VA, 1982.
tt-10 M. E. Batts, L. R. Russell, and E. Sirniu, "Hurricane Wind Speeds in
United States," J. Struct. Dlv., ASCE, 106 (Oct. 1980), 2001-2016.
I
the
American National Standard A58.1, Building Code Requirements for Minimum
Design Loads, American National Standards Institute, New York, 1982.
U-12 K. Rojiani and Y. K. Wen, "Reliability of Steel Buildings Under Winds," -/.
Struct. Div., ASCE, 107 (Jan. 1981),203-221.
t{-13 E. Simiu, "Aerodynamic Coelficients and Risk-Consistent Design," J. Struct.
Eng., l09 (May 1983), 1278-1289.
ll 14 E. Simiu, E. Hendrickson, W. Nolan, I. Olkin, and C. Spiegelman, "Multivariate Distributions of Directional Wind Speeds," J. Struct. Ezg., lll (April
lJ
18-131. In Eqs. 8.2.19 and 8.2.20, U"c,(o,) and V11.r,(a,) are the mean valuc
and the coefficient of variation of the largest lifetime equivalent wind spectl
Urr,(o,), estimated for the structure with angle of orientation cYr as in Eqs.
8.2.10 and 8.2.13. Use of Eqs. 8.2.19 and 8.2.20 inEq. 8.2.2 yields the saf'cty
index of the member being considered in the case where the orientation of thc
structure is unknown. The case where gravity loads are present is treated in ir
manner entirely similar to that shown for structures with specified orientation.
8-l
ty1i11vrrsorr,
J. A. Pctcrk:r anrl .l . lr. (lcnturk,
n5
t't I
vn,,
l) Srrrry, irrrtl A. (i. l);rvt'rrporl. "l'r,',1r, lrrrll Wrrrrl lrrrlrrrt'rl
llcslxltse irrlltrrit':urt'Zorrcs,".l .,\urttt. /)rr',AS('l'., l(lltlrcr l()l(r),.rIII
ll. V.'l
|,1 ,+
l't
(t.8.
t
I
2350.
largest lifetime wind load acting on the rncmbcr givcn that tlrc
angle of orientation of the structure is cv, and/(cy) : probability density functiorr
of structure orientation in the region being considered. It is reasonablc to ls
where Q,@)
ilr
15
1985),939-943.
E. Simiu, S. D. Leigh, and W. A. Nolan, "Environmental Load Direction and
Reliability Bounds," J. Struc. Eng., ll2 (1986), ll99 1203.
*''w
\
a\
PART B
WIND LOADS AND THEIR EFFECTS
ON STRUCTURES
II APPLICATIONS TO DESIGN
CHAPTER 9
BUILDINGS: WIND LOADS,
STRUCTURAL RESPONSE, AND
DESIGN OF CLADDING AND
ROOFING
'l'ltc design of buildings is based on estimates of (l) overall wind effects, which
rrrtrst be taken into account in the design of the structure, and (2) local wind
r'llbcts, which govem the design of components (e.g., purlins) and cladding.
lrr gcneral, the aerodynamic information needed to estimate overall as well as
krcal wind effects cannot be determined from first principles and must be obIrrincd from wind tunnel tests. However, for a number of common situations
llrc acrodynamic information is already available, and procedures for estimating
slructural response which incorporate that information may be employed. This
is lhc case for tall buildings that (1) have geometric shapes that are not unusual
rrcrodynamically or structurally and (2) are not subjected to strong interference
e ll'ccts caused by the presence of neighboring structures. As an approximate
grridc it may be assumed that if the distance between two buildings exceeds
$ix to eight times the average of the horizontal dimension of the buildings,
rrrutual interference effects will be negligible for practical purposes. For more
rrlincd guidelines and an excellent compendium of information and references
rrrt interference effects, see [9-1]. It is noted in [9-21 that a square building
kratcd in urban terrain near a building with similar geometry and dimensions
will perform satisfactorily, regardless of the relative position of the two buildrngs, if it is designed to withstand the loads (including the across-wind loads)
il would experience in the absence of the neighboring structure. See also [9-3,
,) 41.
'l'his chapter is divided into six sections. Sections 9.1,9.2, and 9.3 discuss,
rcspcctivcly, methods for cstirnating thc along-wind, across-wind, and torsional
lcsponsc of flcxible buildings unullbctcd hy intorlbrcnce effects. (Buildings arc
lclbrrccl lo as .flexible il' thcy cxl)r'ricn('r! sigttilicant clynermic arnplification cl:
It'cts cluc to thc acnrdynatnic lottrl lltu'tttutirttts. A rough critcrion is put lirrth
327
328
lllrll l)lN(ili wtNI) t()nl r:;, :,ilil,r ilJt!r\t ilt .,1,{)t]:,t ANt) l)l :;t(.N {}t lror}t il!(,
by the ASCE 7 9-5 Stlrntllrtl l() 51, wlrrt'lr tk'lirrt's lr lrrriltlirrg rrs llt'xrlrlt'il tlrt.
ratio between its hcight ancl Ic:rst lrolizorrlrrl tlirrrerrsion is lirlgcr lllrrr lirru'. or
its fundamental natural frequcncy ol'vibrlliorr is lcss thun I llz.) l)yrrirrrrrr.
amplification effects influence the structural krads ancl can crcatc two kirrtls ol
serviceability problems: (1) occupant discomfort due to cxccssivc buiklirrg :rt,
celerations (see Sect. 15.1) and (2) nonstructural damage due to cxccssivc sr.ry
drift. To avoid such damage, some designers limit story clrifi scvercry, c.g. r,
l/600 at the design wind speed; see [9-73]. The serviceability problerrs .ury
be solved by increasing the structure's stiffness, but in many instancos rrrr
economical complementary solution is to use damping devices. These arc tlis
cussed in Sect. 9.4. Section 9.5 is concerned with overall and local wincl kxrtlr
on low-rise buildings, that is, buildings with relatively low height which, owirrg
to their relative rigidity, do not normally exhibit dynamic amplification cllct'rs
Cladding and roofing clesign for wind loads are discussed in Sect. 9.6.
Note that Ihc vast rnajority of available results based on wind tunnel tcs(irr1t
or analytical turbulcncc rnodcling were obtained under the assumption that tlrr,
atmospheric flow is stationary. In reality some flows, including hurricane llows,
are highly nonstationary. Some efforts to study nonstationary flow effects lrrvt.
been reported recently; see [A2-14] and [A2-15].
9.1
ALONG-WIND RESPONSE
Until the 1960s drag forces used in structural design calculations were spccilictl
on the basis of climatological, meteorological, and aerodynamic consideratiorrs
alone, independently of the mechanical properties of the structure, that is, ol
its mass distribution, flexibility, and damping. It was subsequently recognizt'tr
that for modern tall structures-which are more flexible, lower in danrpirrg,
and lighter in weight than their predecessors-the natural frequencies of viblr
tion may be in the same range as the average frequencies of occurrencc .l
powerful gusts and that therefore large resonant motions induced by wind rrurv
occur and must be taken into account in design.
The resonant amplification of structural response to forces inducecl by rrt
mospheric turbulence was first studied by Liepmann in a classic paper orr tlrr
in 1952 [9-6]. The application of Lieprnlrrrr.:,
concepts to civil engineering structures required the development of nrtxk.lr
representing the turbulent wind flow near the ground. Such models wcrc l)r(l
posed in 196l by Davenport [9-7], who developed on their basis a proccrltrt,
for estimating along-wind tall building response [9-81 . vellozzi an,J ('olrt'rr
developed a modified procedure, in which, in contrast to l9-81, it is rccognizr.rl
that the fluctuating pressures on the windward face <ll'lr hrriklirrg urc rx)l lx.r
buffeting problem published
fectly correlated
to those acting on the lccwarcl lircc l() ()l 'l'lris inrpcr.lr.rr
correlation is accountccl lor in [9-91 by u r-crlrrcliorr llrttor. llowt'ver'. it lr:r:,
been shown that tlwirrg l() thc way irr which llris l:rt'lor.is rrlrplrr.rl. (lrr'llnrt.t'rlurr.
of [9-91 untlcrcs(irttlrlt's llrt'r'esorr:rrrt trlrrplificlrtiorr t'llt.tt l,) l()1, l,)
cctlttrc lilrcslirtr:rlirr1l:rlorr1l wirrrl n'slxrrrst'lrlrst'rl r.r;:rr-rrlr.rlly on l()
Ill.
A pro
i^il lr:rs lrt.r.rr
r'r:,r
329
(ltt' (':rrr:rtlr;rr Slrrrr.lrrrlrl l)t.st1',rr M;rrrrr:rl l() l.ll. Vit.kcl.y srrbst:
"', lutlt'tl ilt
rlu('nlly tlcvclopt'tl :r Ilott'rlrur'slrrrllu lo llr:rl ol l() fil (lurl :rllows" lrowcvcr,
l.r lll()lL'llr:xrltility willr n'spt't'l lo llre clroir't'ol ecrl:rirr rrrclconlkrgical param,lr'rs lt) l3l. An ttltcrl'tt:ttivc rrpptrr:rr'lr is rrst'rl in l() zl-3 1, which utilizcs cquations
,,1 crltrilihriullt anlonll ltolizolrtlrl lirrucs trl crrclr llrxrr.
lrr llrc proccdurcs ol'19-l2l lrrrtl l9-l.ll i( is ussurned that the characteristics
,,1 rlre turbulcnce do n()t viuy wilh hc:ight abovc ground. Actually, according
r,' tlrc rcsults clf moclcrn rnctconlkrgical research, the energy of the turbulent
llrrtltnttionS that causc rcsonant oscillations in tall buildings decreases signifi,.rrrlly at higherelevations (see Sect. 2.3.3). Computerprograms forcalculating
.rl.rrs 1ryip4 response, in which this decrease is taken into account and which
'll,rw therefore more economical designs, have been deveioped independently
iir l() l4l to [9-16].
()rr the basis of [9-14] and 19-161, simple procedures were deveroped in
l't I 7l and [9-18] that account for the dependence of turbulent fluctuations on
ir,'r1llrl, and on whose basis rapid manual calculations of the arong-wind re'tx)nsc can be performed. The procedure of [9-18] is easy to use, and it is
, ()nsistent with specifications in which the mean wind profile is represented
by
tlrt' krgarithmic law. we include it in this chapter for users of such specificarr'rrs. ThiS procedure also applies to elevated structures, such as signs whose
lrotloln side does not reach to ground level.
'l'he commentary to the ASCE 7-95 Standard
[9-l] includes a procedure
.r,l;rptcd from [9-18] by A. Kareem [9-19], which accommodates wind climate
.rrrrl wind profile information expressed in terms of 3-s basic wind speeds and
rlr(' I)ower law, respectively. In addition to being compatible with the format
.rrrtl rcquirements of the ASCE, 7-95 standard, Kareem's version has over the
1rr.r'cdure of [9-18] the advantage of added flexibility with respect to the choice
,'l thc fundamental modal shape. It is available, in interactive computerized
1,rp111, 2s part of the diskette "Developmental computer-based version of ASCE
/().5 Standard Provisions for Wind Loads" [17-5] appended to this book.*
All the procedures mentioned above are based on the assumption that, around
rlr. structure, the terrain is approximately horizontal and that its roughness is
,r':rsonably uniform over a sufficiently large fetch. In practice it may be nec, ',:jiuy to adjust the results obtained on the basis of this assumption by taking
nrt() llccount the effect upon the flow of changes in the terrain roughness upwind
,'l tlrc structure (see Sect. 2.4.1).1 If the topography of the surrounding terrain
, ' urrusual, or if the building is strongly affected by the flow in the wake of
I or lruildings in lrurricanc prrltc
it ir irnprrn:rn( to verify that thc convcrsion factor liom
'('!'.i()ns
t',.r1 lirrsl l() lllcan lrtturly tttcrtrt slx'('(l us('(l t'xplicitly or irrrplicitly in thc calculation pnrccclurc
l'|{ (oll\i\l('ttl wi{lt lltc cottvtitsiott l:rr'lor rrsr'rl irr llrr'Sl:rrrrlirnl to ()blllill tlcsign pcak gusl spcc(ls
lr'rtrl 1|.'1,1r;t\r'titl'('(l r)V\'t lr,il)'( t lilI, Ilt, t\,t1.,
l ,rr lrtliltlirrgs lot:tlttl ort lwrt rlrttrcrr"rorr:rl r,l1', r, rrrrrl t:,r rr;rrrrt rrls :rrrrl orr lrxisynrrttt.llit. lrills, ;r
rrrr;rlctttt'lltrxl lilt:rltul;tlitrl'rnr.trrr rrrrrl ,.1,rrrl rrr,!r..r,.i.,( .,1r.(.(l 1l)s")isilrr.lrllgrl 1t llt(.nS(,lt
/r)'r $l;p1111;1111 l() \l (\('( ScrI .).l ). ('lt,ll)1, r I / rrrrl r!r..l.r'llr. I)r.r't.lplrrrrr.lrltrl ( llrrlrrrlr.l lr.r.,r.rl
\ ( ri l(,ll (tl AS('lr / r)'r Sl:ttrl.ttrl I'tor r',rrrr'. l,,r \\ rr,,l I ,'.',1. I I / ',11 .r1r;r.rrrlr.rl to lll., lr,,,l.
330
lJt,ilt)tN{il; wtNt) t()nt ): ,. t;illl,r.l jltnt ltt
:,t
,{)t,l:;t nNt) t)t r,t{;t.J ()t n()()r
rrJ(,
large neighboring buildings, arralylicrrl pnrcr:tlrur:s becorrc irrirlrplrt'rrlrle llrrl
wind tunnel testing is necessary.
Another assumption common to all thc abovc-rncntionccl pnlccrlurcs is llr:rl
the mean wind is normal to the building face under considcration. Wincl tunrrcl
tests suggest that, in cases commonly encountered in tall-building clcsign pr-lt.
tice, to this assumption there correspond the highest values of the along-wirrrl
response [9-2, 9-20). In the case of a square building, the peak along-wirrrl
response decreases continuously as a function of mean wind direction, frorl ir
maximum value that corresponds to the case where the direction is normal lo
a building facc to about 0.8 times that value when the direction is parallcl to
rr tlirrgorxrl l9-21.
'l'hc gcncrltl li'rttttcwork ttl'lhc aklng-wind response problem
is presented in
St't'l ().1.1. scctiorr ().1.?. tlcscribcs thc procedure developed in [9-lg] lil
cslirrrlrtirrg tlrc rrlorrg wirrrl rcsponsc ol'prisrnatic, oralmost prismatic, structurc:s
lirr whiclr it nrly bc lssurncd thal (l) the fundamental vibration mode shapc is
lrl.rpnrxinraloly a straight line and (2) the contribution to the response of the
sccond and higher vibration modes is negligible. Also described in Sect. 9. 1.2
is a procedure for estimating the along-wind response of point structures, that
is, structures that may be viewed as consisting of a single mass concentratctl
at a height H (e.9., water towers) 19-181. In the procedures described in Sect.
9.1.2, referred to here as simplified, all computations can be carried out man
ually. If the shape of the fundamental vibration mode deviates strongly from ir
straight line, or if the contribution of higher vibration modes is significant, the
use of a computer program is required as indicated in sect. 9.1.3. In Sect.
9.1.4, results of numerical calculations are used to discuss some of the approximations and errors inherent in the models being used.
9.1.1 Basic Relations, Equivalent Static Wind
:
i(2.)
*
x,,o*(z)
(9.1.1)
where x(z) is the mean deflection, and x-o^(z) is the maximum fluctuating
deflection in the direction of the mean wind. It is convenient to express ,r,,,,,"(:)
in the form
r,."*(Z)
:
,t I
At
(,ll(,willt ) l{l
r,,,,,,(.') A,(.:)rr,(.:)
:,1
.ot.l:,t
:l:ll
(9.1.-l)
rrlrr'tt'o,(;) is llre lixrt
nr(':ur s(luiuLr vlrlrrt'ol tlrc:rlolrg wincl accclerations and
A,(.:) is u pcak lirctol', thc virlrrc ol'wlrit'lr is rrsrr:rlly ubout 4.
'l'lrc gus( tosp()lric lirctor is rlclirrctl lrs
(i(z)
I
- I t*"'l(z)
i(z)
lr( rnaximum along-wind deflection can then be written
X,,u*(z)
:
(e.t.4)
as
G{z)i(z)
(9. 1.5)
It rs convenient to define an equivalent static wind load that would induce in
rl,('structure along-wind deflections equal to those caused by the gusty wind.
It l.llows from Eq. 9.1.5 and the assumed linearity of the structure that the
,,lrrivalent static wind load is equal to the product of the gust response factor
i'r
lhc mean wind load.
l'lrc general expression for the mean deflection x(3) is given by Eq. 5.3.1.
as the respective peak
r,rr rrrs (Eqs. 9.1.2 and 9.1.3) are obtained from Eqs. 5.3.8 through 5.3.15,
iii wlrich the general expression for the quantity s"(2, n; (the spectral density
,,1 tlrc along-wind fluctuating deflections) is given by Eq. 5.2.37.It follows
lr.rrr thcqe equations that the calculated deflections and accelerations depend
rr1r,n the properties of the structure, that is, its dimensions, mass distribution,
rr.rlrrrirl fiequencies, damping ratios, and modal shapes, and upon the assumed
iii,':rn and fluctuating wind loads.
llrt' lluctuating deflections and accelerations as well
Loads
The total along-wind deflection may be viewed as a sum of two parts: the mearr
deflection, induced by the mean wind, and the fluctuating deflection, inducctl
by the wind gustiness. The maximum along-wind deflection of the structure ul
elevalion z may thus be written as
X-o-(z)
*
K,(2.)o,(z)
(9.t.2t
where o,(z) is the root mcan square value of the fluctr"rating tlcllcction rrnd K,(;)
is the peak lactor, lho vltlr-rc tll'which is usually irlxrrrt
maximurn along-wirxl lrccclctlrliorr lrury hc cxprcsst'tl ;rs
l to,1. Sirrril:trly tlrr.
', L2 A Simplified Procedure for Estimating Along-Wind
Response
l ,rllrrwing [9-171, a procedure for calculating along-wind response is now pre,r'rrlctl, applicable to prismatic, or almost prismatic, structures for which it may
l', rrssumed that (l) the shape of the fundamental mode of vibration is linear
.rrrrl (2) the response to wind loading is dominated by the fundamental mode.
I lr. lirst of these assumptions is acceptable in a large number of situations of
lr:rt'rical interest such as in the case of typical multistory framed structures
r, 1'.. sce 19-21, p. 4281 or [9-22, pp. 60 and 242]).'the second assumption
,rrll gcncrally hold if the ratios of natural frequencies in the second and higher
i,r,rtlt's to the fundamental frequency are sufficiently large (see Sect.9.l.4).
\1.'. givcn in this scction is a pnrcedure for estimating the along-wind response
,'l lxrirrl slnrcturt:s, that is, structures that may be viewcd approximately as
, ,'rrsisting ol'a sirnplc rrlrss M c<lnccntralctl lrl tr hcighl l/.
ll;tsic Assumptions. 'l'ltc procctlttrc pn'st'rrlt'tl in llris st.t'liorr is blrst'tl on llrt.
l.llr
r1yi11;' :rssrtrtrPl
iolrs.
332
I
tit
.
2.
3.
4.
[l t)tN(i:i wlNt)t()At):i.:;ililt(.ilt|lAt
ilt
:;l
,()N:it nNl)l)l :;l(,N()t tt(xrt
'l'hc bchavior ol' llrr: slnrctrrrt' is lirtr'lu ly t'llrstic.
The f'undarrrontal ttrotlc ol'vibt'lrtiort is rr lirrcirr lirrrctiorr
ol
ground, that is, x{z) : 7.111.
The contribution of the second and highcr vibration uxrclcs
is negligible.
The mean velocity profile is described by the relation
u(z)
:
2.5uxlnZ
- Za
Zo
l0
U(.2):2.5uxln-
z >- za * l0
z < z7-l
lO
l<l
il
tN(i
lrciglrt rrlxrvt'
thc rcsl'rorrst.
(e. 1.6)
(9.t.1\
211
(lrr lils. (). l.(r lrntl t).1.'7, i., i1y, and 2,7 ?rc expressed in meters.)
'l'hc rrsc ol'thc logarilhrtric pnrfile above elevation (z,r * l0) meters implic:s
thc assunrl'rtion ol' horizontal homogeneity of the flow. This assumption mly
not hold ovcr rcgions neara change in surface roughness, as indicated in Secl.
2.4. However, in such regions Eq. 9.1.6-with suitable values of the pararn
cters u*, Ze, and ZaTma! be used to obtain reasonable upper and lower bounds
forthe value ofthe response. Equation 9.1.7 is used, conservatively, on ac-
of the uncertainty with regard to the actual nature of the flow near
building for z < z7 f l0 or so.
count
5.
and
Table2.3.1, and by Eq. 2.3.16.
7.
8.
nl
ollt iwllltrttt ,,t,|il"t
l3J
t:tsc tll wcltlctl slt't'l slltt'ks, ol r'ct'l:utt ptt'sltt's:;t'tl .,l lt( lur(.r., ut ol :,llt( {lt(':,
ol lltt: li:urtctl (trlrc lylx' l() ll, 9 l.ll. lrr irtltlilrolr lo llrt. trrt.t lr;rrl,:rl rl,rrrrPnr1,.
lltc ltct'tlrlyruuttic tlirrtllirrg nrily, irr Plint'iPlt'. ;rl:;' lrt' r;rIr'rr rrr' ;rt r,lrrr 'llrt.
:rt'trrtlynatrtic tllrrtrpirrg, wlticlr rrriry ltt'lp r-t'tlutt'lltt'nr;rl,rrlrrtlt.ol llrr.r{.r()n:url
oscillati<ltts, is associltlctl willr cltrrrtgt's irr (lre rt'l:rlrvt' vclot rly ol llrr. :rrr rvlllr
r(^sl)cct t<l thc builclirrg as thc little r oscilllr(t's ;rlrorrl l(s nr('lut rlt'lorrrrt.tl
lx)stlt()n.
Its clctennination is vcly uncorlllin, irrrtl il is tlrt'n'lirrt';l'rrtlt.rrl to rrt'glccl it irr
st nrctural calculations.
According to [9-691, darnping nrtios llrvc signilicant statistical variability,
r.vith coefficients of variation ol'about 0.4 to 0.8, depending upon building
rypc; mean damping ratios increase with vibration amplitude in accordance
with a power law with exponent ll9 to lllo measurements indicate that 5- to
.]O-story buildings tend to have roughly 60% larger mean damping than building
.vcr 20 stories high, presumably because energy dissipation by the foundations
lurs a smaller relative contribution to the damping of taller buildings; on the
lrirsis of limited observations, it appears that for buildings with more than 20
slories, concrete buildings exhibit only about 3o% more damping than steel
lrrrildings.
To reduce occupant discomfort due to wind-induced building accelerations
(scct. 15.1.1), the damping inherent in the building may be augmented through
thc use of dampers (Sect. 9.4).
ir
The mean velocity U(z) in Eqs. 9.1.6 and 9.1.7 is averaged over a periotl
of one hour.
6. The longitudinal velocity fluctuations are described by Eq. 2.3.2
!rt
The mean and the fluctuating pressures are described by Eqs. 5.3.3 antl
5.3.6, respectively. The expressions for the mean response are thereforc
given by Eqs. 5.3.4 and 5.3.2, and those for the fluctuating response hy
Eqs. 5.3.7 through 5.3.15 (or the equivalent expressions in nondimensional form, Eqs. 5.3.16 through 5.3.28).
The spatial cross-correlations of the fluctuating pressures in the acrosswind and along-wind directions are described by Eqs. 5.3.48 and 5.3.49.
rspectively.
[errain Foughness Parameters, Zs, 26. The variation of mean wind speed
with height is determined by two parameters, the roughness length ze and the
zcro plane displacement z7 (Eq.9.1.6). The roughness length may be inter1r'cted physically as a measure
of the turbulent eddy size at the ground level.
Values of zo suggested for structural design purposes are given in Table 9.1.1
(scc Sect 2.2.4).
ln densely built-up cities (or in forests) rhe buildings (or trees) obsrruct the
llrw near the ground; the mean flow thus begins to develop above an elevation
rr:lbrred to as the zero plane displacement and slightly lower than the average
lrcight of the surrounding buildings (or trees). For design purposes the zero
Pllne displacement may be assumed to be zero in coastal and open terrain and,
il'the values of zo of Table 9.1.1 are used, in built-up terrains as well.
'l'AIILE 9.1.1. Suggested values of Roughness Lengths
'l'crrain
Response Parameters. A brief discussion is now presented of some of thc
structural, micrometeorological, and aerodynamic parameters involved in tlrc
estimation of along-wind response with a view to assisting the structural clc
signer in their interpretation and selection.
Damping Ratio, (,,. Suggested valucs fbr mechanical <lirrrrpirrg nrlios ol'stc:cl
and reinforcecl concrclc I'nrntcs arc 0.01 and 0.02, leslx'ctiv('ly l() Ill l,owcl'
values ol'lhc Itrcchitttit:rrl tlrurtpirrg rrriry llrvc lo bc rrst'tl. lor ('\iunl)l(', in lll('
l'ypc
ol'
It'r'r'lin Coastal"'/'
..
,,(rtr\ 0.(X)-5 0.01
Olrcn'
o.03 0. r0
zn
Sparsely
Built-up
Suburbsb
0.20 0 40
for various Types of
Towns,
Densely
Centers
Built-up
of Large
Suburhs/'
Cities/'
0.80-
L20
"ApPlitirblt'1o srru('lurrs tlirt'tlly t'xlxrst'rl 1. wirrtls blowirrg lirrrrr opcn watcr.
''V:tlrrt's
ol
:t,, 1o
lrt'ust'tl irr t.orrjrrrrt.lrorr willr llrt ;rssttrrrpliott
:t,,
O.
2.00,3.00
334
tttrtt
t)tN(iti wtNI)lo^t l;.::lltt r{.il|rnt ilt
'()N:;t nNt)t)t :;t(iN1)t n()()t
iit
tN(i
rt
Exponential Decay Parameters, C, C..'l'lrt' n:rrnrw llurtl spirlilrl cr()ss ('()r
relation of the fluctuating prcssurcs in tlrr: lrcross wirrrl tlircctiorr (llt|. 5 .1 '16;
is a measure of the extent to which prcssurcs trpplicrl al dillcrcnl poirrts ol'tlrt'
same building face act coherently or at cross-purposes. 'l'hc srrrallcr tlrc vlrlrrt's
of the parameters C, and C, in the expression fbr the cross-corrclation lhc rnon'
coherent will be the action of such pressures and, therefore, thc largcr llrt.
Friction velocity, u*. The friction (or shear) velocity a* is a measure of the
wind intensity over terrain of given roughness. If the mean wind at a specificrl
reference height above ground za is known, u*. can be obtained by using Et1.
9.1.6:
U*:
U(zn)
2.5ln[(zn
(9.l.ltt
-
z)lzo]
.,
In designing tall buildings it is reasonable to use mean wind speeds averagctl
over a period of one hour. In this chapter the symbol u will denote hourly
mean speeds. If mean wind speeds (J' are specified that are averaged ovcr'
periods t different from one hour, the mean winds averaged over one hour can
be obtained by using Fig.2.3.10. For convenience, the information includcrl
in Fig. 2.3.10 is summarized in Table 9.1.2. (Forbuildings in hurricane-pronc
regions, see also first footnote of Sect. 9.1, Sect. 2.4.3, and [9-5, p. 155].)
For values of / not included in Table 9.1 .2,linear interpolation is permis
sible. If the wind speeds are given in terms of fastest-miles u7, the averaging
time in seconds is given by
:
36OOlUr.
(9.
l.tll
TABLE 9.1.2. Approximate Ratios of Probable Maximum Speed Averaged ovcl
Period / to That Averaged over One Hour (at l0 m atrove Ground in Open
Terrain)
t
l0
30
60
lo0
200
5(X) (XX) 3(,(X)
I
(s)
u'lu l.-53 1.41
L12 l2tt I24
l.
llJ
0il{ . Wltjt I nt
,t ,r
)t.|,t
3:15
I r,ln.
,
Syr:r
rl
l, l:rilr
r:;t'ly
llrrrlt
('olrsl:rl
( )lx il
0. tts
l.(x)
I
Iorvrts,
)crrst'ly
rr;r
litrilt
Strlrrrrlrs
Su hr
l
l.5
r
Clcntcrs
-u1r
ol' Largc
rbs
Citics
l.45
I .33
As indicated in chaptcr 2, thc retardation of the flow due to increased terain
causes thc mean speeds over built-up terrain to be lower-for any
,'r\'('il large-scale storm-than the mean speeds at equal elevations over open
r{ rririn. Since wind climatological information is commonly provided
in terms
,'l wind speeds measured over open terrain (generally at airport weather starr.rs), the problem arises of converting this information into wind speeds apf irtrrlrlc to a built-up environment. In Sect. 2.2.5 this problem was shown to
l'{' solved as fbllows. Let u*1, 201 denote the friction velocity and roughness
l, nr',lh over open terrain, and let z* denote the friction velocity over terrain
rr rrlr roughness length 20. For the surface roughness categories
of Table 9.1.1,
.r;r;rrrrximate ratios u*lu*1can be obtained from Table g.1.3. once z* is known,
l/( .:) can be calculated by using Eq. 9.1.6.
r,r111'l111gss
r. This parameter appears in Eqs. 5.3.1r and 5.3.14, which
rrrrlrcats in effect that the expected peak values of the fluctuations will be higher
rl llrc: duration of the storm increases. The assumed storm duration is implicit
lturation of storm,
In meteorological work, the reference height most commonly used in zn
l0 m.
r
Al
l .\lll,l,l '). 1..1. ltirlios it ,ltt ,, lot \':n.iolls Sur-lirr.r. ltoulihrrcss ( ,:rlt,gol.ir.s
response.
On the basis of wind tunnel tests, it has been suggested that it is reasonahlt.
to assume Cy: 16 and C. : l0 [9-13]. The procedures presented in this
section are based on these values. However, as indicated in Sect. 2.3.4, full
scale measurements do not always confirrn this assumption. As shown in Sccl.
9.1.4, the effect upon the total along-wind response of changes in the valucs
of Cu and C of as much as 30% to 4O% is, in general, relatively small (of tlre
order of 5%-10"/"). However, the effect of such changes upon the accelerations
may be considerable. (See also footnote following Eq. 2.3.31.)
!
l.ll
lo/
tor
l.()()
ut" of design mean
1,1,1''"
speeds averaged over one hour, that
Mr:an Pressure and suction coefficients,
is,
I:
3600
c*, cr. The mean pressure and
'.rrt'tion coefficients are functions of the shape of the structure (see
chapter4).
Ir rlrc case of tall buildings with a rectangular shape in plan, it may be assumed
(,,
0.8,
C/:0.5, andCp: C*,+ Ct:1.3.
A"tt:an square value of rurbulent velocity Ftuctuations. The ratio, p, berr't't:rr the mean square value of the longitudinal velocity fluctuations,7, and
rlrt' square of the friction velocity, u?* @g. 2.3.2) depends upon surface rough-
r('ss. ils shown in Table 9.1.4.
i
i\lll,lJ 9.1.4. Approximate Ratio p
(':rlrgories
i ype
ol
Icrlrirr
Ctxtslal
ti
(r 5o
: it"r*
for Various Surface Roughness
Sparsely
Dcnse ly
Ccnters
Built-up
Iluilt-up
)pcn
Suburbs
Subu rbs
ol'Largc
('itics
().(X)
52s
(
4.
t{.5
.l (x)
336
aUlLDlNos: wtNt) ro^l)li,
lirlrt,(;ilrrnr
nr rl,()Nlir , nNr) r)l ,,;r(iN
()r
n()()r rN(i
1f
Expressions for the Along-Wind Response. Using thc basic ussunr;'rtiorrs
listed earlier in this section and relations given in Scct. 5.3, rcsults ol'nurncrical
integrations were closely fitted in [9-18] by simple functions, and cxprcssions
for the along-wind response were developed that are listed in Table 9. 1.5 lirr
buildings with a nearly linear fundamental mode shape (Fig. 9.1.1), and in
Table 9.1.6 for point structures (Fig. 9.I.2).
In Tables 9.1.5 and 9.1.6, h and H are the vertical dimensions shown in
Figs. 9.1.1 and 9.1 .2, b : across-wind dimension of structure, d : alongwind dimension of structure, Zs : roughness length (see Table 9.1.1),70:
zero plane displacement (for practical calculations it may be assumed that
: 0), nr : natural frequency of vibration in fundamental mode of vibration,
ux =. f'riction vclocity, Cp: drag coefficient (Co: C* + C), C*and C1 :
avcrrgc prcssurc coefficient of windward and leeward face of building, respcctivcly, M : L<ttal mass of structure with dimensions b, h, and d in Fig.
9.1.2, z. : hcight above ground, M(z) : mass of building per unit height,
poQ) : bulk mass o1'building per unit volume, f1 : damping ratio, p : mass
of air per unit volume, 0 : coefficient given in Table 9.1.4, T: duration ol'
storm (Z : 3600 s), x : mean displacement at top of structure, 6 : gust
response factor, X-u* : peak displacement at top of structure, oj : rrns acceleration at top of structure, and iu^ : peak acceleration at top of structure.
(tz) M, :
|
+
(13) q*
fl4)t:
0.26bth
- n,h
(4) .f' : L
(15) o,
u+
t1
(5lClxt :*-Z*r(l-cL)
: 1232L4
Qh
(7) N(h: c@)
@ c'?Drc) : c?" + 2c*ctN(f)
(6)
(9)
(10) M(z)
(il)
G
3.55
:
(17) K,
+
c?
0
rJ
6
:
*u,)"
\''
[1.175 1- 2ln(u,T)ltt)
o,
QD x,,,.,.
cl,
0. 1, respectively, which is the case
:
Example Consider
;rlrlc 9. 1.3,
u*lu*1
,,r 'lrrlrlc 9.1.5,
o,,t
Ml
(21) K\. =. I l. l7-s + 2 ln(nit')ltt'
q ( ?1" cf,r./,r
: o.5e
" -.
l
: 1.33, and a* : 2.98 m/s. Then, refering to the equation
Q:9.60 (Eq. l); J :71.83 (8q.2); G :591 (Eq.3);,fr
tt.74 (Eq.4); x, :2.63 (Eq.6); N(,fr) : 0.31 (Eqs. 7 and 5); C'zDJ.(f) :
I I I (llq. 8);x: : 4.34 (8q.9), M(z) : 245,OOO kg (Eq. l0); G : 353 (Eq.
I
: Gi
Cubhq*
'.1:l
a building with h : 200 m; b : 35 m; d :
0.175 Hz h : 0.01; po : 20Okg/m3; C. : 0.8; Cr : 0.5; and
('t, 1.3. The building is located in a townf (zo = | m, see Table 9.1.1). It
r', :rssumed p : 1.25 kglm3, and the fastest-mile wind speed at l0 m above
l,rrrrrrrd in open terrain (zo : g.Ot m, see Table 9.1.1) is UdlO) :78 mph.
Iirorn Eq. 9.1.9, the averaging time forthe fastest-mile wind speed is r =
l(r s, and from Table 9.l.2theratio Ua6lU = 1.25, that is, the hourly wind
'.lrcerl at l0 m above ground in open terrain is U1(10)
= (7811.25) mph =
'/ 13 rn/s and u*1 : 27.81[2.5 ln(10/0.07)] :2.24 m/s (Eq. 9.1.8). From
f') ln; nr
(18)G-r*&?
eol
.1.()\ ( l,let
ctz
W#(f
o
rol.Jt;t
:rlrlcs 9.1.5 and 9.1.6 are in principle applicable only
Nttrnerical
- M,(2rn,l2
Mr(2rnr) \
:;t
if nlhlU(h) > 0.1 and
for most structures. In practrtt'. they may be used even if these conditions are not met, in which case the
r,'srrlts obtained will be slightly conservative.x
f
u,lllU(H) =
Mr(2rn,)'
(19) X*..
bdp1,Q.)
' (-\--:'
ix,t
I r
_
ttu( ; wlt{ I l lt
rall
building with rectangular cross section.
\tpG/Ol + G/
[
r, \./, /
)out*
Cobhq*
u'*=
qr=,,(
n6)u.
,,r
.v,
x,'
x, :
:
# f, Me)22
l
FIGURE 9.1.1. Schematic of
TABLE 9.1.5. Equations for Estimating the Along-Wind Response of Buildings with rrr
Approximately Linear f,'undamental Modal Shape t9-l8l
Zs
n
wnYl
2.,1
/ ,),\ r'-l)Q:z(t-3)ln"h'/
"-l
\
(2) J : 0.18Q1
6ltQ2
(3) G :
I
' \ ptott'tltttt: sitttilltr to -l'ablc ().1..5. irlso
birst'rl on tlrt' wolk ol' 19-lul but adapttrl lo rrsc willr
,1'tlsl spt'etls ltrttl tltc ltowt'r l:tw rvirrtl s;xt'rl prolilr', is irrt'lrrtlctl irr llrc ('ontrrrt.rrl:rry 1o lltr'
Ui('li 7 ()5 Slrtlttl:ttrl l() 5l :rrul is ;tv:rl;rlrlt irr llrr'rlr:,kt'llt ol I I / 5l rrlrlrt'ntlt'tl kr llris lxxrL
'llt:,;tsstttttt'tlllIIlllttlel:tiltlrttlllrrl'rr,t,1r,,rrr,,1,,11,,y11,.ovr'l;rrlisl;trir.t,lpwirttlplirllt.lrsl
K,o,
1
t!,lltl
r'.,
:t
&*
, Scr'l
l.)\/r
.1..1. I ).
1l
I
Al oNo
wtNt) tit
litroNt-it
339
h
\
S
^1
s.,!l
+'lP
€
I
o\
a)
L
a
3
cet- l\5
dll+ c
<l-_
\
€51^
sc
i ;
i
.r'\ "lh - lX + dlr'< € +
-le
ilSSlS
Y s$lss$
='$
--.' vt\
t+
s
ll L)l -ll
ll
ll
"_"
ll
tt
6 ll ll
>-$ '* d J q u tf d g,xF
ra)
a.t ca
\o
+
F- oo o, o .i
*t
+qrt-vt!_!l
il
-Nt
i
-tve*t
ll
l<
6
C.l
N
v v!
()
q
a.l
c.l
.-Y
FIGURE 9.1.2. Schematic of point srructure.
o
: 16,333,300 kg (Eq. 72); q*: 5.55 kg/m/s2 (Eq. l3); x : 0.1g4
l4); o,:0.074 m (Eq. l5); u,:0.114 s-'(Eq. 16); & : 3.63 (Eq.
l7); G :2.46 (Eq. 18); Xpu" : 0.452 m (E^q. 19); dj : 0.058 mls2 1eq. ZOj;
Kt : 3.75 (Eq.2l); and X.u* : 0.218 mts2 1yq.2ZS.
fn
\
ll);
M1
m (Eq.
l.\
I
's
b0
co
(\
o
6
I
b0
\ .s\
rd
tl
3
\I
,61
lltt
A,\
ci
-bt
ul
u^
c'r
I
o- l'-i
+
i' s
- l.\ l\ =r's l\ -s l\)--_
U,.<f q,l;la-\lr;
| ,Slq
v
\O J.
\O O
+
H
^t
,ql
ru's u
n
O
I
3
\'
\
vl
o\
'c;
+
a
\o
o
c.l
a
rd
L
rd
\a
UG
+U
5
!J
p
_r !
{l x -'5
-i {l s ,,
\5
"5"':.-U
q<
c.)
$
n
ll
\o
n
6 jj__
6i r
l,G
rr
ll
ll
q
\
\O =U
F-go O\
e
s
oi
ll
d €
O
3g
9.1.3 Computer Programs for Estimating Along-Wind
Response
For certain structures the assumption that the contribution to the response of
the higher modes can be neglected may not be realistic. Also, it miy be of
interest in cerlain situations to employ micrometeorological and aerodynamic
rnodels different from those incorporated in the procedures of rable 9.1.5 or
l9-5]. In such cases, in lieu of those procedures, a computerprogram must be
used to estimate the along-wind response. The computation of the response
amounts essentially to the evaluation of the integrals in Eqs. 5.3.1, 5.3.2, and
5.3.7 through 5.3.15. computer programs have been developed in which suitable numerical integration schemes are used and in which the specified struclural, micrometeorological, and aerodynamic information is incorporated as
input or in specialized subroutines. A computer program developed by the
National Bureau of Standards is available on tape in [9-14].
9.1.4 Approximations and Errors in Estimation of the Along-wind
Response
Irl this section estimates basctl orr nurncrical calculations are presentcd of errors
itssociatcd with uncertaintios rcgirnlirrg ce:rtirirr lbllurcs ancl paramctcr valucs
ol'lhc lnoclcls cmployccl. 'l'her t'rr['rrlirtiorrs werc r,irrlicrl oul lirr lhrcc typicll
338
tstJtLDtN(i!i: WlNl)
340
l()nl):;, i;lllt,(;ll,llnl
lll lil'()Nl;l , nNl) l)l :;l(iN ()l li()()l lN(i
TABLE 9.1.7. Description rll'lluil<lings St'ltclcrl its (last Stutlits
llt
B
Building
H
I
365
2
150
60
60
3
45
45
(Hz)
D
(m)
45
45
45
i
l) r,
u*
QM
kg/rrrr
0.10
0.20
0.01
150
0.0r
r50
1.00
0.01
150
O
ti
a
il
,a,
.l
I
(t)
I
6J
-:-
u!
UT
wind speed
0.07 m) was assumed to be Uv :
buildings selected as case studies and described in Table 9. 1.7. The
at l0 m above ground in open terrain (zo :
7-5 mph, wherc U7 is the fastest-mile of wind.
I
f*
F
(-)
Contribution of the Higher Vibration Modes to the Response. The root
ol'thc lluctuating deflections and accelerations were calculated for
2 in open and town exposure. The assumed modal shapes in
I
ancl
builclings
are similar to those represented in Fig. 5.2.1. The damping
modes
three
first
the
ratios were assumed to be fr : h: f: : 0.01. Calculations were carried out
: 5.
separately for the casas n2ln1 : 1.2, n3ln1 : 1.5 and n2ln1 :2.5, ry|ry
The contributions of the higher (i.e., of the second and third) modes of vibration
to the response are listed in Table 9.1.8. The contribution of the cross-mode
product was also included in Table 9.1.8. This contribution represented about
half of the amounts shown in columns 1 and 5 and was altogether negligible
in all other cases.
mean squarc
C!
"J
(a
il
{i
0)
z
expression
9\O
oo O\
crr)
tr-:f
"?
lt
s\
--.:
n6l
a
o!
o:/
tl
4)
L
Oa
()+
il
q
lnfluence upon Calculated Response of the Deviation trom a Straight
Line of Fundamental Modal Shape. A convenient means for estimating the
influence upon response of the fundamental modal shape is provided by the
oc!
E
(.) c.)
C)
r
o
o.
X
o,_l*^yI2aO
l*7-lax
IJ]
a)
(9.1.10)
a
.l
derived by Vickery [9-13] on the basis of the assumptions that the power law
(Eq. 2.2.26) holds and that the fundamental modal shape is described as follows:
Q
0)
lr
x(z)
: (;)'
(9.1.il)
il
ON
OJ
*N
"?
ll
s\
n_I
s\
(.)
h@
i
€
o\
where .y is a constant. In Eq. 9.1 .10, o" is the rms of the fluctuating deflections,
x is the mean deflection, Q is a function of geometrical, mechanical, ancl
environmental parametcrs, independent of "y. It may bc assumcd, roughly, that
d can vary bctwccn 0. l0 krr ()pcn cxposurc and 0.40 lilr ccrttlcrrs ol'largc citics.
It Ioll11ws thcn l!rlrr l;.q. 9.1.10 thal lirr ry =. 0. l0 tlrt't'rtlt'ttlittctl rittitls o,/.r
rd
bo
Fl
F
*
a..l
FA
34t
tluilt)tN(i:; wtNt) l()nl ):;. i;lltll(.lllllnl
342
calculatcd assulning
lll
i;l
,ri'
'{)N:;l nNl) l)l :;l(iN ()l ll(){)l lN(;
? : 0-5 irrrtl 7 1.5 tlilll'r by llxrtrl l%' ll'orrr llr:rl
7 : I (i.c., a lirroar llntlarrrcn(al rrurtlitl slttPc). l'ot'rv
Across-Wind Correlation of the Pressures and Along-Wind Response. It was noted in Sects. 2.3 and 9.1.3 that uncertainties subsist willr
regard to the actual values in the atmosphere of the exponential decay coclli
cients C, and C.. It is therefore of interest to estimate the errors in the calculalt'rl
TABLE 9.1.9. Ratios
It, lilr Cl - 4, (', (r..1 (t'rrst. (r). lrrrrl lil. lirrrr. ilrlel,rnc:tlitrlc
irr which
( ' (".wcrc asstllllt)tl t'itlrel tottsl;rrrl llrlrrrglroul tlrc licqucncy cascs
rangc (casc 4)
Exposure
Opcn
I
'l'own
Building
Building 2
Opcn
Towrr
(
)pcrr
'l'own
f,,,
l.(x)
I .(X)
l.(x)
l(x)
0.9r3
097
0.91
o
(,,1
f,,,
0.10
o ()\
0.()
0. l9
o.t)
l
O (Xr
O.(Xr
o()I
o()I
o 8/
f,,,
l(x)
ancJ
beyond
('hangcs in the valucs <ll'C,, anrl (' in thc lowcr-f-requency range
were found
trr lravc little efltct on thc rcsponsc (cascs I ,2, and 3). lf for frequencies near
:rrrtl beyond the tundamental fiequency the values of these parameters are
c,
6.3, Cr: l0 (cases 4 and -5), the total response is approximately 5% to
It)% higher than if C. : 10, q : 16 (cases 1,2, and 3); however, rhe
:r,'t'clcrations increase in rhe case of the taller buildings by 20% to 4o%. If c,
4, ct: 6.4-a situation that may be encountered in moderate winds such
.rs occur during full-scale measurements
of tall building response-then the
r()tirl response is about l0% to 2o% higher than in the case c. : lo, cv :
| (r. while the accelerations of the taller buitdings
are higher by 30% b aon .
I'lrc significant dependence of the exponentiar decay coefficients upon wind
',1rt:cd reflected in Figs. 2.3.5 and2.3.6, and the sensitivity
of the along-wind
;rt'cclerations to variations in the values of these coefficients suggest that caution
r:; in order in the interpretation of full-scale building acceleration measurements
;rntl the extrapolation of results based on such measurements to design situaI l(
)nS.
11.2 ACROSS-WIND RESPONSE
I .ll buildings are bluff (as opposed to streamlined)
bodies that cause the flow
l. 1;nd".ro separation, rather than follow the body contour. Depending upon
()nclitions discussed for certain classical cases in chapter
+, tie wake flow
tlrrrs created behind the building exhibits various degrees of periodicity,
ranging
(
lrrrn
virtually periodic with a single frequency to fully turbulent. In each of
tlr('sc cases' at any given instant, the wake flow is asymmetrical (e.g., Fig.
I 1.3). The across-wind response is due principally to this asymmetry, although
tlrt' lateral turbulent fluctuations in the oncoming flow may also contribute to
t lrt' :rcross-wind lorces.
lixpressions based on first principles for estimating the across-wind response
,l tall buildings do not currently exist. However,
lr;rsctl t)n such infbrmation hrrve
.l
:0.01
:
'rr lo havc ltlwcr vltlttcs ltl low lt'ctqrrcnt'ics:rrrtl lrighcr valucs near
tlrt' lirndamcntal li'cqLrctrcc rr, (cust:s 2, -3, urrtl -5).
empirical information obt:rilrcd fkrm wind tunnel measurements is available concerning the across-wind
r( sl)onsc of tall buildings not sub.jcctctl to inteference
effects, and expressions
[X",o*1y,,/[X",uJo.or
Building
343
wilttl l('slx)lls(' llt;tl t or,'sP,rrrrl lo lxrssilrlt' r.uors ln lltt' v:rlrrt's ol lltcsc
l)irtilttlctct's. 'l'lrt' lrlonli lvrrrtl lt.:;1xlr:;t' ol lruiltllrrl,s l. .1, lurrl .! ilt oPerr trrrtl
l()wil cxl)()sufL:s w;rs llrt'rt'lorr'r':rlr'rrl;rlt'tl st'p:rr:rlt.l-y lor'('
lo. (',, l(r (casc
nificant effect upon the calculated ratio o,/x.
Spectra in the Lower-Freguency Range and Along-Wind Response. ll
was shown in Sect. 2.3.3 that no universal relation exists describing the shapt'
of the spectral curve in the lower-frequency range and that this shape appean
to vary strongly between sites and between atmosphere and laboratory. 'l\t
estimate the effect of this variation, the response of buildings 1, 2, and 3 (scc
Table 9.1.7) was calculated for open terrain and town exposures, using lrr
expression for the lower-frequency portion of the spectrum of the longitudinirl
velocity fluctuations that depends upon a parameterJ,, as in Eqs. 2.3.25. Ratios
[X-o*]y',,/[X."*]0.c,: of the peak response calculated by assuming various valucs
J, to the peak response based on the value l, : 0.03 are listed in Table 9.1.9.
The results of Table 9.1.9 show that the dependence of the peak responsc
on the shape of the longitudinal spectrum in the low frequency range is rclit
tively small, particularly for taller buildings.
It is also noted that as indicated by Eq. 5.3.41 the influence of the spectrrrl
curve shape in the lower frequency range upon the value of the accelerati<llts
is negligible.
,(tl{:,t
,rloltp.
calculated assuming
0.4, the corresponding diff'erences arc about 3'/,,. lL is thus scctl lhll ttttltlcr:tlt'
deviations from a straight line of the fundamental rnodal shapc havc rtrr irtsig.
tnfluence upon Calculated Response of Errors in the Estimation of the
Roughness Length. To estimate the magnitude of the error associated witlr
unccrtainties regarding the actual value of the roughness length, the responst'
<rl'builclings 1,2, and 3 was calculated for coastal, open, suburban, centcr ol'
lowr.r, irntl con(cr of large city cxposures. The zero plane displacement was irl
Irll crrscs irssrrrrrod t<l hc zcro. Thc calculations showed that the sensitivity ol
llrrr lc:srrlts (o cvcn llrrgc crrors in thc estimation of the roughness lengths (e.g.,
50%,) is lrtlcrlbly srrrall (abovc l0%).
n(.ttrl,t.wllJlr lil:,1
l(x)
I
bc:cr11 1111lp1;1;cd
in thc literaturc. Dilltrcnt
t \l)lcrisions arc applicablc ltccolrlirrl', (o wlrcrhcr ()r nol thc rrns vrrluc ol'(hc
:rttrtss wintl tlscillltlirllts irt (lrt'tip ol (lrt'lrrrilrlinll. (,\, t:xcocrls tr r'ri(it';rl vrrlrrt'
'r',,- ll'o, ) o,,,,, l0ck-irr clli't ls l)( ( ()rlr(' :,r1,rrilrr';rrrl. lrtttl llte :tt'lrrss wirrrl lolrrls
.rrr<l
rtst'ill;rlitltts ittt'tt'itsr' rts llrr' \\'rn{l r,l)( ('rl:, r!rr'r(.:l\(. Slrlr'lgli's slrpllll
so llr:rl lot'k irr t'llt't'l:; rl,r r,,r rrr( ur (lurrrr;, llrt.rr :rnlii.ip;rlt.rl lrlr.
'lt'si1'1;1'1;
lrt.
i
l
i'
344
llult-ulNGS: WINU tOADS. 5i1tll,(:ltlllAl
lll r;l'()NSt,
ANIJ DESI(JN
For square tall buildings, oxpcrinlclll;i rr.rpoflctl in l9-20, p.
suggest that it is conservative to assutttc ilrat
? = o.ott
(oPen terrain, zo
T = O.ort
(suburban terrain, zs
trs!
b
= 0.045
(city center, zo
=
=
Ol ll(X)l
lN(i
1).1' A(;llOt;liwlNt) nt
ttll antl l9-241
a
@
:
JA:,
a
h
a
1 m)
2.5 m)
horizontal across-wind dimension of building.
where b
these ratios are largely tentalive.
A
It is emphasized
a
thut
(worst
direction)
r/A
6ycr. Several expressions for estimating o.v are
available in the literature. In all these expressions, the wind is assumed to blow
from the most unfavorable directions (in the case of a square building, normul
to a building face). Vickery [9-251proposed the expression
gyo),(h)
JA
:
^l u(t lf' t
'l;lil
:
(Zrn)2or(h)
(e.2.21
teristics that do not differ drastically from those shown in Fig. 9.2. 1.
The Supplement No. 4 to the National Building Codc of Canada l9-l2l
proposed an expression that may be written in thc fonrt
=
ttiltxttt')
I
ti, i;,,0.(x)5e
=
v
4.1
JA:,
h
3.4
2OOkg/m3
FIG[]RE 9.2.1. Characteristics of
rt(ttl l,'
tt Jlulltt'
I
0.2,31
models
tested in the wind tunnel [9-251.
(e.2.1)
f,,;
Equation 9.2.7 is based upon measurements of the response of building modcl$
with a linp4r fundamental modal shape and with geometric shapes, slenderncsn
raios JA/h, densities, and dampingratios shown in Fig. 9.2.1. h is notcd in
[9-25] that the use of Eq. 9.2.1 should be restricted to buildings with charac.
o,,(ttt
h
f = 0.0t
p
where or(ft) : rrns of across-wind oscillations at top of structure, gy : Pcak
factor expressing the ratio of the peak response to rms response (8, = 4.0), h
: height of building, ,4 : cross-sectional area of building, U(h) : mean wind
speed at the top of the structute , fl1 : fundamental frequency of vibration, f1
: damping ratio, p : air density, pr : bulk mass of building per unit volumc,
n and C : constants determined empirically from wind tunnel measurementi
(n :3.5, C: 0.0006 + 0.00025). The rms of the accelerations at the top of
the structure, or(h), can be estimated by using Eq. 9.2.1 and the relation
or(h)
Pn
l
JA:,
m
1
4.2
h7
VA
Structures for Which 6,
345
+ @
+
o.o7 m)
=
lil\)Nlit
where b : across-wind and d : along-wind dimension of the structure. It can
lrc seen that Eq. 9.2.3 is similar toEq.9.2.2 (where o, is given by Eq. g.2.r)
trxcept that the exponent n : 3.5 is replaced by n :3.3, and the coefficient
(' : 0.0006 + 0.00025 is replaced by c - 0.0006. Equation 9.2.3 is based
on measurements on models similar to those described in connection with Eq.
e.2.1.
we note that unlike [9-25] and (9-121, which do nor differentiare among
Iruildings with the shapes shown in Fig. 9.2.r, [9-74] and
[9-g1] report wind
Irrrrnel test results according to which buildings with square cross section have
ir rnuch greater across-wind response than circular buildings or square buildings
with chamfered comers. Reference [9-81] contains detailed results on wind
e ll'ccts and their dependence on wind direction for buildings of
square cross
scction with and without chamfered corners or spanwise openings.
Expressions in which measured modal force spectra are used follow from
l;.t1. 5.3.32.If Se(n)
across-wind modal force, and the notation
:
orl n,b I
ngefnptU(h)l
'lu&rl- ppuruW
(e.2.4)
is uscd, Eq. 5.3.32 becomes
o,(h)
=
;r:i::,,
,, n,,',r,r,1,
r,u)ttt)hh
f
().2.s\
lll :;l'()Nlil , nNll l)l l;l(iN ()l ll()()l lN(i
llt,ll lllNcs wlNl) l()nl):l, iilllllclt,llnl
tlis(ributccl ovcr llrc bLriltlirlg hciglrl,
that the building has a square shapc in plan, ancl that tho llntlantcrrt:tl rtt<xlrl
lf it is assumed that thc nrass is unilirlrrrly
shape
in linear, then
M, :
(9.2.tt1
\o6bzh
bb
!
O
t<
(Eq. 5.2.6), and
qq
NI
=...=.=.??q?=.?=:.*:
-qen\.1r-s)-oc>no---no-^i
:ta)lnOOo-@-@-€
.tdN--dNN__N_
a,(h) :0.0337
I
I
utn
,e
12
I
ol
--bY
Qn \r
.i
(9.2.1)
oi
rrNN--NNN-J-:
Noto thrrt thc quantitics
i:
I
)-
+r
I
#l''
:=
a
->- -t--.:::
^:
and
. N-nNO€-Qq-rc)d
i = 4.45 x ro 3l#)"
(9.2.e)
of f
based on measurements reported
-€hc>^i
-----N-
*.
-vlq
+
o-
=t
- o 6io
roi
ON
aq
nr
t- -'
-.'--SN6O--O€^OO
r+r;od6ddri6.i+6G;
!
r) d+
I
r€
tl
s{
':oq-5.)-bnon*\-boo
O,O.O^i Os-i .+sssO>+
TT
++
r' 7 r :o :i- r. @
- o o * o * -i -i
a--
N
o
!
oc;
OE
.2A
>q
o€
n€od
N
-oooooc)€oboro€
o-,$€rocjr+Adqjd<t
':e=CLr,cc=bcc.=oooonooi
-io.o^i
N_N+___
-<rN*
xE
€
nog
nr
*r
-cEqEtgoQinonno
o6onnesqidoodrodri
a9
.E
9po
oE
aY-
sfr
ON
ll o
r:
-j-: r
rr
o
ol
,!
ss
\n
nro,
1,
i
-:t"leon€n-b>6.o_
b -a,c-ncn.rd6-;
\!
'1'.. f:q "-' -: q ql1
r,n€a>n+o€.or;
!
il
i.;
--cj -i ++
!
!c
ro
:.r"9ql-:enqE-:,:qq
:rr-ornr€@r@c)r
v
+t
s4
dr;
@
{
d
the case because in rougher terrain the turbulence intensity is higher, which irt
tum causes the across-wind force to have a less peaked spectral density (Fig.
9.2.2), as well as a decreased coherence in the spanwise direction. Note tha(
there are significant discrepancies among values oi i obtained by various rc
searchers. For example, for urban terrain, nPlU(h) : 0.105, and b/h '
118.33, f : O.tO according to 19-261, versus t : O.ZZ on the basis of dalil
from 19-271; for urban terrain, nlblU(h) : 0.105 and b/h : ll4, i : O. tt
zccording to [9-281, versus i : O.l5 on the basis of data from [9-27]. Dilf'cr
ences between values of lgiven by Eqs. 9.2.8 and9.2.9 and those from l()
241, [9-26], t9-271, and [9-28] are also relatively large in several instances.
Numerical studies [9-29] show that, as in the case 0f along-wind responsc.
the contribution to the total building deflections ancl itccclcntlions of t.t.totlcs
higherthan thc funilarlcntal rnodc is ncgligiblc in prirr'tit'r', ttltlcss ll'tc rlt(itls ol
natural frcqucncics in lhc highcr rnodcs to lhc lirntlirlttcttlltl l'n't;trr'ttt'y tttc clost'
^ ^
*_*__N:
:-
for square building models itl
in
Table 9.2.1. Also included in
listed
are
and
[9-28]
19-241, 19-261,19-211,
Table 9.2.1 are values of f given by Eqs. 9.2.8 and 9-2-9.
Table 9.2.1 shows that for any given nlbl(t(h), f is a function of terrain
exposure and the slenderness ratio b/h. For example, the peak values of 7 firr
urban terrain appear to increase by a factor of approximately two if b/h clc"
creases from l/3 to 1/9. Also, according to data from t9-241, 19-271, antl
t9-281, the peak values of lincrease as the terrain becomes smoother. This is
Values
^ '.::':::-::
* ^ ^5-
'-qqEr,:qqEqeoo
' j --
a
a
.;
9r:
oh
io ir o o
E.
't
^rio n * o nFonn€on^i
^i ++,+ri ri ci
-,rrOrd-Ai€n@€O
___N<_N'-;
(9.2.8\
{a
il
t..=-b=cc.=tcco=
14.45 r r.8o)'o-'
tr
o
il
c.r q
o$
n
Ncr€.FNOOrcF-n&Orc
-+$6++^i^i^i-;d+doi
,:t=bcc==L=:o:
d.--hON€-OnO$@
implicit in Eqs. 9.2.1 and 9.2.3 are, respectively,
a
-joi 6i +i^i
rr
f f
$+
o
-:
6
{?-9:ta,-o.o'+m+.i-i
-o
.'o
$ fqNN
6 6
N
o
Qc
6.:
!
i
€
HE
t'J
q
EE
-OO
9?
"d
Lp
-i
--oo
-
q
Ftr
o
O<<
--= -&>e& tro!- .9e
34t
st
O
- fo-
348
LIUILDINGS: WIND LOAI)S, ri I lllJ(
)I
t,i
l^l lll
til,ONt-itr, AND
t)t. StGN
Ol [(X)t
tN(]
n(;t
tor;li wtNt) lil
tipoN$E 349
l lrc tttcarr lrourly wirrtl sperccl at thc top ol' thrr lrrriltling is thcn [t(h) : 2.5u*
lrl(/,/it)), or u(lt) -. 2.5 x 2.98 x ln(2(X)/1.(x)) .tt).4 rn/s. Thc fbllowing
1.0
n,blulltl:0.155: h/h ' l'15.7: i' = 0.075 (Tableg.2.l,
o, : 0.23 m (Eqs. 9.2.7 or 9.2.5); oi, : 0.2g m/s2 (Eq.
tj .'.2). Assuming that the peak t'actors arc gv 3.-5,.g0 4.0,
it follows thit
=
llrr' pcirk across-wind response and accclcration arc L,"* : 0.805 m, l.u^
=
f.rrrlrs irrcObraincd:
riulrrul)iu.r tcrrain),
I I
I
rrr/s2. These values are larger than thc corresponding values of the alongrvrrrrl rcsponse calculated previously, that is X-o^ = 0.452 m andX_*
= 0.21g
rrr/s'.
lrigurc 9.2.3 shows the mean and rms along-wind response and the rms
*t ross-wind response of a 1/400 model of a 64-story building in urban teq4in.
'I'lrt' characteristics of the model
:
were the following: h : 0.658
(l 154 ttt (where,4 is the floor area), n, : 8.3 Hz, n, : 8.49H2, and
^, l"Ji: l,
0.01 (where n and I denote frequencies and damping ratios, respectively)
I'l l0l. Results of wind tunnel tests for the model of a 53-story building are
rlrrrwrr in Fig. 9.2.4 for open and urban terrain t9-13].
I
101
0.01
0.1
|
nb/U(h)
FIGURE 9.2.2. Shapes of f2 curve in open and urban terrains. After A. Kareem,
"Across-Wind Response of Buildings," J. Struct. Dlv., ASCE, 18 (1982), 869-887,
I
hl
1oo
l" l!'
nl#
to unity, that is, substantially lower than those occurring in typical high rise
;l
buildings.
For a square building model with height to width ratios h/b : 8.33 located
in urban terrain, it was found in19-26] that the across-wind response decreaser
from the maximum value that corresponds to wind normal to a building face,
to about 5O% of that value when the angle between the mean wind direction
and the normal to a building face is about l5o. The peak across-wind response
and the peak along-wind response induced by wind parallel to a diagonal ul'
the cross section have approximately the same value; that is, they are appn)ximately equal to 0.8 times the peak along-wind response induced by wind
normal to a building face 19-261.
Numerical Example The building considered in thc trtrnrcricll cxamplc of
Sect.9.l.2 is again assumcd to bc actcd upun by wirrtl colrrrslxrncling to u
fastest-mile spccd al l0 nr abovc gnluncl in opcn lerririrt l//(10) , 7ti rnplt,
Ilt
5
:'I
or/h
2
10
I
1
5
235J1015
U(1.8 h.l
n,,1f A
l'l(,illlll'l 9.2.3. Mcln
rr'trrrl
itkrttg witttl tttttl rrxrl rlrrrrr s(liliuc ol irkrng-win<l lntl lcnrsstlt'llt'cliotts ttl it (r4-sloly ltrrilrlirrg rrrork'l wrtlr tr t'ilt,rrlirr slrapc irr plln l9-.101.
350
BUILDINGS: WIND t.OAlJS.
Slllti(;lt,llAl
lll t;l'ONSt
ANIJ DFSIGN
(]f Il(X]I
!,
ING
it l()lilil()NAl ilt lil'()Ntit
351
'll)
Along-wind
-A>-----
*------2
.r'
--r
/
l()
ry ll',,,i',
/'
I
= 0.01
/;r
Urban exposure
-./Across-wind
5
Arons-win{--9
--/'
/' ../'
-r-'licross-wind
-- -2
r
*F't
i'1
= o'01
5
10
o o.oo17
.f)
v0.04
v0. 055
I
l)
4
lr'2
5
k
5
o 0.001 96
o0.001 92
.0.01
56
.0.0167
v0.0450
v0.0626
5
o 0-00147
o0.00250
.0.0180
o0.01 49
10
64
.o.o1iz
v0.0510
w.0423
vo.
15
oPen exPosure
o0.001
.0.014
t5
' I I ;l:l]ll
(a)
6----g-t'
10
- f], fl ,l
Wind speed ar 47O m above ground (m/s)
o 0.00
20
5
t5
U/f^B
u
o0. 001
/
J
o47t
f
30
50
40
Wind speed at 300 m above qround (m/se)
FIGURE 9.2.4. Ratios of peak along-wind and peak across-wind response to
mean
51015202530
along-wind response for a 53-story building model with a square shape in plan in urban
and open tenain [9-13].
5 t0
oO.00229
r 0.01
t5
25
o 0.001 90
.0.01
60
u/f0B
78
v0.0534
o 0.00234
Figure 9.2.5 19-751 shows the across-wind response in smooth flow, flow
over suburban terrain and flow over urban terrain, for prismatic buildings with
several depth-to-width and damping ratios. The model scale was estimated to
be about 1/600, and for all models the height H, the sectional area BD, aruJ
the density were 0.5 m, 0.0025 m2 and l2Okglm3, respectively. In Fig. 9.2.5,
fs, (J, and h*, denote, respectively, natural frequency of vibration, wind speed
at building top and root mean square of across-wind response at building top'
respectively.
9.3
TORSIONAL RESPONSE
Severe distorti<lns duc to the combincd cllccts ol'ilcl'()rir' wirrtl krittls and lorsional momcnts occurro(l rluring tho 1926 Floriclt ltttt'rit'tttlc itt lwrl Miarni high
.0.0133
v 0.0476
l)"
0
I
l)
40
510152025
u/
f
0r0203040
0+ 5 10 15 ?O
u
o./BD
smoot
h
open
/ f ^,/BD
U
terrai
25
l- 5,to t0, to , to20,
15
0
t/
n
f
'9 "/to
?5
a,/BD
urban area
lil(;t.lRE 9.2.5. Across-winrl rcsponsc of prismatic buildings (circles and triangle intlit'lrtc clamping ratios). Fnrln I'1. Krwli. "Vortcx Induccd Vibration of Tall Buildings,"
.l . Wirul l,)ng. Ind. Affrxl.,4l-U ( I9r)2). Il'7 128.
352
llt,ill)lN(iii WtNl) l()nl )li, lilllll(.1(lltnl
lll
l,l
'{rNlil nNl) l)l :;l('N t)l ll{)(
tr I
)l lN(i
rise structurcs, thc l-5-story llt:alty llrriltling, rltttl lltc l7-s(ory Mtrycl Kiser
(thc tli
Building t9-3 11. Both buildings hacl utrusuillly nitrK)w shapcs in plarr
x
rrr)''l'he:ir'
42
14
mensiois in plan of the Meyer-Kiser Builcling wcrc about
l'rantcs ol'
Structural Systems consisted of steel frames. The two transvcrse cnd
0'60 rrr
of
about
the Meyer-kiser Building experienced horizontal deflections
and
-0.20 m,
resPectivelY.
Following these incidents engineers became concerned with wind-inducctl
torsional edects, as shown by subsequent developments in the literature, irr
cluding a 1939 ASCE report that dealt with such effects in some detail [9-32'
mentioned irr
9-331 .\everrheless, wind-induced torsion of tall buildings is not
codc or
building
U.S.
in
any
rhc 196l ASCE srate-of-the-art repoft 19-341, or
muy
This
deficiency
starrtlartl tlcvclopccl hclirrc the ASCE 7-95 Standard [9-5].
wincl'
against
provisions
of
cxpl:rin wltll itppcll-s l<l havc bcen the absence
inclucctl torsion in f hc original clesign of the John Hancock Building in Boston'
which by virluc ol'its shapc is particularly sensitive to both across-wind antl
torsional cll'ccts.
Torsional cffects are due to the fact that in any individual building the centcr
of mass and/or the elastic center do not coincide with the instantaneous point
of application of the aerodynamic loads. Ad hoc tests simulating these effects
have-been conducted for a number of years on individual building models'
However, until recently, relatively little work has been performed toward thc
development of design information and analytical procedures for use by struc
tural d-esigners.
-by
A first attempt at studying
analytically torsion induced
ort
fluctuating wind loads was reported by Patrickson and Friedmarr
buildings
potentially usefirl
[9-35]. More recently, Safak and Foutch have presented
response ol'
torsional
and
methods for estimating the along-wind, across-wind,
of
sufficie nl
absence
to
the
owing
rectangular buildings 19-36, 9-371. However,
lirt
usable
presently
not
are
methods
the
infonriation on aerodynamic loads,
design purposes.
WinA tunnet and full-scale research studies of torsional response were firsl
reported in [9-26] and [9-38]. Reference [9-26] includes information on wintl
induced torsional moments in an isolated square building model having a heigltt
to width ratio h/b : 8.33 in flow that simulates urban conditions. Accordirrg
to the results of [9-26], torsional moments are largest when the mean wirrtl
velocity is normal to a building face. As the angle cv between the mean wirrtl
velocity and the normal to the building face increases from 0o to 45', the
torsional moments decrease from their maximum value corresponding to o '
0o to about 25% of that value for a : 45". Assuming that the mechanicirl
properties of the model are similar to those of typical high rise structurc-s' il
was estimated in [9-26] that, for a : 0", the peak torsion-induced respottst'
of a corner column is approximately 65% of the peak along-wind rcspotrsc
ol
corresponding to a : 0o. For ty:45", the peak torsion-induccd rcsptlnsc
corrcsptltttliltg
rcsp()nsc
pcak
along-wind
<tf
the
u .o-"1. column is ab6ut 15%
toa:0o.
()lt
Systcmatic wirrtl trrrrrrr:l slrrtlirrs corrtlrtc(etl rrl lhc Illrivt'r'sily ol'Wcslet'rt
I
r
'l
l',lr
)l
lAl
i
il ',1'r
rf
l',1
l;rtl() wel-(' sttlrst'tqttcrtlly tt'Potlt'tl irr l() l()l (o l() .f .tl.' 'llr,'r-,,' :,lrrtlrcs lutvt' lt'rl
l(' llr(' li)ll()wirrg crrrpilit'rrl t'llrtiott lol cslirrr;rlirrl' {lrc Pt';rk lr;r:rt' lortlrrt' /i,,,,1 {/(/r)l
rrrrlttccrl by wincls witlr spc:ctl U(lt) al tllc l()l)
T',,,,,,1U(h)l
:
{rl'i'
ltttttll
ol lltt'
lruilrllrrl,,:
I,q/'/,,,,,,1
t
t(h)lti
(9.3. r)
rvlrcrc ry' is a reduction coefficicnt thut is briclly cliscusscd subscqucntly, gr. =
I l"i is a torsional peak factor, and thc Iincar antl nrrs [-rase torque, T[U(hl ana
1 ,,,,,1U(h)l are given by the expressions
7 wr.tt)l
:
(9.3.2)
o.o38pL4 hn?ru?
I,.-l U(h)l = 0.00167
!Sr
ptonr'rU','r
(9.3.3)
u(h\
If
(e.3.4)
nrL
,
Jlrl
ds
(e.3.5)
At2
ln Eqs. 9.3.2to9.3.5, p is the air density (p = I .25 kglm3), ft is the height
()l lhe buildiflg, trr and f7 are the natural frequency and the damping ratio in
rlrr: fundamental torsional mode of vibration, ds is the elemental length of the
I'rrilding perimeter, lrl is the torque arm of the element ds (i.e., the distance
lrctween the elastic center and the normal to the building boundary at the center
ol the element ds; see Fig. 9.3.1), and z4 is the cross-sectional area of the
l,trilding. Equation 9.3.2 and 9.3.3 are based upon the experimental results
slrown in Figs. 9.3.2 and 9.3.3, in which the ordinates are the reduced mean
;urtl rms base torque,
f, : Tlbfahn?) and o, :
l,'l(
Tn,"(t/zl(pL4hn?i, respectively.
il ll{l,l 9.-}.1.
r'lltt'ttsttlls ol lltt'st'slttrlics rvtrc Lrr'll\ I'r"\rl,rl t,r tlr,
Notrtliorts
,rrrtlr,'r., lrY
l)l \
l:;yttrrrov
354
Fltlll l)lN(ifl:
wlNl) loAllli, $llit,(llllltAl ltl tit)oN$t , ANr) l)trit(iN ot lt(x)t
tN{i
lr
1o
o
6
l
a
cc
o
F
z
t!
10-
o
uJ
4
c
I
ul
2
F
2
l
a
(r
+
\
6
o
f
o
tlJ
4
1
I
)t
o 10 I'
F
a
- '/.
(E
lo
I
4
2
1o-1
o uioo
ur
-
UJ
tr
6
10
-3
8
6
I
/
2
-4
2
to-l
FIGURE 9.3.2. Mean base torque for tall buildings with various shapes in plan (courtesy Dr. N. Isyumov, Boundary-Layer Wind Tunnel Laboratory, University of Western
Ontario).
<d
TH
la
4
10
::::l
"l
4
b
REDUCED VELOCTTY
N
""
I
A^Q
*'/
sc
o 10'
' o u1o, ' o ufo,
R
A
;:;::1"
10-s
I
la
l
EH
OE
|-
6
PS
rG
o
uJ
o
I
10'
s\
8
6
P*
2
HH
100
s-\'
ul
EF
4
HH
8
355
'u
FH
100
ir t oilr;toN^t ilt r;t 'r )Ni it
/
/
I
a\
OE
53
I
2.7
;:;i'.r'
4 AA
2
4 6a
2
10 1
Oo
UT - REDUCED VELOCITY
4 6A
1
1O2
lrl(;URE 9.3.3. Root mean square of base torque for tall buildings with various shapes
rrr plan (courtesy Dr. N. Isyumov, Boundary-Layer Wind Tunnel Laboratory, Univerr.rly ol' Westem Ontario).
The torques 7 and f-. are each induced by wind with reduced speed U, ancl
with the respective most unfavorable direction. In general, the most unfavorable
directions forTand 7-, do not coincide. In addition, in most cases neitherol'
these directions will coincide with the direction of the extreme winds expected
to occur at the site. For these reasons, the coefficient ry' in Eq. 9.3.1 is less
than unity. It is estimated in [9-39] that 0.75 < t!
= I in most cases.
The peak torsional-induced horizontal accelerations at the top of the building
at a distance u from the elastic center can be written as
^
AU
2g7T^.u
=- p6bdhri,
(e.3.6)
where 0 is the peak angular accclcration and r,,, is thc rirrlirrs rll'gynrlion. For
a rectangular shapo with unilorrn bulk mass pcr unil volurrrc
&
b2+d2
(e.3.7)
12
lirl. 9.3.6 was obtained in [9-39] assuming a linearfundamental modal
of vibration.
shape
;rrrtl ncgligible contributions by higher torsional modes
Numerical Example For the building considered in the numerical examples
rrl Sccts. 9.1.2 and9.2, h : 2OO m, b : d : 35 m, U(h) : 39.4 mls, p6 :
.l(X) kg/m3. It is assumed that the natural frequency and the damping ratio in
tlrc lirndamental torsional modc of vibration arenr:0.3H2 and f7: 0.01,
rt'slrcctivcly, and that thc air clonsity is p : 1.25 kglm3.
lirrrrrr Eq.9.3.5. t,: tl(ltl2l{ol4)lb :35 m. Then U,:39.41(0.3 x 35)
.1.75 (tlq. 9.3.4),-rpv.+1 l.11 x 107 Nm (cq. 9.3.2),7,,,,. 139.41 : 1.95
. l0/ Nrrr (lic1. 9.3.3), 7,,,,,, r).2 r 107 Nrn (liq. 9.3.1 in which it is assutttctl
356
rl,
tll,lt t)tN(i:i wtNt) l()nt):;. t;ililt(:l,ilnt
ltt
:,t
'()N:it , nNI) t)t :;t(iN ()t tt(x)t tN(i
= 1,8't :3.lt).'l'hc llcrrk l()lsi()rr rtttltttr'tl ltolizorr(lrl rrt't'clcllrlion ill tlte t()l)
: 35 x ,1212 - 24.1 rrr; is /,t,,,.,,1, -. 0..17 nr/s'. N.)tt. lltirl llris
corner (e,'
exceeds the peak along-wind accclcralions but is substarrtially lcss lhirn tlrc
peak across-wind accelerations calculated in thc previous numcrical oxanlplcs.
According to wind tunnel tests reported in [9-82], fluctuating torsion tlcpcntls
strongly on building cross section, being largest by far tbr triangular builclings,
intermediate for rectangular buildings, and lowest for D-shapecl and diamontl
shaped buildings. Such dependence is not apparent from Fig. 9.3.3.
The peak combined effect of the along-wind, across-wind, and torsionlrl
loads can bc obtained by summing up vectorially the individual peak effects ol'
thcsc krads and rnultiplying the result by a reduction factor (e.g., equal to 0.g)
il
lllll l, Irl\Mt'l ll:i nNl) Vl:,(.()l ll\l;ll(
li/\Ml'lllri
I,l \/lr l .
,|at7
,'.rr{lrtlrltkc lolrtls lrrrtl l() i,nl)lx)rl lttotlttllrt sttlrslltttltttt':' llr.rl lrior,irL'lltt'tl:,:tlrlt'
I'urltlrng spirce.
1l-4-1 Tuned Mass Dampers
llrt' 'l'MD cclnsists o1'a rclativcly srrurll vilrrrtoly sysl('nl (nlrss, splirrg, rrrrtl
,l.rslr1xlt.) attachcd to a structurc whosc viblrrtiotts it is tlcsigrrctl to rrritigatc. It
rr':rs invcntcd in 1909 by Frahm itntl hlts trntil rcccntly bccn usod prirnarily in
rrrt't'hanical engineering systcms. ln thc last dccadc TMDs have increasingly
Tuned dampers consist of a mass, usually of the order of 05% to l% <tl'
the total mass of the structure, that is added to and interacts dynamically witlr
the structure. Inherent in or attached to that mass is a system that dissipatcs
energy during the relative mass-structure motion. Active controls may be usotl
in wind-sensitivc structures, including the Centerpoint Tower,
lirrlrrcy, Australia 19-441, the CN Tower, Toronto 19-451, the John Hancock
! .rryg1, Boston (equipped with dual TMDs designed to control both torsional
,rrrtl lateral motions) 19-46, 9-471, and the Citicorp Center, New York City
l') 113. 9-49, 9-501. Generally, the purpose of the TMDs is to reduce building
rrrot ions insofar as they affect occupant comfort, and the effect of the TMD is
!r()r (aken into account in strength calculations [9-46,9-41].
A schematic view of a TMD operating on the top floor of the Citicorp Center
r:, slrt)wn in Fig. 9.4.1. The mass of the TMD consists in this case of a400r.n q1tn.r"," block bearing on a thin oil film. The TMD structural stiffness is
I'r,rviclcd by pneumatic springs which can be tuned to the actual frequency of
tlr,' building as determined experimentally in the field. The TMD damping is
1'rovided by hydraulic shock absorbers. The system includes fail-safe devices
r. l)roVert excessive travel of the concrete block [9-49]. Additional information
,'n l'MD equipment and control systems is given in [9-46].
'l'hc theory of the tuned mass damper was developed by Den Hartog [9-51]
lrrr tlre system shown in Fig. 9.4.2, with Cr : 0 and a harmonic load F(l).
( lrr thc basis of results given in
t9-5 11, the theory was subsequently extended
rrr l() -521 to include the case where C1 * 0 and F(r) is a random load with
, rrrrstant (white noise) spectral density. In Fig. 9.4.2, Mt, Cr, Kr, and M2, C2,
A , rrlc the mass, damping, and spring constant of the structure and of the TMD,
r, slrcctively.
proposed which by various means ensure dynamic interaction (e.g., springs or
I
which uccounts lirr thc fact that, in general, individual peaks do not occur.
sitttultttttgrttsly. Il'llrc cornhincd elTect so calculated is less than an individr,url
cll'cct, it is tlrc lltlcr tlrat should bc considered in design.
Morc rcccntly, based on acnrdynamic data reported inIg-21, [9-66] presentctl
a rigor<tus dynarnic analysis of torsional moments which takes into account thc
of the distance between the elastic center and the center of mass of tht:
structure; see also 19-831 .
el1'ect
9.4
TUNED DAMPERS AND VISCOELASTIC DAMPING DEVICES
Two main types of device have been employed for the reduction of tall building
vibrations in translation and/or torsion: tuned mass dampers, and viscoelastic
dampers. Devices intended to reduce torsional vibrations must consist of at
least two units located at sufficient distances from the elastic center of thc
structure.
to improve performance 19-76]. Several types of tuned dampers have becrr
pendular devices) and energy dissipation, and which may have more or lcss
elaborate control systems 19-161. In tuned liquid ilampers (TLDs) most of tht:
mass is due to liquid contained in a tank, and liquid motion provides or con
tributes to energy dissipation [9-71 , 9-78]. The latter may be increased hy
placing obstacles in the liquid's motion, for example, cruciform poles [9-791
or floating elements t9-801. In Sect. 9.4.1 we discuss in some detail the dcvicc
known as tuned mass damper (TMD), which illustrates the basic principle
common to all types of tuncd dampers. Section 9.4.2 discusses viscoelastit'
lx ('n cmployed
llect of TMD upon Deflections and Accelerations of Structure. The
, llt't't ol'the TMD can be viewed as being equivalent to changing the damping
,,rrr,, ()l'the original system (not provided with a TMD) from the value f1 :
t ,l:J K tM I to a larger value f" Thus the deflections and accelerations of mass
,'1/, irr thc system of Fig. 9.4.2 can be obtained by calculating the deflections
.urrl rrccclerations of miss M1 in the system with dampinE Cn: ZJX,tt4y"
,i'wrr in Fig. 9.4.3. Using results fiom [9-49] and [9-52]it can be shown that
I
dampers.
For a recent rcvicw of'clarnping dcviccs firr thc t'orrlrrl ol wirrtl-intlucctl
ll3 1041. ltcl'clcncc l9 ll3l rliscrrsst's t'orr{rol ol vilrr.:rti6rrs 6l
tall builclings crlrrsisl irrg ol lrrcglrslnrelrrrcs tlcsil'.lrt'rl t,r rvtllr:,1:urrl wirrrl lrrrrl
vibrations, scc
\,,
2
rv1((troj rv11(rv.,rv
1
rvl) I
cYy)
- a1la]
cvl(pj
*
2cvr')
*
rrr
(9.4.1a)
rr l rt't t'
(().1 Ilr)
9,4
.:o
:
od
Eo
o6
.:
>E
I
r6
3t
I)
DAMPERS AND VISCOELASTIC DAMPING
DEVICES 359
OE
oo
=o
ff
o
IIJNI
I
o
dg
Mt
a
I
U9
c
o
z
I
>|
{)
ts
o
>'
D
g,P
€.:
Ee
z!6
U
FIGURE 9.4.2. Schematic of system equipped with a tuned mass damper.*
JI
o
B
o
z
@1
0)
g
3*
o
O
co
c
9g
o
o
8_E
gE
o8B
Q2
g
A3
U
Pr
D
o
c
o
((tf + b)
(9.4.1c)
t+fze+p)+4f(&z
2ft + 2fzf (l + p)
(9.4.ld)
2lzf
(e.4.rf)
(9.4.1e)
Itt lJqs. 9.4.Ia through 9.1.4f,
q
c
o
2f
ii
-,i.
o-t'
Fa
?t)
EF
3g
9e
s6
!o
g-
tE
>s
.ju
iE
o\ [l
frl
eg
rg
cq
aE
sb
og
A
&a
?n
Eo
8;a
oE
3e
6
E!
! o5
3E-
e:r
4
FIGURE 9.4.3. Notations.
+f iigrrrcs 9.4.2
t<t 9.4.6 arc rcprotluccd l'r<tn Engineering Structure.s, Vol.
tirissiolr ol' thc publishcr, lluttcrwoflh Scicntilic t_td.
358
4, No. 5, with
pcr-
360
BUILDINGS: WlNt) lOADri,
tilllt,(llllllAl
IIfSPONSF, ANI)
lll til(iN (ll llO()l lN(i
p:
MzlMl
(9.4.19)
f :
a2la1
(e.4. th)
:
',: JK1M, (i 1,2)
| : C/(2M;a) (i : t,2)
Forexample,if
col lArili(r DnMl|rN(i
p:0.01,/:0.98,
(e.4. li)
(e.4.
r.i)
:0.01,
J;
fln'=o+0.8fr >fl
Forexample, tf
p :0.01
and f1
(9.4.2)
&
(9.4.3)
2
:
0.01, then
flpt=
361
Displacements of TMD Mass. lrr dcsignirrg rr 'l'Ml) systln, alkrwancc lnust
lrt' lrtadc firr thc clisplitcotttcnts (travcl) ol'the 'l'Ml) nrirss. 'l'ltcsc clisplaccments
;rrc in practicc rclativcly largc. Frlr cxarrrplc, irr lher clsc ol'citicorp Center,
IMI) displacements induced by a stonrr with l l0-yoar rcturn period were
rslirrrated from model tests to bc on thc orclcr ol' 1.00 m.
Lct the displacement of the TMD lnass with rcspect to mass M, be denoted
r' (Fig. 9.4.2). The displaccrrcnt duc to resonant amplification effects only
ol nrass Mlinthe original system (shown in Fig. 9.4.5) is denoted by"r1,s. (It
is t'rrrphasized that.rl,e does not include contributions due to mean or quasistatic
lrxrtlirrg.) Using results from [9-49] and [9-52), it can be shown that the ratio
rrl llrc mean square values of x2 and -r1,s, denoted by xlno^, is given by the
rt'llrl ion
0.033
and fipt=
0.05,
;-:z
'.1nom
(dcnoted by fin'):
,rO,
12
-
ti
Iry
and f2:0.0515, then f,, =
0.03226. The dependence of f" upon f2 is shown in Fig. 9.4.4 for fr : 0.01,
/: 0.98, and various values of p. [9-49]. Note that for each pr there exisrs an
optimal (maximum) value of f" (denoted by l3p') which can be sought by
rcprcscnting 8q.9.4.1a graphically. Alternatively, it is shown in [9-53] that
with ncgligible errors, the following approximate relations can be used tbr
preliminary design purposes to obtain l?Pt and the corresponding value of f2
f1
t)t vt(;t
xi.o
zSPtt
a1(a2c"3-cr)-uso.!
if p1 :0.01, /: 0.98, fi : 0.01, and f2 : 0.0515, then
:
r1,,,,,,,,
13.7. The dependence of x22no^tt2 upon f2 is shown in Fig. 9.4.6 for
(r ' 0.01, "f : 0.98, and various values of p.19-491.
lrrrl cxample,
Ocsrgn of TMDs for Actual Structures. Because buildings are multirlcp,rcc of freedom systems, the model shown in Fig. 9.4.2 is not a rigorous
rt'prcsentation of a building with a TMD. The error inherent in the assumption
thirl the building equipped with a TMD can be represented by the system of
a
lJl
12345678910111213
(2 ("/.)
FIGURE 9.4.4. Dcpcndcncc of t. upon t2 and p. Allcr I{. .l . Mr'Nurrrirra, "'funerl
Mass Dampers in lluiltlings," .1. Strrct. Div., ASC|i. l0.l (l()77). l7t{5 179u.
(9.4.4)
l,'l(il llll,l 9.4.5.
Notations.
362
ll(lll I)lN(;1;
WtNl
) t()nt)li, l;ltt(,(.illilnt
ilt
:;t
,oNt;t nNI) t)t :;t(iN ()t il(){)t tN(i
il
I r rW I Il',1
lll
Jll I )ll'l(
i::
363
)vcrlrll tl:ttttpttt;r. rt'tltrirt'tl :rs lr Iutrt(iorr ol llrt' r,1x't llrctl lt('ln tccurfoncc
irrlr:t'virl ol'tlrc winrl lorrtlirrg (c.ll., lO or l(X) y(',us).
Irnvinrrrrrrcrrl clrirrlrc'(cris(ics itl (lilnllx't l(x':rll()ls (r..g., trir tcntpcrature).
(
lirr (liulpcr tlispl:rt't'rrrt'rrt, :rrrtl rctprisik: damper stiffness.
lir.cqucncics <ll' vibration ol' brriltlirrg (tlrursl:rtiorral and torsional).
S;lrrco availablc
llrt' tlatnpcr design includcs llrc sclcction ol'the material properties (shear loss
rrr'xlrrlus, loss tangent, and thcir toulpcraturc dependence), and the size and
rrrrrrrlrcrof dampcrs; see l9-67,968,9-1O,9-7 1,9-121 fordetails. Buildings
, rluippcd with viscoclastic dampers include the World Trade Center, New
\ olk. and the Columbia Center Building, Seattle.
(I,5
3 4 s 6
t'uf'
e 10 11 12
13
FIGURE 9.4.6. Dependence of
upon f, and p. After R. J. McNamara.
",*',,
"Tuned Mass Dampers in Buildings,"
J. struct. Div., ASCE, r03 (1977), 1785-l79tJ.
Fig' 9.4-2 (where Mr, Kr, and
c'
are equal, respectively, to the generalizccr
mass, the stiffness, and the damping in the fundamental mode of
the building
not equipped with a TMD) was estimated for a particurar structure in
t9-53 i.
According to the approximate estimate of t9-531, the simplified model
of Fig.
9 .4.2 led in that particular case to an overestimation
of the equivalent damping
ratio of the structure by a factor of about 1.2.
It is noted that results reported in [9-54] on the dynamic response of lighr
equipment attached to structures are applicable to thl study of the errors
in
herent in the model of Fig. 9.4.2. These errors are generaily negrigible
for
structures with ratios of frequency in the second mode to fiequency
in thc
fundamental mode of the order of two or larger.
9-4.2 Viscoelastic Dampers
viscoelastic dampers are passive devices that have the advantage of not rc
quiring constant operational monitoring and of not depending on eiectnc
powcr..
Like tuned mass dampers, viscoelastic dampers are used for acceleration rcduction only. The buiiaing damping they achieve can attain 4% or more,
antr
for very large buildings their construction costs were estimated to be about
0.5% of total construction costs l9-lo, g-jll.
The fbllowing l)ctors ncccl to bc considcrccl in thc tlcsigrr 6l-viscgclastit.
dampers*:
'l'l)crsorr:tl (rrrrtttrrrrir':rliorr
lry l)r lr M;tlrrrrrrrli, lM (.otilP;rrr\,. ljl l,.rrrl NlNt 1,Ii.,
LOW.RISE BUILDINGS
Itriltlings with relatively low heights are, as a rule, rigid and do not exhibit
'.ry'nilicant dynamic amplification effects.
As was shown in Sects. 4.6,4.'7,1.3, and7.4, wind loads on any given
.,, trrlrl building or model depend upon several factors, including the characterr'.tr('s of the oncoming flow, model scale, area affected by the wind load, and
r.rlro ol'openings to gross area of the building envelope. Recent work on the
, llt'cl of these and other factors was reviewed for low-rise buildings by Stath{,lx)ulos (see [9-84, 9-85] and references quoted therein).
ASCE 7-95 Standard [9-5] windloading provisions for low-rise buildings
.,r,' bused to a considerable extent on results obtained in wind tunnel tests at
tlrt'(lniversity of westem ontario and Concordia University. Despite the small
.,;rlc at which the tests have been conducted (usually l:200 to 1:2000), it has
l,t't'n the consensus of code writers in the United States and Canada that they
I'r,,r,itle a reasonable basis for codification, with occasional adjustments reflect,,r)' l'osults of full-scale tests or the desire to calibrate new provisions against
, rrsling practice. The tests have confirmed that the fluctuating part of the load
, .rrr in many instances be significantly larger than the mean loid and that, for
,ury givcn storm, peak pressures and the ratio between mean pressures and
lirrt lrurtirlg pressures decrease as the terrain roughness increases.
lir:sults on the influence of geometric parameters have been used to simplify
t;rrrtlrrrcl provisions. It was found, for example, that for buildings with small
lr, rlht-to-width ratios and length-to-width ratios of 1.0 to 3.0, the loads do not
rL pi'nd significantly on length; wind loads increase with building height but
tlr,' tk:pcnclcnce of pressure coeliicients on height is reduced if they are refer, rr, r'tl wi{h rcspect to the velocity pressure at the mean roof height; roof slope
'. rrrr irrrl'rotlitnt paramctcr l9 tt6l.
Wt' rtotc llutl stunclutrl lirrrrrlrls lrrc bcing devclopcd that wrtr.rlcl all<lw thc use
,'l tl;tlrt bltscs oltlrtittcrl lirrtr winrl lrrrncl lcsls, as opposccl lo thc usc ol'rlata
'',ttttttt:tt-it's, wlrit'lr is lypit':rl ()l ( ur('n{ sllrttllutls. Mlrny ol'(lrc silrrplilit'lrtiolrs
ir:,rttlt'rl lo itt t'tult'n( s(:lrrl:rrrl:. \r'orrlll 11,,',.'1i,11.n,, lorr1,,t.l i,,.n,.,..lr"rl, rrtslr';rtl
364
lll ,tt t)tN(iti wtNt )l()nl r:,,:.ltillr
tunt\t lt :;t'()N:,1 .nNl Il)l :;t(il,l ()t l(x)t
l) Itr",t ill(.lrrt tWlilt ) ll)/\l r:;
tN{;
of using conscrvativo otlvcl()l)cs ()l l)r('ssur('rllrl:r, tlc:sigrrcl's woul(l lt:soll lo (ltr.
more economical or risk-consistcllt ol)ti()ll ol'usirrg thc <lrigirral tlu(a corn.
sponding to the set of gcometric paranrcte rs ol'intcrcst. 'I'his issuc is tliscLrsscrl
in Chapter 17.
Tests have also been performed to obtain information on the cfl'cct of'buil.|
ing orroof configuration on the loads. Forexample, it was found that nega(ivr,
pressures are lower on hipped roofs (four-slope roofs) than on gable roofs (twO
slope roofs) t9-871. The ASCE 7-95 Standard incorporares results f'rom [9-tt7l,
as well as results on two-level flat roofs [9-88], sawtooth roofs [9-89], arrrl
multi-span gable roofs [9-90] (see Chaprer l7).
The influence of tributary area on loads can be ascertained by summing rrP
the sirnultaneous pressures (or pressures multiplied by appropriate influencc
cocfficients) at a sufficient number of pressure taps over the area of concerrr,
using thc pncurnatic avcraging technique t9-911. Recent progress in the devcl
opment of dcviccs capable o{' rneasuring local pressures and performig spatilrl
integration ol' pressures is rcported in [9-92], which describes a device witlr
length 55 mm, width 35 mm, and depth 25 mm, equipped with 32 pressurr
measuring ports whose frequency range is 0 to about 2OO Hz.
Architectural features such as parapets [9-93] and roof overhangs, both ol
which are accounted for in the ASCE 7-95 Standard [9-5] (see Chapter l7), as
well as eave details (i.e., whether roof and wall meet at a sharp angle or an'
connected by a curved transition surface) [9-94,9-95], were fbund to influencc
local pressures, in some cases significantly.
For a study of wind effects on mobile homes, see [9-l l2].
3(i5
rlrllt'tr'ltl oticttl;rliolrs I llrt' prl;xrst' ol llrr. rLk t orr:,r:,lt.rrl tlt'si1',lt PtrrcctlLtrc
lrtr'st'ttlctl irt tlris tllrlllt'l rs l() r'litrritt:rlc or rr'rllrtr'srrtlr lrorrrrrrilirrtrtitics.
'l'ltt: t'oltvcltliottitl ltrrrl (lrc lisk t'orrsislt'rrl
tlt':r1',rr prrrt't'tlures havc a number
.l r.t)illnlon s1c:1)s. 'l'lrcsc rrr.t: r.t:vit.wctl irr sr.t.t. ().(r.1, which also includes a
{l(:;('lil)lion ol'tltc slcps tlrirl tlislirrguislr llrc lwo plrcctlurcs. Section 9.6.2 sumrtt:ttizcs rcsults <ll'clcsign upplicirliorrs tlurt illustr-lrte the economic and safety
.rlv;ur(agcs inhcrcnt in (lrc risk consisrcnr pnlccdure. Section 9.6.3 lists a few
l',r:;it' rcf'crcnces on wind cllbcts ol'rrxrfing.
!1"6-1 Conventional and Risk-Consistent Procedures for Designing
(:ladding Glass
I'rrt'ctlures fbr conventional and risk-consistent design of cladding glass entail
rlrt' lirllowing common steps:
l. Obtaining information on the extreme wind climate.
Converting basic wind speeds (e.g., fastest-mile wind speeds at l0 m
above ground in open terrain) into wind speeds used for aerodynamic
reference purposes (usually, mean hourly wind speeds at the top of the
l
building).
Obtaining fiom wind tunnel tests information on the time-dependent aerodynamic pressures acting at various points of the building facades.
t. Converting the information on time-dependent aerodynamic pressures
into equivalent wind loads with standardized time history, that is, loads
whose effect upon the cladding panels is equivalent to that of the actual
time-dependent loads.
9.6
DESIGN OF CLADDING AND ROOFING FOR WIND LOADS
The main purpose of this section is to present a risk-consistent procedure lor
the design of glass cladding subjected to wind loads. The procedure is appli
cable to buildings with specified orientation and requires the availability ol
sufficient (l) wind speed data characterizing the extreme wind climate in tht,
region of interest and (2) aerodynamic pressure data obtained in the wind tunncl
for various zones of the building facades.
The procedure presented here differs from conventional design practicc irr
two respects. First, in conventional practice the design of each cladding pancl
is based on the requirement that the nominal wind load corresponding to ir
specified mean recuffence interval N lusually iv: so years) may not cxccr,(l
a load capacity corresponding to a specified probability of failure p7 (usurrlly
P/: 0.008). Second, in conventional design practice wind clircctionality is trot
taken into account. As shown in scct. 8.1 .2, this cun lc:lrrl to signilicant tlis
crepancies betwccn thc norninal loacls uscd in clcsigrr rrrrtl tlrt':rclrurl kratls. 'l'lrc
safety level ol'(hc cllrtltlitrg clrn llrcrclirrc: hc slnrrrl'ly norrrrrrrlirru :rrrroltg llrt.
vari<lus ztlncs ol'llrt'lrtriltlirrg llrclrtlr:s trrril t1rr9rr.11 ltlr.rrlii;rl lrlilrlirrl,s lt:rvirrli
(r.
Estimating design wind loads using information on the wind climate and
on the equivalent standardized wind loads.
Obtaining information on the load capacity of the cladding panels.
1
Adopting a design criterion relating the design wind loads to the load
tt.
capacity of the panels.
Designing the cladding glass.
lior. additional details, see [9-96].
lxtreme wind climate. The conventional design procedure uses information
of direction. To apply the risk-consistent
rlt'sign procedure, the information needed to characterize the extreme wind
()n oxtreme wind speeds regardless
,lirrralc in regions not subjected to hurricane winds consists of directional larg('sl
yolrly wincl speeds. Such information may be extracted l'rom rnonthly Local
Altlt'rtsttrt'olllrccllrtltlings:rli'lylcvt'llol;lzorrt lorlrrriltlirrll)isgivt.nbyllrcr:tliolr,/rr,,lrr'lrrlt.rr
llt('('xl)('('l('(l ttttlttltt'r ttl lt:tttr'ls llrrrl lrrrl rlrrrrrrl' tlrr' lilr'lirrrr'r)l llr(.slnr(lul.rrrrtl llrr. lpl;rl rrrlrrlpr
,rl ;r:trrt'ls lor llr:rl zolrr.(ol lruiltlrlrll)
366
rrt,'r)rN(i:i wrNr)
r()nr ):;.
rit*r( rrrnl
rrr :,r,()N:ir
. nNr)
r)r l;r(iN
()r ,()()l
rN(i
Climatological l)a(a sttttttttlttics issrrt'rl lr.y rlrt' Nirtiorlrl ()ccrirnit.
rrntl Alrrr,
spheric Administration (sce Scct. .1.4;.'lllre,sc clatl
arc usually rcc.rtlctl .vcr
open terrain (airports) and should be rccluccd
to a comrnon cllvati.n (usrurlly
10 m above ground).
In hurricane-prone regions directional information
on hurricane wind spcctrs
can be obtained by Monte Carlo simulation (see
Sect.
stored
3.3.2) or tiom crat,
in [8-9] (see also [3-71]).
conversion of Basic spe^eds to Aerodynamic Reterence
speeds. Givcn
uf(r},0) (i.e., theiastest-mile wind from direction g ar
l0 m above ground in open terrain), the corresponding hourly
u(h,0), at elevation h over the building site can be estimated mean speed,
by using Eqs.
9'l'6' 9-I.tl, and 9.r.9 and the micro"meteorological paramete'rs
of rabrcs
9.1- l' 9- 1.2. ancl 9- r .r. p','r cxampre if u/(10,
0) : ls mph and the building
has hcight h : 2oo rn and is lrcatcd in a
iown with roughness length upwintr
of thc building z0 = 1.00 nr, U(200,0)
39.4 -1, frJ"
=
in S".tr.
9.1.2 and 9.2). For hurricane wincls, sce Secr. 2.43
"^"-pi",
and fq_S,
p. iSSt.
the basic wind speed
Aerodynamic Pressures on Buflding Facade.s. Information
on aerodynamic pressures is obtained fiom wind tinnel
tests. It r, pr"ro,t"o in terms or.
aerodynamic pressure coefficients defined as
C,(Mi.0^7
: --P(M,.i- 6r1
louzth.
;
t) il( )( )t [J( , l ot t wll{l)
r'.;r lirrrctiolr ol (lrc rtllitrl strt.rrl'tlr ,\10;:rtrtl ol llrc klrtl p(l) (Fig. 9.6. l). This
ol tlre initial strength from probability
,lr:;trrlrutions obtainctl cxpt'r'irrrcnlrrlly, us wcll as thc calculation by numerical
rrrr'llttrtls o1'the nonlirrcitr rclirlion bctwccn the loads p(t) and the normal stresses
,'tlllr.6r, r), tbrasuflicicrrt nurnbcrol'points M1anddirectionsd/of thestresses.
llrt'lpproach is applicd to panels subjected to (1) loads with the time history
/'t /):rrrd (2) constant loads with a 60-s duratiorr, poo, which are commonly used
rrr N.flh American design charts. Probability distributions of the load capacity
,'l rlrc panels are obtained for the loads p(r) (indexed by their mean value
/'(/)) and for the 60-s load pon. Let these distributions be denoted ay Polpal
.rrrtl P2,,,(p6,e), respectively. The 60-s load pfifl equivalent to the load p(r) is
lrvt'rr by the relation
.rp;rlrriqgll cntails Mortlt' (':rr lo sirrrrrlrr(iorrs
p"u?,
:
P;J{P,lp(/)l}
Equivalent 60-s Wind l_9ads. Wind pressures p(.M1,
0r) andthe corresponcl_
ing pressure coefficients CrMi,01) are randomtynuJiu#ng
run"tion., of tirrrc
that depend upon the position M1 and the mean
wind direction 01 (e.g., scc
Fig. 4.7.2.).
The load capacity of glass cladding panels depends
upon the entirc tirrre
..
history of the load. This, dependerce cln^in principre
be tui.cn inkr acc.unt rry
using basic fracture mcchanics relations to clescribc
thc cl'lccl ol'lirtig'c c;*rsctl
by the fluctuating loacl, thal is, thc tirnc-clcpcnrlcrr(
gnrwth irr l5c sizc: .l. llrrws
present on thc srrrlirccs luxl crlgcs ol'lhc;.r:rnr.ls,
:rrrrl llrr.r..orrst.tlut.rr( lilnt,
(e.6.2)
l lrrrs, fbr any point M1 and wind direction 0p, an equivalent aerodynamic coefirt rt'nt can be defined as
:
C},n(Mi.01,1
ors
directional maxima.
367
,l.Pt'tttlt'ltl tlt't'tt';tsr'ol lltr' y'l,r::. -lrt'r11'(11 ,\(/) l() \til l lrc I;rrlrrn'lotrtl is ol-rtainccl
li,rrt lllL: ctttttliliott llt:r( l:rrlrrrt' ('('( lrs wlrt'rr llre tt'lrsiolr stl('ss (r(/), which is in
r', n('r'lll lr norrlincltr-lrrrrt lrorr ol llrr. lo;rtl7r(/), is t.r;rurl lo llrc strcngth S(f), which
(e.6. t)
w.here p(Mt, 0*) is the pressure at point
M1 of the facade, induced by wind
blowing from direction 91 with u ,''"un hourly speed
at the top of the building,
u(h,00; p is the airdensiry;,ay.d cr(\, d1) is the pressure.i.m.i"n,
ar poinr
M; corresponding to lhe_wind direciion'd^. pressure
coefticienrs c,,(Mj. d1 ) arc
recorded as functions of time for various wind
directi"", B- v-Jrious points
of the building facades' including points near corners
"i Measurements
and eaves.
are usually made for angles 0r: k x 15. (k :
1,2,
.. ,)q although
occasionally the increments may be smaller than
15. to alow detection or
t ( )/\t ): ;
pZlru,.
ot
-"
\pU'(h.0*)
(9.6.3)
--------
s(r)
l5
"(t)
i'r'?,
MPa
ih.r,
{
l0
50
I
I
t50
?fl)
1{l,r,,
'q,o'
2s0 300 350 a{t0 450
5{10
/( .)
l"l(Jtlltl,l 9.(r.1. livolrrliorr ol
tt rr:,r,rrr ,,tr,'
l,tr't' rtl 1tl:rss 1ll:rlt'. lr:rilrrrr'
ltr:. ;tt tr!!rt
()( (
.ilrii
I
'11
.,tr(
)(,
n,'tlr r,villr lllrtt' :rl :r lxrirrl orr
'. | (,
')ri
I
lltr
368
llt.,ll
l)tN(it; wtNl) l()nt )l;, :iillt t{.ililt^t ilt:;t ,{)Nt;t nNt) I)t :it(iN ()t il()()t
tN(i
The approach just dcscribcd lrirs so liu'lrt'crr rrscrl orrly irr cxpl1;r.irl6ry ilvcs
tigations t9-58]. Currently a sirnplcr appnr:rch is usctl lirr rlcsign psrl)()scs, 1r
which it is assumed that the actual fluctuating loacl causing lailJro is oquivale,rrl
to a constant load with asmall duration, tpr, and a magnitudc equal t. thc pcirk
fluctuating load averaged over the time ipt , ppr. It is iommonry assumecr thlrr
trp = I s.*
The l-s constant loadprp must in tum be converted into an equivalent 60-s
load p[[. It can be shown from basic fracture mechanics relations that rhc
stresses o6e and onr induced by the 60-s loadpifi and the 1_s load pr1,,
respcc
tivcly, are equivalent from the point of view of their effect on glass if
x
o[,,
60
: oirx
I
(9.6.4)
whcrc rt is thc cxponcnt in thc phenomenological relation describing subcritical
crack gnrwth. For soda limc glass it may be assumed for practic-al purposcs
that n : 16 [9-59]. From Eq. 9.6.4 and the simplifying assumption that thc
load-stress relationship is linear, it fcrllows that
P'fi
=
o.78n,,
(9.6.s)
(e.g.' see t9-601). Thus, in this simplified approach, the equivalent 60-s
aenr
dynamic coeffrcient for any point M1 and wind direction g; ias the expression
c;1,(Mi. or1
=
YP'illtJr)
lpu)th.
(9.6.6)
(,1
nl rl)lN(i nNl) il()()l lNt(, t{rt
t Wll.ilr l{)/\t r,,
:tfi!}
n'lrit'lt ltirs tltc slrntt' lolnr trs lttl. t{. Llt. 'l'lrc rlt.sll'11 lvlntl lt,;rrl:; r.;rrr tlrt.rr.lorr.
lrt' t'slirrrirlctl lrs slrowrr in Sccl. lJ. 1.2, irr wlritlr yri,ilt /'tr. t)t.l:rrr.l {',,,1,{ Ilt. 0Al
'.lrorrltl bc substitutcd lirr p(rrl) arrtl ('(//)(,,({/). rt'spt.t.trvr.ly
'l'lrc cstirnation <ll'clcsigrt wirrtl loruls tlillt'r's irt'tollurlr. lo
14,11,'11,,., tlrt' r'orr
rt'ttlional orthc risk-c<lnsistcn( tlcsigrr pntcctlrrlr'is rrst'tl. lrr corrvt.lrllorlrl tlcsilirr
Irltclicc equivalent 60-s l<lacls witlr :r 50 yr.:trl rnc:ut tccut'r't'rrct. intglv:rl,
1",,,)\ n1(M) are estimated withtlul cortsitlcring llrc cll'ccts ol wirrtl tlircc(iorrality.
l'.r'this
reason the actual mcan rccurrcncc intcrval, N.,,r.tlt'the clcsign load
fiom panel to pancl. ln thc case of panels for which the direction
l,,,ij ,,, varies
,rl (hc most severe extreme winds coincides with the direction of the largest
,rt'nxlynamic coefficient, N"., ir indeed 50 years. However, for most other
l);illcls Nacr exceeds 50 years, in some cases by one or even two orders of
rrr:rgrritude (see Sect. 8.1.2).
A second consequence of not accounting for wind directionality is that any
trvo buildings that are identical in all respects but have different orientations
r'ill cxperience different numbers of panel failures during their lifetime. Indeed,
',ilrcc conventional practice does not account for wind directionality, it will
v rt'ltl exactly the same cladding design for the two buildings even though, owing
t. thcir different orientations with respect to the direction of the most severe
r'\trcme winds, the two buildings will exhibit different degrees of sensitivity
to wind effects.
lior the risk-consistent design procedure it is necessary to estimate the mean
,rrrrl lhe coefficient of variation of the equivalent 60-s largest lifetime load. This
r:, tlone as shown in Sect. 8.2.2 (Eqs. 8.2.7 to 8.2.14), in which pffi, and
| ,,,,,, should be substituted forQ, ancl Vn,, respectively.
Load capacity of cladding Panels. Information on the load capacity of
, l;rtlcling panels can be obtained from manufacturers' charts [9-55, 9-56]. These
Estimation of Design wind Loads. From Eq. 9.6.3 if follows thar the
equiv
p[[ are given by the relation
alent 60-s loads
0i :
l)l l:l{,l.J ()l
o*t
For additional details, see [9-97].
pzl,(Mi,
t
ipcil,,(Mj, 0iu2(h,
0k)
(9.6.1)
*The peal valueprl depcnds
upon the record length (orstom duration) z. commonly
T = 20 min to r hr (full-scale), to which therc corresponds a laborabry record it is assurrrcrl
length 7,,,
T(D,,,/ D)l(u,,,lu ), whcre D,,,lD
u,,lu are the mode I gcomctric and verocity scare, respccrivt.ry
.and,
For structural rcliability calculations it is dcsirable to estimate thc nrcan
xnd standrrd dcvirrti'rr
of thc peak pressurt.Tr,,,. siner.. lirr lny givr,n v:rlrrt./rU).
7r,,r v1rr.it.s 1,,,,,, ra.,,,,1 ,,, ,.....,,,..;. ,,,,,
can be donc fntnl sovcr:tl lccortls wilh lcrrgth 71,,, or hy trsirrp k'r.lrrrir;rrt.s
h:rsul .rr *rrrlrrrr
processcs thcory (sct: lirl. A2.:1.1,
l4 751, l9 (r.11. irnrl cspr,t.i:rlly l1)(rll, llri,.lr r.lrrt:rirrs rrst.lll
practical rcsults).
,lr:rrts include estimates of the standard deviation and of the 0.8 percentage
P'irrt of the load capacity of panels with different sizes for annealed, heat,lrt'rrgthened, and tempered glass.x The charts of [9-55] and [9-56] exhibit
rrrrrlual inconsistencies, and apparent internal inconsistencies have been noted
rrr l() -551 (see [9-57, 9-58, 9-59], which report research aimed at improving
rlrr'sc chafts).
()wing to fatigue effects, the load capacity of glass panels depends upon the
trrrrc history of the applied load [9-57,9-58,9-59]. The load capacities given
rrr l()--5-51 and [9-561 have a standardized time history; that is, they are expressed
rir l('f'rlrs of constant loads with a 60-s duration, denoted by puu.
' Ilrr'o li lx'r(cll(rtgc point ol lhc kr:rtl t lrp:rt ity is thc load to which lhcrc r:orrcs;rontls ;r prolr:rlrility
,'ll;rrlrrrtoll.lJritrrclsouloll,(XX)(sct.Sct.t Al5).Irrlirnrurlionorrkr:rrlst.orrt,s;xrrrrlirrl,1oo1llq.1
I'r,,lr;tlrililit s ol l:rilrrrc is :rv;rillrlrlt. rn l1) 'r(rl
370
tltlt t)lN(it; wtNl) l()nt)ll, 1;iliU(.ililtnt ill :;t'()Niit nNt) Ilt lil(iN ()t il()()l lN(i
trt .triil rrt i:tnDl)lfl(,
/\lll) llt)()lll.lr,
Ir111
t7y11.11)
l()/\l):,
:lIl
Design Criteria. Tho convcntionirl tlt'sigrr lllrcr:tlrrrc uscs lhc litllowing rlesigrr
criterion:
p66(0.008)
>
p"f,,o(Mi)
(9.6.13)
i
wherep[fl.r0(M) is the equivalent 60-s wind load estimated without considcrirr;i
wind directionality effects (see discussion above), and p6s(0.008) is the 60-s
load capacity of the panel corresponding to a cumulative failure probability ol
8 panels out of 1000.
'fhe risk-consistent design procedure is based on the requirement that thc
probability of failure of each panel during the lifetime of the building be lcss
ii
lhan a spccilicd valuc Pr. It is now shown that this requirement leads to a design
critcrion cxprcsscd in tcrnts ofequivalent 60-s loads and of60-s load capacitics.
Considcr thc sal'cty indcx p dclined by Eq. A3.29.It is possible to writc
.
'
L)--
ln(Dn lP'A)
tv",t +
i
:;::iU*"
(9.0.tr1
v;d,1"'
9.6.2. Division of a high-rise building face into
zones of equal glass thick-
rill
i
where p[fl,, and V r"r, are the mean and the coefficient of variation of the largcsl
equivalent 60-sec wind load during the lifetime of the building, the subscript
/, represents the lifetime of the structure in years, andf6s and Vp^, are the mcitll
and coefficient of variation of the load capacity. From Eq. 9.6.9 it follows thirt
B60
of the glass panels may change as a function of elevation, as in the case
John Hancock Building in Boston. In other cases the same glass thickness
rr used over an entire building face or even over the entire building. For
Lrt'1s[iens where wind-borne missiles, including roof gravel, may be expected
trr lrit the cladding, special zones are suggested in [9-62].
Wc denote a zone in which the glass type and thickness is uniform by D, (i
l, ...n).If the conventional design method is used, Eq.9.6.8 must be
,;rr(isfied at all points M1 within D,:
rrt'ss
,,1 the
should satisfy the relation
Pon
>
P'8,(.Mi;exptP(V'?";,,(
Mj)
+
v?,uJt''l
(9.6. r0)
where B is the value of the safety index corresponding to the failure probabilily
PJ (see Eq. 4.3.37). Equation 9.6.10 is the design criterion used for risk
consistent design.
[p6o(0.008)];
The question of the selection of the safety index 0 or, equivalently, of tlre
failure probability Py, is discussed next.
>
max [p"d5o(Mj)]
(9.6.11)
Di
It is possible to estimate the expected number of failures inherent in the design
l';rscd on Eq. 9.6. 11 as follows: Each zone D; is divided into subzones Aii(j
l, 2, n4) over which it may be assumed that the wind loads do not vary
',11'rrificantly.* Using Eqs. 9.6.9 and ,{3.37, it is possible to calculate, forany
1'rvc:rr orientation of the building o7, the safety index 0ii1 and the lifetime prob-
of construction it is necessary to divide the building facackrs
into zones, each characterized by a single type of glass panel. Thus, for alty
given architectural pattern defined by the location and by the height and witltlr
of the panels, the design of the cladding consists of (l) dividing the buililirtlq
facades into such zones and (2) selecting the type of glass (i.e., whethcr rrrt
nealed, heat-strengthened, or tempered) and the panel thickness for each zottt'.
,rlrility of failure P|; of the panels within A,,.
Lct the number of panels within A,, be denoted by nnii. For the building
,lt'signcd by the conventionltl rncthod (Eq. 9.6. 11), the expected number of
;':rrrr:l lhilures in subzonc,4;i rlru'ing lhc lifctimc of the structure with orientation
An example of division into zones of equal glass thickness, suggcslctl irr
t8-111, is shown in Fig.9.6.2. This division rnakcs il possiblc to provirlc
stronger panels al uncl ncar thc cclgcs and cavcs, wlrct'e lretrxlyrrrtttic prcssutl's
are usually Iargcsl . Ilowcvcr-, <ltlrcr possibilitics r:xisl . liot r'xlrttplc, thc thick
rlrr pnttlitc. ,4,, luc tltc ltilrttlrrry :rrt';r:. ol llrr' prcs:rrrrt lirl)s ()r lllc wirrtl lrrnrrt'l rrrodcl <lf the
i'rrlrlirtg (or'. il rt ltilrutltt'y iue:r ('tl('lrrli, lrr'lrrrrrl llrc trrrrlrrt'r ol /),, llrt.porlion ol lltirl tributary
,rr,';r tottlltint'tl itt lltc zottc /),).
Design. For
ease
*A
1
i
372
cv1
lll i;l'()Niil . nNl) l)l :;l(iN ()l ll()()l lN(i
BtJtil)tN(iti: wtNt) t()nt )1i, :;ilt{,(;l('lrnl
isx
nt1i1
:
n1,iiPI1,1
(9.6. I2)
The expected total number of panel failures per lifetime for thc zonc 1); itttrl
for the entire building are, respectively,
s-/
-t
nn:4ntu
(9.6. r3)
-t
L ntr
ny: s-1
(9.6.t4t
I
lixpcricrrcc appcars to indicate that the cladding in any subzone ,4, designctl
in tccortlarrcc with thc conventional method (Eq.9.6.8) is acceptable from a
sal'cty point ol' vicw if thc aerodynamic and climatological data upon which
thc <lcsign was based are adequate. It might then be argued that the probability
P7 corresponcling to the saf'ety index B used in Eq. 9.6.10 may have the value:
Pf :
nllaf
{PLJti}
(9.6. rs)
is the largest of the values P!;.
However, such a choice of Py for use in risk-consistent design might bc
imprudent. The authors believe that it is reasonable to adopt as a design ob'
jective an expected number of panel failures per lifetime for the entire building
where mnx;.1,,
tph|
n7
:
l)l :,1(ir! ()t ct
nt
)tilN(i nNt) il(x)t til(, t()t I VVllJt, trr/\t ): ,
:ll],
irily onc lrirrrt:l tlrrrirrg llrc lilt.tirrrt.ol llre lrrriltlirrli rs lj
tt,lrt,,. wlt.r,,. rr,, rr rlrt,
Iol:tl lruttttrcr rll llttltt'ls ol'llrc lrrrilrlirrg. 'l'lrc s;rlcty iirtlt.r is
1i tlrcrr t,rrlr.rrlrrlt.tl
rrsirrg IJq. 43.17' itrttl lltc clatltlirtg lirr clrch r,ont l), is
tlcsrlirrt'tl lry tlrr.rrsk
( ()nsistcnt pntcorluro in accorclarrco
willr lit;. ().(r. lO:
lpooli
>
maxT;, {7ri,il,,( M)cx1.tlf}(V),,,,1M;
t
V,:,,,,,,)t,,1 (9.6. l7)
A c:rmputer program fbr thc crcsign .r'cradding by Eq. g.6.11 in
c.nluncti.n
with Eqs. 9.6.16 and 43.37 is rcfbrcnccd in
t9-611. Illustrative results obtained
lry using that program are prescnted in Sect. 9.6.2.
9-6.2 Economic and safety Advantages of Risk-consistent
Procedure
Design
I
Iir illustrate the potential
i
advantages of the risk-consistent design procedure,
rt'sults of computations taken^from
[9-61] are presented for a 200-ir tail uuitaing
rcpresented in plan in Fig. 9.6.3. rt was assumed that
the building is located
in lerrain with uniform roughness in all directions (zo
1.00 m; anO
there
i'c no neighboring structures influencing the building aerodynamics.that
Aerody_
rurrnic pressure coefficients obtained in the wind tunnel
were
l
:
extracted from
li
l
max
nl
(9.6. r6)
Indeed, the conventional design procedure, ignoring as it does wind directionality effects, can be viewed as providing sufficient safety levels for all buildings,
regardless of their orientation. This can be interpreted as meaning (l) that thc
expected number of failures rej inherent in the conventional design procedurcr
is icceptable even for buildings with the most unfavorable orientation cv1 antl
(2) that if the conventional procedure is used, building with more favorable
orientations are overdesigned.
If Eq. 9.6.16 is adopted as a design objective, the probability of failure <tl'
*The failure condition for each panel of a zone is defined by the event Pa - P"d < 0. Notc thirl
these events are not in all cases statistically independent, since the loads induced on vatittttr'
panels, and in some cases the load capacities ofvarious panels, may be correlated. Howcvet,
Eq. 9.6.12 holds regardless of whether the failure events are independcnt or not. This ctn lrt'
shown by considering thc simple cxample of n,, coins. Let lailurc dcnotc the occurroncc ol
"heads." The expectation o1'thc numbcr of failurcs that would occttt il lltc rt,, coins wt:rc losst'tl
once is 1, : ll2nt,.This is truc rcgarcllcss ol'whcthcr thc luiltrt' ('v('trls ilr(' irrtlt:pcrttlcnl (lts itt
the case ofcoins liaving c:rt'lt lrn intlcpcn<lcnl rttotion) or grt'r'li'clly toltt'lrtlt'tl (lrs itl lltc tltsc ttl il
set ol'n, coins, lixcrl onkr :r wciglrllcss holrrtl wilh :rll lltt' "ltt';rls" ott llt( :i:tlll(' sitle, so llt:tl
llilurc 11l grrt: coilt worrltl r'rrllril lirilrrn'ol:rll llrc rr,, toitts) Nolt llr;tl rrlrrlr'lltt t'r1x'tlit(iotts ol
Ir, Wottkl
?
ltt.lltC Slrtrrt'irr llrt lwo r'lrsts. lltL'sl;tttrl:ttrl (l('vl:llllrll'. ttlttlrl ttol
\
e
o.
dr
\
lil(;lJltl,l l).(r..1-
| )rrri.rr,,rrrrr'.
ol lrrrrlrllrrl,
rrr pl:rrr
;
llt,ll l)lN(il; wlNl) l()nl ):; 1;lltu(.ililtnt ttt t,t,()Nl;l nNl) l)l 1;l(iN ()l tt{)()l lN(i
374
I
'l':rlllt'tJ.I.].
19-601. 'l'hc wincl clinratc was assunrc(l kr lx'tlt'lirrctl by thc tl:rlrr ol
forwhich summary statistics arc givcn in Itig. -1.4.1. 'l'ho lucadc:s wcrc tlivitlerl
into zones of uniform glass thickncss in accorclancc with liig. 9.6.2. lt w:rs
assumed that the cladding consisted of anncaled glass pancls with dirncnsions
1.8 x 1.8 m. The information on the load capacity of thc pancls was lakcrr
from [9-56]. Approximate typical prices per unit area of panels with various
thicknesses were obtained from glass distributors. These were used as a basis
for performing estimates of the nominal cost of cladding glass inhercnt in :rny
given design.
From an inspection of Fig. 3.4.1 it is apparent that the wind effects arc not
cqually severe for the parallel faces AD and BC (or AB and DC) of the building.
shown in Fig. 9.6.3. Nevertheless, as noted earlier, the conventional design
mcthod would result in this case in identical designs for those faces. It is also
clcar that the severity of the wind effects on the various faces depends upon
the orientation oi of the building. Again, this is not reflected in the conventionrrl
method, which results in identical cladding designs regardless of the buildirrg
orientation a;.
The cladding of the building shown in Fig. 9.6.3 was first designed in
accordance with the conventional method. The nominal cost of the cladding so
designed was estimated to be $361,000 for the entire building. Using the pnr
cedure described in the preceding section, the expected number of panel failurcs
per lifetime inherent in the conventional design was estimated for various buikl
ing orientations cy;. The results of the estimates are shown in column 2 of Tablc
9.6.r.
Also shown in Table 9.6.1 (columns 4 and 5) are nominal costs of claddirrg,
designed by the risk-consistent procedure on the basis of the following desigrr
objectives: the expected total number of failures per lifetime is equal (l) to tlrc
vahe n! of column 2 (see column 4) anil (2) to the value n]: 12.0, whiclr
corresponds approximately to the most unfavorable orientation of the buildirrp,
(see column 5).
ll I I I ll l.l(.l
37s
('ottsitlt't lltr' tlt's11'111. lr:r:.r'rl orr llrt' lusl ol llrt'st' lwo olrlt.r'lives- lt is scclt
tlr.rl irt (ltis t':tst'lll('('(()n.int,s lrtltit'vr'tl :rlt'ol lltt'oltlt'r'ol'5'/,, kl l0'/,,.
ll,lv1'v1'1', llrc lrrt'l llr:rl llrt' r'orrvt'rrliorr;rl rlr'sip,rr is :rcce:lltlrhlc to building in.;,,'t (iolt trullrorilit's, lt'1'.:rrrllt"ss ol llrt' lrrrilrling or.icrrtlrliorr, irnplics [hat in the
r rr'\\, ()l ll'tcsc uutlrol'ilit's srrtlr :r rlt'sigrr is srrllit'icntly sal'c cvcn in those cases
rrlrr'n'lho builcling olit'rrltr(iorr is rrrrllrvol'rrblc; as notcd in Sect.9.6.1, this
,'lr:,t'r'vlrtion lcacls to tlrc rrtkrpl iorr irs ir tlcsign objective of an expected number
,
'l l;tilttt'cs pcr lil'ctintc apptrrxinra(cly cqual to the largest of the estimated values
,,', \l
1,2, . .. , 8). A cornparison between columns 3 and 5 of Table 9.6.1
.lr.ws 1[i11, for buildings with favorable orientations, the use of the risk-con.r'.tt'rr( design procedure can then result in significant savings (in the case
, i:rnrirrcd here, almost 25%).
As stated earlier, the results
of Table 9.6.1 were obtained for building fa,.r,k's divided into zones in accordance with Fig. 9.6.2. As indicated in
l') (rll. similar conclusions hold for designs in which the glass thickness is
,
"ilslilnt over an entire building face.
(;.3
'r
Effects of Wind Loads on Roofing
l.'t t crrl material on the perfbrmance
of roofing in strong winds shows that wind
high suctions, which may induce peeling failures, panel failures,
.rrgrlxrrling member failures, or system failures, and (2) scour of roof gravel
l') ')li l. A procedure for the selection of gravel size and parapet height to avoid
,,r:rvt'l scour (displacement) and, more important, gravel blow-offfrom the roof
,. I'rolroscd in [9-99] (see also 14-63, 4-eD.
lrr rrrcas of the roof where calculations indicate gravel blow-offwould occur,
rlrr' usL: of concrete slabs instead of gravel is recommended. Alternatively, the
,,r.rvt'l should be fully embedded using a double-surfacing technique [9-100].
It,r:,t'tl on obseruations of roof behavior during hurricane Hugo, it has been
,i ( ()n)nrended that for buildings less (more) than 13.7-m high, parapet heights
,.ur (lruse
(l)
i',
:rr lcast 0.3 m (0.6 m) [9-101 ,9-102].
l;or wind-related information on mechanically attached single-ply systems,
,,, lt) l03l; metal edge flashings, see [9-109] and [9-ll4]; asphalt shingles
.'rr, I tlrcir attachment, see [9-104, 9-105]; roof fasteners, see
[9-ll0]; looselrr,l rtrol'insulation systems, see [4-83, 4-84,9-lll]. Studies of wind effects
,'rr trlt'nxrf.s, including pressure distributions on gable roofs and around indi' ',irr;rl [ilcs, are summarized and referenced in [9-106]; see also [9-107]. It was
TABLE 9.6.1. Nominal Costs of Cladding for Various Designs
Conventional Practice
Risk*Consistent Procedurc
Nominal Cost.
Orientation
Number of
Panel _
Failures, nf
(l)
Nominal
Cost ($)
(2)
(3)
from Col. 2
($) (4)
2.5
I 1.9
36 r .000
330.(XX)
2u0,(xx)
16l .(xx)
1.15.(XX)
.14.5 .(XX )
Ltl
.l(r I .(XX)
t.15.()(x)
.1t30.()(x)
.l(, I .(XX)
t,15.(XXt
r.l5.(xx
Building
0"
45"
90"
t3.5"
l.)
I
Design Based
on Value Di
Nominal Cosl
.
Design Blsctl
:
on Valuc ri
12.0 (:f) (\)
)
I,
'rrntl that aluminum shingles behaved poorly in hurricane Andrew, while sprayrl 11'11 polyurethane fbam roofs performed outstandingly
[9- I 08].
r 1 r1
III FERENCES
') |
lt.
1.. Wirlllrrw, "lttlt'tlt'rt'ttr r' ;rrrrl l'roxirrrily lillccls," in llitrtl l,.rcitttl Viltnr
lutt',\." Il lior l., l (r rl 1, $1y1i111','r'Vt'rllrg. 11..* y,r1l,, lr)rtl
tiutt,\ ()f ,\!ntt
376
9-2
9-3
9-4
9--5
T. A. Reinhtlld ct al., Mutrt ttrnl li!u('tu(ttin,q litn't,t tttrtl 'litttlttt,,t rtrr tr'lirll
Building Model oJ Square Cro:;s-St<'tittrt otr tt Singlc Mtxltl, itr tlrr Wtrkr rtl rt
Similar Model, andinthe Wake oJ'a Rcctangular Mtxlcl,l{cport VI,l l1 7() ll,
Department of Engineering Science and Mechanics, Virginia Polytcchrrit. l1
stitute, Blacksubrg, VA, March 1979.
J. Blessman and J. D. Pierce, "Interaction Effects in Neighboring Tall lirrihl
ings," Proceedings Fifth International Conference on Wirul Engint,aring, ltl
Collins, CO, July 1919, Yo1.2, Pergamon press, Elmsfbrd, Ny, l9tt0.
J. A. Peterka and J. E. Cermak, "Adverse wind Loading Inducecl by Atl.jlccrrr
Buildings," J. Struct. Div., ASCE, 102, No. ST3 (March, 1976), 533-,54U.
()
ll. W. l,icprtrann, "OntheApplicationof
I
9-8
9-9
9-10
9-l
I
9-12
9-13
ilt IIiltt.t(
')'O
') 'f
ri r.)
'l
.)\
'(r
J
t,
.'
l,
'11
')
)()
l0
!)
ll
t970.
9-15 C.
Soize, "Dynamique stochastique des structures dlancdes soumiscs
charges du
vent,"
Revue Frangaise de Mdcanique,60
(1976),57
9-16 G. Solari, DAWROS: A Computer
9-17
Program .for Calculating Aktng-Witttl llt'
sponse oJ' Structures, Pubblicazione dell'Istituto di Scienza dcllc Cilstnrziprrr.
Serie IV, No. l, University of Genova, Genova, Italy, lgtt l.
E,. Simiu, "Revised Procedure firrEstimating Along-Wind llcsponsc," .l . Strttt t
Dlv., ASCE, 106, No. STI (Jan. l9tl0). I 10.
9-18 G.
Solari, "Along-Wintl llcsponsc Ijstiuurlion: ('krsc:tl Iiolrrr Solrrtion."./
Strucr. I)iv.. nS('lr. lOtl, No. S'l'I (.lrrrr, l()lil).22.5 1.1,1
.
9-19 A. Kiu'ccrn (llcrsorr:rl
torrrrnrrrrictrliorr, l()()(r),
')
ll
()
l
l
irrrr
65.
W. C. Iltrrty atttl M. ll . l.ltrlrirr,.;lt irr. l)trt,tttrir'.s t,l :;tnt('tut'(,t. Prcntice-Hall,
lirrglow<xrrl ('lill.s, N.l. l(Xrl
. A. BlLrlno, N. M. Nt'wrrr:rrk. :rrrtl l, ll. ('ornirtg, Design of Multistory
Ilcinfttrcr:d Crttt'n'lt' Ifitiltlittg.r lttr I'.rtrtlttltrttkr Motions, Portland Cement As-
.f
I
l)nrblcnr." .1. Atntraut. .St.i. 19, l2 (Dec. 1952), 793-800,822.
A. Ci. I)avcnport, "'l'hc Application oi'statistical Concepts to the Wind Lo:ul
ing to Structurcs." Prrx'. Insr. Civ. Eng.,19 (1961), 449-472.
A. G. Davcnport, "Gust Loading Factors," J. Struct. Div., ASCE, 93, No.
Windload Program," NTIS Accession No. Ptr_294157 lAS, Computer Progrrrrr
for Estimating Along-Wind Response, National Technical Information Servir'c,
Springfield, VA, 1979.
I
irlion. ('hit'irgo.
|(t(r
I
.
G. T. Taoka, M. llogrrrr, Ir. I(lrrrrr, unrl R. H. Scanlan, "Ambient Response
olSome Tall Structurcs," .l . Srruct. /)iv., ASCE, l0l, No. ST1, Proc. Paper
StatisticalConceptstotheBullttirrpi
9-14 E. Simiu and D. W. Lozier, "The Buffeting of Structures by Strong Wintls
:IT
l'. A. Itosltlr, .lrt l'.rIt'titrrr'tttttl,\lrtrlt t,l tlt. lit\l't't,rr'1tf tt,\'tlutu'('l'ti.tttt to
(rti. lilrtrrlly ol (it:rrlrr:rlt Slrrtlrt':;. Urrrvt:r'sity ol'Wcstcrn
Wittrl ltnttl.lil,W'l'll
sot
1995.
ST3, Proc. Paper 5255 (Junc 1967), I l-34.
J. Vellozzi and E. Cohen, "Gust Response Factors," J. Struct. Dlv., AS('li,
94, No. 5T6, Proc. Paper 5980 (June 1968), 1295-1313.
J. w. Reed, wind-Induced Motion and Human DiscomJbrt in Tall Buiklingr,
Structures Publication No. 310, R7l-42, Department of Civil Enginecrirrpi.
MIT, Cambridge, 197 l.
E. Simiu, "Gust Factors and Along-wind Pressure correlations," J. strucr.
Div., ASCE, 99, No. ST4, Proc. Paper 9686 (April 1973), 173_783.
canadian structural Design Manual, Supplement No. 4 to the National Builtl
ing Code of Canada, Associate Committee on the National Building Codc rrrrrl
National Research Council of Canada, Ottawa, 1975.
B. J. Vickery, "On the Reliability of Gust Loading Factors," in proceeditr,q*
of the Technical Meeting Concerning Wind Loads on Buildings and Structun':;.
Building Science Series 30, National Bureau of Standards, Washington, l)(',
I
()rttlrrio, l,orrtlolr. ()rrlirrio, (':rrr:rtl;r. l(l(rH
ASCE 7-95 Standard on Minimum Loads for the Design of Buiktings orul Otht'r
Strutturrs, Amcrican Society of Civil Engineem, New york,
96
;
tlt,ilt)tN(ili wtNt) l()nt )t;, :;ilt(,(.1,n/\t nt :;t'()N:it nND I)t 1it(iN ()t lr()()r rN(;
i)
'f
t-l
15
l05l (Jan. 197-5), 49
.55.
C. S. Kwok and W. H. Melbourne, "Wind-Induced Lock-In Excitation of Tall
Structures," J. Srruct. Dlv., ASCE, 107, No. ST1 (Jan. l98l),57 72.
B. J. Vickery, "Notes on Wind Forces on Tall Buildings," Annex to Australian
Standard 1170, Part 2-1973, SAA Loading Code Part 2-Wind Forces, Standards Association of Australia, Sydney, 1973.
T. A. Reinhold and P. R. Sparks, "The Influence of Wind Direction on the
Response of a Square-Section Tall Building," Proceedings Fifih International
Conference on Wind Engineering, Fort Collins, CO, July 1979, Pergamon
Press, Elmsford, NY, 1980.
J. W. Saunders, Wind Excitation of Tall Buildings with Particular References
b the Cross-Wind Motion of Tall Buildings of Constant Rectangular CrossSection, Doctoral Thesis, Dept. of Mechnical Engineering, Monash University,
Victoria, Australia, 1975.
A. Kareem, "Across-Wind Response of Buildings," J. Struct. Diy., ASCE,
108 (April 1982), 869-887.
A. Kareem, "Wind-Excited Response of Buildings in Higher Modes," J. Struct.
Div., ASCE, 107, No. ST4 (April 1981), 701 706.
A. G. Davenport, "The Response of Six Building Shapes to Turbulent Winds,"
Phil. Trans. Roy. Soc. London, A269 (1971),385-394.
F. E. Schmit, "The Florida Hurricane and Some of Its Effects," Eng. News
Record,97, t6 (Oct. 14, 1926),624-627.
A. Smith, "Basis of Design fbr Hurricane Exposure," Report of Committee
308, Proceedings, ACI, 27 (1931),903.
"Torsional Efects of Wind in Buildings,'' in Wind Bracing in Steel Buildings,
Sixth progress Report of Subcommittee No. 31, Committee on Steel of the
Structural Division, Reports, June 1939, pp. 988-996.
"Wind Forces on Structures," Trans. ASCE,126,Part II (1961), 1124 1198.
C. Patrickson and P. Friedmann, "Deterministic Torsional Building Response
t<r Winds," J. Stnrct. I)iv., ASCE, 105 (1979), 2621-2637.
r)
l(r I). A. Foutch and Ij. Srrlirk, "'lirrsional
r)
\l
(,
ll'i (i. ('.
Vibrations of Wind Excited Symmetric
Struclurcs," .1. Wirttl l,.,trg. lttt!. .,1t,nnlvr.7 (198 l), l9l-201.
l). A. Iirutclr arrtl li. S:rlrrk. " lirlsrorr:rl Vihrution ol' Along-Wind Excitcd
.l . I'.,tr,q. NIt', lr ltrr'. AS('lr, 107 (l9lll), 32l 337.
. ll, M. l)i.ltrlro. ;rrrrl l\4 lcrv, " lolsiorurl l{csJronsc ol lligh llisc
lltriklirrlls." .1 . ,\tnrr'r. /)ry Ali('l', I0l { l{)/1). l()/ .ll(r.
Slrrrctrrrcs,"
ll;rl1
378
tlttll l)lN(ii' wlNl) l()nl ): ;.
:;lllll(.lllll/\l lll
:'l '()l'll;l . nNl
I l)l :il(;N ()l
;
g-39 G. L. Grcig, 'llnttnl
rttt l,..ttitrrtttt t,l ll'trttl lrttlttcul I)\'tttttrtit' llttrltrt'tttr'lltll
Buildings, Mastcr's'l'hcsis, l)cpl11 . ol lingirrccr-ing, IJrtivr:lsily ol Wt'slt'ttt
Ontario, London, Ontario, Scpt. l9tto.
9-40 G. Lythe and D. Surry, "Wind-Induced Torsional Lrtads on'l'all lltriltlittgs."
J. Wind Eng. Ind. Aerod.,36 (1990), 225 234g-41 N. Isyumov, "The Aeroelastic Modeling of Tall Buildings," Pnx'cctling.s ttl
International Workshop on Wind Tunnel Modeling .for Civil Engin.ccring A1t
plication, Cambridge Univ. Press, Cambridge, 1982'
g-42 N. Isyumov and M. Poole, "Wind-Induced Torque on Squarc and Rectangulilr
Buil<ling Shapes," Proceedings Sixth International Conference on Wind I'tr
ginrtring, Illscvier, Amsterdam, 1984.
9 4.1 .l N Yang antl Y. K. Lin, "Along-Wind Motion of Multistory Building"'./.
I'ttg. Mrclt. /)ir,., ASCti, 107 (198 l),295-307.
9 44 "'lowcr-('ublcs l-lurrtllc Wincl, Water Tank Damps lt," Eng. News Recrtnl,
l)cc. 9, 1971. p. 23.
9-45 "Lcad Hula-Hoops Stahilizc Antcnna," Eng,. News Record, 197,4 (July 22.
1976), 10.
9-46 K. B. Wiesner, "Tuned Mass Danrpcrs to Reduce Building Wind Motion""
Preprint 3510, ASCE Convention and Exposition, Boston, April 2-6' 1979.
9-41 "Hancock Tower Now to Get Dampers," Eng. News Record, October 30'
1975,
p.
11.
9-48 N. Isyumov, J. Holmes, and A. G. Davenport, "A
9-49
9-50
I
9-52
9-5
9-53
9
54
Study of Wind Effects lirr'
the First National City Corporation Project-New York, U.S'A.," Universit.t'
of Western Ontario Research Report BLWT-551-75, London, Ontario, Canaclir,
1975.
R. J. McNamara, "Tuned Mass Dampers for Buildings," /. Struct. Dir'.,
ASCE, 103 (Sept. 1977),1785 1198.
"Tuned Mass Dampers Sway Skyscrapers in Wind," Eng. News Record, Attg.
18, t977, pp. 28-29.
J. P. Den Hartog, Mechanical Vibrations, McGraw-Hill, New York, 1956.
S. H. Crandall and W. D. Mark, Random Vibrations in Mechanical Sys/arr,r.
Acadcnric Press, New York, 1963.
R. W. Lutt, "Optimal Tuned Mass Dampers for Buildings," J- Struct. Div .
ASCE, 105, (Dec. 1979), 27 66-2'772.
A. Dcr Kiureghian, J. L. Sackman, and B. Nour-Omid, Dynamic Responsa ttl
Lig,ht Equipment in Structures, Report UCB/EERC-81i05, Earthquake Engi
neering Research Center, College of Engineering, University of Californirr.
Berkeley, April 1981.
9-55
9-56
9-57
9-58
lll ll lll ll'
I
PPG Glass Thickness Recommendations to Meet Architccts Specified l-Mintttt'
Wind Load, PPG Industries, Pittsburg, 1981.
LOF Technical Information-Strength of Glass Under Wirul Loarl.s, Publicatiorr
ATS-109, Libbey-Owens-Ford Company, Tolcdo, OH' 19u0.
W. L. Beason, and J. R. Morgan, "Glass Failurc Prediction Motlcl," .1. Stntt t.
Eng., ll0 (Fcb. 1984). 191-212.
D. A. Rccrl irnrl li. Sirniu, "Wincl l-oarling anrl tlrc Slrcrrgth ol (ilrrss."./.
SIrutr . l')r,q.. I l0 (April 19t34). 715 129.
I
')'r()
li. Silrritr lurtl l) A litt.tl, "liinli orr l(rrr;1 li.:,1::rrrrl tlrr.l\lrrrlrlrrr;,ol (,1.111111111,
(ilrrss Stlr'lrg,tlr lry tlrt' Wcilrrrll l)islrilrulrrrr. " I't,,,.r.,!rrt.r:t lU
t lA! .\\tttlt,,trttrrt
tttt l'rrilxtltilistir-Mt'tlttxl.s itt tltt'lllt't'ltrtttir'.s rtf ,\'rtlttlt ttttl ,\'ttnt /r//, t.,!il.r( Llrplsr.
.lurtc l9 21, l9tl,1, N (' l,irrrl :rrrtl s. l11'1'rvr.rrz (r.rl:, ). Syrrrrrl,t'r Vr'rl:r1,. Nt.ry
') (()
J. A. Pclcrka antl .l . lj. ('erttt:tk, ll'irnl Iirtrtrr'l ,\trtrll,rtf ,ltltttrrtt ()l.litt'lltti!tlirt,q.
York.
l9lJ-5.
I)cpartl)lcnt
I
t)
(rl
ol'(livil
l')rrgilrccrrng, ('olor';uLr Sl:rlt'Ilnivclsity, Iioll ('olli1s, N9v.
978.
E. Sirniu ancl A. Iiilotti, "Wirrtkrw Class Facadcs as Structural Systems: An
lmprovcd Rcliability-Bascd Dcsign procedure," proceedings International
conference on strut'tural saJbty and Reliabitity, May 27-29,1995, I. Konishi
M. Shinozuka (eds.), Kobe, Japan.
Beason, and p. L. Harris, "Designing for windborne
Missiles in Urban Areas," J. Strucr. Diy., ASCE, 104 (1979), 1149_1760.
') (r I
N. J. cook and J. R. Mayne, "A Refined working Approach to the Assessment
of wind Loads for Equivalent Static Design," J. winrt Eng. Ind. Aerodyn.,6
fl980), t25_t31.
') (rl J' A. Peterka, "Selection of Local Peak Pressure Coefficients for Wind Tunnel
Studies of Buildings," J. Wind Eng. Ind. Aerodyn., f3 (19g3), 477_499.
() (r.5 E.
Simiu, "Modern Developments in Wind Engineering, parl 4,,, Eng. Struc.
s (1983), 273_28r.
') (r(r A. Tallin and B. Ellingwood, "Analysis of rorsional Moments on Tall Buildings," "/. Wind Eng. Ind. Aerodyn 18 (Aprit 1985), t9l_195.
't (t/ P. Mahmoodi, "Design and Analysis of viscoelastic Vibration Dampers for
Structures," in Proceedings, world Innovation week Conference INovA-73,
and
')
(rl J. E. Minor, w. L.
rr
(rlJ
Eyrolles, Paris, 1974.
P. Mahmoodi, "Structural Dampers," J. Struct. Div., ASCE,95 (Aug. 1969),
t66t-1612.
') (t) A. G. Davenport
') /()
't ll
't l .'
') l\
and P. Hill-carroll, "Damping in Tall Buildings: Its variability and rreatment in Design," Building Motion in wind, N. Isyumov and
T. Tchanz (eds.), American Society of Civil Engineers, New york, 19g6.
D. F. Sinclair, "Damping Systems to Limit the Motion of Buildings," BuiLding
Motion in wind, N. Isyumov and r. Tchanz (eds.), American Society of civil
E,ngineers, New York, 1986.
c. J. Keel and P. Mahmoodi, "Design of viscoelastic Dampers forcolumbia
center Building," Building Motion in wind, N. Isyumov and r. Tchanz (eds.),
American Society of Civil Engineers, New york, 1986.
P. Mahmoodi and c. J. Kccl, "performance of Viscoelastic Structural Dampers
{or the columbia ccnlcr l}rrilrling," Builtling Motion in wirul, N. Isyumov and
'l'. ]'chanz (cds.), Arrrc'rt'irrr S.t'icly .l'civil E'ginccrs,
New york, 19g6.
J.
13.
Skilling, 'l'. 'l't'lr:rrz.
N
rsvrrrr.v.
I'. L.h. iuxl A. G.
Davcnporl, ..lix
pcrilncntal Stutlics, Slnrt.lrrr;rl l)r'sl1,n :rrll lirrll St.:rlt, Mc:lrsurcrrrcrrls lirr llre ('o
lrrrrrhi:r Scltlirsl ('ertlt'r," IIttrl,lrrtrl llrtlrttrt rtt llitt,l, N. lsyrunov trntl'l'. lt.lr:rrrz
't ll
(ctls.), Arrrc|it'lllr Srx it'lV ol ( rr rl I il1,ilr,.r.r:.. Nt'rv Y0tk. l()lJ(r.
W. ll. Mt'llxrlllll(', "'l'tttlrrrl, rr,, ,rrr,l llr, l ,.r,lrrr;, lrrl1,t. l,lrt'rrorn(.n()n,"
l'.tty. ltttl. .h'nul...l') ( lr)tf tt l ' r' l
J ll trrr!
lltJll l)lN(il; wlNl) l()nl )l ;, :;lllll(.il,il/\t
ilt
:;t
'()N:;l . nNt ) t)t :;t(;N ()l tt()()t lN(i
9-75 H. Kawai, "Vortex Inclucctl Vilrtirliorr ol 'l'trll litriltlirrgs,"' .l . llirrtl I,,,rt,g. lrrrl.
Aerod.,4l-44 (1992), ll7 128.
9-76 S. Yamazaki, N. Nagata, and H. Abim, "'l-uncd Activc l):rrnpcrs Instlllctl irr
the Minato Mirai (MM) 2l Landmark Tower in Yokohama." .l . Wirul I')t,g.
9-17
Ind. Aerodyn., 4l-44 (1992), 1937,1948.
L. M. Sun, Y. Fujino, B. M. Pacheco, and P. Chaiseri, "Modclling ol thc
Tuned Liquid Damper (TLD)," J. Wind Eng. Ind. Aerodyn.,4l-44 (19()2),
l 883- 1 894.
9-'78 T. Wakahara, T. Ohyama,
K. Fujii, "Suppression of Wind-Induced Vi
bration o1'a Tall Building Using Tuned Liquid Damper," J. Wind Eng. Irul.
Atnxl.vn.. 4l-44 (1992). 1895 1906.
919 'l'. []ctla. ll.. Nlgakaki, and K. Koshida, "Suppression of Wind-Induced Vi
lrration by I)yltarnic l)anrpcrs in Towerlike Structures," J. Wind Eng. Inrl.
t
lt rrlt ltrI',
')
().) ('. Sollit'c
't
t)I
tr
t)'l A. P. Robertson, "Ell-ccl ol li:tvt's l)r'lrril orr Wirrtl l)rcssurcs
it
t)5
't
rxr
tt
t)J
and
Acnxl.yrr.. 4l-44 (1992), 1907 1918.
9-ll0 Y. 'l'aurura, R. Kousaka,
9-81
antl V. J. Modi, "Practical Application of Nutation
Darnper fbr Supprcssing Wind,lnduced Vibrations of Airport Towers," J. Wilul
Eng. Ind. Aerodyn.,4l-44 (1992), l9l9-1930.
K. Miyashita et al., "Wind-Induced Response of High-Rise Buildings-Effccrs
of Corner Cuts or Openings in Square Buildings," J. Wind Eng. Ind. Aerodyn..
s0 (1993), 3t9-328.
9-83 M. Islam, B. Ellingwood,
9-84
9-85
and R. Corotis, "Dynamic Response of Tall Build
ings to Stochastic Wind Load," J. Struct. Eng.,116 (1990), 2982-3002.
T. Stathopoulos, "Low Buildings," in Wind Loading and Wind-lnduced Struc
tural Response , State,of'-the-Art Report, Committee on Wind Effects, American
Society of Civil Engineers, New York, 1987.
T. Stathopoulos, "Evaluation of Wind Loads on Low Buildings: A Brief His
torical Review," in A State of the Art in Wind Engineering, Ninth International
Conference in Wind Engineering 1995, Wiley Eastem Limited, New Delhi,
I
995.
9-86 A. G. Davenport, D. Surry
and T. Stathopoulos, "Wind Loads on Low-Risc
Buildings: Final Report opf Phases I and II, Parts I and 2, BLWT-SS8-1977.
Boundary Layer Wind Tunnel Laboratory, University of Western Ontario, Lon
don, Ontario, Canada, 1977.
9-87 D. Meecham,
9-88
9-89
9-90
9-91
"The Improved Performance of Hip Roofs in Extreme Winds
A Case Study," J. Wind Eng. Ind. Aerodyn.,4l-44 (1992),1717,1726.
T. Stathopoulos and H. Luchian, "Wind Pressures on Buildings with Steppcrl
Roofs," Canadian J. Civil Eng., 17 (1990), 569-571.
T. Stathopoulos and P. Saathoff, "Codification of Wind Pressure Coellicicnts
for Sawtooth Roof.s," J. Wind Eng. Ind. Aerodyn.,4l-44 (1992), l1Z7-fl't\.
T. Stathopoukrs and P. Saathoff, "wind Pressures on Rool.s of Vari6us Gc6rrr
etries," J. Wind Eng. Ind. Acruxlyn., 38 (1991), 213 284.
D. Surry and'l'. Stathopoukrs, "An ltxpcrirncnltrl Applr:rt lr lo llre licorrorrriclrl
Mcasurcrrrcrrl ol'Sllirlilrlly Avcnrgctl Winrl Lo:rtls," .l . Irt,l .lt,trnl .2 (lgjli,l
385 197.
ltrrtl .l . M:try, "Stlttttll;tttt'otts N4r':r:,rr, rrr,'rrt., oi I,lrrr lrr:rlrrrl, I'rt.ssltres
Usrrtg l'iczort'sislivr' Mrrllit'lrirnrrcl 'lrirnstllr( (.r:, ;r:, AgrPlll',1 to Atlrroslllrr'l.it. Wirrtl
'f'unltcl 'l'csts," .l . Wirttl l,,tt,q. lrttl. ..lr,rrnl\tt . $(r ( l(r()\), / I X(r.
It. J. Kincl, "Worst Sucliotts Nt'rrr lill't':; ol lil:r{
.1. Wind Eng. Ind. Acnxl.\,rt., .ll ( l()l.lli). .)51 .l(rl
liool lirps with
l)arapcts,"
ovcran Industrial
Building," J. Wind lhg. ltrtl. Aotrlt,rr., -llt ( 199 l), 325 333.
'f . Stathopoulos ancl H. l.rrt'lriiur, "Wind-lnduccd Pressures on Eaves of Low
Buildings," J. Wirul l,.ng. Irtrl. Acrtxlyn., 29 (1988), 49-58.
E. simiu and E. M. Hcnclrickson, "Design criteria for Glass cladding Sub-
jected to Wind Loacls," J. Struct. Eng.,ll3 (19g7),501_518.
E. Simiu and E. M. Hendrickson, "Wind Tunnel Tests and Equivalent l-Min
Loads for the Design of Cladding Glass," J. Wind Eng. Ind. Aerodyn.,29
(1988),49-58.
'tt)l.i
')
9-82 D. I.
Beneke and K. C. S. Kwok, "Aerodynamic Effect of Wind Induccrl
Torsion on Tall Buildings, J. Wind Eng. Ind. Aerodyn.,50 (1993), 271-280.
:lBl
')
J. E. Minor, "Perlormance of Roofing Systems in Wind Stoms," Proceedings
of the Symposium on Roofng Technology, National Bureau of Standards and
National Roofing Contractors Association, 1977, pp. 124-133.
(x) R. J. Kind and R. L.
Wardlaw, "The Development of a Procedure fbr the
Design of Rooftops against Gravel and Scour in High Winds," Proceedings of
the Symposium on Roofing Technobgy, National Bureau of Standards and National Roofing Contractors Association, 1977, pp. ll2 123.
100 T. L. Smith and J. R. McDonald, "Roof Wind Damage Mitigation: Lessons
from Hugo," Professional Roofing, Nov. 1990, 30-33.
') l0[ T. Smith, R. J. Kind,
and J. R. McDonald, "Hurricane Hugo Tests the Performance of Aggregate Ballasted Single-Ply Systems," ProJessional Roofing,
Aug. 1992,29-34.
't 102 T. Smith, R. J. Kind, and J. R. McDonald, "Hurricane Hugo II: Testing
the
Performance of Aggregate Ballasted Single-Ply Systems," ProJ'essional Roofing, Sept. 1992,32 38.
') 103 T. Smith, "Mechanically
Attached Single-Ply Systems," Profbssional Roofing,
Mar. 1992,14.
') 104 D. E. Shaw, "Better Uplift for Asphalt Shingles?" Professional
RooJing, Mar.
1993,30 32.
() 105 T. Smith, "Asphalt Shingles: The Importance of Corect Attachment,"
Professional Rrxfing, Dec. 1992, 54.
') 06 C. Kramer and H. J. Gerhardt, " What are the Effects of Wind on Tile Roofs? "
1
Professional Roofing, June 1993, pp. R7-R10.
') 107 c- Kramer and J. H. Gerhardt, "wind Loading in permeable Roofing Systems," J. Wind Eng. Ind. Aerodyn., ff (1983), 34j-358.
l0l]
r.
Smith, "Hurricane Andrew: A Preliminary Assessment," ProJessional Roof')
ing, Oct. 1992, 58.
') l(X)
R. McDonald, P. P. Sarkar, andH. Gupta, "Wind-lnduccd Loadson Mctal
Irdgc F'lashings," Wirul I)r,qitrecring, Prccecding.s, Ninth Inttnuttiotutl Crn
.li'rcrrt'c tn Witul I,.rt.rlirtr't.r'irt,q. Vol. l, pp.69 ltO, Wilcy litrstcr-n l,t(1., Ncw
.1.
I
)t:llri.
') Il0 li. A.
llirsk;rr';ttt:ttttl
() Ilull
I r':rltt:rliort ol'llool liitsl('n('ls trnrlt'r I)vrr:rrrrit'
382
9-11I
9-ll2
UUILDINGS: WIND LOADS,
Slllt,(; lt,llAl lll I]PONSF, AND DESICN
()l
lttX]l lN(i
Wind Loading," Wind Enginccrittg, l'nx'ttdirt14s, Ninth lntanuttiotuti (Iur.li,rence on Wind Engineering, Y<tl. 3, pp. l2O7*1217, Wilcy B,astcrn Ltrl., Ncrw
Delhi.
Y. Sun and B. Bienkewicz, "Wind Loading and Resistance of Lr>osc-Laid llrxrl'
Paver Systems," Wind Engineering, Proceedings, Ninth International Ctrtli,rence on Wind Engineering, Yol.3, pp. 1255-1266, Wiley Eastem Ltd., Now
Delhi.
R. D. Marshall, Wind lnad Provisions of the Manufactured Home Constru*
tion and Safety Standards-A Review and Recommendation for Improvemenl,
NISTIR 5189, National Institute of Standards and Technology, Gaithersburg,
MD,
t'
CHAPTER
1O
1993.
9-l13 R. Zhang and Q. M. Feng, "Vibration of Tall
Wind," Proceedings, Third International
Buildings under Turbulcnt
Conference on Stochastic Structurul
Dynumics, San Juan, Puerto Rico, Jan. 15-18, 1995.
9-114 T. Smith, "Hurricane Hugo's Effects of Metal
Tech.,2 (1990), 65-10.
Edge Flashings," Int. J. Rooling
SLENDER TOWERS AND STACKS
WITH CIRCULAR CROSS SECTION
slcrrtlcr towers and stacks are designed to withstand the effects of both alongwrntl and across-wind loads. The along-wind response can be estimated by
usrrg, the computer program of [9-14]. Simplified methods may be used if
irlrlrnrximate estimates of the peak along-wind response in the fundamental
rrrrxlc are sought. Since the gust response factor G depends only weakly upon
llre lirndamental modal shape (see Eq.9.1.10), it can be calculated by using
'l'irblc 9.1.5. The peak
along-wind response is then obtained from the relation
,\, ,,,,,^(Z) : G7{z), where.rl(z) is calculated using Eq. 5.3.4 (with t : 1).
'l'hcse procedures must be used with appropriate values for the average
rvrrrtlward and leeward drag coefficients C, and C1. For slender towers and
rt;rcks with a circular shape in plan, it may be assumed in all cases that c, :
( l. so that
the total drag coeffici ent Cp : C, . Information on the magnitude of
( j, irrrd its dependence upon Reynolds number, surface roughness,
and aspect
rrrtio is provided in Sect. 10.2.2.
sc:vcral procedures for estimating across-wind response are currently avail*lrlc. Among these, the procedure developed by Rumman [10-l] has been
rvrrlt:ly applied to the design of reinforced concrete chimneys. The basis of this
luoccrlure is largely intuitive. Nevertheless, it appears that the results of its
{rl)l)lication have been satislactory in practice.
ll is generally agreed that thc kriuling ancl response models inherent in Rumlll;tlls' procedure are not cntirerly crtttsistcrr( with aclvances made over the last
lrvo tlccaclcs in the ficlds tll' ttticttrtttt'lcotrrlogy, ircrrxlynamics, and acnrclastic-
tty. According to llG2l tltis corrkl irr t't'rlirirr silrlrtiotrs lcacl to thc unclcrcsti-
itutlitttt tll'thc acr<lss-wintl tcslxrttsc, lrrrrlit'rrlirrly in llrc scconcl nurtlc rll'vil'rr-ir.
ll(ttt. l)nrcctlurcs in wlticlt tllolt'rrrlvnrrt't'rl rrplrrrrirt'lrcs irrc tttilizcrl wcrc
383
384
:;t f Nt)t
developcd in
tt l()wl ltl; nNl) i;ln(.hl; wl lll{illl{:tll nlt (;ll()l;l; l;l (;ll()N
[0-31,
arrd by Vickerly, l]rrsrr. :rrrtl
('lalk in
[10-21 arrtl
r()
ll0
ztl ttr
\vlr(flc t) is ail tlerrsity
,.lt"vrrli9rr r.,,,. ('t is lltc
t.lt.virli6n e.'l'hc litr-cc
tlO-el.
The ESDU procedure [0-31 is bascd on a rrodified vcrsiort ol' lhc lrxrtlc:l
consisting of Eqs. 6.1.4 and 6.1.5. It considers two response rcgions, trnc itt
which the forces associated with the motion are ignored and another in whiclr
the effect of these forces is taken into account. The response of the structtllt
is estimated for each of these two regions, and the structure is designed fbr thc:
higher of the two responses. It is suggested in [10-4] that a drawback of thc:
ESDU procedure is the lack of a natural transition between the two response
regimes, which introduces an element of arbitrariness in the application of thc
r
ilr rMMAt.t
:,
t't rr'r.t trt
rttt
381=r
1p 1.J.5 kg/rrr'), l/(,,,',) is llrt'r'ttllt:tl wrtttl sllcctl itl
lill coollicicrrl, irntl /)(l') is lltt'tli:trrtt:let ol sttttclttrc itl
/'ir(:) is assruttcrl to l'rc pt'rtr't'll-y toln'littctl splrrtwisc.
lit'lcrcrrcc Il0-IIsuggcsts 2,,, : lt' wltcrrt: /r is tlrt'lrciglr( ol strttctttrc' ()thcr
r.li'rcrrccs suggcst 2.,., : 2l3h to -5l(r/r I l0- l0l, or t:,,, 213h ll0- I ll- 1'hc wind
,,1x.ctls U(2,.,) producc at elcvatioll 2,,, votlcx shotldirtg with licquencies equal
Ir) lhc natural f'requencies of thc structtlrc, so
U(2",)
: -I niD(2",) (t : 1,2,...)
(10. r.2)
proccdure.
'l'hc Reynolds number at elevation
Thc proccdurcs dcveloped by Vickery and coworkers imply in effect ther
lirllowing appnrach. A nominal response is calculated which coresponds trl
thc assurnption that acroclastic effects do not occur. The actual response is thcn
obtained through rnultiplication of the nominal response by an aeroelastic correction factor which varics continuously over the entire range of possible aenl
elastic effects. The derivation of that factor is explained in some detail in Sect.
6. 1.2. Because of uncertainties inherent in them, these procedures should bcr
used with caution.
Structures that are light in weight and have low structural damping (e.g.,
ceftain steel stacks) could experience unacceptably severe aeroelastic effecls
unless provided with aerodynamic or mechanical devices for the alleviation ol'
across-wind motions. Some of these devices have proven to be quite effective
and are routinely incorporated in the design of steel stacks.
This chapter describes Rumman's procedure (Sect. 10.1) and the procedurcs
developed by Vickery and coworkers (Sect. 10.2). These procedures are applicable to isolated structures.* Also presented in this chapter is informatiott
on aerodynamic and aeroelastic devices for the alleviation of the across-wintl
response (Sect. 10.3).
I
zn,
is calculated as follows:
I
G."
:
(10.1.3)
61,000 U(2",)D(2",)
inm/s andD(zn) inm). ForGe ) 3 x 106orsoitisusuallyassumed
0.220 to 0.25. From Eqs. 5.2.8, 5.2.10,5.2.14, and 5.2.16 it follows
tlr;rt lhe peak deflection for the structure excited in the ith mode may be written
tU(:.,.,)
:',
":
.i)
i
ir
Y,(zt
:
#?
D2e",)
\',o3ffi4
(10.1.4)
r,u,
rvlrt'rc y;(Z) is the ith normal mode of vibration, f, is the damping in ith mode,
i
*, :
Io,
(10.1.5)
m(z)Y?Q) dz
for design purposes [0-U.
According to [10-11] it was determined from observations that tall reinforced
( ()ncrcte chimneys with constant or nearly constant diameter do not appear to
, \l)cricnce unacceptably large motions if their Scruton number c;, defined as
PROCEDURE
In this procedure it is assumed that towers or stacks with a circular cross sectirllt
are subjected to a sinusoidal force per unit length with amplitude
Fo(z) :
iCrpLI2(2",)D(z)
-/
(l0. r. r)
2Mi
f, .
\
,itz.l ,tz
n
(10.1.6)
PD'(Z,,\
.Jrr
*If
several stacks are grouped in
I
il
t,
llrc generalized mass, and m(z) is the mass of the structure per unit height.
I .r rcinforced concrete chimneys ratios C1lfi = 13-16 have in many instances
r:,
Irr't'rr assumed
10.1 RUMMAN'S
t,
a row, buffcting fbrccs associatcd with vo(cx shctlclirrg tutt
r'; lrrlgor than filur.
'I'lrc pcak m()tncnt l( trrry clt'vlrlron.'ts rlotttitt:tlt'tl lty crlnlribtttiorrs tltttr ltr
rrrt.llitrl litrccs.'l'hcrrclirrt', rlt'rtolttt!: llrr'rt':rl':tr'tt'lt'tltliotl lt( clt'vitlioll.l lry
cause the response of stacks located downwind of thc first structurc in thc row lo bc irs higlr ir:
fourtimes the responsc ol'an iclcntical but isolalcd stack. l-irrri(txl tlulrr ort lltc rcsponso ol grottlx'rl
stacksareavailablcinllO(rl,ll024l,ll0-25l.liorallxrnrtrghlcvit'wrtl ittlirt'ttutlionutttl lilct:tlttrc
on interference antl proxilnlly cllccls on cylintlrictrl s(lll(lllr('s, scc l() II
),(.t), tlrC pclk tntttttcttl:tssrtt't:rlt'rl rvtllr
&.i
llrt'tllt tttotlt'ol
vilrr':tlitttr
is
386
:jLLNt)ftr towilri
ANt
) litA(:Ki; wt ilt oiltoUt An (in()tili :it (iil()N
trL,(z) =
trt(2,,)Yi(2.)(z.t
J.
*
t():'
( 10.
z.) dz.,
t.7)
l,l
t(x;l l)t,lll
1;
l)l vl lol,l l) ltY vl(;hl lty ANI) (;0w(lilt(t
nl;
387
Atterrrllts to ohlilin srrc'lr irrliu'rrraliorr lhlrtt wintl trrrtrtcl t('sls (c.9,.. irt ll0 l2l),
ilrI llr:norillly rrrrstrcct:sslirl, owirrg (o scvclt scirlc cllc'r'is. l,or llris rcilsott i( ltas
lrt'crr pointcd out that wincl tunncl sintullliotts ol'lltt: irctrrss-wintl ltchavior ol'
.,lt'ntlcr structurcs with circular cK)ss sccliort tttttlcr wittrl klads cannot bc uscd
lor tlcsign purposcs unless caretully intcrprc(orl irr tlrc light ol'acroclastic thcory
ph
J(Li({
=
12rn1)2
\ *(r,)yik)kr o1
;rrrrl ol'clata obtained f-rom
z)
dz1
(
full-scalc tcsts
ll0-lil.
l0. r .n)
10.2.1 Basic Approach to Estimation of the Across-Wind Response
The shear force at elevation Z1 ma! similarly be written
Si(21;
= (2rn;)2
Io,
*r',r',r,,,
or,
as
(10. r.9)
Example Consider a chimney with constant circular cross sectigrr
for which D : 17.63 m, h : 193.6 m, and n1 : 0.364 Hz [10-9]. It is
assumed CLlh : 15, y{zlh) : 17lh)t67, m(z) :58,000 kg/m for z < hlL,
m(z) :41,000 kg/m for z ) hl2, and S : 0.22. From Eq. 10.1.5, M, =
1.87 x 106 kg. The critical velocity in the first mode of vibration is u(zu,) ..
29.15 m/s (Eq. 10.1.2). The corresponding Reynolds numberis G" : 3.4 x
107 6q. 10.1.3). The peak."rpon*i at elevation zis y,(7) : O.s\(zlhllbT nt
(Eq. 10.1.4), and the peak moment at the base is Jll(O) = l.l7 x 106 kNrn
(Eq. 10.1.8). If it is assumed fr : 0.02, then the Scruton number cr :4.3
(Eq. 10.1.6).
Lt'l o',1)""(z) denote the rms value of the nominal across-wind response at elev;rtiorr z in the ith mode of vibration. The rms value of the actual across-wind
r('slx)nsc at elevation z in the ith mode is denoted by ou;(z). The following
rrlrrtion holds:
Numerical
o,i(z)
:
(r,
.
,,--r;"
o]?
(z)
(10.2.1)
ivlrt'r'c fr is the structural damping ratio, far is the aerodynamic damping ratio,
;rrrrl l(,/(i I (",)lt'' is the aerodynamic correction factor in the ith mode.
Estimation of Nominal Across-Wind Response. The nominal across-wind
r('slx)nse is obtained by subjecting the structure to the across-wind aerodynamic
loirtls it would experience if it were at rest. No aeroelastic effects are taken
rrr(o account, and the only damping that affects the motion is the structural
rl;rrrrping.
10.2
It
PROCEDURES DEVELOPED BY VICKERY AND COWORKERS
was mentioned earlier that these procedures may be viewed as, in effcct,
estimating the across-wind response in two phases. First, a nominal responscr
is calculated by assuming that the structure is acted upon by the across-winrl
aerodynamic loads it would experience if it were at rest. The nominal response
therefore does not reflect any aeroelastic effects, since the latter involve loads
associated with the motion of the structure. The actual response is obtaincrl
through multiplication of the nominal response by a correction factor that ac
counts for the aeroelastic effects. The approach used to estimate the nominll
response and the aerodynamic correction factor is described in Sect. l0.z.l.
Information on the requisite aerodynamic and aeroelastic parameters is proviclctl
in Sect. 10.2.2. Approximate expressions for the across-wind responsc arr
given in Sect. 10.2.3.
It is emphasized that, although the procedures presentccl in this section 1re
conceptually advanced, they yield results that may bc rrnccr(1in t9 within irl
least 3O%. This is the case in part because the structurirl tllrrrrgling is in lrosl
cases poorly known. In addition, much of thc availlblc irrlirrrrrirtion conccrnirrg,
the aerodynamic ancl acr<lclastic parillnctcrs (soc Set't. l0.l.l) is orrly lcrrltrliver.
ln a turbulent flow the structure at rest would experience a superposition of
tw() ilcross-wind loads. The first of these two loads, due to vortex shedding in
tlrr' wake of the structure at rest, is denoted by L{2, r). The second load, due
lo tlro lateral turbulence in the oncoming flow, is denoted by L2(2, r). The load
L1(;-, t) can be written in the form
LrQ, t)
:
)pC{2, t)D(z)U2(z)
(10.2.2)
,,o that its spectral density is
Sy,(2,
n) : l*pDQ)Ut(z)l2Sc,Q, n)
(10.2.3)
Atcrrrding to [10-2], measurements indicate that the spectral density Sgr(2, n)
lrc represented by thc bcll-shaped function
, :rn
nt'{},'
"'
J
n|,,,,,"^n[ I
l-:-vull]
(to
2'4)
u,ltctc rt tlcnolcs lhc lir:t;trcrrt'y, r,i l.i lltc votlcx slrcrllling l'rcqucncy givcn by
llrc tclrrliort
388
sr t NDFn
towlnri
l
ANI) s |ACK!; wt il
1.0
4
=
10.'l
=
0.1 83
(:il
tcul Ar1 cno$ri st c iloN
t0? Ilt(x;l l)lllll r; l)tvl loPIt) llY vloKl ltY
I,
nr
ttS, (n)
)(t)
tp()1Jl'!17.1
r'(i,
I
ANI) oowolrKElr$
/)
389
(r0.2.8)
I(2.\
Spcctral and cross-spcctrul irrlonrurtion on thc lateral velocity fluctuations a(r)
rs givcn in Scct.2.3.-5. lnlilrrrration on the aerodynamic parameters B,3,
cl
t'i"t, o, and Cp is givcn in Scct. 1O.2.2.
'l'he mean square valuc of the nominal response induced by each of the loads
/,1(/) and lo(t) can be estimated as in Sects. 5.2.7 or 5.3.2.The mean square
virluc of the total nominal response is equal to the sum of the mean square
vrrlrrcs of the responses due to the loads L(t) and Z2(r). However, because
tlrcsc loads are uncorrelated, the peak value of the total nominal response is
It'ss than the sum of the individual peak responses due to Z,(r) and LzQ).*
0.1
0.01
10 2-
10-1
1
ttD
U
FIGURE 10.2.1. Power spectral density of lift force coemcient c. measured on Hamburg television tower. From H. Ruscheweyh, "wind Loadings on the Television Tower,
Hamburg, Germany," J. Ind. Aerodyn.,
I
Estimation of Aerodynamic Correction Factorl!,lftt + e"ill't'. One of the
tlrlliculties that arises in the estimation of the aerodynamic correction factor is
llur( relatively little reliable information is available on the structural damping
rrrtios f;. Ranges of values fi suggested in [10-7] are listed in Table 10.2.1.
(1916),315-333.
TABLE 10.2.1. Suggested Structural Damping Ratios
"
SU(z)
D(z)
(10.2.5)
s is the Strouhal number, and B is an empirical parameter that determines
Type of Structure
Structural Damping Ratio
Unlined steel stacks and similar structures
Lined steel stacks
0.002-0.010
0.004-0.016
0.004-0.020
Reinforced concrete chimneys and towers
the
spread (bandwidth) of the spectral curve. This model is compatible with results
of full-scale measurements (Fig. lO.2.l).
The cross-spectral density ofthe load L{2, r) can be expressed as [10-41:
Sr,(zr, zr, n1
:
Sl,2Q1, n)St/,2(22, n)Rs(21, 22, n)
Ro(zr, zz)
:
cos(2ar)exp(-arz)
r:
.rl- _ 1-21
-l4l
D(a) +
(10.2.6)
(r0.2.7a)
I
D(72)
(r0.2.1b)
The parameter a in F,q. 10.2.7 a is a measure of the decay of the cross-spectral
function Sr,(4, zz, n) with the distance lz, - zrl. Associated with the parameter
a is a correlation length z which is a measure of the spanwise length beyoncl
which the force fluctuations are no longer correlated.
The lift force Lr(t) is the projection on the across-wind direction of the drag
force induced by the resultant of the mean velocity U(2.) and of the lateral
turbulent velocity u(2,, t).In large-scale turbulence this lorcc hus an anglc ol'
attack with respect to the along-wind direction equal kt t,l L/, irrrtl its pr<r.jccti<ln
on that direction is
r'lt is of interest to estimate the extent to which the effect of the load Lz(r) is significant from a
1rr;rtlical point of view. Using the information of Sect. 2.3.5, it can be verified that the lateral
vr'krt ity fluctuations differ from the longitudinal velocity fluctuations as follows: (1) the ordinates
rrl rlrc spectral density at high frequencies are largerby 33% forthe lateral than forthe longitudinal
llrrr'trrations, (2) the area under the spectral curve is lower by 50% for the lateral than for the
l,rrr1'.itudinal fluctuations, and (3) the exponential decay coefficients are lower by about 33% for
tlrr' l:rtcral than for the longitudinal fluctuations. Calculations then show that the peak nominal
,rr rrrss-wind response due to l4(r) is of the order of 5O% of the peak fluctuating part of the alongrvrrrl rcsponse or, roughly, abottt 25% of the peak total along-wind response.
It lirllows that if the ratio between the along-wind response and the nominal across-wind
rr'rlx)nso estimated without accounting for IaQ) is small, then taking L2Q) into consideration will
lrirvc rr negligible effect on the magnitude of the nominal response, particularly in view of the
l;rcl rrrcnlioned earlier that L'(t) and Lr(t) are uncorrelated and that their peak values are therefore
rrot rrtltlitive. On the other hand, il'thc ratio of along-wind to nominal across-wind response is
lrrplr, lhc design will be govcrncrl by thc along-wind response regardless of whether Zr(l) is
,t((r)untcd fbr or not. Finally, il'lltc lirlio rrrxlcr considcration is close to unity, accounting for
/ ,(l) wrruld incrcase thc pcak notttitt:rl irt't-oss wintl rcsponso hy ahout 25% if Llt) and Lr(r) wcrc
i ()r('lltcd; howcvcr, sincc this is rrol llrr' trrst', lhc irrcrcrrsc will only bc of thc ordcr ol' l0 to
lr',?,. lirrthcsc roasons,1oa lilsl nl)l)roxrrilll()n, llrc lirt'c inrlucctl by latoral turbulcncc lluctuir
trrlns nlity bc ncglcclcrl, ttttlcss lltc clilnuulr'(l ;x'rk rrkrn;i wintl:rtttl irt'toss-wirttl tespottse lt:rvr'
trl)l)r1)xinritloly lhc sirrrrc virlrrc, itt wlticlt r'irrf lltr ir|ilrrs wirul rcs;xrrrst' slr()rrltl lrt'itrr1;rrrcrrlcrl lry
rorrpllrly l0%.
390
SLENuEn towfti.ri AND sIACKsi wt ilt oill(:r,t An cn()ri$ sfcTtoN
t0
The approach to the estimation ol' thc ar:nrdynarnic darnping ralio 1,,; is rlc=
scribed in some detail in Sect. 6.1.2. Inlbrmation on the acroclastic parartrcter
K,6 needed to estimate l,; (see Eqs. 6.1.36 to 6.1.38) is summarizcd in Soct.
t0.2.2.
? I'll()(il l)lllll li
* =,.[,,.r.,
,(x)71
l)t vn
',,r,,(r)
ol)fl)
nY vloKl ltY nNl) c()woltKt
_,rl]
(n,.= 2
x r0('
The purpose of this section is to provide information on the drag coefficicrrt
Cp, the Strouhal number S, the rms lift coefficient Czytt2, the bandwidth parameter B, the parameters describing the spanwise correlation of the acrosswind load, and the aeroelastic parameter K"s used to calculate the aerodynamic
damping ratio f,,,.
6t,OOO U(dD(z)
(10.2.9)
where u(e) is the wind speed at elevation z in m/s and D(z) is the outside
diameter in meters; upon the turbulence in the oncoming flow, upon the aspect
ratio h/D(h) , where h is the height of the structure and D(h) is the diameter at
the tip and upon the relative surface roughness klD of the structure, where ft
rirlio as follows:
(10.2.12a)
(to.2.rzb)
h is the height of the structure and D(h) is the diameter at the tip
t3l).
wlrrrro
llo
RMS of lift (
irrc suggested
is the height ofthe roughness elements. For steel stacks and reinforced concretc
chimneys and towers l0 3 < kld < 10-5 [10-7]. It is assumed herein that
klD vaies only within this
range.
o.otr
(, - hl
(10.2. t0)
where Cf, is the value of the drag coefficient taken from Fig. 4.5.5c. Fronr
elevation h - D(h) to the top of the structure the drag coefficient may bc
assumed to have the value Co : 1.4 Ci for all structures regardless of aspcct
ratio (see [l0-l3l). The main effect of turbulence in the oncoming flow is to
decrease the Reynolds number corresponding to the onset of the critical region
defined in Fig. 4.5.2.
strouhal Number. The following values of the Strouhal number
in [10-13] (see, however, [4-86] and Fig. 4.4.4):
C7''". The following values of the rms
purposes (see [10-13]):
(see
lift coefficient
G,.<2x105
(t}.2.l3a)
2x10s1Ge72xtO6
(10.2.13b)
I
/ ft\ 12)
+0.03515 + log,o(=)l
"',"\D/l) t
Drag Coefficient Ce. The dependence of Cp upon Reynolds number anrJ
surface roughness is represented in Fig. 4.5.5c for cylinders with aspect ratior
hlD(h) > 20. Forstructures with aspect ratios l0 < hlD(h) < 20it may be
assumed that up to the elevation h - D(h) the drag coefficient has the valuc
Co: c'o[t -
(r0.2.nc)
lirrr2 X lOs < 61" < 2 x 106 thc vortcx shcclcling is random, and the Strouhal
Eq. 10.2. I I b corrcsponds to the predominant frequencies of
tlrt' lkrw in the wake. In Eq. 10.2. llc the coefficient c depends upon aspect
The aerodynamic and aeroelastic parameters depend upon the Reynolds num.
bcr
:
391
rrrurrbcr given by
10.2.2 Aerodynamic and Aeroelastic Parameters
G..
lti
G" > 2
I
x
106
(10.2.13c)
lrr
liq.
10.2.13c the coefficient d has the expression
= 12
d-
<
h
olny
< tz
(10.2.14a)
(lo'2'14b)
I lrt' lilt coefficient also appcars to depend significantly upon turbulence intenrrty. However, little inlirrrrrirlion on this dependence is available to date.
are suggestcd
S:0.20 G"<2x105
0.22 <,S < 0.4.5 2 x 105 ? (11,, :- t v l0('
Bandwidth Parameter
(10.2.11il)
(l0.2.llh)
B.
l{cli.r'r'rrt'tr l0-4 suggcsts that
ll
00t( , )tt'
,,,
(
r0.2. r5)
392
.LENDER
towflis
AND !.itAoK:i
Wt
llt (;i,(;(,t An (i'oss tiE(;iloN
r()
I
I'ir( xri
L,'
where u'is the mean squarc valuc ol' krrrgitudinal turbulcncc lluctuations
the mean wind speed. According tu ito-e1, for practicar purposes
'rrtl
it rrruy
be assumed B = 0.18 for all flows.
uis
At:\
(10.2. t6b)
obtain Koo:
U
-,.nt)
t
", (r t
*",
(+,)
u- < o'85
(t0.2.t7a)
<
(10.2.r7b)
o.8s
{,.,.0
t.o =
t.t
(10.2.17c')
",(rr,-,t)
,,=[,<r3
(10.2.17d)
",(oou -,rtt),,
o
=
nti
393
(tO.2.ZOa)
*
(10.2.20b)
(t0.2.21)
lro
J-rt.t
D(h)
(ro.Z.z*a)
| 1.0 - 0.(X |/ 12.5 - *n \I
D(h\/
\.
\
r
< 12.5
==
D(h)
(to.2.22b)
lir;rrirtions 10.2.20 reflect the fact that if the wind speed at 10 m above ground
ir, rclatively low the atmospheric turbulence may be weak. This can lead to a
t onsiderable enhancement of the aeroelastic effects (see Fig. 6.1.10).
t0.2.3 Approximate Expressions for the Across-Wind
Response
l'lrc across-wind response in the ith mode of vibration may be estimated as
o.55a,
=
12
oo:\J
(10.2.16u)
Aeroelastic Parameter Kro. on the basis of tentative information from
[10-4] and [10-13], the foilowing approximate expressions may be used to
I
U(10 rrr)
lr.u
lIto u(lonr)= 12r
at:oe+o2|'"r,,,(f) *rl
spanwise correlation Parameters. For Reynords numbers G" > 2 x
rot
it may be assumed that in Eq. 10.2.7 the coefficient a : l/3 and that to
this
value-there corresponds a correlation length L
D t10-41. For G" < 2 x
l0'. I = 2.5 [lO-14]. Then, using the noration =g : LID,"
^ : (z.s G."<zxtos
" [ ,.0 G" > 2 x ro5
ilunt li Dfvf I Ol'il) try vr(;Kr ny ANt) (;()w()llK[
{,<
{,<
1.84=t
r
84
ori(z)
:
t?t''y,(:r)
(r0.2.23)
Y,(z)
:
gyioyi(z)
(10.2.24)
Byi
:
C2t/2 _
sr
(10.2.17e)
r2ln(36oon;)1,,,
*
,ffi;,g,
I
ll/2
lisC2 r/2 |
snom'r
L(r, + r",li
0o.z.2s)
(t0.2.26)
I
ph
Si(z)
(to.2.t7tl
= 12rn;\2 ), mk)Y/z)
(r0.2.27)
dz1
ph
SlLik)
where
A1
(
at:
:
t.O
1.8
[r o
I
1
(10.2. tn)
Q(L2A3Q4
G"
(
lOa
< G" <
ros < G"
toa
(10.2.
los
19n)
(r0.2.19h)
(10.2. t9c)
= 12rni\2 ), ^Q)r,(r,)kt - d
dzt
(10.2.28)
rvlrcrc oni(Z) is the rms of the deflection at elevation z in the ith mode of
vrlrlrrtion, t?''' it the rms of the corresponding generalized coordinate, li(z) in
llrc ith rnodal shape, )i(z) is thc pcak deflection in the ith mode of vibration,
,r;,., is thc peak factor, a1 is llte rtirtural I'rcqucncy in the ith mode in Hz,
,i,,,,.,"' is thc rms norrrinirl ge rrcrirlizr'tl crxlrtlinalc in thc ith mode (which
tollr:sponds to thc rcsprrtsc t'sl itttrrlt'tl lry itssrrrnirrg tha( no acroolaslic cfl'ccts
j-
394
SLENDEII
lowEllsi ANI) silAcKli wl lll (illl(;trl All (;liosis
t0,,
SFCTI()N
-t
occur and that the motion is all'cctcd Only by structural damping), li/(f,
in
thc
l")\t,, is the aeroelastic correction f'actor, f, is the structural danrping
ith mode, f,, is the aerodynamic damping in the ith mode, si(z) and sT[;(z) arc.
respectively, the shear force and the bending moment at elevation z due to thc
acrbss-wind response in the rth mode, and m(z) is the mass of the structure pcr
unit length.
To estimate the across-wind response, expressions a19-ne^ eded for the rms
of the nominal generalized coordinate in the ith mode, tiu.,i"'. and the aentdynamic damping in the ith mode, f,,. These expressions are given below
r"purut"ly for (1) structures with constant cross section and (2) tapered structuies. In both cases the expressions are valid only for relatively small ratios
or;(h)lD(h), firr e xample 3% or less (to which there would correspond negligiblo
valucs <rl'thc sccont.l tcrm within the bracket of Eq. 6.1 .22).It is noted that,
in practicc, thc tlcsign of a structure will be acceptable only if the ratios or(h)/
D(h) inhcrcnt in that dcsign arc indced small.
Ntmerical Examplo ('onsidcr the chimney described in the numerical exirrrr;rlc ol'Scct. 10.I (h : 193.6 m, D : 17.63 rrr, n1 : O.364Hz,y(zlh) :
(.'lltltl'7, m(2.) :51i,000 kg/m for z < hl2, m(z) :41,000 kg/m for z > hl
). Mr : 1.87 x tO6 tg;. It is assumed fr : 0.02, klD: 10-s, and zs :
ll0-5 rn. We seek the response in the first mode.
Assuming tentatively that 3 : 0.22, the critical wind speed at elevation
thl6: 161.3 m is u...r :0.364 x 11.6310.22 :29.16 m/s (Eq. 10.2.31),
to which there corresponds a Reynolds number G..:3.4 x 107 >
tlit1. 10.2.9). The aspect ratio is hlD = ll. It fbllows that
3=
lt,
.lt
Structures with Constant Cross Section. The following
approximatc
proposed in
were
10.2.1
in
Sect.
described
expressions based on the approach
vitzt dz
=
LY#t
:
1
2
x
l0('
lc, lO.2.l2b)
(Eq. 10.2.16b)
(Eqs. 10.2.13c and IO.2.l4b)
44.7 m
r
ffi''':
[10-e]:
,z
(Eqs. 10.2.
0.178
S:1.0
4t'' = 0'143
I
ffi
I'n(x:t l)unl ri t)[vLLopEu By vtcKERy AND cowoRKERS 395
u(10)
(Eq. t0.2.29)
0.115 m
> 12 -m
(Eq. 10.2.31
and, 70.2.32)
S
*El,
\oo
r?u, o,]'''
(r0.2.29)
K'o(l) = 0.465
(Eqs. 10.2.17c, 10.2.18, 10.2.19c,
lO.2.2Ob, 10.2.21, and 10.2.22b)
Soi
=
-#
*'""' \oo'?u> o'
(10.2.30)
1.25 kglm3), Mi is the generalized mass in the
where p is the air density (p
ith mode (Eq. 10.1.5), and D is the outside diameter. The critical wind specd
corresponding to the ith mode of vibration has the expression
-
fl,D
It
vcr,r - _L
lnt l0/zn)
tnr(y6ieru"''
ancl Sccl. 2.4.1).
EtD
-
0.130 m
(Eq. t0.2.26)
(Eq. 10.2.25)
/
\
l7
o,rk):or:o(re:..1
/
(10.2.32',)
1I(,(0)
.
:
1150
x
l-67
) ^
\167
) ^
106 Nm
.Eq.to'2.23)
(Eq'to'2'24)
(Eq. 10.2.28)
Notc that the results of the calculations depend strongly upon, in particular,
llrt' rrssumed value of the structural damping ratio fr . Had the value f1 : 0.01
lrt'err appropriate, the rcsults oblainccl would havc been larger than those obr;rrrrctl in this example by a lirclor ol'l(0.02 - 0.(n43)/(0.01 - 0.0043)lt'' =
I
'l'iltlc 2.2.1
(Eq. 10.2.30)
Ytz):o'st(u:.01
where h is the height of the structure in meters ancl z1y is lhc: rottghncss lcngllt
in meters for the tcrrain that detcrmines thc wind pnrlilc ttvcr' lltc uppcr half
of the chimncy (scc
-0'0043
(10.2.3 r)
Information on the structural damping ratios f; is given in Table 10.2.1. lnformation on the parameters, 3, Ctt'', S, and Krs is given in Sect. 10'2'2'
Note that in Eq. 10.2.2Oathe speed U(10 m) corresponding to the ith modc irr
m)
:
8lr:3'94
S
u(lo
f"r
()().
fapered Structures. 'l'lrc lirlhrwirrp. rrlrlrlrxirrrirlc cxprrssions hirsotl orr lhc
irlrprlach dcscribccl itt
Sert't
.
10.,'.1 wr'rc ;ttulxrsr'rl irt
ll0
()l:
396
st LNDEn
towl
fitAcKli wint (;lt(;t,t An cno$li sfcloN
HS AND
t?o^.,(2",)t'' =
O.o I 6C i.t
t
)
JJt
/2
(
r)/42M,Ptt2(2",)
. - 0.lD(2",) dD(dl
A - d, 1,,,
: -#
J,
(ffi)["#]'
",
r0.2.33)
(t0.2.34)
v?(z)
az
(r0 2
3.5)
where the notations of Eq. 10.2.29 are used, Do : outside diameter at basc,
is the elevation corresponding to the critical velocity
2,.,
lJrr(Z",)
u(z; 2",)
: ryd,
ln(zlzo)
:
Urr(Zn,)
ln(zn,lzo)
(10.2.36)
(10.2.37)
Since, as in Eq. 10.2.26,
- L )'"
*;/t"' (\fi * hik"')/
it follows that the maximum response in the ith mode corresponds to the max.
imum value taken on by the function
F,(2",)
:
Da(2",)yi(2",)
{P(z")Ki
r
(,,(2" )f\t/z
vltoPf t) tlY vl(]KillY
ANt)
cowonKElls 397
:
365.8 m, outside
rlirrrrrotcr at thc baser /)o .17.8 rrr, outsidc diarneter at the tip D(h) : 72.6 m,
t'trrrstant tapcr (i.c. , dl)(r,)ldr. - lDo - D(h)llh), fundamental frequency n1
0.252 Hz ll0-91. ll is assurncd that the fundamental modal shape y(zlh) :
t.'lhf .the mass pcr unit lcngth m(z) :180,000(l - O.9zlh) kg/m, the strucIrrrul damping ratio in thc first mode f1 :0.01, the relative surface roughness
rrl lhe structure klD - lO-t, and the roughness of the terrain z0 : 0.008 m.
We: seek the response of the chimney in the first mode of vibration.
Assuming tentatively S = 0.2, the critical speed U". > 0.252 x 12.610.2
15.9 m/s (Eq. 10.1136), to which there corresponds G" > 67,000 x 15.9
". 12.6 = 1.3 x 107 (Eq. 10.2.9). The aspect rario is hlD(h) = 29.0. k
krllows that
S
=
(Eq. 10.2.11c)
0.23
4', : o.l5
and e6 is the roughness length for the terrain that determines the wind profile
over the upper half of the chimney (see Table 2.2.1 and Sect. 2.4.1).
t',{r1''':
t)t
Numerical Exampla ('orrsitlcr ir chirnncy with hcightll
p Da (2..,) y,(2.,,,)
lrlz,,1
t.i(2,,)
l0r I'tt()t:f t'{[lt ii
(10.2.38)
To determine that value, it is in practice necessary to calculate F(2",), and, in
particular, loi(Zr,), for a sufficiently larger number of elevations O I 2", < h.
As pointed out in [10-8], if the structure is very lightly tapered (i.e., if
dD(a)ldzl,:,,.. and therefore p(2",) is small-see Eq. 10.2.34), rhen the approximations on which F;q. 10.2.33 is based are no longer valid and Eq. 10.2.33
ceases to be applicable. In that case the chimney is assumed to behave as if it
had a constant outside diameter D equal to the average diameter of its top third
[0-9], and Eqs. 10.2.29 to lO.2.3I are applied with the same values of rhe
parameters E, C'rt'', and S as those used in Eq. 10.2.33. In practice, it is
therefore necessary to calculate both the value of the rcsponsc yiclde<t by Eqs,
10.2.33 and 10.2.35 and the value yielded by Eqs. 10.2.2q ancl 10.2.31. lt
follows from [0-21 that the response to be assunrccl lor sinrctrrrirl rlcsign purposes is the smal.l,er <ll'thcsc tw<l valucs.
S:1.0
Mr : 3.3 x
4r : 1.0
az : 0'9
q+ : 1.0
(Eqs. 10.2.13c and lO.2.l4a)
(Eq. 10.2.16b)
106 kg
(Eq. 10.1.5)
(Eq. 10.2.19c)
(Eq. t0.2.21)
(Eq. 10.2.22a)
l'lrc coefficient a2 (Eqs. 10.2.20) depends upon the wind speed U(10 2",).
As mentioned earlier, the function 4(2",) must be calculated for a sufficiently
lrrrgc number of elevations zei to obtain the value that maximizes the response.
Wr: show here calculations for z"r:365.8 m and 2",: 182.9 m'
lior 2", :365.8 m, U".(365.8) : 13.70 m/s (Eq. 10.2.36) and U(10; 365.8)
9. I m/s < 12 m/s (Eq. 10.2.37). Therefore az : 2.0 (Eq. 10.2.20a). It
tirrr be verified that f"1(365.8) = -0.0065 (Eqs. 10.2.17 and 10.2.35), and
/,1(365.8) = 1.6 x 106 ma (Eqs. 10.2.38 and 10.2.34).
ltor zn, : 182.9 (n, U,, : 27 .61 m/s (Eq. 10.2.36), U(IO; 182.9) : 19.16
ttr/s ) 12 mls (Eq. 10.2.37), az : 1.0 (Eq. l$.z.2$b), f"tQ82.9) : -O.OO42
tlitls. 10.2.17 and 10.2.35), F(182.9) = 4.6 x 106 ma 1Eqs. 10.2.38 and
10.2.34).It can be verified that the largest value of Fr(2",), and therefore the
lrighcst response in the first mode occurs for 2", = 182.9 m. It follows that
{i,,,,,. r(182.9)'/2
:
o.Q6 m
t',08Lg1t'': 0'079 r'
o,tk)
: o.ole( ' \' ln
\.l(r5.tl /
(Eqs. 10.2.33 and 10.2.34)
(Eq. 10.2.26)
(Eq. 10.2.23)
sltNuEtl lOwillli
ANI) :i r n(;Kli wr il
l
oil rct,l
) n.r
36s.8 /
\ ==
Sltr(O) : l29O x 106 Nm
Ytk) :0.304(
The response
Al
l 0lrolili ril o l l()N
ilt
(Eqs. 10.2.24 dnd t0.2.25)
(Eq. 10.2.28)
will now be estimated by using Eqs. 10.2.29-10.2.31. Thc av'
of the top third of the chimney is D : 16.8 m. ll
erage outside diameter
follows that
I22-tr2_
i,,n,. r' =
0.035 x0.15
ra ,.
*,
:0.0625
x
1.0 1.25
x
16.8t
q2l - j-n.*
/ .. ^365.8\r'2
( 16.8 5 /
m
Uu: l8'4-
(Eq. 10.2.31)
t2-m
(Eq.10.2.32)
S
1.25 x 16.82
t^,: --(0.9
3.3 x 10"
= -0.0039
x
il vtAiloN ot vollll x tNt)t,clrtj ofi(il
t.AiloNS
'l'lrtr ltclicitl sltitkc syslerrr consisls ol'throc thin rcctangular strakes with a
prtt'h ol'onc rcvoluliorr itr .5 ilialrrctcrs and a strake (radial) height of 0.10
rllurrclcr (to 0. l.l (liilnlertr:r lor vcry light or lightly damped structures) applied
rrvcr lhc t<tp 3301, b 40%, ol'thc stack height. The effectiveness of the system
rs rrot impaired by a gap of 0.005D between the strake and the cylinder surface
llt) 16l. Referencc ll0-l7j reports the remarkable results obtained by using
llris system (with 5-mm thick strakes, 0.6-m strake height, and 30-m pitch) in
llrt.case of a 145-m tall and 6-m diameter steel stack (Fig. 10.3.1).
lr<rr Reynolds numbers 6le 12 x l}s or so, in flow with about 15%
(e.g.,3% to 5% of the diameter), the vortex street reestablishes
'ilnl)litudes
ilscll', and the aerodynamic devices become ineffective t10-301. It is noted that
llrc strakes increase drag, as shown in Fig. 10.3.2 t10-181 .
Shrouds can also be effective in reducing the coherence of shed vortices. A
r,thcrnatic view of a shroud fitted to a stack is shown in Fig. 10.3.3. Results
S
u(ro) >
At
Irulrrrlcnce intensity, helical strakes were found to reduce the peak of the acrossrvrrrtl resonant oscillations by a factor of about two, as opposed to a factor of
llrorrt 100 in the case of smooth flow [10-23]. It appears that the performance
ol s(rakes can be unsatisfactory in the case of stacks grouped in a row [10-28,
l(l 291. Also wind-tunnel and full-scale tests indicate that for large vibration
(8q.10.2.29)
m
:t
365.8
0.55)5
(Eqs. 10.2.30, 10.2. l7 c, 10.2.18,
lO.2.l9c, 10.2.20b, 10.2.21,
10.2.22a)
/
orlk)
:0.0?e(#J)
-
Y(z)
: o'm+(fr)
12
t
fltr(o)
:
l28o
x
(Eqs. 10.2.23, t0-2-26)
r2
lo6
'
Nm
(Eqs' t0'2'24,10'2'2s)
(Eq. 10.2.28)
It is seen that in this case the response
yielded by Eqs. 10.2.29-10.2.31
ix
approximately the same as that obtained by Eqs. 10.2.33-10.2.35.
10.3 ALLEVIATION OF VORTEX-INDUCED
OSCILLATIONS
Aerodynamic Devices
A common method of alleviating vortex-induced oscillations is the provisiorr
of "spoiler" devices that destroy or reduce the cohcrcrrcc ol'thc shcd vrlrtice:lr
[10-26, 10-27]. Of the various types of such dcviccs, ort('ol lhc rrrost oflcctive
,lrr'wt'ylr, "lrLrll-Scalc Mclsrrrcrrrcrrls orr Slccl ('hinrncy Stacks," ,l . [ru|. Aerodyn.,l
is the helical strakc syslcm lirst clcscribctl in ll0:1.51.
t I t, /(r
l,
l(;llltll
10.3.1. Stccl clrirrrncy witlr hclictl strakes. From G. Hirsch and H. Rus-
). l4l
147
.
400
SLENDEII tOwr..n..j ANt) StA(;KS
-o
Wt
lil C[l(]tjtAU
CROSS SECION
ilr
E
o
!
J,
401
0 0.52D to 0.070D).
.E
r.o
c
o
!
T/D =
O.12
T/D =
O.06
Mechanical Devices
!
Srrch devices include hydraulic dampers and tuned mass dampers (TMDs).
()a
E
s
.,rrlrstirrrlially rcducod witlr only thc top 25'I, ol'lltc tnorlcl hc'i8,ht shroudcd. The
rrrrrsl cll'cctive shrouds wcrc l()und to bc lhrtsc with l gap width w = 0.12D
;rrrtl an open-area ratio between 2O%' antl .l(r%, (with lcngth of square s :
E
,9
.o
.9
ilr Nor
ol wirxl tunncl cxpcrinrenls rclx)ilo(l in ll0-l(rl slrowctl thirt oscillutions wcrc
1.5
o
a
il
0.5
Plain cylinder
o
O
o
G
O
105
106
Reynolds
107
numberl)ttl
FIGURE, 10.3.2. Effect of strakes on drag coemcient. From L. R. Wooton and C,
Scruton, "Aerodynamic Stability," \n The Modern Design of Wind-Sensitive Stru*
tures, Construction Industry Research and Information Association, London, U.K,,
1971, pp.65-81. By permission of the Director of the National Physical Laboratorl,
U.K., and the Director of the Construction Industry Research and Information Association, U.K.
'l'hc use of hydraulic dampers to reduce vortex-induced oscillations is dislrrssod in [0-19]. An example of such an application is given in [0-17],
wlrich mentions the use of three hydraulic automotive shock absorbers installed
;rl 120" angles in a plane view between a 47-m high stack and a separate
tr'ucture at the 18-m level.
'l'hc tuned mass damper (TMD) consists of a secondary vibratory system
iruirched to the structure and located near its top (see Sect. 9.4). If excited by
lrulnonic (or quasiharmonic) oscillations of the structure, the TMD will vibrate
rrr opposition to these motions and thereby reduce the amplitude of the structural
rrslx)nse. The basic theory of the TMD is discussed in [10-20]. One of the
Irrs( tuned mass dampers used in a large structure was designed for the Centerpoirrt Tower in Sydney, Australia. The mass for the damper was in this case
pnrvided by the water tank of the tower t10-2U. Further applications of TMDs
trr rcduce tower oscillations are discussed in 19-791, |0-221, and [13-91].
REFERENCES
T-,f
l()
I
lo
I
I(
I'
Lo.o,
)
.l
l{),1
W. S. Rumman, "Basic Structural Design of Concrete Chimneys," J. Power
Div., ASCE, 96 (June 1970), 309*318.
B. J. Vickery and A. W. Clark, "Lift or Across-Wind Response of Tapered
Stacks," J. Struct. Div., ASCE, 98, No. ST1 (Jan. 1972), l-20.
ESDU, Across-Wind Vibrations of Structures of Circular Cross-Section in Wind
or Gas Flows,ltem 78006, Engineering Science Data Unit, London, 1978.
B. J. Vickery and R. I. Basu, "Across-Wind Vibrations of Structures of Circular Cross-Section, Part 1, Development of a Two-Dimensional Model for
Two-Dimensional Conditions," J. Wind Eng. Ind. Aerodyn., 12 (1983),
49-'73.
FIGURE 10.3.3. View of shroud fitted to a stack tl0-161.
From D. E. Walshe and L. R. Wooton, "Preventing WindInduced Oscillations of Structures of Circular Section."
Proc. Inst. Civ. Eng.,47 (1970), l-24.
lo
5
lo
(r
R. L Basu and B. J. Vickery, "Across-Wind Vibrations of Structures of Circular Cross-Section, Parr. 2, Development of a Mathematical Model for Full
Scalc Application," ./. Wirul Eng. Ind. Aerodyn.,12 (1983),75-97.
I). .1. Vickery, "Across-Wintl Buft'cting in a Group of Four In-Linc Modcl
(lhinrncys,"
lo
/
l.
Wirul lit,q. ltul.
Atnxlyn.,8
(198
l), 171-19?.
Ilasu und l]. .l . Vickcry, "A ('orrrparison ol'Modcl arrtl Irrrll-Scrrlt: lltr
lrrvirrr irr Wintl ol 'lirwe ls:rtttl ('lrirrtrtcys," I'rrtt't,t,tlitrgs Witul 'littrttcl Mtutt'ftff:-'-''
R.
l.
402
SLENDEn tOWEltS AND STACKI] Wl
lll
clllOULAll (il|OSS SECIION
llr
for Civil Engineering Applir:utiotrs, (iaithcrsburg, MD, April l9tl2, Clrrthritlge
Univ. Press, Cambridg,
B. J. Vickery, "The Aeroelastic Modeling of Chimneys and Towcrs," Prrrceedings Wind Tunnel Modeling for Civil Engineering Applications, Gaithcrsburg, MD, April 1982, Cambridge Univ. Press, Cambridge, 1982.
B. J. Vickery and R. I. Basu, "Simplified Approaches to the Evaluation ol'thc
Across-Wind Response of Chimneys," Proceedings 6th International Conl'crence on Wind Engineering, March 1983, Gold Coast, Australia, in J. Wi.nel
l0
14
1982.
l0-8
l0-9
Eng. Ind. Aerodyn., f4 (1983), 153-166.
Maugh and W. S. Rumman, "Dynamic Design of Reinforced Concrcte
Chimneys," Journal Am. Concrete 1nst., Sept. 1967.
l0-ll G. M. Pinfbld, Reinforced Concrete Chimneys and Towers, Viewpoint Publications, Scholium International, Inc., Flushing, NY, 1975.
10-12 K. C. S. Kwok and W. H. Melboume, "Wind-Induced Lock-in Excitation ol
Tall Structurcs," J. Struct. Div., ASCE, f07 (1981), 57-72.
l0-10 L. C.
I. Basu, Across-Wind Response of Slender Structures of Circular Crost
Section to Atmospheric Turbulence, Vol. I, Research Report BLWT-2-1983,
University of Western Ontario, Faculty of Engineering Science, London, Ontario, Canada, 1983.
l0-14 A. G. Davenport and M. Novak, "Vibration of Structures Induced by Wind,"
Chapter 29-II in Shock and Vibration Handbook,2d ed., C. M. Harris and
C. E. Crede (eds.), McGraw-Hill, New York, 1976.
10-15 C. Scruton, Note on a Device for the Suppression of the Vortex-Excited Oscillations of Flexible Structures of Circular or Near Circular Section, with Speciul
Reference to lts Application to Tall Stacks, NPL Aero Report No. 1012, Nutional Physical Laboratory, Teddington, U.K., 1963.
10-16 D. E. Walsh and L. R. Wooton, "Preventing Wind-Induced Oscillations of
10-13 R.
Structures of Circular Section," Proc. Inst. Civ. Eng., 47 (1970),
l-24.
l0-17 G. Hirsch and H. Ruscheweyh, "Full-Scale Measurements on Steel Chimncy
Stacks," J. Ind. Aerodyn., l, 4 (Aug. 1976), 341-347.
10-18 L. R. Wooton and C. Scruton, "Aerodynamic Stability," in Modern Design
of Wind-Sensitive Structures, Construction Research and Information Associu=
tion, London, 1970.
10-19 A. Brunner, "Amortisseur d'oscillations hydraulique pour chemindes," Jtttrr'
nles de I'Hydraulique, 8, Part III, Lille, France Q9e).
lO-20 J. P. Den Hartog, Mechanical Vibrations,4th ed., McGraw-Hill, New York,
1956.
lt," Eng. News Reunl,
187,24 (Dec. l97l),23.
10-22 R. H. Scanlan and R. L. Wardlaw, "Reduction of Flow-Induced Vibrations."
10-21 "Tower's Cables Handle Wind, Water Tank Damps
in Isolation of Mechanical Vibration Impact and Noise, AMD,
Vol. l,
Soctittrt
2, ASME, New York, 1913,35-63.
10-23 I. S. Gartshore, J. Khanna, and S. Laccinole, "Thc lill'cctivcncss ol'Vtttlex
Spoilers on a Circular Cylinder in Smooth and'l'rrrbrrlcttl likrw," in |liud
Enginee ring, Procccdings o1' thc Filih Intcrnationll ('ottlcrclrcrr, lirtrt Collitut,
CO, July 1979,.1
.li.
Ccrnrak (ccl.), Porg,trlrrttlt lltcss,
()rlirnl,
l()110.
il lil
NCt
$
403
W. lllnerrkllnl) iul(l W. lllurlrrcr., ..'l'r.iutsvcrse Vihrirliorr llclrirviorol.(.vlirrtlcr.s
in [.inc,".1. Wittl l'.,ttg. ltul. Arnxl.yrt.,7 (l9l{l)..]7 5.1.
lo .15 H' Ruschowcyh, "l)rrrblcrns with ln-l,irrc Stlcks: lixpcricncc with lrull-Sc:alc
Objccts," Eng. Srrut'r., 6 (l9tt4), 340 143.
l() 2() M. Zdravkovich, "Review ancl Classilicltion ol' Various Acrotlvnarnic and
Hydrodynamic Means for supprcssing V.rtcx Shctlding," J. winct Eng. Intl.
lll
l7
l0llt
l{l
l9
Aerodyn., 7 (1981), 145-189.
M. Zdravkovich, "Reduction ol'Eft'cctiveness of Means for suppressing windlnduced Oscillation," Eng. Struu., 6 (19g4), 344_349.
H. Ruscheweyh, "straked In-Line steel stacks with Low Mass Damping,"
-/.
Wind. Eng. Intl. Aerodyn, 8 (1981), 2O3-21O.
H. Ruscheweyh, "Dynamische windwirkung an Bauwerken," Bauverlag,
Wiesbaden, 1982.
l0 |0 H. Ruscheweyh, "Vortex Excited vibrations," tn wincl-excited vibrations
Structures, H. Sockel (ed.), Springer-Verlag, New york, 1994, 5l_g4.
oJ'
il.r l)l ti(;llll,ll()N ()l wlNl) lo^t)tN(l
CHAPTER
11
As usual, il is eonvcrricrrl lo tlcscrifu (lrc l)lcssurr:s in tcrrrrrs
ir nlcAn and a lluclttltittg prrtl .
ol'lhc
405
surn
of
11.1.1 Mean Pressures
'l'hc rlcan pressurc at a point clolinccl by thc hcight above ground z and the
irrrgular coordinate d (Fig. I l.l.l) can be expressed as:
pk.0l :
jplCne.
ilu'(d + CpiU2(H)l
(11.1. 1)
(p = 1.25 kg/m3), U(z) is the mean wind speed at
t'lcvation z in the undisturbed oncoming flow, Co(2,0) is the mean external
wlrorc p is the airdensity
is the height of the tower, and Coi is the internal pressure
cocllicient. Based on results of full-scale measurements, [11-2] suggests Cpr =
0..1.t'The following tentative relations, based on wind tunnel and full-scale
rrrilsurements, have been proposed for the external pressure coefficient Co(2,
0r I I l-31:
f
HYPERBOLIC COOLING TOWERS
rcrisure coefficient,
Much research into the wind loading of hyperbolic cooling towers has been
conducted following the wind-induced collapse in 1965 of three out of a group
of eight cooling towers at the Ferrybridge Power Station in England tll-ll,
Principal areas of investigation have been (1) the spatial distribution and the
variation with time of the wind loading on the tower surface and (2) the response
of the tower to wind loads, including the dynamic effects induced by fluctuating
wind loads. This chapter summarizes and references the principal results cur'
rently available in these two areas. * These results are presented in Sects. I I ' I
and ll.2 for towers that are not significantly affected aerodynamically by the
presence of neighboring structures. Information on groups of cooling towers iB
F1
Cr(2, 0)
=
coQ,o)
- I-
Cok,0)
:
|
B sinc
(r
fr)
(ll.l.2a)
0<0<0b
(11.1.2b)
0>
CoQ,06)
B=1I
: oo
0
04
(y)'"
0u
(l l.l.2c)
+ AC,
(11.1.2d)
presented in Sect. 11.3.
11.1
DESCRIPTION OF WIND LOADING
Wind-induced pressures acting on a tower are determined by the characteristictt
of the oncoming flow, the tower geometry, and the features of the tower surface. In addition the pressures depend upon the Reynolds number of the flow,
which is in most cases of the order of 107 to 108 for the prototype, and hy
about two orders of magnitude smaller in the wind tunnel. On account of thil
dependence it has been necessary to complement wind tunnel test by full-scale
measurements.
FIGURIi I l.l.l.
*The authors would likc lo acknowlcclgc thc valuablc cottlribttliotts to lltis t ltitDtt:t lty l)tolcstrttl'
D. A. Rccd.
404
'tWirxl
lttrrrre
l
rncilsurr:nrcllls <;trolcrl
llyltcrltolic ctxrlittg towcr-noliltions.
itt
rrt (r' irr rrrrvcnlc:rl lowcrs; lltrrl is, ( ),,
ll
l .t l sttltpcst lltrrl sornt'wllrl lrigltt:r
t ut r'vr'tt (l (r
()
irtlt'rtritl l)l('sslrr('s
lr
406
HYPEIIUOLTC
C(X)UN(i tOWl ilri
r
r I t)t li(;t
ilt ' I t( )N
()t
wtND I c)Al)tNCi
407
maxCo
N
o
()
o
o
4
6 8ro-2
2
4
6
810-t
l,'l(iURE 11.1.3. Distribution of pressure coellicient Cr,. After H. propper and J.
Wt'lsch, "Wind Pressures on Cooling Tower Shells," in Wind Engineering, proceed2
rtt,tl,s
rf' the Fifth International Conference, Fort Collins, CO, July 1979, J. E. Cermak
Pergamon Press, Elmsford, NY, 1980.
(rtl.),
ROUGHNESS COEFFICIENT k/a
ll.l.2. Approximate pressure difference ACo as a function of roughnesa
coefficient kla for towers with 36 to 144 ribs. After H. J. Niemann, "Wind Effects on
Cooling-Tower Shells," J. Struct. Div., ASCE, f06 (1980), 643-661.
FIGURE
C-
lnB
ln[sin 90(0"/01)]
(11. l.2c)
ll :
where
height of tower, ACo is a function of the ratio kla of the rib height,
k, to the distance between ribs, a, represented in Fig. 11.1.2, a is an exponent
characterizing the mean wind profile (Table 2.2.2), and the angle 0 is expresscd
in degrees. The angles 0o,0r, and06 are represented in the schematic pressure
distribution diagram of Fig. 11.1.3. Values forthese angles are given in Fig,
ll .l .4a (based on full scale measurements on the Schmehausen tower) and in
Figs. ll.l.4b and 11.1.4c (based on wind tunnel measurements) [ll-20].
Numerical Example Assume that, as in the case of the Martin's Creek towcr,
: 127 m, kla = 0.02, and cv = 0.17. We seek the values of Cnk,0) lbr
:
z 95.4 m and 0 : 35',70", and 97".
H
From Fig.
ll.l.2,
LCo
- 0.65. For z:95.4 m, Fig. ll.l.4q yields 0,,
= 97".Itfollows thatB :2.1 , C:2.14 (Eqs.
= 35",0r ='loo,and06
1l.l.2d and ll.1.2e). From Eq. 11.1.2b, Cp(95.4 m,35") = 0, Cr,(95.4 rrr,
70') - -1.1, and Cp(95.4 m,97") = -0.38.
values of the external pressure coefficient c,, at thc towcr throat obtainotl
from full-scale measurcmcnt by a numbcr of invcstigirl()rs ltro sh<lwn in lrig.
I 1.1.5. Note that the values obtained for the tower of [11-6] differ appreciably
Irorn the other sets of values. This is due to the absence of ribs on the external
srrrlace of that tower. Note also the agreement to within about 15% between
tlre values obtained in the numerical example and the values measured on the
Mrrrlin's Creek tower at the throat elevation z : 95.4 m [1]-2]. Figure 11.1.5
llso shows an example of differences between values of e obtained from a
rt't of wind tunnel tests on the one hand and full scale measurements on the
ollrcr.
11.1.2 Fluctuating Pressures
sllcsscs induced by fluctuating pressures are usually comparable in value to
\llcsses induced by the mean loads. The purpose of this section is to present
r lcscriptions of fluctuating pressures for use in the estimation
of tower response.
Atltlitional information on fluctuating pressures is presented in [1]-5] and
lr 22l.
RMS of Fluctuating wind Pressures. The rms of the fluctuating wind pres',rrlt's. o,,(2. 0). may bc wrillcrr irs
ot,Q,,0l \1t{'i,8.,0\ll;(:.1
(ll.l.3y
p is lhc air dcnsity, l/(.:) is llrc rrrt'rrrr wirrtl slrerertl trl clcvirtiorr;, irrrtl
t'i,l:.,0) is an cttrpilit:itl llttclttttti!lg prt'Hriur(r r'ocllit'icrrl. Alltrrrrltls lo rclirlc
tvltt:ro
408
HYI)FRBO|
tc coot tNG rowFil$
lr I l)l !i(;llil'ilt)N ()t wlNt)
0o
ot
0o
t()nt )tN(i
409
Numbrlr
ol rlba
Weisw€iler 52
v Martin's Croek 84
oscfrnehausen 144
- - Wind tunnel
Aco = 9.6g
d =0.13-O.17
A
(a)
600
900
Ret
Ro
o Maomin
8.bx
1O
-3
2.2x1O-2
5.4x
1O
6.5x
1O
'
7
1x1O8
1
1-6
1-4
1
1-2
1
2.3x1O-2 4-6x1O7
-11-4
1O 5
11-7
'|
1200
1
500
1
.6x
800
ti
(b)
750
I
I
1000 1250
I
I
0
01
0o
1.0
0b
lrl(;uRE 11.1.5. Mean pressure coefficient around the throat section of hyperbolic
r'rxrling tower for four full-scale data sets and one wind tunnel set. After Tien-fun Sun
;rrrtl Liang-mao zhou, "wind Pressure Distribution around a Ribless Hyperbolic Coolirrg'fower," J. Wind Eng. Ind. Aerodyn., f4 (1933), l8l_192.
Aco
= 9.7t
d = o.18
* o.u
[\
(c)
0
25o 5Oo 75o jOOo
1250
0.4
0
FIGURE
ll.l.4.
Angles 0o,
0,,
and 0u
(after tll_201).
,'n'
l' ,,r \
0.3
!q\r, 0) to the turbulence intensity of the oncoming flow have been reported
and
Il-3]
[11-8]. According to
itt_:1,
Ci,tz,
'
ol = 1.8 o'
u(zt
C;
in
(11.1.4)
where o, is the rms of the longitudinal
velocity fluctuati.ns. Thc variation
Ci'Q,0) with d at the elevation of the throat is
'r.
sh<lw' ftrr.tlrccr scts of mcir-
surements in Fig'
. 1 .6 tr.r
ation depends upon thc ratio
'
?l)
krir,
According k)
whcrc
r r r -3 r
I r r r r i, rhis varit is rhc hciglrr'r*ror. lrrt. r.irrs irnrr /) is
.-a7/
0.2
\
\
0
-.-.-
l.-1.-
:-
0.1
0
30 60 90 t20 150
Full-scale
[1]_8]
Full-scate [1 1-9]
Model [1 1-10]
180
Degrees
lrl(;IIRli ll.l.6.
t'rxrling towor.
F'lLrctrrlting
l)ltssllr
c'rrcfiicicnt arouncl thc throat
ol'a
hypcrbolic
410
il t
HYPERBOLIC COOLING IOWI-I1!i
?.o
1.0
k/D = 5.4x1O-a
NIN
9lv
ol o
.t
nro=r.r-to-lFl
^U-C
olo
goo
600
.l
2oo
k/D =4"5x lO
1
4ll
ir,
LO
k/D=2"Ox1O-3
1500
)l()Al rlN(i
lfr
-\
h/l) =4 br ll)
k/l):
o
3Oo
WtNt
o.5
'
o
licltll'li()N(l,
k/D:o
,/-))
1.5
--l
/--k/D=5x10 "
Olo
t)t
o
300 600
800
900
1
200
r
50.
1
o
800
0
0
FIGURE 11.1.7. Ratios Ci(2, 0)lci,k,0) for towers with various roughness param"Wind Pressures
eterc klD at elevation z - 0.7 F1. From H. Propper and J' Welsch,
" in Wirut Engineering, Proceedings of the Fifih International
CO, July l97g' J. E. Cermak (ed'), Pergamon Press' ElmsCollins,
Conferenit, F.rt
on Cooling Tower Shells,
ford, NY,
1980.
4.O
3.O
1--L0
Ii
a
-\
= O.27
= O.18
2.O
1.O
o
lower
the diameter of the tower at the throat. Note that the coefncients Ci are
120'
<
<
0
60'
region
in
the
towers
smoother
for
the
than
for the rougher
(Fig. 11.1.7).
Spectra of Fluctuating Pressures. The following expression for the spectra
o
3oo 600 90o 12Oo
.tsoo
18Oo
0
l,'l(;URE 11.1.8. Parameters a,(0), bo(0), and B,,(0). From H. Prcipper and J. Welsch,
"Wind Pressures on Cooling Tower Shells," in Wind Engineering, Proceedings of the
liilih International Conference, Fort Collins, CO, July 1979, J. E. Cermak (ed.),
I't:rgamon Press, Elmsford,
NY,
1980.
of fluctuating pressures was proposed in [11-3]:
nS,(2, 0. n)
(11.1.s)
o?Q,0)
l.
Windward region (0
So(2,0,
Z',0', n)
:
< 100',9' < 100'),
R,,(2,
where
{,
n)Ry(O,0', n)S}/z(2,0, nlsto/212,,0,, n)
(11.1.e)
| a'@\
d(0): - ,
(1
'YpQ):[m]"'"'
(11.1.7)
1.1.6)
2
= 100', 0' = 100'),
SoQ, 0, z',0', n) : R,(2, z', n)R,(O,0',
Leeward region (0
nySltz(2,0, n1stotz1z,,0,, n)
(1
xog)
:
| .,
/ D\''''ltuttqt n,
lb;'"(o)
(;/ | ,^
(11.1.tt)
where n is the frequency, the parameters a(0), bs(0), and Bo(0) are given in
Fig. 11.1.8, a is the power law exponent (Table 2'2'2), D is the diameter at
thJ throat, and Ii is the integral scale of turbulence (Sect' 2'3'2)'
Cross-spectra ol Ftuctuating Pressures- According to lll-l2l' quadraturc
spectra are negligible; that is, the cross-spectra arc ltlr pntclicitl prlrposcs cquill
to the co-speitra. Thc lirll<lwing rclations wcro pf()porictl irr lll-l2l lirr the
cross-spcctra <tl' lho pn:ssurc lluctuations:
1.1.10)
At'cording to [l1-12], cross-spectra of pressures on the windward region, on
llrc one hand, and pressures on the leeward region, on the other, are negligible.
'l'his is a simplifying assumption that is not entirely consistent with
results
re lxrrted in
1-3].
[
ln Eqs. ll.l.9 and 11.1.10,
:
R,(0, 0', rt)
R1(),0'. tt\
R,,(2., z.' .
z)
exp(
(il.l.lt)
(lr.r.r2)
-p,i)
cxp(- Az.fz)
(',(0,0'\R(10
*
0'1, n)
(ILI.1.1)
412
HYPERBOLIC COOLING TOWEI|ti
il 1'
c2(0,0')
600
c2(0,0',
0 =12Oo
y(s, 0, t)
_\- I
m
9=
correlation coemcients c2(0,0'\. From H. pnipper and J. welsch,
''wind Pressures on cooling Tower Shells, " in wind Engineering, proceedings of tha
Fifih International conference, Fort collins, co, July 1919, J. E. Cermak (ed.),
Pergamon Press, Elmsford, NY, 1980.
exp(- tszf|)
nlz
- z'l
u(6)
(1 1.
l. l4)
(11.1.15)
-
l0 0'l
'
360'
tnD'
i,
u(6)
(11.1.t6)
ESTIMATION OF TOWER RESPONSE
ttl thc cstirnation ol'towcr rcspolrse lurvt. bcrrlr prlpgsctl,
For towcrs that cxhibit no sigttilicanl rcsor)iull irrrrplilir'rrtiorr cllircls, lll-71
Several approachcs
qi,,,(t)sin
mTly^.,(s)
(11.1.7)
tt'lxrrted in [1]-13].
ln [l1-14] and Il-15] finite element methods of analysis are used in con[rilction with step-by-step integrations in the time domain. one advantage of
srrclr an approach is that it can accommodate nonlinearities and changes of the
|hysical properties of the structure during the loading process. Time histories
,l lluctuating pressures used in this approach can consist of measured data, as
rrr lll-l4lx and Il-15], or can be simulated by Monte carlo methods from
rlrcctral and cross-spectral information. More recently, ARIMA (Auto Regresrrvtr Integrated Moving Average) methods have been used for representing
llrrc(uating loads in the time domain tll-161. Time-domain solutions, though
grotcntially useful for research purposes, are costly and may be impractical for
rorrtine design.
where P1 = 7, 0z = ll, 0t = 25 U1-121, U(6) is the mean wind speed ut
the gradient height 6 listed in Table 2.2.2, and Cz(0, 0'), as obtained in
ll 1-31, is given in Fig. I I . I .9.
11.2
+
wlrcrc s is the distance along the meridian, 0 is the angular coordinate, r is the
lntrc' qm.i and q'.,, are the time-dependent symmetric and antisymmetric genrlirlized coordinates for mode m, i, respectively, and j^.i3) is the vertical modal
slrrrpc, An attempt to use a spectral approach to estimate the response was also
ll.l.9.
it:
[q.,,(r).os m0
18Oo
60.
- o'1, n1 :
413
lol tlrc ntcritlional irnrl cirr,'rrrrrlt rcrrlirrl t'otrt-lrrtions ol llrc
llrtt'itraling prcssttros to obtain tho variarrecs ol'llrc rrrt.r'itliorrrrl, cilt'iiruli'lcrrliirl.
;rrtl n<lnnal displaccmcnts ol'tho towcr shr:ll.
'l'his approach was superscdcd hy
lll l2l, whiclr cnrpl,ys u spcclrirl irp=
lrloitch in which models of spcctra artrl c:nrss-sprrctrit ()l'ptt:ssiu'c lluctuir(ions
(st'c Scct. ll.l) are used to obtain thc spoclnrl rlcnsilics <ll'thc rcsponso by
rrrclh<rds fundamentally similar [o lhosc ol'Sccts. 5.2.7 anrJ -5.3. 'l'hc spcctral
irppnrach is applicable to towcrs ltlr which resonant amplification effects are
rrlinificant, as well as to towers which-as is most commonly the case-are
r.rrlliciently stiff that resonant amplification effects are negligible. In both cases
tlrc calculations can be carried out by using a computer program similar to that
lrslcrl in [9-14], but modified to account for differences in geometry and in the
rrrotlcling of pressures, as well as for the fact that a typical response of the
Itrwcr, rather than having the form of Eq. 5.2.1, is written as
- 1.0
R(lo
ot lr)Wl tt ttt tji,(lNlit
t'rrtPltlys cxprcssirttts
c2 ( 0, 0',
FIGURE
t silrMAlt()N
spectral methods, as developed in [1]-12], were applied in [1]-4] to study
llre rcsponse of typical reinforced concrete towers with ratio HID :2.0 (D :
rlrrrnrctcr at throat). The results obtained indicated that the resonant amplifi.irlion cffects contributed less than 57o to the total response. A typical diagram
ol thc ratio N11lq,at the stagnation point is shown in Fig. ll.2.l for U(e,n.ou,)
45.4 mls (N' I is the mcridional stress, q, : (ll2)pUr(z,n^,o,), p is the air
(l('tlsily, and U(2,1.,,,,,) is thc rncan wind speed at the clevation of the towcr
llrroirl). It is secn that in this clsc tlrc pcak total responsc dillbrs insignilicantly
rlk'r':tttsr: tttcirsutul tl:rlrr wclr. irvtriltthh. ully lor
lll(' tllroill sccliotr,
ol llrt. hrrrrlr rr rrrllorlr.
tlr:rt llrr' verlit'irl tlisllilrrrliolr
il
wirs itsstrrlt.tl rrr
lll l4l
414
il ?
HYPERBOLIC COOLING TOWERS
FStiMAiloN
ot
towt tt nt
fit()Nst
41S
180
160
't
40
't20
o
o
6E 1OO
*Bo
FIGURE
N1 1,
ll.2.l.
Ratios
of meridional
*")o",," '
stress,
at stagnation point to dynamic pressure, 4r,
at elevation of tower throat. After H.-J. Nie-
mean
mann, "Wind Effects on Cooling-Tower
Shells," J. Struct. Div., ASCE, 106 (1980),
&3-66t.
soo
Nrr/gt
1000
(meters)
from the peak quasi-static response (obtained by neglecting resonant amplifl'
cation effects). The latter is approximately twice as large as the mean response,
It is shown in [11-4] that for the type of towers studied therein, the design
may be based on an equivalent static pressure
Pk. 0) : Cok.O)qokl
(11.2.l)
where, in open country,
7-10.23
qp(z) =
'(;)
lipu2(to)l$
(rt.2.21
o
6 = 280 m, p is the air density (p = I.25 kg/m2), U(10) is the hourly
mean wind speed at 10 m above ground, and @ is a factor accounting for
resonant amplification effects (1 < d < 1.1).
where
An equivalent static pressure approach is also included in [11-17], in which
the expression for the equivalent pressures is consistent with the format used
for dynamic pressures in the American National Standard A58.1-1972 tll-181,
Reference tl1-171 recommends the use in this expression of aerodynamic coef'
ficients obtained from wind tunnel or full-scale tests, and of a gust loading
factor to be determined by a dynamic analysis.
The use of a single gust loading factor implies that the stress amplification
due to wind gustiness may be considered for practical purposcs to be the samg
at all points of the towerand forall types of stress. As shrtwu in Ill-231, thiti
assumption is not necessarily correct in all cascs.
Itl(;uRE 11.3.1. Tower locations at Ferrybridge c Generating station. From J. Arrrritt, "Wind Loading on Cooling Towers," J. Struct. Div., ASCE, f06 (1990),623_
()41.
'll
416
11.3
ftr
ilyt't rtll()t t(: (.()()l lN(i l()wt ttl'
ilt IIllt
, .,.'. '
((
\.....-,))r
/
GROUPS OF HYPERBOLIC COOLING TOWERS
Wind-induced stresses in the tower shclls can bc considcrably trrorc scvclc irt
the case of groups of towers than for isolated structures. This was httrttt: ttttl
by the behavior during the November l, 1965, stormx of thc cight lowcls ttl
the Ferrybridge C Generating Station (Fig. 11.3.1), three of which collapst:tl
while five survived. The inquiry of Il-l] indicated that failure was duc lo
large tensions in the windward face of the towers. On the basis of wind tunne I
tests and of infbrmation on the design of the towers, it was estimatcd irr
tll-l9l that the mean hourly wind speeds at l0 m above ground, U(10). rtl
which failurc of thc towers could be expected to occur had the values showtt
'l'Alll,l,l I l.-1.1. listinratcd Wind
il l-1.)l
Tuwcr
u(10)
IA
t9.
t
Speeds Corresponding to Tower Failures (m/s)
\i:yi/
.lt)t, ,lr.
I .1/ (trl
/
..\
(( ))t
\z/
,*"/
l*, nn
|"u
I
4l I
(( '"))
|
l{},)
m; r\;
tl5
iliI
"'iil
;"'
ee q =
177 dB
2
rn,iil;,7
loA
,---\
(o)'
I
/ Yli";^"
Tin""/
l,l(JURE 11.3.3. Ratios of stresses amplified by interf'erence effects to corresponding
(d, is the diameter at throat: d3 is the diameter at base). After
ll L Niemann, "Reliability of Current Design Methods for Wind-Induced Strcsses"
rn Nutural Drafi Cooling Towers, P. L. Gould, W. B. Krdtzig, I. Mungan, and U.
wirtck, (eds.), Springer-Verlag, Berlin, 1984.
'.rresscs on isolated tower
IB
2A
28
t9.l
19.1
23.4
3A
21.5
3B
23.8
44,
21.6
4l]
21..1
ru'l'able ll.3.l. The wind speeds U(10) during the storm were reported to rise
llrrn about l8 m/s to about 20 m/s. The reported sequence of tower failures
rv;rs fbund to be consistent with the results of Table tl.3.l [ll-19].
It is noted in [1-19] that higher mean and fluctuating loads ofien occur
t'lrcn the wind blows through a gap between upstream towers. Details on
,listributions of mean fluctuating pressures on the surf'ace of towers placed in
tlrc wake of other buildings or in groups are given for specific configurations
rrr lll-l9l on the basis of both wind tunnel and full-scale measurements, and
of full-scale tests.
to interlerence eflbcts can also occur in the case
,rl lrairs of cooling towars (Fig. I1.3.2). Laboratory data on such amplifications
.rre shown in Fig. 11.3.3 for various wind directions and distances between
rrr I I l-81 on the basis
Stress amplifications due
llr('towers in a pair (max n11 is the maximum hoop, max n22 is the maximum
rnt'r-iclional tension, min n22is the maximum meridional compression, max ln12l
r:; thc maximum shear force). It is seen that in some instances the amplifications
,rri'considerable (over 3O%).k is noted in Il-23] that cooling towers can also
I't'uclvcrsely affected by the presence of adjacent buildings within a power
pl;urt.
REFERENCES
FIGURE 11.3.2. Cooling towers, Lin-rerick Gencrating Sllliotr, l.irrrcrick,
Pcnrrsyl
ilt
vania. Courtesy of Philadclphia Elcctric Cornpany, l,irrrclick ( ie rrt'r'rrlirrg Strttiott.
il.1
tThc approxitultlc nlciur witttl tlittt'lion is sltowrt irr lrrli II I
I
Ilcport of'thc Cotttrtritt,'t'rl lrrtluiry intrt thc Cttllrtlt.sc t2l (ixtlitrg'lltvt.'rrs trl
l"trr.l,britlgt otr Mttrtlttv, I Not't'ttrlrcr /9fi.5, ('cntrul l')lcctl-icily (icnerirlirrg
Illolrtl, Il.M. Slitliottrrrv ( )llicr'. l.otttlott, l9(r(r.
N. .l , Sollcnlrcrgt'r. l{ ll S,:rrrl:rt. :rtttl l). l). llillirtglorr. "Wirrtl l,o:trlittl' :ttttl
llt's;xrrtst'rrl ('txrlttrl, lrtrltr:,. ' .l ,\trtttl.1)lr'., AS('lt, l(Xr (l()li0). (rOl (r'l
418
ll-3
HYPEntsoLtc
cooltNc town*i
trt
H. Pnipper and J. Wclsch, "Wirrtl
l)russurcs on C)rxrling'lirwcr Shclls." irr
Wind Engineering, Prcceedings ol'thc Filih lntcrnational Conl'croncc, liort (lol-
ll2l
lins, CO, July 1979, J. E. Cermak (ed.), Pergamon Press, h,lnrslord, Ny,
1980.
Il-4
l1-5
H.-J. Niemann, "Wind Effects on Cooling-Tower Shells," J. Struct. Div.,
ASCE, 106 (1980), 643-66t.
J. F. Sageau, In Situ Measurement of the Mean and Fluctuating Pressure Fie lds
around a 122 Meters Smooth, Isolated Cooling Tower, Electricitd de Francc,
Direction des Etudes et Recherches, 6 quai Watier, Chatou, France, Sept. 1979.
1l-6
T. F. Sun and L. M. Zhou, "Wind Pressure Distributions on a Ribless Hyperbolic Cooling Tower," Proceedings 6th International Conference on Wirul
Engineering, Gold Coast, Australia, inJ. Wind Eng. Ind. Aerodyn., f4 (1933),
ll-l
S. H. Abu-Sitta and M. G. Hashish, "Dynamic Wind Stresses in Hyperbolic
Cooling Towers," J. Struct. Diy., ASCE, 99 (Sept. 1973), 1823-1935.
1l-8
J. F. Sageau, Caract€risation des champs de pression moyens et fluctuants d
la surface des grands airorefrigdrctnrs, Electricit6 de France, Direction des
18r-r92,
Etudes et Recherches, 6 quai Watier, Chatou, France, July 1979.
1l-9 H. Ruscheweyh, "Wind Loadings on Hyperbolic
Towers," J. Ind. Aerodyn., I (1976),335-340.
l-10 A. G. Davenport
Natural Draught Cooling
and N. Isyumov, The Dynamic and Static Action of Wind on
Hyperbolic Cooling Towers, Vol. 1, Research Report No. BLWTI-66, Univ.
of Western Ontario, London, Ontario, Canada, 1966.
l1-ll M. Pimer, 'Wind Pressure Fluctuations on a Cooling Tower, J. Wind Eng,
Ind. Aerodyn, 10 (1982), 343-360.
ll-12 M. G. Hashish and S. H. Abu-Sitta, "Response of Hyperbolic Cooling Towers
to Turbulent Wind," J. Struct. Div., ASCE, f00 (1974), 1037-1051.
l1-13 M. P. Singh and A. K. Gupta, "Gust Factors for Hyperbolic Cooling Towers,"
J. Struct. Div., ASCE, 102 (1978),371-386.
1l-14 P. K. Basu and P. L. Gould, "Cooling Towers Using Measured Wind Dara,"
J. Struct. Diy., ASCE, f06 (1980), 579-600.
1l-15 R. L. Steinmetz, D. P. Billington, and J. F. Abel, "Hyperbolic Cooling Tower
Dynamic Response to Wind," J. Struct. Diy., ASCE, f04 (1978), 35-53.
ll-16 D. A. Reed and R. H. Scanlan, "Cooling Tower Wind Loading," in proceedings of the 4th U.S. National Conference on Wind Engineering Research,
Department of Civil Engineering, University of Washington, Seattle, July 2629, 1981, Vol. 1, pp.254-261.
ll-17 Reinforced Concrete Cooling Tower Shells-Practice and Commentary, ACI
334, lR-71 , American Concrete Institute, Derroit, Michigan, 1977.
I l-18 American National standard Building code Requirements for Minimum Design
Loads in Buildings and Other Structures, A58.1, American National Standards
Institute, New York, 1972.
ll-19 J. Armitt, "Wind Loading on Cooling Towers," J. Struct. Dly., ASCE, 106
(1980), 623-64t.
ll-20 J. Welsch, Der Einfluss des Windprc{ils aufdia stu!ix'lt,rt Witullnttttspruthutt
gen von rotationshypcrfutlischcn Ktihllurmschulcrr, l,clrlsltrlrl I, lnslitut liir korr.
struktivcn Ingcnicrrrhau, Ruhr-Univcrsitiit llochutrr. l t.lrnrruy l t)ll{.
I
ll 22
ll73
l
tit N(l
ll
419
I). A. l{ecrl irrrrl li. Sirtriu, "Wintl l,oarls rur ('rxrlinp'lttwt't's," l)rrrlt Slrrlc ol
tho Art l{cpol't on Wintl lill'ccts ort Sintcltttcs, ('ontntillr't'ott Witttl lilli'c'ls,
Arncrican Strcicty ol' (livil linginccrs, l9ll4.
Y. Kawarabata, S. Nakac, and M. Haracla, "Srttuc Aspccts ol'thc Wilrtl l)csigrr
ol'Cooling Towcrs," J. Wind ling. lrul. Aanxl.yrt. l4 (l9tl3), 167 lti0.
H.-J. Niemann, "Reliability ol'C,'urrcnt l)csign Motlrods lir Wind-lnclucotl
Stresses," in Natural Drrsught Cutling'l'owers, Procccdings tl'thc 2nd Intcrnational Symposium, Ruhr-Bochurn, Gcrmany, P. L. Gould, W. B. Kr:itzig, l.
Mungan, and U. Wittek, cds., Springer-Verlag, Berlin, 1984.
x
l:,
CHAPTER 12
(or'1r,i1111'1s1, (lrc slrit'lrlirr11 tlt'pt'rrtl:; orr llrt' rttttttlrt't lrlrtl slttrciltg tll'
tlrc tt'ttsscs 1ol ginlcrs).
'l'lrc slr:r;lc ol llrc rrrr-:rrrbcrs, tlurl is, wlrt'llrcr tlrc tttctttbcr are rounded or
lurvc sltrrp ctlgcs. Iiorccs orr lorrrrtlt:tl rrctttbcts dcpcnd on Reynolds numbcr 61" ancl on lhc rougltttcss ol lltc tttotttbcr surface (see Fig. 4.5.5). For
trusscs with sharp-ctlgctl rttcrttbcrs the elI'ect of the Reynolds number and
<ll' the shapc ancl surlacc rtlughness of the member is, in practice, negligible.
The turbulence in the oncoming flow. As noted in Sect. 4.5, the effect of
turbulence on the drag force acting on frameworks with sharp-edged members is relatively small in most cases of practical interest ll2-2, l2-5,
12-141. A similar conclusion appears to be valid for frameworks composed
of members with circular cross section in flows with subcritical Reynolds
members. For this reason, and owing to scaling difficulties, in most cases
wind tunnel tests for trussed frameworks are to this day conducted in
ltltsri('s
.
.
TRUSSED FRAMEWORKS AND
PLATE GIRDERS
smooth flow [l2-l , l2-5, 12-6].
o The orientation of the framework with
Trussed frameworks subjected to wind loads have routinely been used
in struc_
tural engineering applications for more than a century. Nevertheless, the
statc
of knowledge conceming the effects of wind on thii type of structure is still
imperlect and provisions concerning such effects included in various
standards,
codes, and design guides are in some cases mutually inconsistent
and in clisagreement with experimental data ll2_ll.
For any given wind speed, the principal factors that determine the wind loacl
acting on a trussed framework are:
o The aspect ratio \, that is, the ratio of the length
of the framework to its
width. If end plates or abutments are provided,lh" flo* around the frame_
o
work is essentially two-dimensional, so that for aerodynamic purposes thc
length of a framework may be considered to b" infinit".
The solidity ratio @, that is, the ratio of the effective to the gross
area of'
the framework-x For any solidity ratio s the wind load is-for practicar
purposes independent of the truss configuration, that is, of whether
a
diagonal truss, a K-truss, and so forth, is involved.
o The shielding of portions of the framework by
other portions locatecl
upwind. The degree to which shielding occurs depends on the configuration of the spatial framework. If the framework consists of parailcr
*The effective areas of a plane
lruss is the area of the shadow pnrjccrctl by ils rrrcrrrhcrs .. :r
plane parallcl to the truss, thc pnricction bcirrg norrnal kl thl{ plrurc. 'l
lrt' 1,1i;.r :rrr.;r .l :r pl:r'c
truss is thc arca ctlntlinctl within thc oLrtsitlc conlorrr ol' (lurl tlrrss. Ilrt. ,lll,.r.tir,,.
lrrt.:r lrrrtl llrt.
gross area of-a spalial It:ttttcwrtrk itte rlclirrctl, rcsyrr't'livcly, ;rs llrt. r'llr.r trrr.
rrrr.;r .rrrrl tlrt. g.rss
arca
420
ol'ils upwintl
Ilrcc.
42 I
respect to the mean wind direction.
l'his chapter reviews the aerodynamic behavior of trussed frameworks and plate
liirders, including single trusses and girders, systems consisting of two or more
yrlrrallel trusses or girders, and square and triangular towers. Test results are
oltcn presented from several sources with a view to allowing an assessment of
llrc errors that may be expected in typical wind tunnel measurements. Throughrrrr( this chapter the aerodynamic coelicients are referred to, and should be
rrscd in conjunction with, the effective area of the framework, A1 .
Wind forces on ancillary parts (e.g., ladders, antenna dishes) must be taken
into account in design in addition to the wind forces on the trussed frameworks
llrcmselves Il2-1 , l2-l7l. Drag and interference effects on microwave dish
irntcnnas and their supporting towers were studied inU2-271. Drag coefficients
lirr an unshrouded isolated microwave dish with depth-to-diameter ratio 0.24
wcre found to be largest for angles of 0 to 30 degrees between wind direction
rrncl the normal to the dish surface, and are almost independent of the flow
Itrrbulence (Co = 1.4). For a single dish the ratiofo between the incremental
total drag on the tower due to the addition of a single dish and the drag for the
isolated dish depends on wind direction, and it is higher than unity (as high as
1.3) for the most unfavorable directions. This is due to flow accelerations
rrrtluced by the dish. As more dishes are added at the same level of a tower,
irrlcrf'erence factors are still greater than unity, but tend to decrease as the
nrrrnber of dishes increascs. According to |2-271 an empirical formula for the
inlcrl'crcnce factorgivcn in ll2 2l'll is loo krw by a factorof more than two for
t't'rlirin wind directions; utr ltlte I'rutlivt' lirrrrrttllr is proptlscd in l12-271.
Akrng-wincl cl'l'ccts on (()w('r'r.i rrr;ry lrt't'slinr:rtt'tl lry rrsirrg procctlurcs suclt
:ts wcrc rliscussccl in Sccl. 9.2.1 'l'lrt' tlt'r,t'lopttrt'rttlrl ('()lnl)ulcr bltsc:cl vt:rsitttt
ol'tlrc AS('li 7 ()5 Stlntliu'tl ;lnrvisrorrs u:,('r,:,u( lr rr pror't'tltrt'lirr lltrxiblt'l()w('r's
(ll7 5l stc tliskt'lttr:tppt'rtrlt'tl lo lltrs lrool.)
422
ilttit;l;t t) tnnMt
w()t
il\ii nNt) t,l /\il (,ilil)t
rr:;
t
l;,t
:;lN(,ll
Ref-erence U2-291 rop()ns lirll-scirlc nlcirsurcnrL'nts ircconlirrg (o wlriclr ircross
wind effects on square towers with anglc rrrcrrrbcrs arc conlllaritblc (o rrlorrg
wind effects. It proposes a semiempirical proccdurc lirr cstintating lolsiorurl
effects, which are due largely to the presence of eccentrically locatod antonnir
dishes.
For studies on wind effects on cranes and guyed towers, see [12-l-5 1,
|2-161, and [12-17] to 112-261, respecrively.
12.1 SINGLE TRUSSES AND GIRDERS
Figure 12. I . I summarizes measurements of the drag coellicient C$) tor a singlc
truss with infinite aspect ratio normal to the wind. The data of Fig. l2.i.l
were obtained in the 1930s in Gottingen for trusses with sharp-edged membcrs
l2-2, 12-31,* and in the late 1970s at the National Maritime Institute, U.K.
(NMI), both for trusses with sharp-edged members and trusses with membcrs
of circular cross section.i It is seen that differences between the Gottingen antl
the NMI results for frameworks with sharp-edged members do not exceed l5/,,
or so. For single trusses normal to the wind and composed of sharp-edgctl
members, ratios Cg)(},)/Cg)(\ : o) of the drag coefficients corresponding ro
an aspect ratio
\,
-^
Q
i
d
O
E
o
o
o
b0
ta
'.
--.._._--.
o
o
0.8
.+
=- t-l
_-
Fig. 12.1.2 tl2-31.
Drag coeflicients C!j) reported in [12-5] for trusses normal to the wind.
composed of sharp-edged members, and having aspect ratios l/6 < \ < 6,
are listed in Table l2.l.l. Also listed in Table l2.l.l are values C$)(X : *l
obtained from the drag coe{ficients of [12-5] through multiplication by thcr
appropriate correction factor taken from Fig. 12.1.2. It can be seen that dil'
are shown in
ferences between the values
C!'(\ :
lllllllllllllll ---*
oo) based on U2-51 and the corresponding
*References
tl2-21, 112-31, and [12-41 are available in English as Building Research Estahlislr
ment Library Translation No. IT2202, Building Research Station, Garston, watford, U.K.
lThe NMI measurements for trusses with
members of circular cross section ret'erred to in tlris
chapter were carried out at Reynolds numbers 104 < 61" < 7 x 101 [12-61.
+Figures 12.1 .3 and 12.4.5 to 12.4.8 are reproduccd
with pcrmission oI CIDEC'I'(Corniti Irrtcr
national pour le D6veloppement et I'Etude clc la Construction Tulruluirc) |ront llirul [iprct,.s rtrt
UncladTubular Structurcs, H. B. Walkcr (cd.), Constrado Publit:rtiorr l/75. ('onstructiorrirl Slccl
Research and Develttptttcnl Orgirnization, ('nryrlon, ti.K., 1975. Ilrr.y ;rrt. lr:rst'tl irr plll orr
rcscarch work carrictl oul by ('ll)li("1'irrrtl n:ylrrlr:tl irr ll2 ltl ln(l ll.'(rt
Angle-sectionmembers
0.6
on the one hand, and to an infinite aspect ratio, on the othcr,
Flachsbart U2-2, 12-31and NMI values of Fig. 12.1.1 do not exceed 20%.
Figure 12.1.3 [12-7] summarizes results of tests on trusses with membcrs
of circular cross section (x : -) conducted in the subsonic wind tunnel lt
Porz-wahn, Germany [12-8] and in the compressed air tunnel of the Nationar
Physical Laboratory, U.K. tl2-101.+ Note that for Reynolds numbers G" <
lOs the drag coefficients in Fig. 12.1.3 differ by about 5% or less from rhcr
corresponding results of Fig. l2.l.l.
A framework whose solidity ratio is 6 : I is a solid plate (or a girdcr).
o
1.0
0
0.1
0.2
))
I
norno section members
[
nectaneutarmembers
0.3
0.5 0.6
0.4
Solidity ratio
01
08
ro
Drag coefficient C$) for single truss, \ : o, wind normal to truss.
linrm R. E. Whitbread, "The Influence of Shielding on the Wind Forces Experienced
lry Arrays of Lattice Frames," in Wind Engineering, Proceedings of the Fifih Inter'
rttttional Conference, Fort Collins, CO, July 1979, J. E. Cermak (ed.), Vol' 1, Pery'rrrrron Press, Elmsford, NY, 1980, pp. 405-420.
lrl(;URE
l2.l.l.
l'lrc drag coefficient coffesponding to wind norrnal to the plate can be obtained
l2.l.l and 12.1.2. Additional information on the aerodynamic
lrrrhavior of rectangular plittt:s is givc:n in Sccts. 4.5 and 4.6.
ll was shown in Scct.4.(r llr;r( llrc lrt'rorlynlrtttic lirrcc norlnal t<l a rcctangular
plirtc with aspect ratirl \ - 5lrt lO is lrrlllt'l wltt'lt llte yitw:tttglc'r'is rv -'1O"
lrrrrn Figs.
tlurl il'lhc winrl is nortttirl lo llrt'pl:rlt'1l'r1i
'1.(r.
l). llowt'vt't, lirr lrrrsses willr
lltr'lrolizorrl;rl,rrr1'1, l',1\\{(rt lll rrr,.rr rrtttrl rlttr'rlloll.ttrrl llr ttotttt;tl lrt lltt'
,,l,lfl:]/,,*;rrrglt'is
Illr..,{isttJ lltnMl w()l tKli nNl) I't n il (,ilil)t il:;
r:':,r'nllr:;()l
lltl
,:;1
,l :,ntlll{)l
l'l All (;llll)lll::
4?:i
,i''
03
0.5
1/^
FIGURE 12.1.2. Ratios C!j,)(\)/Ctj)(\
TABLE
l2.l.l.
(2) Clj)(X
:
. .
^
o.)
wincl nonnal to truss [12-3].
I | ' I'l
3
1
Drag Coefficients for Simple Trusses
o.t4
(r) c8, (:
: o),
u)-
1.40
+ 5%
- 1.45
o.29
1.54
-
+
o.77
0.4'7
5%
1.65
1.21
-
+
5%
l.l8 + 5%
t.45
-
1.35
1.0
1.28
I r*
*2.l0
I,
s
I
I llll
6 7 E 9
I
2
105
I I t l' | | ll
a
4
l(;URE 12.1,3. Drag coeflrcient C!i)forsingle truss with
: o,
ll
'
s 6 7 8 9106
Rg
I
2
members of circularcross
wind normal to truss [12-7] (courtesy Comitd Intemational pour le
l)('vcloppement et I'Etude de la Construction Tubulaire, and Constructional Steel Re-
',r'tlion, X
'.,'rlch and Development Organisation).
*Reference [l2-51.
cB): clj)(v, *
solidity ratios { < 0.4 or so the maximum drag occurs when the wind
normal to the truss Il2-21.
12.2 PAIRS OF TRUSSES AND OF PLATE
is
GIRDERS
We consider a pair of identical, parallel trusses, and denote the drag coefTicicnt
coresponding to the total aerodynamic force normal to the trusses by
C9@), where a is the yaw angle. Forbrevity, the notation C|Q): C!'zr)it
used. The cases where the wind is normal to the truss (a : 0) and wherc rv
* 0 are considered in Sects. 12.2.1 and 12.2.2, respectively.
12.2.1 Trusses Normal to the Wind
Two parallel trusses normal to the wind affect each olhc:r lrc:nrtlynirrrrically, srr
that the drag on thc upwincl and on the clownwincl trrrss will lurvt'rlrirg cocrlli
cients VrCtj)and V,,Cff), rcspcctivcly, whcn: f ilj) is tlrt'rlr;r1, 1.1y,.11;,.1t'rrt lirr.;r
single truss n<tnnlrl lo lhrr wirrtl trntl, in gr:ncrirl \lrr / ,1,, / I ll lollows tlurl
(t2.2.r)
vrr)
liigure 12.2.1 shows values of Vr and Vyy, repofted in [12-41 for three types
truss, all with sharp-edged members and infinite aspect ratio, as functions
,,1 thc solidity ratio d, and of the ratio between the truss spacing in the alongrvirrd direction, e, and the truss width, d. Values of Vy and Vn, also reported
tn ll2-4], for four types truss of truss with sharp-edged members and aspect
r:rtio \ : 9.5 are shown in Fig. 12.2.2. On the basis of the data of Figs.
l.l.2.l and 12.2.2, [12-4] suggested the use for design purposes of the con',('lvative values C(3)lCr:) given, for eld > 1.0, in Fig. 12.2.3.
l{ocent measurements conducted at the National Maritime Institute, U.K.,
rNMl) on trusses with infinite aspect ratio are summarized in Fig. 12.2.4.
l(r'lercnce l2-6 suggests thc lbllowing approximate expressions based on the
r,'srrlts ol' Fig. 12.2.4:
.l
,r,"
''(:;)'''
'
lrrt (rttsst's willr slrlrlll t'rlgt'rl tttt rrrlr,'t:'. ;ttttl
lirr0 < (, <
0..5
(
l
:.1.1)
PAIRS OF TFUSSEB AND OF FiI ATF (tiItDFItIt
TRUSSED FFAMEWORKS AND PLA'TE OIRDERS
til
+lLr-*iFt
1.0
0.8
1.0
o.4
---":91::
lou l
{:,'
4Em
f
,p
.^
0.430
\f.
'--)' -v-llostt
0.4
--r'
/t
I
"t/nt'arP:0
.
{
0.8
0.6
==270!2t-",-ffir'o
=
---;>o
= o 427
!t-*;;iJ=;>-+--a---:;-
o = 0.545
o.2
-_arv-'
*,
,.a"=9.545
L--
0.2
0
2.0
1.0
zt.tzazzO'
'{----y'"t
,
-o.2
3.0
4.0 6.0
e/d
1.0
(c)
2.0
3.0
4.0 5.0
FIGURE 12.2.1c. Factors V, and Vnfl2-4).
e/d
(a)
c'B'
c*:z-Qe
4rt
q=O.404
0.8
a_ _ ,-z ae
0.6
_b
___
= 0.234
'P
--, - -- -a---tr..
.^
;,r;;;;
= n LnA
:
T
.
'y,z--x''---x-'-^-__---Lu--------b
$NN4
0.4
,\r'-o
ot
(r2.2.3)
\t/
'P=O 234
1.0
:
, or/
fbr trusses composed of members with circular cross section. The nominal
solidity ratio $" in Eq. 12.2.3 is related to the actual solidity ratio as shown
in Fig. 12.2.5.
Figure 12.2.6 shows ratios CB)|C$) for trusses with sharp-edged members
and aspect ratio X : 8 t12-11.
a'
o.2
0
-o.2
-t'
-.r
2.0
-!!1'-"t9-r-YT!9:':-'-'-"
c= 0 411a--:-----------?o
-o-t"oer@
-,- "\- ------o!{-l --- + - - --+
--'-r',*r@
--./ 1-.-.s:j-i-*7=;
- -,2":irl_ _ "- _ _) - - o- - - - -'e- - - - - 4- - - -' rvlodetc),r=0627
/ito''-
3.0
e/d
v
(b)
FIGURE 12.2.1a,b. Factors V' and VilIl2-41.
1.0
FIGURE
n.2.2.
odgccl menrbcrs,
2.0
.l
0
4
0
rl
lr,0
6,0
7.0
8.0
Factors V, nnrl V,, lirr lirur selr ol'two purullol trussos with sharp9.5, wind nortttal tr tntsl:,s ll2 41,
\ :
428
TRUSSED FRAMEWORKS ANI) PI
AII (IIIII)I N!]
l?
!
PAltls
of
Ilil,Jssl
ti
ANt)
ot I'l Alt oillt)t
429
nfi
2.2
2.L
2.0
1.9
1.8
r.7
CB)
cg)
1.6
1.5
r.4
*t,,lttt
*,,
1.3
I
1
r.2
_^[
,(U
1.1
0.3 0.4 0.5 0.6
t{l
0.7
a
lel
FIGURE 12.2.3. Approximate ratios c$) lC$) proposed for design purposes by Flachsbart |2-41.
Examples:
1. Consider a truss with sharp-edged members, solidity ratio @ : 0.1g,
spacing ratio eld: 1.0, and aspect ratio \ : oo. From Fig. l2.l.l,
cS) = 1.70 according to both Flachsbart and the NMI tests. From both
Flachsbart's and the NMI t.e^lts, C$ttC$\: Vr * Vrr
1.5 (Figs,
l2.2.la and 12.2.4a), so C$) = l.7O x 1.55 = 2.65. =
Note that according to Fig. 12.2.3-p.loposed by Flachsbart as a deliberately conservative design chart_C(]tlC')t = 1.83, which exceeds the value based
on Figs. 72.2.1a and l2.2.4aby about2O%.
2. Consider a truss with sharp-edged members, solidity ratio @ : 0.46,
spacing ratio eld: 1,.Q, and aspect ratio X = 9.0. Approximate values
of drag coefficients C$), ratios.CSrtCg, : Vr * V,,, anO corresponding
calculated drag coefficients C$), based on the Grittingen tl2-41, NMI
[12-6] and western ontario [12-5] information, are listed in Table 12.2.1.
It is seen that while the difference between the values c$) based on
|2-41 and [2-5] is abott l2%, the corresponding values Cfi) are vir-
tually identical in this case. Note also that thc clilibrcncc between thc
values Cf;) based on [12-61, on the one hancl, arrtl orr l12-41 or l12--51,
on the other, is about 25%.
-0
0
0.1
02
0.3
0.4
05
0.6
07
0.8
q
(o)
lflGURE 12.2.4. Factors Vr and Vtr for two parallel trusses with (a)
rrrcmbers and (D) members of circular cross section,
sharp-edged
\ : o, wind normal to trusses.
lirom R. E. Whitbread, "The Influence of Shielding on the Wind Forces Experienced
lry Arrays of Lattice Frames," in Wind Engineering, Proceedings of the Firth Interrttttional Conference, Fort Collins, CO, July 1979, J. E. Cermak (ed.). Vol. l. Pergamon Press, Elmsford, NY, 1980. pp. 405-420.
12.2.2 Trusses Skewed with Respect to Wind Direction
Wc now consider the case in which the yaw angle is cv * 0. For certain valucs
a the effectiveness of the shielding decreases, and thc drag cocfliciont
flll'}(cv) characterizing thc total l<rrcc normal to thc trusscs is largcr ilrln lhc
valuc C!]). (Rccall that, by dclinition, Cll\fq : C'i|'.)
Ilati<rs rnax lC(,ltkyl )/('If ) rcporrccl in ll2-51 lirr lnrsscs wilh sharp ctlgtrl
ol'
430
TRUSSED FRAMEWORKS AND PLATE OIRDEFE
0.35
0.30
o.25
0.20
0.15
0.10
0.05
tl,ltt
0.3
FIGURE 12.2.5. Equivalent solidity ratio
0.4
{.
o.7
0.8
for trusses with members of circular
cross-section and solidity ratio d. From R. E. Whitbread, "The Influence of Shielding
on the Wind Forces Experienced by Arrays of Lattice Frames,' ' in Wind Engineering ,
Proceedings of the Fifth International Conference, Fort Collins, CO, July 1979, J. E.
Cermak (ed.), Vol. l, Pergamon Press, Elmsford, NY, 1980, pp. 405-420'
0.2 0.3 0.4 0.5
0.6
i
e
r
)
(b)
T
i/
FIGURB 12.2.4. (Continued)
"t"'---
tt\ /
!iv
ir
members and aspect ratio
eld: l.o, O :0.286,
versus
CptC$)
:
X
:
i
8 are shown in Fig. 12.2.7.. For example, for
and X
:
8, the ratio max {C?'tol}lc8't
--'--a=0286
=
1.77,
1.59 (Fig. 12.2.6).
l,*-*--*- --'*---*
if
--- -----+---
e=
0'464
l4t
rc\
**--o____.o\\\
c=
0.773
12.2.3 Pairs of Solid Plates and Girders
Figure 12.2.8 shows the dependence of the factors Vl and V11(see F;q.12.2.1)
upon the spacing ratio eld for a solid disk and for three girders normal to the
wind [12-4, l2-lll. For certain values of the horizontal angle cv between the
wind direction and the normal to the plates the ratio C\\@ltCllt rnay be larger
than unity. For example, for a ptate with aspect-ratio \ = 4 und spucing ratio
eld:0.5, if 40' z .r < 65", then Cl3)(a)/C\3' 1,20 Il2"ll.
=
FIGURE 12.2.6. Ratios C!l)/C!i) ftrr tnrsses with sharp-edged members, \ : 8, wind
normal to trusses. From P. N. Gcorgiou rrnd B. J. Vickery, "Wind Loads on Building
ljrames," in Wind Engin,eering, Pnxvcding,r ol' the Fifth Intcrnatbnal Crryfercnce,
Fort Collins, co, July 1979, J. Il, ('ennpk (ctl,), Vol, l, Pcrgamon Prcss, Elmslirrd.
NY, 1980, pp. 421-433.
431
452
TRUSSED FRAMEWoFKS AND PLATE GIFDERS
I2,3
MULTIPLE.FFAME ARRAYS
433
TABLE 12.2.1. Drag Coefficients Based on the Giittingen, NMI, and Western
Ontario Studies
a----l
Flachsbart
NMI
Westem Ontario
t2-4
t2-6
t2-5
1.5x0.95=7.43o
0.23+O.92=1.15b
1.7x0.95=1.62"
12.7'
cgtcg'
1.29',t
1.30r
CB'
l.l5xt.43:r.64
2.08
1.65
References
cg)
0.6
"Figs. 12.1.1 and l2.l .2.
brig.
12.2.2.
'Figs. l2.l.l and 12.1.2.
'|Eq. 12.2.3a or Fig. 12.2.4a.
.Table 12. l. l.
---r
A --+til
xtffitta=2.0
+l
[lrt
-t"dt4 Ud=L3.6 +
#ei
"BtW, l/d= 9.5--c--
trig. t2.2.6.
i
il
tl
tl
!l
ill
-0.2
_.
':-A--
\l'--t'
. \rQ \
----
-^--'(t::lo
-o----:1.5
a\..
\
\\.r\io.'a--:
"\lr..\oi)..t*..-- :
1.0
-..
\.\-
\+
8.0
tt^
- --_
FIGURE 12.2.8. Factors V, and V,, for two parallel solid plates (girders) [12-4,
tz-rtl.
----
+
Data concerning the effect of bridge decks on the aerodynamic forces acting
on pairs of plate girders are available in [12-12].
=0.50
\-o
=
oj
7.0
1.0
i\\t\t'
o
2.0 3.0 4.0 5.0 6.0
e/d
-=
'\.----.=i;'\
\\\o
-:
\
rttt
-0.4
-^--
\'\l.
t
o;;i:.-----i
\\
\
1.0
FIGURE 12.2.7. Ratios max {C|@)llCg) for trusses with sharp-edged membeni, )\
: 8. From P. N. Georgiou and B. J. Vickery, "Wind Loads on Building Frames,"
Wind Engineering, Proceedings of the Fifth International Conference, Fort Collins,
CO, July 1979, J. E. Cermak (ed.), Vol, 1, Pergamon Press, Elmsford, NY, 1980,
pp.42r-433.
12.3
MULTIPLE.FRAME ARRAYS
'l'he first attempts to measure aerodynamic forces on multiple frame
arrays were
roported in
and [2-6].
[2-l]
For frames normal to the wind, thc drag coefficients for the first, second
... n-th frame may be writtcn ns VrClj), Vrcltt
V,,Ct|, where c[j)is
lhc drag coefficient fora single l'ruttte nonnul to thc wind. Thc clrag cocllicicnt
lilr the array of frames nomral lo tlre winel is tlrur
cl'j' =
clj'tvr
*
v2
r
I
V,)
(
12.3.
l)
434
TRUSSED FRAMEWoFKS AND PI ATF oInDET:IFI
\fj
I2,4
SQUARE AND IIIIANGULAH TOWERS
435
(,1 :
Factors
1,2, . . . , n) for arrays tlflthrce, four, and five parallel trusses
with sharp-edged members and infinite aspect ratio are given in Figs. l2.3.la
and l2.3.lb for spacing ratios
0.5 and
respectively [12-6J.
Drag coefficients C$) for the same arrays are shown in Figs. 12.3.2a and
12.3.2b tl2-61. Also shown in Figs. 12.3.2 are measurements of c(p tor trusses
with infinite aspect ratio and members with circular cross section 112-61.
eld:
lst
eld: l,
Frame
-{
Frame
Symbol
o
configuration
.......-.=
-----+r
1.0
o
0.9
-
tt
it
l
TI
L-J
rirtl
12
3 45
s
+
I\
,ltl,Qtt
tltn
05
t
)@
OC
il
0.2
nl
0.3
0.3 0.4 0.5 0.6
0.7
(b)
o e/d
A
02
= 0.5
FIGURE 12.3.1. (Continued)
1.0
o 2.0
+ 3.0
te
01
Ratios
4.O
o"2
0.3
0.4
0.5
u.b
0.1
(a)
FIGURE 12-3.1. Factors'rj (i : 1,2, . . . , n) for arrays of r parallel trusses (n =
3,4, and 5) with sharp-edged members, X : oo, wind normal to trusses. (a) Spacing
ratio eld: 0.5. (b) Spacing ratio eld: 1.0. From R. E. Whitbread, ..The Influencc
of Shielding on the wind Forces Experienced by Arrays of Latticc Frames," wind
Engineering, Proceedings of the Fijlh International ConJbrence, F<lr1 collins, co, July
1979, J. E. Cermak (ed.), Vol. l, Pergamon Press, Ehnslirnl, Ny, l9tt0, pp. 4051
420.
cfltc$)
measured
:
in [r2-rl for trusses with
sharp-edged members
and aspect ratio x
8 ar9 shown in Fig. 12.3.3. As pointed out in Sect. 12.2,
the drag force normal to the trusses doei not reach a maximum
when the trusses
are normal to the wind, but for some yaw angle cy ;e 0. Ratios
max
lc?@>l tc\) measured for the trusses just iescribed are shown in Fig. 12.3.4
lt2-tl.
12.4
SQUARE AND TRIANGULAR TOWERS
As pointed out earlier, thc aenxlynarnic crrcflicicnts givcn in this
chaptcr aro
in all cases referred to, ancr shourcr be usotl in conjurition with,
thc clJ.ectivc
irrca of thc fiamework, A.s. For nqunre rrnrl triunguliir lowcnt,
zll is rhc cll.ective
illt,lilit l) lltnMl w()llKli nNl) I't n ll (,llll rl ll:i
436
l:,4 :i(.)llnlil nNI) ilInil(ilil/\t I towt il,.
4:ll
-r---I
r-@)
4.4
4.8
4.0
4.4
3.6
4.0
-1
I
cg)
-D
n:5
n:4
oAngle ser:tiorr
mernbers
x Crrcular-sectron
members
3.6
J.Z
1.6
,6
2.8
2.0
2.4
L6
2.O
-\_lr:2
t.?
-*--*---)5*
"
0.8
16
tt-l
r2
--\!
--x-n:2
x\--x--x-n:
08
0.4
0
-ra_
0 0.1
0.2
0
3 0.4 0.5 0.6 0 7
a
(a)
0.8
I
I
I
{\
A
- ___. _ ----.
-- -L' = 0.286
a
,
I
I
OM
0 0 1 0.2 0.3 0.4 0.5 0.6 0.7
0,u
a
(b)
FIGURE 12.3.2. Drag coefficients C$)forarrays of n parallel trusses, 1 : oo' wintl
normal to trusses. (a) Spacing ratio eld:0.5. (b) Spacing ratio eld: 1.0. Frorrr
R. E. Whitbread, "The Influence of Shielding on the Wind Forces Experienced by
Arrays of Lattice Frames," Wind Engineering,, Proceedings of the Fiilh Internationul
Conference,
,
FortCollins,CO,July 1919,1. E.Cermak(ed.),Vol. l,PergamonPress.
Elmsford, NY, 1980, pp. 405-420.
area of one of the identical f'aces of the tower. The influence of wind gustiness
on the tower loading and response can be determined by using the methods firr'
estimating along-wind response discussed in Chapter 9'x
For information on guyed tower response and design, see [4-10], 14-111,
and I l2- l1l ro 12-261.
12.4.1 Aerodynamic Data for Square and Triangular Towers
'l'ho rcsults of wincl tbrce measurements on square towers can be expresscd irr
tcrrns of the aeroclynamic coefficients C,v{cv) and C7(a) associated, respectively,
with the wind force components N and Z (N = 7) normal to the faces of tlrt:
*The width of the structure used as an input in these methods should be equal to thc actual witlllr
of the framework. This ensures that the lateral coherence of the load lluctuations is lakctt ittlo
account. On the other hand, the depth (along wind dimcnsion) ol'thc ll-atttcwork slurtrltl lrr'
assumed to be equal to zero in order not to ovcrcstinlatc thc lavorahlc cllcct ol tlte itlortg wittrl
cross-correlations ol thc fluctuating loatls (scc Scct. 4.7.4). lrirrlrlly, thc itrc:t ol lltt' li:ttttt'wotl
per unit height at any givt:rr clcvlrliorr. uso(l to cslirnirlt: llttr ttterttt ittttl llrt' llur'ltlltliltli tlr:rg lirt t s.
shoulcl bc cqual to lhc clli'r'livr':rr('it l)cr rrlrit lrt'ig,hl :tl lltltl t'ltv:tliort
e,d
IGURE 12.3.3. Ratios c!j)/c!j) for arrays of five trusses with sharp-edged members,
\ - 8, wind normal to trusses. From P. N. Georgiou and B. J. Vickery, '.Wind Loads
trn Building Frames," wind Engineering, Proceedings of the Fifih Intemational conli,rcnce, Fort Collins, CO, July 1979, J. E. Cermak (ed.), Vol. l, pergamon press,
If
lilmsford, NY, 1980, pp. 421-433.
tower (Fig. 12.4.I) and in terms of the aerodynamic coefficient Cp(a) asso-
ciated with the total wind force Facting at a yaw angle cv : tan-r (Z/N). Note
that Cp(cv) : tcfo") + C2r1a11t/2, since, as indicated earlier, all aerodynamic
eoemcients are referenced to the effective area of one face of the frame-
work, Al .
For a triangular tower (which has in practice and is therefore assumed here
Io have equal sides in plan), the results of the measurements can be expressed
irr terms of the aerodynamic coemcients Cp(cv) (Fig. 12.4.2). The aerodynamic
t:ocfficients Cr(0") and CF(60') correspond, respectively, to wind forces acting
in a direction nornal to a side and along the direction of a median (Figs.
12.4.2e and 12.4.2b).
Measurements of loads on a tapcrcd square tower model with sharp-edged
rttcmbcrs, aspect ratio \
= oo. lrrrrl soliclity ratio averagcd over the height clf
lhc k)wcr d
0. 19 (rangilrg lnlrr y'r O. 13 irt (hc hasc to @ : 0.47 at tltc
=
(ip) wcrc rcportccl in lhc I().]Os lry lr.:rlzrrrryl rrrrtl Sirilz ll2-131. Ilrrtil rcrcc:ltlly
(ltcsc: Ittoasurclncnls ltltvc lrt't'rt llrt' l)ur( rl);rl s()ur'((' ol'rllrlrr ()n s(luiu1' l()w('t'ri.
'l'ltc: crlcllicic:nls (i(rv), (',(rv). ;rttrl {', 1rv) olrl;unt'rl in
llJ l.\l ltl' listt.tl lil
v:rriorrs irrrgltrs rv in 'l'lrblt' I J .l I
Ittt-rv
45" tltt' v:tlttt's ol (''1,(,r ) .rnrl ( r(,r) :,lrottltl lrt' r't1rr:rl, lr:l Porrrlt'rl
qf
438
I?.4
TRUSSED FRAMEWORKS AND PLATE OIRDERS
SOUARE AND II]IANGIJIAII iOWrIIH
439
$
-L
']l
^t^
"flti
;l
t.^
\;'...
\'6.\. \^
u-\*i..:--o -i3:r o-{-_
-V- -' J-d-=g zi- ___A
Eo
o=30'
-. - - - --{
'ts\'i-*
'x.-_;atji.-_-::1
NI
EI
(a)
(b)
(c)
FIGURE 12.4.2. Notations.
;;l;-:l=Tr- -----:
1.0
0.5
0
1.0
a.t"."r, of five trusses with sharp-edged
8. From P. N. Georgiou and B. J. Vickery, "Wind Loads on Building
Frames," Wind Engineering, Proceedings of the Fifth International Conference, Fofi
Collins, CO, July 1979,1. E. Cermak (ed.), Vol. 1, Pergamon Press, Elmsford, NY'
FIGURE l2.3.4.Ratios max {Cf,'1ultClt,
members,
\:
1980, pp. 421433.
out in [12-131, the 4% difference between these values in Table 12.4.1 is due
Itr measurement errors. Note that the value C^(0") :2.54 is close to the values
inf'erredfrom [12-5] and [12-6], whichare, respectively, C,,(0") : Ct)
= I.5
x 1.73 :2.60 (as obtained by linear interpolation for 6 : O.I9 and eld
:
1.0 from Table 12.1.1 and Fig. 12.2.6), and C"(0") : Cg)
= t.7(0.93 +
0.58) : 2.57 (Eq. 12.2.1, and Figs. 12.l.l and 12.2.4a). Note also that while
(hc largest tension (compression) in the tower columns is caused by winds
rrcting in the direction a : 45" , the largest stresses in the bracing members
trccurfora:27".
Measurements of forces on square towers with sharp-edged members
oo) were more recently conducted at the National Maritime Institute,
(\ :
u.K.
(NMD ll2-141. coefficients cr(0") and ratios cp(u)rcp(O") based on these
nrcasurements are shown in Figs. 12.4.3 and 12.4.4, respectively. Note, for
cxample, that for 6
= 0.19, Cr(0.) = 2.60 (Fig. 12.4.3), versus Cr(O") :
2.54, as obtained in [12-13] (Table 12.4.1). The agreement is less good for
lhc ratio cF(45")/cF(O"), which is about 1.12 according to Fig. 12.4.4, and
rrbout 1.40 according to the data of rable 12.4.1. As shown subsequently in
llris section, data on square towers composed of members with circular cross
scction suggest that the NMI results are more reliable than those of
[12-13].
'I'ABLE 12.4.1. Aerodynamic Coefficients: C"(o), C.(a), and Co(o) for
St;uare Tower with f = 0.19 and ), = o [12-13]
(t
0"
9"
(,v(tr)
2.54
)11
2.54
0.19
2.76
J,05
J..r0
(', (c)
FIGURE 12.4.1. Notations.
('1"Qx)
18"
a
270
36"
45"
2.97
3.0I
0,7{)
1,36
2.84
2,05
3.50
2.60
2.49
3.60
f,r
l;'4 l;()unl rl nl.Jl) ilttnil{irt t\il t1)wt tt,.
441
Lr(0")
08
0.2
o
o
Angle members-smooth flow.
Angle members-turbulent flow.
+
Square shaped members-smooth flow
FIGURE L2.4.3. Drag coelficients Cp(O") for square tower with sharp-edged membcrr
measured at National Maritime Institute, U.K. From A. R. Flint and B. W. Smith.
"The Development of the British Draft Code of Practice for the Loading of Latticc
Towers," Wind Engineering, Proceedings of the Fifih Intemational Conference, Fctrl
Collins, CO, July 1979, J. E. Cermak (ed.), Vol. 2, Pergamon Press, Elmsford, NY,
1980, pp. 1293-t304.
- - n 1?,
e
= 0.535
5
1
5
6 7 8
9
2
105
3
4
s
6 7 8 9106
4z
2
FIGURE 12.4.5. Drag coefficients Co(0') for square tower with members of circular
r'mss section [12-7] (courtesy Comitd International pour le Ddveloppement et I'Etude
tlc la Construction Tubulaire, and Constructional Steel Research and Development
(
)rganisation).
cr@)
cr(0.)
Sguare Towers Composed
fion.
I oo:
15' 30'
45'
FIGURE 12.4.4. Ratios Co(a)/Co(O') for square tower with sharp-edged mcrnlrt'rr
measured at National Maritime Institute, U.K. From A. R. Irlint and B. W. Srrritlr.
"The Development of the British Draft Code of'Practicc lirr tlrc Loltling ol'Lirtlit'c
Towers," Wind Enginct'ring, Pnx'rcdings rl tha [,'i.lilt ltttt'rttttti,ttrrtl ('rtr.li,rt,rrcr,, ltrrt
Collins, CO,.luly l9l9,.l .1,1. Ccrrrrlrk (ctl.), Vol. 2, ll'r1'.:rrrron I'n'ss. lllrrrslirrtl, N\',
1980,
440
pp. 1293 I304.
of Members with Circular Cross
Sec-
Figures 12.4.5 and 12.4.6 [12-7] represent, respectively, proposed aero(l_vnamic coelicients Cr(O') and Cp(45') as functions of Reynolds number Ge
lirr towers with aspect ratio }, : oo, based on recent wind tunnel test results
rcported in [2-8] and [2-9]. The values CF(45') of Fig. 12.4.6 may be
regarded as conservative envclopes that account for the loadings in the most
rrnlavorable directions. Rcsul(s ol'tcsls conclucted at NMI in both smooth and
trrlbulent flow at Rcynokls rrtrnrlrt't.s (11,. : 2 x 101 li)r solidity ratios @ :
rx') rrurtt'lt thc r.rrrvcs ol'ljig,. 12.4.-5 urrcl
O ll , O : 0.23, ancl <,f - 0. I I (^
ll.' l.ll
\ llrt' r;rtrr' (; (,1i")/('/ (O") is r'plsltlcl.lrfly t'lost.r'
to LI
l.'l rrlrt'tt'trl rrr'lrrlrlt' I-).l I 'l'lris worrlrl lr.rrrl lo
,ottlirnr lltc hrrxrrl vlrlirlity ol llrt NNll rr',,rrll . on .,(luju(. 1()\v(.1\ rvrllr slr:rrP
t'rlgt'rl lttclttlrt'r's rlisr'rrsscrl t';ultt'r ln llu', '.( ( lt'n
11.4.6 to within ab<lul 5%, or lt':i:;
lhat lirr 0 < ,h .
thitlr lrt lltc virltrc
N<rtc
1',
442
c
Inul;l;l l)
I nn Mt w(
)t
rh:,
n
l\| ) t,t n
l
(
,l
l,t
*
tr.
l;'.1
:;(Jl ,nt
il
nt..ll
| iltlnt.t(,t,t nll l()Wl tt:;
4A.-l
n(45)
5
1
5 6 789
10s
7
r
4
s 6 r I 9106
q,
2
TIGURE 12.4.7. Drag coe{ricients c1.(0") and co(60') for triangular tower
with membcrs of circular cross section [12-7] (courtesy comitd Intemation"al
pour le D6veloppe_
rncnt et I'Etude de la Construction Tubulaiie, and Constructional
Steel Research and
I)evelopment Organisation).
3
1
5 6 r s
9 1S
2
3
4
s 6 r I
9
10.
!/te
?
FIGURE 12.4.6. Drag coeflicients Cr.(45') fbr square tower with members of circulal
cross section [12-7] (courtesy Comit6 Intemational pour le Ddveloppement
et I'Etuilc
de la Construction Tubulaire, and Constructional Steel Research'and Developmcnt
0rganisation).
Triangular Towers composed of Members with circular cross section- Figurcs 12.4.1 and 12.4.8 [12-7] represent proposed aeroclynamic cocr'ficients c/.(0") = cr(60') and cp(30") as functions of Reynolds number
61"
fttr t<twcrs with aspect ratio X : oo, based on measurements reported in
tl2-81, It2-91, and [12-101.
FIGURE 12.4.8. Drag coefficients C,.(30") for triangular t()wcr with rncrrrbcrs ol
circular cross section ll2-71 (courlcsy c.miti rrrtcrn.ti..:rr prrrr lt' r)t:vt'r.ppcrr*rrr
t.r
I'Etude de la Constnrctitln'ftrbulairc, lrnrl ('onslrucliorr:rl Slt.r'l ltt.st':lt.lr rrrrtl l)t:vt,l
opmcnt Organistrl iorr ).
I i,,(30')
'1.0
444
ilttjlitit t) |tnMt w()nKt;
t
nNt) r'r n r {;lu)l rr:i
REFERENCES
l2-l
l2-2
P. N. Georgiou and B. J. Vickery, "Wind Loads on Building lilrrrrcs." l4lirrrl
Engineering, Proceedings of the Fifth International ConfcrcnL'c, Forl Collins,
CO, July 1979, J. E. Cermak (ed.), Vol. l, Pergamon Press, Elrnslirrd. NY.
1980, pp. 421-433.
O. Flachsbart, "Modellversuche iiber die Belastung von Gitterlachwcrkcn durclr
l. Teil: Einzelne ebene Gittertniger," Der Stahlbau, T (April 1934).
Windkriifte.
65-69.
l2-3
l2-4
Loading on Opcrr
Framed Structures," Proceedings Third Canadian Workshop on Wind Engi
neering, Vancouver, April 198 l.
12-6
l2-7
l2-8
l2-9
R. E. Whitbread, "The Influence of Shielding on the Wind Forces Experienccd
by Arrays of Lattice Frames," Wind Engineering, Proceedings of the Fililt
Internatktnal Conference, Fort Collins, CO, July 1979, J. E. Cermak (ed.),
Vol. I, Pergamon Press, Elmsford, NY, 1980, pp. 405-420.
Wind Forces on Unclad Tubular Structures, H. B. Walker (ed.), Constradrr
Publication l/75, Constructional Steel Research and Development Organiza
tion, Croydon, U.K., 1975.
G. Schulz, The Drag of ktttice Structures Constructed from Cylindrical Metn
bers (Tubes) and its Calculation, CIDECT Report No. 69/21, Drisseldorf, Wcst
Germany, 1969 (in German).
G. Schulz, International Comparison of Standards on the Wind Loading ol'
Structures, CIDECT Report No. 69/29, Drisseldorf, West Germany, 1969 (irr
German).
12-lO R. W. F. Gould and W. G. Raymer, Measurements over a Wide Range tl
Reynolds Numbers of the Wind Forces on Models of lnttice Frameworks, Nlt
tional Physical Laboratory Sc. Rep. No.5-72, Teddington, U.K., May 1972.
l2-ll G. Eiffel, ln Rlsistance de I'Air et l'Aviation, H. Dunod & E. Pinat, Paris.
l9l 1.
12-12 J. M. Biggs, S. Namyet, and J. Adachi, "Wind Loads on Girder Bridges."
Transactions, ASCE, f2f (1956), l0l-113.
12-13 D. Katzmayr and H. Seitz, "Winddruck auf FachwerktLirme von quadratischcrrr
Querschnitt," Der Bauingenieur, 2lI22 (1934),218-221 .
12-14 A. R. Flint and B. W. Smith, "The Development of the British Draft Coclc ol
Practice for the Loading of Lattice Towers," Wind Engintt'ring, Prot'cctlittgl
of the FiJih International Conferenr:e , Fort Collins, ('O..luly l()79, J. U. ('cr
mak (ed.), Vol.2, Pergamon Press, l9fl0, pp. 129.1 llt),I
12-15 J. F. Eden, A. lny, and A. .1. Bullcr, "('rlrncrs itt Slorrn Wirrrls." l'.'rr,q. ,\trttt't..
3 (r981). r75 rtto.
lt
N(.t
:;
445
l.) l(r .l . li. lltlcrr. A..l . lhrllt'r. rttttl .l . I'rtlit'ttl . "Wirrrl lurrnt'l 'lcsls orr Mtxlcl (-'mrrc
Slrttt'ltrcs." l'.rr,q. ,\trrtr't ., 5 (l()8 ]), .ll"i() -)()S
I ) l1 (i- A. Savitskii, (lrlt'rtltttittrt.t litr' ,,ltttt'ttrrtr ltt.tttrllrttiotr.s, 'l'cchnical Translation
Il llt
l -l-
O. Flachsbart, "Modellversuche iiber die Belastung von Gittedachwerken durch
Windkrdfte. l. Teil: Einzelne ebene Gittertreger," Der Stahlbau, T (May 1934)
13-79.
O. l.'lachsbarl, and H. Winter, "Modellversuche iiber die Belastung von Git
tcrlachwcrkcn tlurch Windkr:itte. 2. Teil; Rdumliche Gitterfachwerke." Drr
Srultllnu, tt (April 1935), 57-63.
l2-5 P. N. Gcorgiou, B. J. Vickcry, and R. Church, "Wind
nt I r
19
'l"l'79-52040, prrblishcrl lirl llrt' Nrrliortirl Sticnec Iioundation by Amerind Publishing Cb., Ncw l)cllri, lgllf , :rv:ril;rlrlc liurr National Technical Infbrmation
Scrvicc, Springlicltl. VA 22 l(, I .
V. Kolouick, M. Pirncr, (). Iiischcr, and J. Niiprstek, I{ind Effects on Civil
Enginecring St rudur(s, Elscvicr, Amsterdam, 1984.
R. J. McCaffrcy and A. J. Hartmann, "Dynamics of Guyed Towers,', J. Struct.
Dlv., ASCE, 98 (1912), 1309-1323.
12-20 J. W. Vellozzi, "Tall Guyed Tower Response to Wind Loading," proceedings
Fourth International Conference on Wind Effects on Buildings and Structures,
Heathrow, September 1975, Cambridge Univ. Press, Cambridge, 1976.
l22l R. A. Williamson, "Stability Study of Guyed Tower under
Ice Loads,"
-/.
Srruct. Div., ASCE, 99 (1973),2391-2408.
12-22 J. E. Goldberg and J. T. Gaunt, "Stability of Guyed Towers,,' J. Struct. Div.,
ASCE, 99 (1973),'741-1 56.
l) 23
l)-24
R. A. Williamson and M. N. Margolin, "Shear Effects in Design of Guyed
Towers," J. Struct. Diy., ASCE, 92 (1966),213-260.
F. Rosenthal and R. A. Skop, "Method for the Analysis of Guyed Towers,',
J. Struct. Div., ASCE, f08 (1982), 543,558.
Brown and J. W. Melin, Guyed Tower Program Listings and
l)-25 D. M.
(Jser's
Manual, Technical Report sponsored by United States Coast Guard, U.S. Depaftment of Transportation (Contiact DOT-CG-52604-A), J. W. Mellin and
Assoc., Urbana,
lL,
1975.
1226 A. G. Davenport and B. F. Sparling, "Dynamic Gust Response Factors for
Guyed Towers," J. Wind Eng. Ind. Aerod., 4l-44 (1992),2237-2248.
l) 27
ll28
J. D. Holmes, R. W. Banks, and G. Roberts, "Drag and Aerodynamic Interf'erence on Microwave Dish Antennas and Their Supporting Towers,', J. Wind
Eng. Ind. Aerod.,50 (1993), 263-2'70.
rnttice structures: Pan 2-Mean Fluid Forces on Tower-like space Frames,
Engineering Science Data Unit, ESDU Data Item 81028, 1988 (rev. ed.).
l)29 K. Hiramatsu
and
Action of Wind,"
K. Akagi, "The Response of Latticed steel rowers to the
J. Wind Eng. Ind. Aerod.,30 (1988), 7-16.
CHAPTER 13
b,it'
-o
tsEp
;i=
o
qs cJ u>
!!^
> :::
zv=t
SUSPENDED.SPAN BRIDGES,
TENSION STRUCTURES, AND
POWER LINES
b
t
ri
I
t
N.
I
t!
{
Structures that consist of or depend for their integrity on cables or membranes
may exhibit an increased susceptibility to wind effects. Notorious examples arc
the Brighton Chain Pier and the original Tacoma Narrows Bridge (Figs. l3A
and l38). The purpose of this chapter is to present information and references
conceming such structures, including suspension and cable-stayed bridges, ca-
t
&
d
{
\
o
R
t'
,s
)*\
sI
\
and power lines.
9)
t
[,
I
SUSPENDED-SPAN BRIDGES
(i.e., suspension and cable-stayed) bridges must be
designed
to withstand the drag fbrces induced by the mean wind. In addition such bridgcs
are susccptiblc to aeroelastic effects, which include torsional divergence (or
latcral buckling), vortex-incluced oscillation, flutter, galloping, and buffeting
in thc prcscncc of silf--excitecl fbrces. The study of these effects is possiblc
only on the basis of infbrmation provided by wind tunnel tests. Various typcs
of such tcsts arc briefly clescribed in Sect. 13. 1. I . Procedures for analyzing thc:
susceptibility of suspended-span bridge decks to aeroelastic effects and pertincnt
design considerations are presented in Sects. 13.1.2 through 13.1'5'
It is noted that the action of wind must be taken into account not only lirl'
the completed bridge, but for the bridge in the construclion stagc as wcll. Irr
'l'tr
general, the same methods of testing and analysis ap;rly ilr (ltt' two cltscs'
wirrrl.
t(:lltp()rlll'y
decrease the vulnt3rability of thc parfially cornplotr:(l bt.irlp,c lo
ties ancl damping rlcviccs arc uscrl. Also. lo tttittilttizt'lltt'rt:;l'
446
ol sl trrttg wittrl
q
(
ble roofs, fabric structures (air-supported or otherwise subjected to tension)'
Suspenclcd-span
!
{t
jt
l,
i
:i
E,
t'
I
t;
\
\
-J
.l
d
I
t
w
*
w:
$
,#,
N
$
ai*
--
-Ez-
JV
'34)
'od.,
ttt
13.1
^
i*
,1,}
r
#,
io€.5
ar.o 6}EO
cn--"!3
s
6-|:..
'=o\
-a@t
.'!
! c
Or5Cs
N ^ U_
L
A
os.=U
91|,=
/r
c9!Y
!,XgH
e^a)
zAv*
P
e.?
3-o
=Yaoa
.-9V
,r!
! ^
=
-!d*
,@H+
aJ r
Coo
v
H p.i
.:5bt
*-a
(.)e3_.
F 3;3
e
:,-E
b05 h."'tr@.=t
m,=A
or3-o(a
" -a i.A'
-i!!;
beed
O!d;
!FAS
U c.)V
li
g'^Fi
-9
rn4=. H 9'-,
<-a2a
4=&R
.U
gd
l.-E
a)i,-ti
tu .1\
il
444
t;U:;ltt Nt)t t)r;t,nN tit rilxit
i; il
N:,t()r..r
:,rtr,(.l,nr :;
nNr) r,()wr tr
lNt
*
r:r
:
YW!,,,.,,.,.
&"
qsaiid:,,,'i;,'
,!q4$.,i.,
r
:;U:;l
'l Nlll t):;l'nN lllillxit li
449
ll''1,1. ltti*.'..';l
.iirg$rr,:-rrrip ..,,r.;{.
.,
id
Flutter of the Tacoma Narrows Bridge, November 10, 1940 [13-1,
loading, construction usually takes place in seasons with low probabilities ol
occurrence
of severe storms.
Aeroelastic phenomena may affect, in addition to the deck, bridge toy_or'
hangers. and cables. Problems relatecl to the design of these. or similar. clc
ments are dealt with in Sect. 13.1.6.
lflGURE 13.1.1. Model of Akashi Strait suspension bridge (courtesy of T. Miyata,
Yokohama University, and
M. Kitagawa, Honshu-Shikoku Bridge Authority, Tokyo).
13.1.1 Types of Suspended-Span Bridge Wind Tunnel Tests
The following three types of wind tunnel tests are currently being used to obtain
information on ihe aerodynamic behavior of suspended-span bridges.
l.
l'ypically a support structure consisting of taut wires or tubes, or of a fine-wire
t'ltcn&rl, supports the geometrically simulated deck structural form. Usually
lirndamental vertical and torsion modes are simulated. The model is enveloped
of the full bridge. In addition to being geometrically
similar to the full bridge, such models must satisfy similarity requiremc-nts
pertaining to mass distribution, reduced frequency, mechanical damping, arrrl
shapes of vibration modes (see Chapter 7). The construction of full-britlgc
models is thus elaborate and their cost relatively high. The usual scale of srrclr
models is of the orderof 1/300, although scales of l/100 have been usecl irr rr
few cases [3-l] to [13-6]. A view of a full-bridge model in a wind tunncl is
sholvn in Fig. 13.1.1.
2. Three-dimenskna.l. partial-hridgc mtxlcls. ln rrrtxkrls ol'tlris typt: thc rrririrr
span (or occasionally lrull'ol'il) is rrrrxlclcrl in rrrr t't'orurrrrir':rl ;rlrpnrxirrurliorr.
Tests on models
i1 three-dimensional simulated boundary-layer flow in the wind tunnel.
'3. Tests on section models. Section models consist of representative spanwisc sections of the deck constructed to scale, spring-supported at the ends to
:rlkrw both vertical ancl torsional motion, and, usually, enclosed between end
plrrtcs to reduce aeroclynrrtrtic c:ntl cllbcts (Fig. 13.1.2). Section models are
t'lrrtivcly inexpensivc. 'l'lrt'y t':rrr lrt' r'onslnrctctl to scales of the order of l/50
to ll25 s<l that thc tliscrcp:rrt'ics lrt'lwt'r'rr lirll-scllc and modcl Rcynolds numlrt'r''r'irrc srrrallcr lluru irr (lrc t'rr:rc ol lrrll lrlrrllit'lcsls. Scclion trttlclcl.s are quite
lry
ll,ol lr tlistrrssir)il ()l lt('ylt()l(ls ililrillr'r
',rrrrtl.rttl\ ri ilIrtt ilr{ [l',. sct ('lt:t1tlt'ts'1 itrrrl 7.
lit,lil,l Nl)l l)til'nN lililtxit :;, il Nl,t()t'J :;lltl,(iil,1il:i.
ANt
) t,()wt n ilNl
**
i,l,l;l'l
l:| I
i
tllrl lr l;l'nll
llllll)(il
',
451
o.lr
(lt
04
t
0.3
0.2
,v
(rlr'r1)
0.8
0.6
0.4
0
o.2
,0
C,
-0.8
CM
-o.2
-0.4
useful for making initial assessments, based on simple tests, of the extent ttr
which a bridge deck shape is aeroelastically stable. Finally, section models
have the important advantage of allowing the measurement of the fundamental
aerodynamic characteristics of the bridge deck on the basis of which comprchensive analytical studies can then be carried out. These characteristics includc:
a. The
steady-state drag,
lift,
and moment coefficients, defined as:
vD
.o
-
wrB
C," :
,
?M _
,-,
L,
lpu'n
i,urB,
-0.8
F--Q--n
-2.4
-0.6
FIGURE 13.1.2. Section model of the Halifax Narrows Bridge (courtesy BoundaryLayer Wind Tunnel Laboratory, University of Western Ontario).
+nl
- l.t)
B
1a
-30-20-10 0 10 20 30
a
(deg)
-30 -20 -10 0 10 20
a (deg)
FIGURE 13.r.3. Drag, lift, and aerodynamic moment coelficients for
Tacoma Narrows Bridge [13-l].
30
replacement
Tacoma Narrows Bridge [13-1] and in Fig. 13.1.4 for a proposed streamlined box section of the New Burrard Inlet Crossing t13-81'
aerodynamic coefficients. These coefficients characteize
forcls acting on the oscillating bridge and are discussed
self-excited
the
in Sect. 6.5.2. Examples oimotional aerodynamic coe{ficients F1,f , .4f
(t : 1, 2, 3, 4)for various types of bridge decks are given in Fig' 6'5'3'
questions pertaining to the laboratory ditermination of H,f , A! are teviewed in [13-9] and [13-88].*
c. The Strouhal number S (see Sect. 4.4)'
b. The motional
(l3. r. r )
(13.1 .2)
(13. r.3)
where D, ,L, and M are the mean drag, lift, and moment per unit span,
respectively, p is the air density, B is the deck width, ancl U is thc nrcarr
wind speed in the oncoming flow at the deck clcvalion. 'l'lrcsc cocllicicnts
are usually plotted as functions of thc anglc rv lrt'lwt't'rr tlte lrorizrlrrltrl
plane and thc planc of thc briclgc clcck. ('ocflicicrrts ('t,. (', . irrrtl ('p, rrrl'
shown in Fig. 13.1.3 lirr lhc opcrt lntss britlgt' tlt'r'l' ol llrt' rt'lrlirt't'rrrcrrl
1g.1.2 Torsional Divergence or Lateral Buckling
I-ateral buckling of a bridge deck may be viewed as that condition wherein,
mogiven a slight deck twist, the drag load and the self--excited aerodynamic
divcrgcncc
Thc
ttlrsional
instabilitydivergence
r-ncnt will precipitate a torsional
phcn<lmcnon has been analyzccl in Sect. 6.4 in thc casc ol-a lwtl-tlittlcttsiottltl
r,l' rrr.rt, rct.t:.1 slrrtlias t.rx.llit.it'rrls // ] ;rrrtl ,1 f' hlrvt: bccn irrclrrtlgl; :tlso, tlr:t1' r't'lltlt'tl t rx'llit icttl:
l, 2, l, 4) lltvc lrt't'tl iltltrxlrtr ''rl (:;t't' Ii; l I l 4l :rrrtl I I l ltt1l)'
t,',!' li
452
st,sPFNI)H)t;t)AN Slillxit
li, llNlit()N tilnuott,nl t;, ANt) t'()wt n ilNt
1.20
tttt
t:l I lit,lit'l Nt)t t) l;t'nN Bnil)(it l;
I
tt
453
(:lcnrcnls ()l lhc nritltrx ('1 :rrc tlt:rrolc:tl by r';; arxl rcprcsotr( thc anglc ol'twist
(yi irl .r' - .r'i irttlrtccrl lry lr rrrril lolsional nr()nlcnt acting at x : xi.
Lct lrr) rcprcsurl tlrr colurun rrratrix of the angles of twist a;. In matrix
nolll ion
J--/<.-".-"-/
\J
*rl-
CD
ttlr
0.0
{cv}
CL
rl
tttt
0.0
ttlt
: Cr{M}
(r3.1.4)
{M} represents the column matrix of the torsional
x : x;. These moments can be written as
where
at
moments M1 applied
-.t0
_.40
u, : )pu2B2tt,Cr1ai)
.20
*.80
Al, is the span length associated with point xi. The problem is now
susceptible of solution by iteration on Eqs. 13.1.4 and 13.1.5. First it is assumed o; : 0 for all j and M1 are calculated from Eq. 13.1.5. Inserting these
rcsults in Eq. 13.1.4 yields a column of values cy;; reinserting these into Eq.
13.1.5 develops new moments, and so on. The process will converge for any
chosen velocity less than the critical divergence velocity that conceptually is
approached in an asymptotic manner by the iterative method suggested.
The process is simplified, however, in the case where Cr,(o) can be approximated by a linear function
-.30
- 10.0
where
-1 .20
0.0
a
tttt
0.0
d
CM
A
Handrails
-
Q No handrails
0.0
tt
dC,
Cy(u) =
da
guardrails
-
no guardrails
(13.1.6)
where CMs
:
Cu(O). Using the notation
trtt
1: -I oU'B"A,L,
lnd assuming A/,
: A/
lrlt
0.0
0
FIGURE 13.1.4. Drag, lift, and aerodynamic moment coeltrcients for proposed deck
of New Burrard Inlet Crossing [13-8]. Courtesy of the National Aeronautical Establishment, National Research Council of Canada.
structure. In this section the analysis of Sect. 6.4 is extended to the case of a
tull bridge.
The data needed for the analysis are the experimcntully rrrr::rsrrrcrl rngrncnl
coefficient C7a@) and the torsional flexibility matrix (',.oIthc tlt't'k. l.ct.r', arrcl
x1 G, i : 1,2,. . . , N) ilcn<ttc valttcs tll'tho c<xlnlirrirlr' r irkrrrlq tlrt'splr1. 'l'lrc
for all i yields
{o\
10.0
:
c,
| (dcM
* arr]
,Ld" "
I
ac^,c.lI
{o) :
lr, -n
licluation 13.1.9
(13. 1.7)
2'
p
-.20
-10.0
cro
-a-F
-.10
-.30
(13.1.s)
Cr{Cuo}
(r3.1.8)
(13.1.9)
will havc inlirritc (lolsiorrllly divergent) solutions when the
tlctorrninant
1," ",','n''''l
'
(r3.r.r0)
454
lltjljl'l Nl)l l)l;t'AN nnllxil :;, ltNt;t()N tiilil,{:t{,ilt :;, ANt) t,()wl ti ilNt
:
;l,l;l 'l Nll,l t):it 'nN ilhil){:t:
Equation 13. l.l0 yiclds a sct tll' c:hllrrctclislic valucs 7r ol' whiclr llrr: lrrr.gr:sl
: pc corresponds to the lowcst vclocily IJ - IJ,. lilr torsional divcrgr:ncc:
f
l
ltt2
U-:l----:--l
' lp,pB'LL)
455
1r
(r3.1.il)
1) 1.8 rr trrin,r,
In general it is found that only torsionally weak bridges incur the actual
danger of torsional divergence/lateral buckling at wind speeds attainable irr
practice. It should also be noted that for many bridge decks the moment inducctl
by the horizontal wind is negative (i.e., it twists the bridge deck so as to creatc
a negative angle of attack, the wind then approaching the upper side of thcr
deck). Such decks are not highly susceptible to torsional divLigence at wintl
speeds in the usual range; however, if the slope of the curve dCTalda vs. o is
positive, a thcorctical torsional divergence is still possible.
13.1.3 Locked-in Vortex-lnduced
Response
open truss sections generally "shred" the oncoming flow to such an extenl
that large, concerted vortices cannot occur and vortex-induced oscillations ol'
the deck are weak. However, in the case of bluff deck sections of the box-or
open box-type, instances of severe vortex-induced response are known to havc
occurred.
one such instance is cited in [13-10]. To reduce the oscillations, fairings
were added to the section as shown in Fig. 13.1.5, which includes results ol'
wind tunnel measurements. It is noted that in this case the water surface is
close to the underside of the projected prototype and could thus be expectcrl
to affect significantly the flow around the deck. For this reason the water surface
was also modeled in the laboratory.
Additional examples of streamlined bridge deck forms are shown in Fig.
Velocity (m/s)
FIGURE 13.1.5. Vertical amplitudes of vortex-induced deflections for various bridge
deck sections of the proposed Long creek's Bridge
[13-10]. Courtesy of the National
Aeronautical Establishment, National Research council of canada.
13.1.6.
Analytical Procedures for Estimating the vertical vortexJnduced Response. Under the action of the mean flow and of the shed vortices, the moclcl
section will be subjected to a self-excited and to a vortex-induced lift. wirh
notations used in Sect. 6.5 and assuming that the vertical and torsional modcs
arc uncoupled aerodynamically, the equation of motion of the section will htr
mfi + z(1,a]t + aihl
:
) ou'n
L,
["rt," * ,,,,in ,,1
(t3.1.t2)
where o is the voftex-shedding circular frequency and 11,f and c1.v are coclli
cients to be determined. If the model is given some initial vcrtical cle{irrrnali91.
its response will have the form
h:
(ho
+ h,e l'')sin(at I
4t\
(t.l.I.n)
l,'l( Jl
lltl,l l.l, l.(r.
St
rr.,rrulrrrr.,l
I
rr
rrly,1.,;,.,
l,
lor
rrr:.,
456
riiltr,cluHEs, AND powEn ilNtti
suspENDED-SPAN BRtDGES, rr Ntit()N
fit.
where fte is the steady-state amplitudc, rf is a phasc anglc, ancl 7 antl fi, arc
constants identifiable from the experimental observations. It can then casily bc
shown that
47maf,h6
Lrt/
-
PU:B
(13.1.r4)
llr(x) 1,,,,,,
Ht:,+[n3-"']
At lock-in 0) =
(13.1.
ls)
0)h.
The dimensionless quantities Cry and HI are applied to the prototype bridge
in the following manner. If fo is the assumed mechanical damping ratio of the
prototype, the total (aerodynamic plus mechanical) damping in the prototype
case can be written as
To:lr-*'f
(13.1.16)
t)
:
hr(x)q(t)
Mi|
I2Tpoflr + ,lqrl
c,r1 is the circular frequency of the chosen mode and
generalized mass of that mode:
*,
: I: hllxym61 ax
(t3.t.2r)
m[li
then
I1f
+
2l6aph
+
..lh]
h2\ h
: ou'nxnf (t -e -l;.
B'/ U
(13.1.22)
and e become the aerodynamic parameters. These are presumed to be
Chapter 6.
The steady-state amplitude
ho
in a manner similar to that described in
of a bridge deck section model is then given
- +s,,lt''
_
B-'l"1ur ,HT
as
(t3.r.23)
I
where S., is the Scruton number defined as
(m
c
ru
--_LOB2
(t3.1.24)
(13.1.18)
The coefficient F1f may be viewed as the value obtained at low oscillation
M, is thc
amplitudes by any one of the several identification schemes employed to obtain
l'lutter derivatives. If the steady-state (vortex-induced) amplitude he is also
rneasured
in a section model test, then e is given by
. Hr - 45,,
e:+&nrBfHf
(13. r.19)
L is the span ol'tlre pnrtotypc bridgc.
The maximum amplitudc at vortex-induccd rcsonancc is llren givcrr hy
where m(x) is the mass per unit span and
a
following (Van der Pol) form:
: )pu!nc,,[J'a,r"r axl sin1,,r + 4;
In Eq. 13.1.18,
(13.1.20)
account of strong nonlinear effects.
An altemate, nonlinear model (see Sect. 6.1.1) may also be employed. If
the description of section activity as given by Eq. I3.I.l2 is modified to the
(13.1. l7)
q{t) is governed by the following equation:
hlx)
The accuracy of the above procedure is acceptable only if the difference
between the mechanical damping ratios of the model and of the prototype is
small. If this difference is large, the procedure may become inapplicable on
evaluated from section model tests
The prototype being assumed to respond in an early bending mode ft1(x) according to the relation
h(x,
+U,rli,,
ilt)ol s
_eu(ne_rv
lr(:)l
I \2 / 1."^ mrofif,
:
number for the bridge deck, and that
pU .?n{:, r ll; 1,,(.t) ,/.t
:
til ,AN lll
For example , if h(x) is a half sinc wavc ovcr the span of a bridge with
uniformly distributed mass, the del'lection at the span center is
n6Al3, fl.p: apl2r,,4 is the
net area of bridge deck projected on
a vertical plane normal to the mean wind (per unit span), S is the Strouhal
where U,
I t;t,t;t,l Nl)l l)
Altcrnatcly,
(13.1.2.s)
if Hf is not obtaincd lrcm a low-amplitudc
irrstcacl thc m<ldcl is alkrwcrl to
oscillllc
cl<lwn lnrnr an
inititl
rrrotlcl lcst, lrrrl
lalgcrittttplittttlc
458
:;t,tit,t Nt)l t):;t 'nN
trl
lt)(it :;. l
N:;t()N
Auto a stcady, lockcd-in stal.o ol'nrclrsuretl anrl)lilu(lc:
be determined fiom
KHT
t
mlfio
:-__10-+
2pB'l
*,
1;ulr{:tlu i;. nNt) t'()wt n ltNt :;
11;,
tlrc vlrlrrc
I
ol'//'f'
rrury
'
B'
]
( 13. r
S is the Strouhal number and
R,, being dcfined as
(13.1.21)
t,
4sB2.l,cl,-nlnil
(
13. I .28)
the response amplitude ratio of first to nth cycles of am-
-r
(usually a simple, low-frequency one) must be considered as well as the probable nature of the spanwise correlation of the lock-in forces. Referencc
ll3-941has considered these parts of the problem.
The sectional equation of motion is Eq. 13.1.22:
where
+ 2tp6h + @iht: )ou,nrcu,
(' - ,U;)L o3 r.2er
it is further assumed that
h : h(x, t)
:
pI/ltlKl/i' x lt
,p(x)BtG)
being the single dimensionless mode of frequency
(13.1.30)
co1
:
{o cos
u--uL
B
x;l
<,l/
i
113.1.34)
(13.1.35a)
: I e4$)I(x)d-r
Jrpon
L
(13.1.3sb)
The strength of vortex-induced forces is dependent upon the local oscillation
amplitude of the structure; there is also a loss in their coherence with spanwise
separation. For example, Fig. 6. 1.2 depicts the correlations between local
lateral pressures separated spanwise along cylinders displaced vertically sinusoidally with different relative amplitudes.x A general review in [3-94] suggests that under such conditions an appropriate correlation loss function can be
approximated by selecting/(x) to be the mode shape <p(x) itself, normalized to
unit value at its highest point. For example, with a mode representing a halfsinusoid over a span L, f (x) may be estimated as
f(x):sin7
(13.r.36)
is
(13.1.3 r)
At steady-state amplitude, as noted earlier, the damping energy balance per
will be zero, a condition that defines the vortex-induced
cycle of oscillation
amplitude
*otol"'
at the Strouhal frequency, that is, where
2trSU
]t ilt
c,:
[ 'P2(x)f(x\(tx
'
Jrpon
L
c,-
responding to locked-
in vortex shedding and {(t) the corresponding generalized coordinate. This
assumed to undergo the purely sinusoidal oscillation
t@
I
r.r'(r){r(/)l{(r).p'r(r) f (x)th (13.1.33)
7fX
<p(x)
il'n fl
where 1 is the generalized full-bridge inertia of the mode in question and
The information given in Eq. 13.1.23 is applicable to the section model
only. To extrapolate it to a full bridge, the oscillatory structural mode involved
mtti
:
IE +zfuri + r?,t]:]pun3mufrc,- eczt2lt
nh; I At-h6,)
plitude decay (Eq. 6.1.14).
)
in which /(x) is a l'unction atltlitionally inscrtcd to account for spanwise loss
ol'coherence in thc vortcx-rclatcd forces.
If integration of the left-hand side is extended to the full bridge, integrating
the right-hand side of Eq. 13.1.33 spanwise results in
a is given by
(x--lllr
I
.26)
where
K :2trS
:;1,:;l't lll )l
lr( r)/l'9'( rylj.(/) tJi,,r1,l; tcr;,lrlr
t)
l6rrs
j I
,, -)lcr'r
q{r
'l
rCaHf
(13.1.37)
]
(13.1.32\
If /z from Eq. 13.1.30 is inserted into Eq. 13.l.Zg unrl tlrc n'srrlt rrrrrltil-rlictl hy
Bp(x), the action of thc scction r/x <11'lhc slnrclrrrt: :rssot'i;rlctl witlr sp:rrrwisr:
point x is sccn to hc rlcsc:rihccl by thc: clrlutrliorr
whcre thc Scruton numbcr is rlcrlirtul
rrs
tllcsttlls tlrrirlitlrlivcly sintillu kr llrrrsc ol l;r1' lr L) lr:rvr"lrt't'rr rt'lxrrlt.rl lirr \(luill('
I
I
l
()61.
l)r
t:]nr, ilr
I3,I
SUSPENDED-SPAN BRIDGES, TENSION I;IIIIJCIURES, AND POWIF LINES
460
s,r:h
For the case of a sinusoidal mode the values of C2 and C4, respectively, are
Hence by
(13.1.39a)
:o 33es
(13. r.39b)
cq: I, ,*'
Tf
nl
:
natural frequency
:
sluss
-^;tt-
^:ffi:88.sy
pL
l - Jo| ^n"'dx
<p
:
sin
rxlL,
I:
mfiLlz
,
(trrB.
Eq. 13.1.37,
I
c"n! -
4s--lt'2
eC+Hf l
'r
^lto.+zul x 1.r87 -4(0.03059)lr/2
:21'
(417r)(0.33es)(1.187)
I
so that the predicted peak-to-peak amplitude is 2lsB :
observations at the site reported in [13-1], for modes
amplitude "could hardly have exceeded 3 ft."
2.35
ft. From visual
of this type the double
a Full-Span Bridge
Theory. The flutter phenomenon was studied in some detail in Sect. 6.5 under
the assumption that two-dimensional geometrical conditions hold. In the case
of a fuIl-span bridge, the deformations of the deck are functions of position
along the span so that this assumption is no longer valid. A generalization of
the results of Sect. 6.5 to the case of the full-span bridge is presented herein.
An example is included.
Let h(x, t), p(x, r), and u(x, t) represent, respectively, the vertical, sway,
and twist deflections of a reference spanwise point .r of the deck of a full
bridge:
slug ft2,
I:
:0.03013
_l
13.1.4 Flutter and Buffeting of
p
' :0.002378
461
0.66H2
B:39ft
For
:
su -L
An example [13-941 will be drawn from the historic Tacoma Narrows case
1940. This bridge underwent considerable vortex-induced disturbance prior
to its demise by torsional flutter [13-93]. Pertinent data, forexample, relative
to an 8-noded vertical mode of this bridge are
of
BRIDGES
(m (0.0025)(88.5)
l. :
ntl't,:
,E: 4O,,'A..u;: 0.030586
,
Cz : 0.4244, C4 : 0.3395 ("standard" values)
5,,
(13.r.38)
fL "rxdx
cz: Jnsin'TT:0.4244
SUSPENDED.SPAN
N
h(x,t): j:Z tt,14ng,1t1
0'0025
(13.1.40a)
I
N
Interpreting data from Ref. [13-94]
K
:
U
:
Ba
U
: 3.1343
35.2 mph
Hf : l.l8l
e
Then
:
a(x,
4l'l
I
:
at lock-in
51.6
p(x, t)
:
a;(x)t(r)
(13. r.40b)
I
N
(13.1.40c)
4rn,{;4ffi,{r)
fr
-
at lock-in
at lock-in
t): i:I
where h,(x) , p i@) , ui@) are respectively the values of the ith modal deformation
fbrm at point x of the deck and {,(t) is the generalized coordinate of the ith
mode.
If 1r is the generalized iner1ia ol' thc
motion for that mode is
IlEt
t
2l'ie,L,
lull bridge in mode l, the equation ol'
| *it,) -
et
(r3.r.4r)
462
sust,t Nl)t t)til,nN
ilnlxit i;, il
Nr;t()N
liiltt,oil,nt t;, ANI) t,()wl tt ltNl
;*
:l
where f, is the damping ratio tll'tho ilh rturrlc, o; is its mdian nalrrrlrl lictprerrcy,
and Qi is its generalized force, delincd by
I
Q,
: I
J deck
f(Lo"
+ L)hiB +
(Do"
+ Dr)pB *
(Mon
* M)ail tk
(t3.t.42t
In the expression for Q, the following definitions of forces per unit span at
x hold:
section
Aeroelastic (self:excitation) forces under sinusoidal motion:
r*:)ou'alr*of
Lr*
ratB]
u*:)ou"'l*f
Lu*
xe;ff + x'eto + K'U*]
o*:)ou'alxrf
+ K2H{a +
er+ Kpf B+
+ Kzpla +
K,r|.|,)
K,fp'f
(13.1.43a)
(13.1.43b)
(13. 1.43c)
:,t ,:;t ,t Nt)l
tlrrtttititt, ttr itplltrrpt
M,
:
Do:
:
puzBz
?rr,oT - (*)
u@,
)ou'alzc,
\
")il
t#l
t)f
UI
cxplicitly in the buffeting fbrce expressions. In the example to be presented
below, these do not happen to be important forces, though they could be in
certain specific cases.
In whgl"f.qllews only a.si11gtg;ggde approximation to the total response will
be postulated. This ttinO'of isdffiption is justifiable from observation of the
lact that typically just one prominent mode will become unstable and dominate
the flutter response of a three-dimensional bridge model in the wind tunnel.
Clearly multi-mode response can also occur. This somewhat more complicated
problem has also been treated in the literature [13-86] t13-9U. On the other
hand, the mode-coupling forces of the wind are usually not strong compared
to those of damping. This problem will not be pursued here.
Following the 5-i5rgle-mode folm of analysis, any mode i may be considered
in Eqs. 13.l .41 . The corresponding modal forms are then introduced into Eqs.
13.1.40 to 13.1.44. This results, when all but those flutter derivatives shown
ure ignored as of lesser imponance. in
frequency
t
.-
,pu'B''t
(KB
ffi
wf cr,o, +
* j*.0 [LbhiB -t D6p;B *
P'f
Gpp,
+ Atc,,.,fti + x'.e{c,,;,)
M6u;l dx
(r3.1.4s)
(13.t.44b)
in which
(13.1.44c)
in the
:
(r3'r'44.)
Note;' In the force expressions above, it is assumed that there is no interaction
between the aeroelastic and the buffeting forces. This circumstance is partially
compensated by measuring the aeroelastic forces under conditions of turbulcncc
[3-89].x Furlher the sectional buffeting forces are written in a form that cx
presses their dependence both upon time-independent gust components antl
steady-state force components, this again being partially accounted for by as
sessing the "static" force coefficients at their mean values under turbulcnl
flow. Modifications to these expressions introducing indicial lift-growth lirnc
tions can be made t6-971. These lead,
ttrltrtilltttrr'r' ltrrrt liorrs l(r ()71 (lrlrt lypiclrlly tlcpict a clirrr-
Irtive or do n<lt inlnrclucc iurporlanl crnrrs, as in an cxample to follow. Further,
sclf:developed local, or signuturc, turbulence efl'ects are also not represented
Q,
w(x- r) l
463
irtutiott witlt ittcrcrrsrrrg llet;rrt'nt'.y ol llrc lirlt'r: lcvcl liirrrr lltut ol'thc stcadyslatc lilrcc. Itr wlritl is tlist'ussctl subset;uurlly lhc ccluivalcnt ol'unit aerodyrtatnic adtni(llnce: is lrrcilly irssrrrrrr:rl. 'l'lrcsc ussunrptions arc usually conserv-
Buffeting forces:
I " f u(x-tl (49!
Lo
+
+
" : :2'pU-B l2C,
| -- U
\ d.t
t;r|,,'
:i
t):;t 'nN tlt ltxit
or power
spcctr.rrl
*Turbulence was fbund ttl havc a stntng llvorablc cll'cct on (hc llrrttt r vckrt ily lurtl lhc rcslxrrrsr.
to vortex shcdding of a sccliort trurdcl ol'{hc Quincy l}ritlgc. :rntl orr tlrt llrrrlt'r vckx'ity lirr lr lrrll
model ofthcl,ion's(iirlcllrirlgr:rrcrrsstlrcllrrlr':rnl irrlt'l (Vlrntorrvtr)llt$\l:st.r':tlsolll lO,ll
Gqq
:
f
lq,
,..*n?olf
:
h,, Pl or a;f
(t3.1.46)
Because of the linear nature of the resulting equation of motion, it may be seen
that under this formulation the conditions of system stability are independent
ol the buffeting lorces.
The system equation
t:i + Z^yioioti -f
<,sioti
_
Qio(t)
Ii
(13.1.47)
rnay bc rcwrittcn with a rrcw ll't:tlrrcncy oig, a new damping ratio "y,, anrl
huli'cting firrcc Q;7, dcfinctl. r'cspcctivt'ly, hy
.r;;' ,,'
"',!,,"
,','II(;,
.,
(
ir
l.l.1.,1r{)
484
suspENDED-spAN BRIDGES, TFNritoN ,.ilnuctuRES, ANt) powrFt uNr s
27iaio
:
Qiilt)
:
2(iai
- ff
*
).,,n[Luh,n
it is then ne-cessgry thqt "yr < 0; this
""
*g9S-,,,*11"1instabilitv crirerion
For instability
Hf
Go,n,
+ pf Gp,p, + Atco,o, --
-t M6a;l
D6piB
dx
#,1, . +
utrftPt Nt)t t) llt,AN tthilxll ti
( 13. r
M:2cu+!+c;Y9't)
,.'U"'U
(13.1.50)
leads finally to the sinsle-
I
I) :2Cr'r'f
4nyor,h, + pf Gp,t,, + Af G,,,,",1 (13.1.49)
lr
)ou'n\
13
oto,,,,)t, (t3.l.sl)
At,,4f
have been
(13.1.54c)
Defining two new functions p(x), rl,@),
p@):2[Cyh;(x) * Cpp;(x) +
in which only the important flutter derivatives F1f, Pf ,
.s4b)
*(x): (CL+
Cp)h;(x)
+
Claai@)]
C'1aai@)
(13.1.55a)
(13.1.ssb)
the integrand of Eq. 13.1.53 becomes
retained. An a.ssumpJion inherent in this criterion is that the flutter derivatives
retain full coheidiiCe among spanwise sections. The effect of reduied ioiierence
can be seen qualitatively as analogous to a reduction in the values of the
Lh1
r
Dpi
t
Ma;
: er;\4f
+
{(x)W;
(13.1.561
Gq,q,'
In practice the flutter derivatives I1f and Pf are most often negative in
value,* while,4f may take on positive values for advanced values of reduced
velocity
2n
(13.1.52)
The effect of the flutter derivative A{ (an "aerodynamic stiffness" effect) is,
in many practical cases, almost negligible. This reflects the relative magnitudes
of the larger structural, versus the aerodynamic, stiffness for typical bridges.
For buffeting analysis, the generalized force may be rewritten
/l\
The method of solution adopted here will be to seek the power spectral
density of the bridge deck deflection. This is partly motivated by the fact that
the power spectral densities of the wind components u(t) and w(t) are known
or can be reasonably estimated from the results of research.
Defining the Fourier transform of {, as
Ei(<at
with;
as
ax
: ouzB2tl [.rr * DP, + Ma;|7
,,, Jo*.
\i )O,rt,t
(13'1.s3)
lr',0
: JJ,
-
<,sz
:
fT
l* J, tiu)e-i"tdt
tne Fourier transform of the response equation for {, becomes
+ 2i,pioc,rlfi :
ry
J.".*
[rt"l
Wf + t@)-*f]+
(13.1.s8)
where
L
*Pf
:
2c,- 4-'')
U
+
(c!-*Cr)t w(r' t\
(13.1.54a)
Multiplying Eq. 13.1.58 by its complex conjugate and by 2lT, we obtain, in
going to the limit T --+ @, the result
y?t(r?o - r')' + (2tio:ioa)2lElf
may be obtained by equating the expressions for the drag
, : /r\
\;)
,: / l\
\r/
Ncglecting thc tcrm in
(13.1.s7)
/2, thc rcsult is
p(u
-
n)BCD
[ [,,,. ..,dx"dxr,
:_ ( eurnrty'
\ ,L ) l.,l " tx,,' xb'') r t
ou2Bxp,i,
,-
Pl = *-2ClK.
whcrc
(13.1.5e)
SUSPENDED-SPAN BRIDGES, TENEION STRUCTUHES. AND POWEF LINES
TI (r", xo, @):
j* '7fit
x
+
lg@)u*(xb, a)
Since the power spectral density
o) -t
A,>u(xo,
Sg,g,(c,r)
rlt(x,)w(x", a)l
a)l
rl,@)w*(xa,
of {; is defined
S-(n)
(13.1.60)
as
)
:
13,1 gUSPENDED.SPAN
-
BRIDGES
336zul
UU
(13.1.6sb)
+ l0(nz/lJ)5t''l
where z* is the friction velocity defined by Eq. 2.2.5, ru is frequency, and z is
deck height.
In calculating.lg,4, as in Eq. 13.7.62, it can be observed that the following
types of integrals require evaluation:
(13.1.6r)
l* ;#f
: j f e@)e@"-ctxa-xut/t++
*, : J f ,r,<*.1,r,<rol"-ctx'-x6ttt ++
R,
we find that
,*[(' - (fr)')' * (,,
:
lolt2g2tlz
I
+
"'
I
ll
JJ
deck
I
Zt
*e)'],,,,,r,r
<'t)
st*,(')
xt,
a)l++
'' I I
03.1.62)
in which the cross spectra Su, and S,, have been neglected. (While limited
data presently suggest that this is a conservative assumption, knowledge of
these quantities in applications can improve accuracy.)
From this point on, the distributed cross power spectral densities of z and
w will be assumed to take the real forms (neglecting their imaginary components)
I cl*" - xnl I
= sr(co)
"*ot--l
I cl*' - x'l
s*(xo. x6, c,r)
= s,(co)
"*PLx6. c'r)
/
{13'l'63a)
I
(13.1.63b)
2onl
U
:
2OOzuzx
qt + 6OrzlU)P
ptttzt,x?o
[1
The variance of
-
{;
(<,r/c,r;s)2]2
+
f2y,(alc,:,o)12
{ReS,+
nrs,}h
03.1.67)
is
"?,
: J;
(r3.1.68)
s4,4,(n)dn
which, for example, can be approximated with the aid of the formula (see Eq.
5.3.39),
[-
Jo
s(,r)an
tl -
(nlnsy2lz
+
[271;nlns)]2
l* g1r1a,
:_ Jo
o,t)u" *'
rno_s@o)
(13.1.69)
4,y
Referring to Eqs. 3.I.65, we find that
J; S,(n)dn
(13.1.64)
z being the frequency of mode i.
According to chapter 2, power spectral densities of z and rv in the atmosphere may be approximated by the expressions (Eqs. 2.3.21 and 2.3.33),
S,(n)
:
I
where C is a constant (see Chapter 2) satisfying
5nl
_<c<_
U_-_
(13.1.66b)
so that, finally,
le@')e@6ts"(xo' x6'
t!t(x.)rlt(x)5,,@o,
su(xo.
(13.r.66a)
(13.1.65a)
:
J; S*(n)dn =
6u2*
(13.1.70a)
l.7uzx
(13.1.70b)
so that a buffeting calculation f<lrmula is obtained for the variance
of {;:
"i,:l#)'[^,[ t#*6u2* . orl*Y*tr,,.lJ#
(r3.r,7r)
*+
468
liL,st,t NL)LD:;t'nN tlnt{xit
l;, n Nl;t()N liilt(,(:tt,t u li.
nNt) t,()wUt ilNt l;
i;t,t;t
on,@)
:
h;(x)Bog,
(13.1.72u)
oo,(x)
:
p,(x)8o4,
(13.1.126)
o,,(x)
:
a;(x)og,
(13.1 .72c)
0. ti
0.,'1
0.2:
0.0
metric, L :
lateral, V :
:
symmetric, ,4S
:
antisym-
metric torsion).
Flutter. The torsional aerodynamic damping coefficient ,4f exhibits a pronounced change of sign with increasing velocity, indicating the possibility ol'
single-degree torsional flutter (Fig. 13.1.8). Mode 7 was selected as the mosr
vulnerable to flutter instability (Fig. 13.1.9). It is the torsional mode with both
the lowest frequency and greatest Gn,r, value. Experience has shown (e.g., in
rl.o
FIl
- 0.2
-0.4
0
4
6
B
IrlnB
10
I'IGURE 13.f.8. Aerodynamic coefficients A! (i : 1,2,3,4), Golden Gate Bridge
(courtesy of Dr. J. D. Raggett, West Wind Laboratory, Carmel, CA).
the original Tacoma Narrows case) that the lowest antisymmetric torsion mode
is typically the most flutter-prone in long-span bridges. In the Golden Gate
Bridge case this mode is practically a complete sine wave along the main span,
with a node at center span and practically zero amplitude on the two side spans.
The pertinent parameters in this case are, in the units* kip, ft, s:
: 0.005 (arbitrary choice)
1z : 8.5 x loe lb ft s2
p : 2.38 x l0 6 kip ft-a s2 : 0.002378
: 0.002378 slugs/ft3
fz
Ib ft-a
s2
(lolden Gate Bridge
l
Frequency
7.0
Gn,n,
G,,
Gon',
8.03E-05
I
0.049
L
2.62E-16
3.33E-01
0.087
ASVI
'7.398-t5
t.tlE-15
3
o.lt2
L
3.25E-01
1.728-14
3.09E-01
l.24E 02
0.129
0.140
SV,
1.90E-01
I .91E-01
7.828-t4
1. 16E- 14
2.438-14
3.44!,-0_L
5.58E- l4
3.87E- I 3
6.611;.-12
3.32F.-02
l.29lr
I (X)
50ti
2.11tt.-Ot
2
5511
0l
. I lt
o
.5
1
0.t64
0.t92
n s'l'r
8
o.t91
S'l',
()
l0
Type
2
4
3
-7.0
z
TABLE 13.1.1. Frequencies, Types of Modal Forms, and Modal Integrals for
+'
0.0
A;
./..
{
vertical, and Z: torsion. Values of the modal
integrals G,/;,,,, suggest the importance of the mode: in Table 13.1.3 the largest
in each category (i.e., vertical, lateral, torsion) is underlined. The most pronounced modes are mode 6 (vertical), mode 1 (lateral), and mode 7 (antisym-
"'I
Ar
-+
l
'l'hc vibration nxrdcs and ficquencies of the bridge, together with their modal
in(cgrals (),,,,,, warc obtaincd for the first eight modes with the results given in
Modal lirrtns arc suggested by the notations S
\,
0. (i
Example In this example parameters of the Golden Gate Bridge are employed.
A I :50 scale section model was used to obtain flutter derivatives H,t and A!
(t : l, . . ., 4). A set of these derivatives forzero-degree wind angle of attack
in smtxrth flow is prcscnted in Figs. 13.1.7 and 13.1.8 [13-97].
l3.l.l.
4ti9
l.o
Then from Eqs. 13.1.40 thc standanl rlcviatiorrs arc ohtairrctl
Tablc
'l t{t)t II:,l'nt.t |ililtr(it ,,
.)
t2
1.25F.-14
I I7,r r lil
FI(;URIt 13.1.7. Acnxlyrlurric cocllicicrrts //1" (i l. I, t,,l) ( iolrlt'lr ( iirlt' liritlgt.
ol'l)r. .l . l). l{lrggell. Wt:sl Wirrrl Lirlxrr':rlory. (':rlrrt.l, ('A)
(cttutlcsy
rl kip
Iil.
lO(X)
lb lirrcc; I lb lirrtt'
l .l.lli Nr'ulrrrr'..
r'
t.'
.)
ltls
()
lil
rrr/s
tl),ll,i
I3.I
471
SUSPENDED.SPAN BHIDOES
B=90fi
I : 6451 ft
troro, -
L.z
')_9
The flutter criterion in this case reduces to
* *<-
u
t1 1
-
{)
o9
Fqtr
4hlt
pB"Goro.,l
or
At
oo
o.6
3
O-
From the graph
(with
n
-
nt
:
forel'6ig.
-
13.1.8) the corresponding reduced velocity value
0.192 Hz) is
o
o
U
1:
()
o8
o-tzt
4'32
which corresponds to a critical laminar-flow flutter velocity of
iotrd
>E
!t=
U,,:
oe
g.x
hou
!r+
6)A
(){:
5r
trx
fE9oq
0)
V:E
rqE
IF
-.r
&e,
\JE
-4
Fr6
o
rt
b
o
o60ao
(9
ASVr
SV'
ASTr
ST'
i
Frequency (Hz)
2
4
0.0870
0.1285
7
0.19t6
8
o.1972
These additional data were used
lus
0l
At
(soqouD luourocqdslp leed
ol lead
:
:
z:
zo
'
470
81.9
*
TABLE 13.1.2. Generalized Inertia of Full Bridge for Four Modes
Mode
6\O
ftmikm
- : 50.9 * :
Bufteting. Four modes, mainly active over the main span (see Table 13.1.1),
were examined, as listed in the following table:
'ro
r\C!
:74.65
(4.32)(0.192X90)
4144
ft :
main span length
0.02 ft
220
fl = dcck hoight
U
2.5 ln(z/r,')
l0
e
I, 0b ft
15
.71
6.15
8.50
8.59
s2)
472
13,1 SUSPENDED.SPAN
SUSPENDED-SPAN BRIDoES, TENSIoN STnUoTUHES, AND PoWER LINES
:
Ct :
Co
with
0'34
0.215
K
(*tn''i"" = P +7 +'2r2{l+e-K\
ft:,,,
{
I\t r: c+
Cu:0
dC,,
d"
The modes involved were assumed to have the forms of simple sinusoids:
:
h6
/
sin(
?r"r
\
lrsl
,
/2nx\
a
" sin(\lrt /I
' /nx\
otsvt : as sin[
" \/rrl
:
hnsv,
&6
R(c)
:
R(c)
: (T)' O,,'
"O
.rn\
++
e3.t.77a)
(t3.1.77b)
I\t *: c+
. /2rx\
:
ll
,i,2f:,i,zfr:,-Ctx'-xut
fw':p:vp.W.#
,^ /
Using these approximate expressions, we obtain the following results:
(13 178a)
(r3.r.78b)
We now have the two following forms of Eq. 13.1.71:
For a purely vertical mode,
For a vertical mode
Ra: 4c,htRc)
(t3.t.73a)
Rv :
(13. r.73b)
(CL
*
Co)z n?AtC>
4
:l#l'l<",'r\t#
+ (cL
For a tonional mode,
nr:
+C2naf,ng\
J"J,t" ffi
: (T)' or-'
e-cv"-',,,,
t.r",.))V
03.r.7s)
For a purely torsional mode,
4
TXosinffi
Trxn
t c)z "?,lt# *
(13.1.74b)
where, for symmetric modes,
Rrcl: f f
+ 6".)
(13.t.74a)
n, : c,ja{n1cy
R(c)
(r3.r.76b)
with
I
QASVT
(r3'r'76a)
1V;;y
and, for antisymmetric modes,
: -0.lll
hsr,
BRIDoES 47s
++
(r3 r.75a)
(
r3. r .75b)
:lutill'f,rr*,,,'lY#
+
+ ou'*)
(c'vai)2lt#? * t.r,'.))V
(13.1.80)
Using the data, the results of Fig, 13. 1.9 ure culculated. In each casc ni6 is the
natural trequency of the mocle in qucntion.
474
t;t,til,t Nt)t t) lit'nN nnlxit :i. n Ni,t( )t.l i.llrt(;llill
:;
nNl
) t,()wl l ltNl
The calculations just prcsontc(l arc inten(lc(l lo hc illuslralive
.
*
Iil I
:
l)ct;rils
rrrrtl
approximations may difl'er sorncwhat accorclirrg t.o thc tlcsigncr's .jutlgrrrcrrt.
Dependence of Aeroelastic Stability upon Bridge Characteristics.
aeroelastic stability of a bridge is controlled by several factors:
l.
'l'hc:
Geometry of the bridge deck. Unstable shapes include solid girdcr or
"H-section" types of deck form; open-truss deck sections with closed,
unslotted or unvented roadways; and certain very bluff cross sections.
On the other hand, stability is enhanced by streamlined forms and by
open-truss sections that contain vents or grills through the roadway sur-
flce.
o.l'vibrution rf the bridge. High torsional frequencies tend
to cnhancc stability. Examples of torsionally stiff shapes are closed torsion box scctions, rlr dccp trusses closed by roadway and wind bracing
to constitute a latticed tube. On the other hand "H-sections" are torsionally weak. Stability is also enhanced if the torsion-to-bending fre-
2. I'-rcquencits
quency is high.
Mechanical damping of the bridge. Aeroelastic stability is clearly enhanced if the mechanical damping ratios of the bridge are high. We also
mention the possibility of enhancing the aeroelastic stability of a bridgc
by vibration reduction devices. Such a device, consisting of tuned mass
dampers (TMDs; see Sect. 9.4.1) provided with disk brakes and not
requiring any power source, was installed on the 1939 Bronx Whitestonc
Bridge t13-1081.
4. Deck inertia. Heavier systems increase the flutter threshold.
13.1.5 Galloping
The susceptibility of a bridge deck to galloping can be determined by inspecting
the plots of the lift and drag coelficients C7 and Cp versus a (e.9., Figs. 13. 1.3
and 13.1.4). The condition for incipient galloping instability is (see Sect. 6.2):
dCt
da
+cD<0
(13.1.8 r)
of large-amplitude across-wind galloping of suspended-span bridges havc:
not bcen reported to date.
It follows from Eq. 13.1.81 that avoidance of deck shapes with regions ol
strongly negative lift curve slopes is conducive to stability.
Cases
:;lll,l'l
lllrl
Ir'.1'l\lI
Illlllr:l
475
ll srlst't'pt ilrilrly (() v()tl('\ tntlttt't'rl vibt;rlrorr r,.; lr prnlrlt'rrr. out'ol llrrr't'ly1x.s
ol solrttittlt ctttt, itt gcnt't;rl, lrt'rrst:rl. l;ilsl, llrr. slrlllrr.:.s ol llrr- rrrt'rrlx't t:rlr lrt'
ittcrcasccl so lhal llrc o.ilicirl witttl vcloci(.y t'xr't't'tls llrt' vt'lrx rllt's llr:rt rrril'lrl lrt.
t:xltoctccl to occur clurirtg the: lili'ol llrt'strrrtlrrrr'. l'o t'rrlt'rrl:rtt.llrt't'lilicrrl
vclocity U,,, thc lirllowing rr.:llrlion is rrsrrl:
u,,
:
tt,l)
s-
(13.1.82)
whcre n I is the fundamental frequency of vibration in the across-wind direction,
/) is the across-wind dimension, and S is the Strouhal number of the member.
Ilccause the dimension D of an individual member is small compared to the
integral scale of the atmospheric turbulence, it may be assumed that the member
bchaves aerodynamically as if the flow were smooth so that the Strouhal number
can be taken from Table 4.4.1.
Second, devices may be used that spoil the coherence of the shred vortices.
llelical strakes and shrouds of the same design and with the same proportions
rrs indicated in Sect. 10.3 may be employed on circular members. Figure 13.1.10
shows a spoiler device consisting of staggered fins that was successfully used
{o suppress the oscillations of a pipeline suspension bridge [10-23,13-48]. This
tlcvice would not be effective if the member were exposed to winds blowing
lrom any direction (as would be the case if the member were vertical) rather
than from just the direction parallel to the plane of the fins. Figure 13.1.11
shows perforations
in the web of an l-section
member that reduce vortex-
induced response and galloping if the wind direction is normal to the web but
wcre found to aggravate the galloping problem if the wind direction is parallel
lo the web.
Finally, in certain cases tuned mass damper (TMD) devices may be emlrloyed. The principle of these devices was discussed in Sect. 9.4. An example
ol' a TMD used to control the oscillations of bridge I-beam members is described in detail in [l3-49]. The device consists of a cantilevered rubber-shank
;#-
13.1.6 Structural Members
Towers and bridge members of circular, squarc, I or H scclion lnay bc sus
ceptible to wind-induced vibrations, particularly urrrlcl llrt':reliorr ol'shcrl vor.
tices.
Itl(illltl,l l-1.1.10.
Sl:rg,picrt'tl
lrrr.,,rrr;r lrrlrr'lrrrt':,rr:,1x'rrsiorr lrlitllit'II t.lli, lO.ltl
476
:il,t;t 'l Nl)t t):;t,nN tnil)ct :i, llN:il()tt :iililr{jil,1il li. nNt) t'()wt il ilNt
U
U
U
U
FIGURE
13.l.ll.
Perforated web
of l-section
member.
pendulum weighted at the lower end. The weight employed may be of the order
O.l5% or more of the weight of the structural member.
To reduce vortex-induced oscillations of individual cables such as those in
cable-stayed bridges or the deck hangers of suspension bridges, cable-to-cable
ties, friction or hydraulic dampers, or TMD devices may be employed. In cases
in which the oscillations cannot be prevented, fatigue-free cable terminations
rnay have to be used to avert damage at the supports.
Mitigation measures may also be necessary to reduce large-amplitude vibrations (0.6 to 2 m double amplitude) that were observed in cables of cablcstuyccl bridges under the combined action of rain and wind. Wind tunnel studies
cstublislrcd that the vibrations are due to mechanisms that include the formation
ol't[:strrhilizing rivulets at the upper part of the cable [3-105]. The use ol
t':rbles with protubcrances was found to be effective in suppressing the vibra-
<tf
l
iorrs
.
'llrt' rrriligirlion ol' bridgc tower oscillations by means of TMD and tunctl
lrrlrritl rLrrrrpcr ('l'l,l)) dcviccs (see Scct. 9.4) is described in [9-79].
13.2
rr:'
:
TENSION STRUCTURES, POWER LINES, AND POLES
13.2.1 Cable Roofs
Vilrr':rlions ol' crrblc-srrl.rpotlctl nxll,s arc causctl prirrcipirlly by btrll'clirtg lirrccs
tlrrt: lo irrcitlcrrt trrrtl slrrrcttrr-c inrlrrcctl lurbrrlc:rtcc. ll is lil.t'ly llr:rl llrrttr:r'(srrll.
ll tJ:,1()N:illll,{:ll,l tl :,, l'0Wt n ilUt '. /\l.lt) l,{rl l..
477
cxcilctl rlst'illitltolt) ol r';tlrlt' trrols is t'irt't'. sirrtt' ruoi{ tool :ilnt( lul(':. (l() u()l
pcrtrrit erttrlttglt tlerllcclion lo itttlttt:c signilit;rrrt tlr;rrr1',t'r. rrr llrr' ;rt'r,rrlVnilnl('
lilrccs.'l'hc rturgrritrrclc: ol tlrc htrllctirrg lirrct's tlrrr lrt'rrrvt'slrp,;rlt'rl rrr wrrrtl lrrrurt'l
tcsting ol'acroclastic or rigitl nurtlt:ls. lrr llrt' lrrllt'r t'lrst' lrxrtlrrrll Irrnt'tiorrs lo lrt.
used in dynamic studics can bc: tlt'vt'lopt'tl l.nrrrr llrt'n'r'otlt'rl llrnt'tlt'grt.lrrltrrl
pressures.
Unwanted vibrations will rtol occrrr il lhc cirblc nxrl'is sullicicntly still'.
Stiffness is achicvcd by thc pnrvisiorr ol'sullicicnt wcight, fbr examplc. in the
form of precast concrctc r<xrl'pancls, by prctcnsioning of cables; and/or by the
provision of a stifl'ening systcm of tensioned cables with curvature opposite to
that of the main, load-bearing system. In double-curvature roofs, the loadbearing and stiffening cables form a network-in most cases, orthogonal. Unless carefully designed, such roofs may exhibit serious vibration problems that
have, in the past, necessitated the provision ofadditional ties and the lubrication
ofcable intersections to reduce noise caused by cable-to-cable friction. In single
curvature roofs, stiffening cables may be provided at some distance underneath
the load-bearing system, as in the case of the well-known Utica, New York
auditorium. The two layers of cables and the vertical members joining them
fbrm elements with considerable stiffness that prevent the occurrence of any
significant wind-induced oscillations.
Recent studies on wind effects on cable roofs are reported
l
3-s3].
I
in [3-50]
to
13.2.2 Air-Supported and Tensioned Fabric Structures
Long-span fabric structures, especially of the air-supported type, are a relatively
new architectural and engineering development Il3-541 to [13-58]. In many
instances their design has been based on rudimentary representations of the
wind loading tl3-581. Attempts to develop more realistic and elaborate wind
loading criteria or wind tunnel modeling procedures are reported in [13-59] to
I l3-631.
13.2.3 Power Lines and
Poles
'l'he design of power lines requires the estimation of drag forces and the pretliction and/or mitigation of wind-induced vibrations.
A comparison between drag coefficients on standard aluminium conductors
with a steel core and trapezoidal wire conductors is reported in [13-98]. Wind
tunnel tests showed that for wind speeds higher than 85 km/hr, drag coefficients
Iirr trapczoidal wire concluclrlrs urc srnirllcr than for standard conductors. RefLrrcnce ll3-991 reports thut wintl lrnrrcl rrrrrl lit'ltl lcsts rlrr 3.(16-rrr long slanrlanl
t'rtttcluctors yicldcd sirrtilur tllrrp, t'ot'llit'icrrls. liir'ltl nr('lrsurcnlcnls ol' swing lrrr
glcs antl insulaltlr lirtccs sltowt'tl llr:rl rvrrrrl lortt's t':rlt'rrl:rlr'rl lry lrssrrrrrirrl', rrrri
lirrlrr wirrtl lrllrtls hitsctl ()tt rn(';r:rur('{l tlr;r11 trx'llt( l('nl:. ()v('r('sltnritl(' lltt' lrtlu:rl
ll.l-99, l3 l(X)l A lrkt'ly ctlrl;111;s111111 ltr'r,.;rl lt'rr:,1 rrr l|:rt, rr tlrt'rrrr
1lt'r'li'r't sPirliirl coltcrcrtt't' ol llrr' ;r,'r,r,lyniunr( l,r;rrl:, I I | / l
lirtcc:s
I
478
lit,r;t 't
Nl
l
t) r;t'nN nt
lilxit
:, l
f..t:,tot.t
r,ll,{ nll
:;. nNt) t,()wt n nNt :;
'l'hr: rrtaill viltl'atiolr
;ttrrlrltrtns
ol lorrp, sp;rrr t'irlllt:s:rntl Jllwt:r lirrt.s:rle irs
lirll slllrn glrlkrpiltg, rrrrtl subsp:rrr wirker intlucctl
galloping. These problems arc briclly discussocl bclow. I,or aclcli(iorurl irrlirr
mation and studies on wind effects on power lines, sec ll3-641 kr ll3-7 ll, rrnrl
sociated with voncx-shodding,
ll3-1011 to [13-103].
VortexJnduced Oscillafions. Vortex-induced or Aeolian oscillations in long
span cables are generally caused by winds with speeds of the order of 2 to l0
rn/s. The oscillations generate packets of narrow-band random waves arriving
irl lho cirblc supports. Since the cable is not perfectly flexible, the waves causr:
oscill:rlory honcling strcsses near the supports that result in fatigue damagc,
rlrlt'ss lrrrr(c:ctivo rrrcasurcs are taken [13-72, 13-73, l3-],4]. In the casc ol'
slr':rrrtk'tl wiri.:s lirtigrrcr tlirrnagc can be produced by shear-induced friction, which
:rlli't'ts rrr:rirrly tlre irrrrt.r wircs.
APPtrtrtcltcs Ltsctl to prcvcnt fatigue failure include the provision of speciul
cttsltiortccl suppotls that allcviate the bending stresses and applications of thcr
Ir-urctl rrrass clarnpcr ('l'MD) concept such as the classical Stockbridge dampcr
ll3-7-5, 13-16, 13-771. The Stockbridge damper (Fig. 13.2.1) consists ot'ir
roactivc countcr-vibrating mass with a fairly wide band of frequency possibilitics. 'l'hc cllbct of the mass is to suppress to a large extent the last half-wavc
(ncarcst thc support) generated by the cable oscillation. Like all TMD deviccs
(scc Scct. 9.4.1), the Stockbridge damper is not an energy dissipation device:
to any apprcciable extent; it is, instead, an anti-resonant spring-mass devicc.
stockbridgc dampers or similar devices can be readily purchased for a wide
nrngc ol' spccific applications.
Full-span Galloping. Full-span power line galloping occurs most characru.
istically when ice forms on conductors and creates a new surface contour tlr:rt
is pnrne to galloping tl3-781 to [13-80].
ilt II nt N(;t :;
479
Mclttts ol ltllt'vt:rlrttl' llrt'l';rllopirrg ol'powcl lirrt.s lrirvr: ilrclutle:tl rrrcltirrg 6l'
ict: by cilt-ryirrg lril'111'1 ('un('nls lr'rupoliu'ily irrrtl llrus hcrrling lhc clblcs, instal
llrtion ol'g:rlltlllirrg w:tttrirtll scns()rs localr:rl al sul)p()rt towcis in rcgigns whcre
cablo icing is knowrr lo tlrkc pl:rco, urrti-galklping acroclynamic {evices designed
kl spoil thc l<lcal llow, and tuncd rlass clampers at the center of cable apun..
subspan Galloping. Subspan galloping, or wake-induced lateral galloping
(see Sect. 6.3), has occurred repeatedly in grouped or bundled conductors
I l3-8ll to [13-84]; for a recent review see [9-ll. countermeasures have included (l) detuning the various cables in a bundle from each other by means
of special spacers and (2) increasing damping by providing energy-diisipating
spacers or by lowering the cable tensions, a measure that results in an increase
of the inherent self-damping of the cable. None of these solutions has been
f'ully effective. In particular, some highly complex and costly spacers with
articulated and spring loaded arms have been found to be unsaiisfactory.
conceptually simpler-although again costly-solutions have included a large
increase in the number of spacers used between supports so that subspan lengths
are cut down and the corresponding frequencies are raised, and a continuous
twist of the conductor bundle from support to support, which breaks the spanwise coherence of the vortices shed in the wake of the windward conductor.
The continuous twist solution has been judged, so far, to be impractical for
application in the field.
Poles with Partial lce coating. Experiments on circular cylinders with approximately uniform coverage by ice or snow over about a third of the circumf-erence (Reynolds numbers based on diameter 50,000 to 500,000) indicated
that such coverage can create strong susceptibility to galloping motion, pafiicularly for ice or snow thicknesses of about 3% to 6% or tfie
rytinaer diameter
ll3-1061. The research of t13-1061 was motivated by massiu" loss", of poles
with partial snow coverage in an Aleutian island wind storm
[13-107].
REFERENCES
l3-l
l.\
\,
2
ll -l
ll4
F. B. Farquharson (ed.), Aerodynamic stability of Suspension Britlges, parts
Bulletin No. ll6, university of washington Engineering Experiment
I-v,
Station, Seattle, 1949-1954.
C. Scruton, "Sevem Bridge Wind Tunnel Tests," Sur-veyor, 107, No. 2959
(Ocr. 1948;, 555.
C. Scruton, "Expcrirtcnt:rl Investigation of Aerodynamic Stability ol suspclsion Bridges with spccill llct'crence to proposed sevcm Briggc.,, l,rpr. Irr.st.
Cir.,. l'.ng., l, Pull l. No. 2 (M:rr. l9-52), lg9_222.
A. Hirai, I. ()klrrrclri,:trrrl
lr'l(;tilllt
l.].2.
l.
Stoc'kbritlge tlrrrrr;x.r
l.
Miyrtl:r, "On thc Bchlviolol'srrspt'rrsrorr llritlpr.s
Acri,r," irr l't., t ttlirt,g.t rtl rlta lnttrtutri.rttt! ,\\,rrtltrt.tit,tt rtrt ,\'rt.s
J)t'tt\i.)tt llritl,4'.t, l.:tlrorrlotio N;r( i()nitl tlc lirrgcrrlrllilr ('ivll, l,islr11rr. l()(r(r,
rrrrrlcr wirrtl
pp. 24()
2.56.
l;tJlil't Nl)l I)l;l'nN llllll)(il :i, llN:;l()N :;lllll(:l(,lll
48O
I
i;' nNl) l'()wl ll llNl
lilttltrr-, ittttl ('. li. l'. llowert' y' '\trttlt'ttl
Ilritlgt'luring lircctitttr ttnd (ltnltltlitnt,llc.lxttl
Wintl Action on u Suspcnsion
No. BLWT-3-69 with Appendix BLWT-4-70, F-aculty of Engincorirrg Scicttce:.
university of western ontario, London, Canada, May 1969 and March 1970.
W. H. Melbourne, West Gate Bridge Wind TunneL Tcsl.r, lntcrnal Rcporl,
13-6
Department of Mechanical Engineering, Monash University, Clayttln, Vic
toria, Austrialia.
l3-7 A. G. Davenport, 'The Use of Taut Strip Models in the Prediction of thc
Response of Long-Span Bridges to Turbulent Wind Flow-Induced Structural
Vibrations," in Proceedings of the IUTAM-IAHR Symposium on Flow-ln
ducecl Structural Vibratktns, Karlsruhe, West Germany, 1972, Springer-Verlag, Berlin, 1974, PP. 373-382.
13,t3 R. L. Wardlaw, Static Force Measurements of Six Deck Sections for thc
I'n4xxr:d Ncw Burrard Inlct Crossing, Report No. LTR-LA-53' NAE, Na{ir>nal Rcscarch Council, Ottawa, Canada, 1970.
l3-9 R. H. Scanlan, Rct:cnt Methotls in the Application of Test Results to the Wirul
Dcsign o.f' Long, Suspended-Span Bridges, Report No. FHWA-RD-75-11-5'
Fecleral Highway Administration, Office of Research and Development, Washington, DC, 1975.
13-10 R. L. Wardlaw and L. L. Goettler, AWind Tunnel Study of Modifications ttt
Improve the Aerodynamic Stability of the I'ong's Creek Bridge, Report No'
LTR-LA-8, NAE, National Research Council, Ottawa, Canada, 1968.
13-11 Y. K. Lin, "stochastic Analysis of Bridge Motion in Large-Scale Turbulent
Winds,' ' in Wind Engineering , Proceedings of the Fifth Intemational Conference, J. E. Cermak (ed.), Pergamon Press, Elmsford, NY' 1980'
t3-12
13-13
13-14
l3-15
l3 16
l7
13-18
l3-19
W. H. Lin, "Forced and Self-Excited Responses of a Bluff Structure in a
Turbulent Wind," Doctoral Dissertation, Department of Civil Engineering,
Princeton University, 1977.
R. H. Scanlan and R. H. Gade, "Motion of Suspended Bridge Spans undct'
Gusty Wind," J. Struct. Dlv., ASCE, 103 (1971)' 1867-1883'
D. B. Steinman, "Aerodynamic Theory of Bridge oscillations," Proc. AscF),
75, 8 (Oct. 1949), ll47-1184.
D. B. Steinman, "Aerodynamic Theory of Bridge oscillations," Proc. ASCF),
76, I (Jan. 1950), 154-158.
R. H. Scanlan and A. Sabzevari, "Experimental Aerodynamic Coelficients irr
thc Analytical Study of Suspension Bridge Flutter," J. Mech' Eng' Sci',ln
stituti<rn ol'Mcchanical Engineers, London, ll, 3 (June 1969)' 234-242'
R. H. Scanlan and J. J. Tomko, "Airfoil and Bridge Deck Flutter Derivil
tives," J. Eng. Mech. Div., ASCE,97, No. EM6 (Dec. 1911), l7l1-l'737
R. H. Scanlan, J. G. Bdliveau, and K. s. Budlong, "Indicial Aerodynartrit'
Functions fbr Bridge Decks," J. Eng. Mech. Div', ASCE' 100, No' EM4
(Aug. 1974), 651-672.
R. H. Gade, H. R. Bosch, andw. Podolny, Jr., "licccnt Acnxlynarnic sltttl
ies of Long-Span Bridges," J. strucr. Div., AS('li. 102, N0 S'l'7 (.lrrly 197(r),
llllcksburg,
1.1,2
I
H. W.'l'iclcttt:trt, itrttl li..l . Mitltt't. Iltr' l,'r.tiurrttl Ilr'sltrtrt.st'
lltitl,qr Stilli'rtin,q'l'nt.s.t Mrnltl lrt ltrtltttltrtrt ttrtt! lo rtrr lllt
:l
481
1974.
'f. A. Rcinholcl, H. W.'l'iclctrrrrtr, rrtrtl li..l. Mahcr' Dccay of the Wake of a
Btsx (iinlcr, Roporl No. Vl'}l l1'7.5 lt3, l)cpartment of Engineering Science
and Mcchanics, Virginia l)olylcchnic lnstitute and State University, Blacks-
burg,
1975.
Reinhold, H. W. Ticleman, and F. J. Maher, Wake Study of a Suspen'
sion Britlge stillbning Tnzss, Report No. VPI-E-75-17, Department of Engineering Science and Mechanics, Virginia Polytechnic lnstitute and State University, Blacksburg, 1975.
G. Roberts, "The Sevem Bridge. A New Principle of Design," Proceedings
ofthe International Symposium on Suspension Bridges, Laboratorio Nacional
de Engenharia Civil, Lisbon, 1966, pp. 629-639.
C. Scruton, Experimental Investigation oJ'the Aerodynamic Stability of Sus-
13-22 T. A.
13-23
ll-24
pension Bridges, Report No. 165, National Physical Laboratory, Teddington,
u.K.
ll-25
l.\-26
l.\-21
1948.
C. Scruton, Experiments on the Aerodynamic Stctbility of Suspension Bridg,es:
Results of Tests of a Fult Model in Horizontal winds , Report No. I 85, National
Physical Laboratory, Teddington, U'K., 1950.
D. E. Walshe , The Aerodynamic Investigation for a Box-Type Roadway Deck
for a suspension Bridge Proposed for the Humber Crossing, Special Report
No. 012, National Physical Laboratory, Teddington, U.K., 1968'
D. E. Walshe, The Aerodynamic Investigation for the Suspended Structure of
the Proposed Bosporus Bridge, Special Report No. 020, National Physical
Laboratory, Teddington, U.K', 1969.
1.1,28 A. Hirai and T. Okubo, "On the Design Criteria Against Wind Effects for
Proposed Honshu-Shikoku Bridges, ' ' in Proceedings of the International symposium on suspension Bridges, Laboratorio Nacional de Engenharia civil,
l\
29
l0
I
]
I
l .ll
I
t 12
1299-13',t5.
1320 T. A. Rcinhokl,
pl tt Stt,sltt,tt,titur
N(:l
.\'tt.'unt ()lt.\tttr'lr', llr'1xrr1 No. Vl'l li /'l .lli. l)r'p:ttltttctll ol lingincclilrg Sciclcc 11tt Meclrirrrics, Virgirriir l'olylt't lrtric lrtsliltrlc alrtl Statc Univcrsity,
A. G. Davcnpo(, N. Isyutttov, I)..1 .
3-5
13
lll lllll
'
II iI
Lisbon, 1966, pP. 265-272.
M. Ito, "On the Wind-Resistant Design of Truss-Stiffened Suspension
Bridges," in Proc. second (lsA-Japan Research seminar on wind Effects on
Structures, Kyoto, 1974, University of Tokyo Press, 1976, pp' 285-296'
N. Shiraishi, "on the Aerodynamic Responses of Truss-Stiffened Bridge Sections in Fluctuating Wind Flows," in Proceedings of the IUTAM-IAHR Symposium on Flow-lnduced structural vibrations, Karlsruhe, west Germany,
I 972, Springer-Verlag, Berlin, 197 4, pp. 40 I -405.
I. Konishi, N. Shiraishi, and M. Matsumoto, "Aerodynamic Response Characteristics of Bridge Structures," in Proceedings of the Fourth International
Conference on Wind Effects on Buildings and Structures, London, 1975, Cambridge Univ. Press, Cambridge, 1976, pp. 199-208T. Okubo, N. Narita, and K. Yokoyama, "Some Approaches for Improving
Wincl Stability f or Cablc-stayed Girder Bridges," in Proceedings of the Fourth
Intcrnutitnal (\rqli'rrtrt'c ott Wirul I')ffctts on Buildings and Structures, London, 1975, ('rrrrrbritll',c (lttiv. I'ttss, ('rrrrrbritlgc' 1976' pp' 241 249.
Y. Nlkulrrtrlr tlul l. \'oslrrrrrr:r, "llitt;tty lilttllct ol'sttspcnsion Bridge Deck
St'cliorrs, .l . l'.tr11. NIr'tlr l)tv n S('lr. 102' No lrM'1 (Atrg' 1976), 6U5 700'
482
l3
1;t,t;t't Nt)t
I)l;l'nN ltllll)(il l;,
llN:;l()1..1
nt t t llt Nct
1;llll,(.lt,lll :;, nNl) l'()wl ll llNl
"Arurlysis ol Acrocluslit' ()st'illrlitttrs ol
Mr.rlti l)irncnsional l)nrcctlurcs," irr /)zr
by
Nonlincar
Structurcs
Long-Span
ceedings of the Founh Inlcrnationul Confcrcru:c tn Wirul Ellcct.s ort liltiltlirt11.t
and Stucture,r, London, 1975, Cambridge Univ. Press, Carttbt'idgc, 1976,
pp.215-225.
(198.1,).
13-35 M. Ito and Y.
t3-31
I
Engineering, ETH-Hcinggerberg, Ziirich, 1982.
R. L. Wardlaw, A Preliminary Wind Tunnel Study of the Aerodynamic Stabilit,ofFour Bridge Sections for the Proposed New Burrard Inlet Crossing, Rcpon
No. LTR-LA-3 l, NAE, National Research Council, Ottawa, Canada, 1969.
R. L. Warcllaw, "Sornc Approaches for Improving the Aerodynamic Stability
ol'f]ritlgc lLoatl l)ccks," in Pnx:ecdings of the Third International Conferenct
ort Winrl lifli't't:; ott Builtlings urul Slructures, Tokyo, 1971, Saikon, Tokyo,
13-52 I. Elashkar and M.
Leonhardt, "Latest Developments of Cable-Stayed Bridges for Long
Spans," Bygningsstakiske Medd., 45, 4 (1974), 89-143.
H. Loiseau and E. Szechenyi, Etude du comportement airo€lastique du tablicr
tl'un pont d haubans, T.P. 1975-75, Office National d'Etudes et de Rc
chcrchcs Airospatiales, Chatillon, France.
J. Roche, "Les Mdthodes d'etude a6rodynamique des ponts i haubans," irr
Proceedings oJ the Conference on Cable-Stayed Bridges, Paris, 1974, pp.
7s-86.
D. Olivari and F. Thiry, "Wind Tunnel Tests of the Acroclastic Stability ol
the Heer-Agimont Bridge," Tech. Note 113 von K:irmiin lnslitutc lirr Fluitl
Dynamics, Rhode St. Genbse, Belgium, 197-5.
R. C. Baird, "Wind-Induced Vibration ol'a Pipc l.irtc Srrspcrrsiorr llritlgc lntl
its Cure," Trans. ASME,77 (Aug. 1955), 797 t't(tl
H. P. A. H. Irwin, K. R. (itxrpcr, irnrl ll. 1.. W;rlllrrw, .llt!tlittttirtrt rtl Viltnt
lion Alt,vtrlttr':; to (lntlntl Witul ltrtltttt'tl Viltntli,,rr rtl I llt'ttrrr Iirt.tt lllt'trtlu'r',s
Hanging Roofs,"
13-53
l3-48
.
13-49
J.
Wind. Eng. Ind. Aerodyn., 13 (1983), 395-406'
Novak, "wind Tunnel studies of cable Roofs," J. wind
Eng. Ind. Aerodyn., f3 (1983), 4O1-42OH. Yamaguchi and L. Jayawardena, "Analytical estimation of structural
damping in cable structures," J. Wind Eng. Ind. Aerodyn',41-44 (1992)'
t96l-19'72.
T. Herzog, Pneumatic Structures, Oxford University Press, New York, 1976'
13-54
l3-55 Practical Applications for Air-Supported
l3-56
l3-57
13-58
13-59
Structures, International Conference
held at Las vegas, oct. 1974, Canvas Products Association International, St.
Paul, Minnesota, 1974.
"Technics: Fabric Structttes," Progressive Architecture, 5l (1980), 110t20.
A. Morrison, "The Fabric Roof," Civ. Eng., 50 (1980)' 60-65.
Air-supported Structures, State of the Art Report, American Society of Civil
Engineers, New York, 1979.
M. Horcic, Windbelastung und Berechnung des Spannungs- und Vetfor'
mungszustandes im zylindrischen Tell von Traglujlhallen mit besonderer Beriicksichtigung des Konstruktionsmaterials (H. von Gunten, Referent, H. H.
Thomann, Korreferent), Eidgencissische Technische Hochschule, Zldlich,
19'74.
13_60 B. V. Tryggvason and N. Isyumov, A Study of the wind-lnduced
Response
of the Air Supponed Roof for the Dalhousie University Sports Complex, BLWT-
l3-6t
13-M F.
13-41
394.
l3--5
.
13-46
lltl
E. Kimoto and S. Kawamura, "Aerodynamic Behavior of One-Way Type
Nakamura, "Aerodynamic Stability of Structures in Wincl,"
IABSE Surveys 5-20182, International Association for Bridge and Structtinrl
l()12, pp.93 I 9.+0.
l3-3tt ll. 1.. Wardlaw, A Wirul Tunncl Study of the Aerodynamic Stability of thc
Pn4toscd PasLrt-Kenncwick Intercity Bridge, Report No. LTR-LA-163, NAIl
National Rcscarch Council, Ottawa, Canada, 1974.
13-39 H. P. A. H. Irwin, Wind Tunnel and Analytical Investigation of the Responst
of'the Lions' Gate Bridge to a Turbulent Wind, Report No. LTR-LA-2 10,
NAE, National Research Council, Ottawa, Canada, 1978.
13-40 C. Ostenfeld, G. Haas, and A. G. Frandsen, "Motorway Bridge Across
Lillebaelt. Model Tests for the Superstructure of the Suspension Bridge," in
Proceedings of the International Symposium on Suspension Bridges, Laboratorio Nacional de Engenharia Civil, Lisbon, 1966, pp. 587-608.
13-41 K. Kloeppel and G. Weber, "Teilmodellversuche zur Beurteilung des aen)dynamischen Verhaltens von Bruecken ,' ' Der Stahlbau, 32, 3 (1963) , 65 19
13-42 K. Kloeppel and F. Thiele, "Modellversuche im Windkanal zur Bemessung
von Bruecken gegen die Gefahr winderregter Schwingungen ," Der Stahlbau,
32, t2 (1961), 353-365.
13-43 F. Leonhardt, "Zv Entwicklung aerodynamisch stabiler Haengebruecken,"
Bautech, 45, l0-11 (1968), 1-21.
l3-45
483
pl'tltt ('(),nnrt,(lt,t1, Iiltrrt'Ilritl,qt,l,rtlrtr':tloty lt'cltrricrrl ltclxrrl No. l,'l'R-l,n '
194, NAl,, Nrrliott:rl ltt:st:itrclt ('ottttcil, ()(tawir, ('anada, 1976.
l-l--50 'l'. Mutsurrrolo, "An lnvcstigation on thc Rosponsc 9f Pretensigned One-Way'l'ypc Suspcnsion Roof,s to Wind Action," J- Wind Eng. Ind. Aerodyn.' 13
34 'l'. Miyata, Y. Kuho, irrrtl M. llo,
13-36
l;
13-62
l3-63
SS7-1977, Boundary-Layer Wind Tunnel Laboratory' University of Westem
Ontario, London, Ont., Canada, 1917.
B. V. Tryggvason, "Aeroelastic Modeling of Pneumatic and Tensioned Structures,,, in wind Engineering, Proceedings of the Fifth International conference, Fort Collins, CO, July 1979, I. E. Cermak (ed.), Pergamon Press,
Elmsford, NY, 1980.
H. P. A. H. Irwin and R. L. wardlaw, "A wind Tunnel Investigation of a
Retractable Fabric Roof for the Montreal olympic stadium," in wind Engineering, Proceedings of the Fifth International Conference, Fort Collins CO,
July 1979, J. E. Cermak (ed.),'Pergamon Press' Elmsford, NY, 1980.
R. J. Kind, "Aeroelastic Modeling of Membrane Structures," in Wind Tunnel
Modeling for Civit Enginccring Applications, T. A. Reinhold (ed.), Cam-
l3-64
bridge Univ. Press, Crttttbt-idgc, 1982'
J. C. R. Hunt and l). .l . W. ltich:rnls. "()vcrhcad-Linc Oscillations and the
E{1cctof Aero<Iynarnic l):trttPt'ts," I'r'tn.Irr,st. Iiltt" ling.' Il6(1969), 1869
l.l 65
.l
1814.
M. l). l{owllrllorr, "Mr'lt'oroloy'11';rl ('otttlilituts Associittctl
wi(lr thc Ilull-Sp:rtr (itrlloPing ( )st rll;rltort:, ol ( )r,'r'rltr';ttl 'l'l:tttstttissiotl l.irlcs,"
.
('. R. Hun{
I'ttt<'. ltr.st.
ancl
l'.lt't
.
l'.ltr,g., 120 ( l() / 1). ti
/'l
li
/('
.t
484
t;ut;t,t Nt)l t)lil'nN ulltxit 1;, IIN:;t{}N :;ntt,(:l,nl i;. nNt) t'()wt
l3-66
V..1. lJrzozowski and It.
13-67
13-68
ll.
il ilNt
llrrwks. "Wrrkc Intluc:ctl liull SPan Instrl)ilily
of Flexible Transmission Lines,"
J. Struct. Div., ASCE, f04 (1978), 763-769.
13-70 "Device Reins in Galloping Power Lines," Eng. News-Record, Nov.
t978,
I
ll1)
l3-73
13-14
13-15
13-76
()l
Bundle Conductor Transrnission l,incs," AIAA J., 14 (lgj6), 179 Iu4.
A. N. Hoover and R. J. Hawks, "Role of Turbulcncc in Wakc-lrrtlucctl (irrl
loping of Transmission Lines, " AIAA J . , 15 (1917) , 66-70.
A.S. Richardson, Jr., Dynamic Load Model study for overhead T'ransmit-siotr
Lines, HCPIT-2O6312, U.S. Department of Energy, Division of Elecrric Energy Systems, Washington , DC, 1977 .
13-69 A. H. Peyrot and A. M. Goulois, "Analysis
l3-7
lll lllll ll( | ';
I
p.
30,
17.
A. (i. l)avcnporl, "Gust Rcsponse Factors for Transmission Line Loading,"
i^ lilitul l')tgitrtrrittl|, Pnrccdings of the Fffih International Conference, Forl
('ollins, ('O, l()79, Pcrganron Prcss, Elmsford, Ny, 1980.
. S. ('irrnrll, "l,ahoralory Studies of Conductor Vibration,,, Trans. AIEE,
55, -5 (Mly 193(r), -543 547.
lr. ll. Irarquhamon and R. E. McHugh, Jr., "wind runnel Investigation ol'
Cl.ntluckrr Vibration with Use of Rigid Models," Trans. AIEE,75, part 3
.l
(Oct. 1956), 871-878.
J. s- Tompkins, L. L. Merrill, and B. L. Jones, "euantitative Relationships
in Conductor Vibration Damping," Trans. AIEE,75 (Oct. 1956), 879-896.
G. H. Stockbridge, "Overcoming Vibration in Transmission Cables,,, Electr.
World 86,26 (Dec. 1925), 1304-1305.
R. G. Sturm, "Vibration of Cables and Dampers-I, il," Electr. Eng., SS
(
r936), 455-46s, 673-688.
13-77 R. A. Komenda
and R.
L. swart, "Interpretation of Field vibration Data,"
Trans. IEEE, PAS-87, a (April 1968), 1066-1073.
S. Richardson, J. R. Martuccelli, and W. S. price, ,,Research Study on
Galloping of Electric Power Transmission Lines, " in proceedings of the Symposium on wind Effects on Buildings and structures, vol. 2, National physical
Laboratory, Teddington, U.K., 1963, pp. 6ll*686.
A. s. Richardson, "Design and Performance of an Aeroelastic Anti-Galloping
l)cvice," IEEE Summer Power Meeting, Chicago, Conference paper No. 68,
13-78 A.
13
19
('t)-670-PWR.
It i'i I
li'rt'rrt'e I):rpcl No. C74 016-2, 1974.
Ir
ri.)
K. ll. ('txrpcr anrl R. L. Wardlaw, "Aeroelastic Instabilitics in Wakcs," in
I'nnttrlirtgs tl tha'l'hinl Interncttional Confereru:c on Wintl lifli,tt.;.n Builtl
rttul ,Structun's,'lirkyo, 1971, Saikon, Tokyo, l()72, pp. (r.17 (r5.5.
l{. 1,. Wardltrw, K. R. C<xrpcr, ancl R. H. Scurrllrn. "()lrsr.r.vrrli.ltri on tlt(.
l)rrrlrlcrrr ol' Srrbsparr ()scilla(ion ol' lluntllcrl l)owt,r ('orrtlrrt.r'rs." I)MI,:/NAt,.'
Q. llttll. l()7.1( l). Nirliollrl Ilcrscirrch ('orurt.il. ( )lt:rw:r. (';rrr;rrl:r
itr,q.s
l] til
lt. ll. St.;rrrlurt,,4 W'ittt!'litttttrl lrtIt',tti!:tttt!,t'rtrlr, lltr' lt'trtlIrtrrtrrrr '\trtltrltlv
rtl Btttttllttl I'tntt'r I'irrt'('t'tt(ltt(l('r'\ lttr Ilttlr" ()ttr'1"'' ' l';rrl Vl l{t'1rotl No
l.'l'R-l,n l2l NAli, Nlrtiorlrl l{t'scirrt'lr ('ottttt tl. ( )llirw:r. ('itruttl;t, l()/'l
ll tt5 R. 1,. Wilnllilw, "liltltt(:r ittttl 'lot'siottitl lrrs(irlrrlily''' rt ll'rtrtl l"ttrtt'rl I'rltrtt
tiotts 0l slructurcs,lf . Sockel (etl.1, Splrrrllcl Vt'r.llrg, Ncw Yolk. l()()1.
l3-86 R. H. Scanlan, "On Iiluttcririltl lhrlli'lirrg Mccllitttistrts irt l.rtng Sprrrr llritlgcs"'
Probabilistic l"nginttritr,q Mtt lttttrit"r, -l ( l9ttl{)' 22 21 '
l3-87 A. Jain. N. P. Joncs, antl l{. tl. Scanlan, "F'ully Couplcd Bul1'cting Analysis
of Long-Span Briclgcs," in wind Engineering, Proceedings, Ninth International Confercn('c, pp.962-91 1, Wiley Eastern, New Delhi' 1995'
13-88 L. Singh, N. P. Jones, R. H. Scanlan, and o. Lorendeaux, "Simultaneous
Identification of 3-DOF Aeroelastic Parameters' in Wind Engineering' Pro-
l.l tt4
New
ceedings, Ninth International Conference, pp.972-981, Wiley Eastern,
Delhi,
13-89 R. H.
13-90
13-91
13-93
1995.
Scanlan and
W. H. Lin, "Effects of Turbulence on Bridge Flutter
Derivatives," J. Eng. Mech. Div., ASCE, 14 (1918), 713-133'
P. P. Sarkar, N. P. Jones, and R. H. Scanlan, "Identification of Aeroelastic
Parameters of Flexible Bridges,'' J. Eng, Mech.,l20 (|994), |718_1742.
..Coupled Flutter and Buffeting
A. Jain, N. P. Jones, and R' H. Scanlan,
Analysis of Long-Span Bridges," J' Struct' Eng' (forthcoming)'
N. P. Jones, "Aeroelastic Analysis of Cable-Stayed
(1990)' 219-297'
Eng.,116
Bridges," J. Struct.
K. Y. Billah and R. H. Scanlan, "Resonance, Tacoma Bridge Failure, and
13-92 R. H.
Scanlan and
l3-g4
Undergraduate Physics Textbooks," Amer. J. Physics,59 (1991), ll8-124.
F. Ehsan, "The Vortex-Induced Response ofLong, Suspended-Span Bridges,"
Doctoral Dissertation, Depaftment of civil Engineering, Johns Hopkins Uni-
13-95
versity, Baltimore, MD, 1988.
M. Novak and H. Tanaka, "Pressure correlations on vertical cylinder,"
Proceedings, Fourth International conference on wind EJfects on structures,
13-96
l3-g1
1968.
M. l). Rowbr)tk)m and R. R. Aldham^Hughes, Sub-Span Oscillation: A Rcvit'n, ..1 l,)ri.stittlg Knowledge, Report No. 22-71(SC)-02, Central Electricity
l(cseirlt lr l,ulronrtorics, Lcathcrhead, Surrey, U.K., 1971.
() Nrgol :rrtl (i. J. cllarkc, "conductor Galloping and control Based orr
'lir'si.rurl Mc:chanisnr," IEEE, Power Engineering Meeting, New york, con"
485
l3_98
13,99
Heathrow,U.K',pp'221-232,CambridgeUniv'Press,Cambridge'1972'
R. H. Wilkinson, "Fluctuating Pressures on an Oscillating Square Prism,"
Aero. Quarterly,32, Part 2 (1981), I: 97-110; II: 111-125'
J. D. Raggett , section Model wind Tunnel studies, Golden Gate Bridge, west
Wind Laboratory Report' Carmel, CA' 1995'
..Response
M. Tabatabai' S. G' Krishnasamy, J. Meals, and K. R. Cooper,
of smooth Body, Trapezoidal wire overhead (compact) conductors to wind
Loading," J. Winrl Eng. Ind. Aerodyn', 4l-44 (1992),825-834'
L. Shan. L. M. Jenke, and D. D. cannon, Jr., "Field Determination of
(1992)'
Conductor Drag Coelficients," "/' Wind Eng' Ind' Aerodyn" 4l-44
835,846.
l3-100 N. G. Ball, c. B. Rawlins, and J. D. Renowden, "wind Tunnel Errors in
Mcasurcments ol'P<lwcr C<tnductors,''
(1992),847
J.
Wind Eng. Ind. Aerodyn.,
4l-4
t151.
Y:rrr;rgrrchi. "'l'ltrcc Diltlcnsional Bchavitlrof GalIeping irr'l'clccornrrrrrrrir':rliorr (':rlrlt's ol liigrrn: ll Scc(ioll," .l . Wind ling- Irul-
l.l l0I y.
Iruiino, M. llo. irrrtl
Il.
A(r(,(l\ttt., -l(l ( 191{t'l). I /
.)('
t
.,r.il! ilillxil ti, lFNtil(lN rillil,t;il,n] l;.
ANt) l,OWl il llNl li
f.l-102
'l'nttt.sttri.s:;iort
Litrt, lIqli.rnrt.r,
liltrA
ll,trrtl lrtt!ttr.ttl (.otttlttt.tot. M(,lit,,t. li,ltl<l
DL- 100-4, ljlcctr.ic:al l)owcr ltcsrrirrclr
lrrstitutc, .]4
l2 Hillvicw
Avcrtuc, Itrrlo
Alto, CA 94304. tgig
l3-103 M' E' criswe, and M. D. Vanderbirt,
Reriabirity-based Dasil4rt ..r..r,n,t.srnis
sion Line structures, E.RI EL-4793,
Electricj po*", n",iurch rnstirurc,
3412 Hillview Avenue, palo Alto, CA
94304, l9g7.
CHAPTER 14
l3-104 G' H. Hirsch, "Damping
y,:(.ti;{;o
Measures to control wind-Induced vibrations,,,
in
vibrations of structures, H. sockel ("d
a;;;;Vertag,
),
New
l3-10-5 M. Matsumoto, N. Shiraishi, and
H. Shirato, .,Rain-wind Induced vibrations
.l'Cables of Cable_Stayed Bridges,,, J. lVind
Eng. Ind. Aerodyn., 4l_44
(te92) 2011_2022.
I I l0(r 1"' H' Durgin' D' A. palmer,
and R. w. white, ..The Galloping Instability
o| Ice Coated poles," J. wind Eng. Ind. Aerodyn.,
4l-4 (rgg2), 675-6g6.
l.l 107 Ir. H. Durgin (personal communication, 1995).
l'l loti
'Harmonizing with the
wind," Eng. News Record., oct.2,
l'l 109 P. p. Sarkar, New lcrentification
Rridges ro wind,.D^epartment of
Ilaltimore, MD. 1992.
1944, pp.
OFFSHORE STRUCTURES
Methods Appried to the Response oJ.Flexibre
civil Engineering, J.h;. H;;;il-- university,
wind loads affect offshore structures during construction, towing, and in
ser-
vice. They are a significant structural design factor, especially ii the case of
large compliant platforms, such as guyed towers and tension leg platforms.
wind can also affect the flight of helicopters near offshore platiorm landing
decks il4-1, 14-2, 14-31, as potentially dangerous conditions may be created
by flow separation (see Sect. 4.3) atthe edges of the platform. Let the horizontal
distance between the upstream edge of the platform and the upstream edge
of
the helideck be denoted by d, and let the depth of the upstream surface-pro-
ducing the separated flow be denoted by l. on the basis bt *ind tunnel tests,
it has been suggested that the elevation ft of the helideck with respect to the
upstream platform edge should vary from at least h
= 0.2t if d = 0 to at least
h = O.5tif d = tIl4-21.
This chapter includes information on wind loads on offshore structures of
various types (Sect. 14.1), and on the treatment of dynamic wind effects in
the
case of compliant structures (Sect. 14.2).
14.1 WIND
LOADTNG ON OFFSHORE STRUCTURES
Methods for calculating winrl loacls on offshore platforms are recommended in
lrrlrorirtory irnrl lirll-scalc mcasurements indicatc that
thcsc mcthods may, itt sttlttt' ittsl:rnt't's, lurvc scriorrs lirni(utions, particularly
instllar as thcy d<l nol it('('()lrrtl tor llrt'l)r('scn('c ol'lili lilrcrrs, rrrrtl irccolll
ll4-4lto tl4-8]. Howcvcr,
irlsullicicntly--rlr ttol :tl itll lirr slrit'ltlirrli rrrrrl rrrrrlrr;rl irrtgr.li.r.t.rrcc t'llr'cls. I;6r.
cxiuttplc, accrlrling lo wirrrl lrrnrrr'l (t.:,1 rt.:;rrlls olrl:rirrt.tl lirr lr jltt.k rr;r (st.ll.
clcvlrlirrg) 1ll:rllirrlrr Il4 ()1, llrt'rttt'llrorl:, ol Il.l .ll:rrrtl Il.l 5lovt.n'sl ilrurlt.wintl
447
488
()t I liil(
)nt
1;
il tt,o t(,nt
*
s;
kriuls on.jack-up units by at lcast .)-5,/n. 1q,;1;"r,rtcs bascd on lirll,scalc tluta lirr
Ittl atrchorcd scmisubmersible platlirrnr ll4-l0l suggest that thc rtrolhotl 6l'
f l4-5f ovcrprcdicts wind loads by as much as loo%. It has therclirrc bcconrc
c()llllll()ll practice to obtain design information on wind forces on platlirrnts
lirrrrr laboratory tests. Most tests provide data on mean, as opposed to gusting
krirtls. ln using such data the effect of gustiness should be accounted fbr by
:rrurlyticirl ffleans (see for example, Sect. 14.2). Possible Reynolcls numbcr
e llccts shoulcl also be assessed with care.
'l'lris sr:ction briefly reviews a number of wind tunnel tests
conducted for
st'rrrisrrbnrc:rsihlc units and for a large guyed tower platform. Wind tunnel test
irrlirrrrrirtiorr on.jack-up units, on jacket structures in the towing mode, and on
tw() tyl)cs ol'concrete platform is available in [14-9], t14-lll to [14-14],
ll,t
\51, ll4-191, and [14-40].
$
o
o
14.1.1 Wind Loading on Semisubmersible Units
A
st'lre tttltl
ic view of the model of a semisubmersible unit used for tests reported
in l14 l-51 is shown in Fig. 14.1.1.*
'l'hc sirlc lbrce and heeling moment coefficients are defined
ti
0)
ir
.o
a
as
o
CY: ,
-Y
tpU'(5O)A,
(14.
l. l)
-o
E
4)
CK:,
"K
>pU'(50\A.H,
(14.1.2)
whcrc Y is the side force, K is the heeling moment, p is the air density, U(50)
is thc mcan wind speed at 50 m above sea level, .4" is the projected side area,
:urtl H, is the elevation of the center of gravity of ,4". Coefficients CY and CK
lu'c obtaincd separately for the overwater and for the underwater part of thc
rrnit. 'l'hc overwater coefficients reflect the action of wind and should be obt:rirrtrtl irr a llow simulating the atmospheric boundary layer. The underwater
t'rx:llit'icrrls account for hydrodynamic effects and should therefore be measurecl
irr rrnilirlru ll<lw.
liigrrlcs l4.l .2 'tntl 14. 1.3 show values of CY and CK measured in [14-l-51
Ior tlrt' t'rrst' ol'trn upright dralil 7r,,. : 10.85 m (corresponding, for the unit
lrt'rrr1, 11111fr'.;t'tl. (o:r tlisplirccrncnl'of lJ,729 tons). As noted in [14-151, thc
l!urlx)sr'ol tltr'lt'sls lirr lhc undctwater part is to determine the elevation of thc
r't'rrtr'r ol rt'lrt'liorr (i.r:., thc point of application of the resultant of the undcrw:rtt'r lort'cs) lirl thc l'r'cc-lkrating unit. For an anchored or dynamically posirlrlittrt's l'l.l.l tlrlorrglr
()v(rlrrrrrirrP,
14.1.(r arc cxccrptcd
o
o
o
()
V)
t
D
tu
lrom E. Bjerrcgaar{ :rrrtl S. Vt:lsglu1r, ..Wirrtl
lllli'(l()rl:tSctnisttbtttcrsiblc,"PapcrOTC3063, Pnx'raliu.rit,()llshort'l't:c6nokrgy
('orrl('11'tr((', lLtttslott,'l'X, May l()71'1. ('opyright 197t3 Ofl,shrlrt:'li.t.lrrrolo;,y ('orrlt'rt.rrt.t'.
Ilrcrr|li1',lrttlr;tli71.11 rstlrctlcplhrtlirtutrclsionolthcrrnitirrllrt.t.vt.rrlrr'r'lrlrllitrprr(c.g..li1.
:ttt:tlt1,,lt'ol lrcr'l y'r O").
'Ilrc tlispl;ttt'tttt'tt1 is lltt'vrtlttntr'ol w:rlcltlisllllrct'rl by
tlrc illrnrr,r:t'rl p;nt rrl llrr'rr1t
I
i
489
()l l.(;l t()nt
490
llilt(,(;tt,nt
*
I
t4
r wtNl) t()nl)tN(i
()N ()t tlit t()t
il 1iilil,(:tUnt li
49'l
1
0.5
0
-
0.5
WIND
-1
,
TM0o:6.4 3 m
0.5
u
"
"
0
0,5
9.00m
10.8
5m
15.25m
1
sponding to 100-knot beam winds tl4-151.
ft'f(illf{ltl 11.1.2. Values CY and CrK as functions of wind direction
:rrr11li's
ol lrt'cl
y'r
rlt at different
lirr the overwater part [14-15].
tionerl rrrrit llrc ccnter of reaction is determined by the anchoring forces or by
lltc tltnrslcrs I l4- l5l.
liigrrrc 14. 1.4 shows estimated values of the heeling forces induced by 100krrol lrcrrrn winds* for various values of the upright draft T1as" and of the angle
ol' lrccl <f . The elevations of the center of action of the overwater (wind) force
lrrtl ol' thc ccnter of reaction on the underwater part are shown in Fig. l4.l.5.
It is sccn that as the angle of heel increases the elevation of the center of action
ol'(ltc wind force decreases. This decrease is due to lift forces arising at nonzero
irrtglcs ol'hcel
FIGURE 14.1.4. Wind heeling forces corre-
@.
The heeling lever is defined as the ratio of the overturning moment to the
displacement of the vessel. Values of the heeling lever for 100-knot beam
winds, obtained from the wind tunnel tests of [4-15], on the one hand, and
by using the American Bureau of Shipping method U4-41, on the other, are
shown in Fig. 14.1.6. (The displacements listed in tl4-l5l for the 6.43 m,
9.00 m, and 15.25 m drafts are 12,740 tons, 16,963 tons, and 19,495 tons,
respectively.) It is seen that for large angles of heel the differences between
the two sets of values are considerable. This is largely due to the failure of
[4-4] to account for the effects of lift. It is noted in [14-16] that the largest
overtuming moments are commonly induced by quartering winds.
In the tests of [14-15] and [4-16] the water surface was modeled by the
rigid horizontal surface of the wind tunnel floor. Following the method de-
,l
0,s
lDistonce obove
2SJVoterline (m)
U
0
deg
0,5
Tggo=$,1,
1
\-."- -
"
"
"
I6
9.00 m
10.85m
15.25 m
$,1 -db*'w't"
\-90
\di
360 deg
Itl('lliltl'l 14.1.-1. Valucs C)'and C'Kas functions of direction ry' at dill-crcnt
y'r lirl lltc rrrrt[:rwllcr pan ll4-l-51.
anglcs ol'
lrrt'l
-'-e"\.
.-'y''--'-
rWitttls blowirtp', rtkrttg lltt' :rxis Y (lrig. 14. l. l). Wind blowing rrkrrrl', tlrr. :rrrs \ :rn' 1.li'r1'tl (g
(or lxrw) wirrtls. Wirrtls wlrost.tlirt:t.liorrs ltisccl lltc:uty',lr.s lrr.tu't.r.lr,rrr'., .y:rrrtl y:rr.r.
tt'lr'trr'tl lo :rs rlululrrirrlt, wirrtls.
,,1
;rs lrt':ttl
lr'l(ltJRll, 14.1.-5. Illcvlliort ol'ccrrlcl ol :rt'liorr
ol wirxl lorccs lrtxl corrcslxrrxlirrl', ccrrle r ol
\l
lrt'(iorr orr llrc rrrrtlcrwrrlcr lllrrl
ll.1 l5l.
rt'
*
I4 I WINI) IONI)IN(i ON OI I SHOIII] SIHUCTUBES
492
Modol
Tesls A B S
{ts-- ---- 6.43m
2 DRILLIN.
RlGs
BENorNc sHoE
- {- - - --- 9.00m
--)(-- ---10.85m
----o-.15.25m
utu
493
/
-tt2oo ToN (1a1 Ms)
CLUMP.WEIGHT
€ffiffi
l. '3o'
HEELiNG LIMIT
I
22" (ra2shm)
0'
5o
10o 15' 20'
DtA.
PEBIMETER PILES
Angle of heel
frl(;lJltl,l 14.1.6. Wind heeling levers obtained from wind tunnel tests
64"
and from Amer-
I
(1372mm) OlA. MAIN PILES
l
12OO' (oAOm) ANCHOR
CABLE - 6" (l27mm)
OIA. COATED
icirn llrrrcuu ol'Shipping (ABS) method [14-15].
scrilrr:rl
in ll4-171, tests reported in [14-18] were also conducted by placing
llrc rnoclcl in a tank filled with viscoelastic material up to the level of the wind
Iturrrcl flxrr. This facilitates the testing of models of partially submerged units.
Itclbrcncc [4-18] also includes results of tests conducted in the presence ol'
rigirl ohstructions aimed at representing water waves. The results revealed that
wuvcs could increase the overturning moments substantially. This suggests thc
ncccl lirr improving the simulation of the sea surface in laboratory tests.
'l'hc acrodynamic testing of the Ocean Ranger semisubmersiblex is reportcd
DIA. ANCHOR PILE
tower platform. (A schematic view of the platform, installed in over 300 m of
water in the Gulf of Mexico, is shown in Fig. 14.1.7 |4-241; see also Fig.
14.1.8.) The mean wind profile created in the laboratory matched closely both
the power law:
in ll4-3t)1. The problem of combining hydrodynamic and wind loads was
by conducting I : 100 scale aerodynamic models in turbulent flow
lltxrr with rigid waves, and using lightweight lines to apply the measurccl
rrrcirrr rrrrl lluctuating wind forces and moments to a l:40 hydrodynamic rnodcl
It'stt'rl irr contlilions sirnulating those experienced during the storm. Additional
wintl lrrrrrrt'l tt:sts ol'scrrrisuhmcrsiblc units are reported in[14-14] and [14-l9l
Io I l.l ))1. I l/t 401. lnrl I l4-411.
<182snm,
FIGURE 14.1.7. Schematic view of Lena guyed tower platform. From M. S.
Glasscock and L. D. Finn, "Design of a Guyed Tower for 1000 ft of Water in the
Gulf of Mexico," J. Struct. Eng., ll0 (1984), 1083-1098.
u(z')
;rtltlrc:sscrl
ovcl
f2"
:
/ - \l'12
urrol
(
(101)
14. l .3)
rr
14-1.2 Wind Loads on a Guyed Tower Platform
l{t'lr'r't'rrt't' ll,X 2ll prcscnts rcsults of wind tunnel measurements on a l:120
st';tlt' rrotk'l ol'llrc ovcrwittcr part of a structure similar to Exxon's l-cna guyctl
r'lltt'()tt'itrt lllrrrgcl llrtl citltsizctl olt licbntatry 15, 1982 in Hitrcurirr l'rcltl. Illr krrr sorrllrt:irst ol
Sl .hrlur's Nt'wlirtlxllittttl. itt it slonrt with l7 rrr lo 20 rn wavc heiglrtr, lrrrrl l.)o hrrr/lr to IlO krrr/
Itt wrttrl s1x't'rls. ll w:ts lltc l:rr11t'sl sttbtttt'tsibk: oll,slrore tllillirrl', plirlllrrrr rrr llrr'rror|r1.,1(r rrr lrililt
lrottt Lrt'l lrt ogrt'ltlirttts tlt't k rttttl witlr l2O rtt ktttp. pottlrxrrrs. All ol llr, li l , rlu' rrrcrnlx rs wt.r(.
losl rtt lltt' :rtr'trlr'nl
i
and the expression for sustained winds (i.e., winds averaged over at least one
minute) recommended by the United States Geological Survey [14-7] for use
within the Gulf of Mexico:
rJkt:
uoo)
/ -
-
(:r-:,)
\01128
(14.1 .4)
where z7 : 2.2 m and z is the elevation above thc still watcr lcvcl in rlclcrs.
The airlwater boundary was modeled by the rigicl horizonlirl surlircr: ol' llrc
wind tunnel floor. Forcc irnrl rrxrrncnt cocliicicnts wcrc tlclirrctl llv n'llrliorrs ol
thc typc
('t
J"
llrtl'1
11,;,'1,'
(l,l I 1)
I
()t I lill()l11
494
1;lllt,(;l{,lll li
l.l I wlNl) l()nt)lN(i
()N ()t
t:iu()nt t;nl,clt,lil l;
495
Y
CD, CMD
Drilling Derrick (2 t
x
+224.9'
-tI
Derrick Structure (2)
-
+
163.5'
Flore Boom
"A Wind Tunnel
Investigation of Loads and Pressure on a Typical Guyed Tower OlTshore Platform,"
Paper OTC 4288, Proceedings, Offshore Technology Conference, Houston, TX, May
1982. Copyright 1982 Offshore Technology Conference.
FIGURE 14.1.9. Notations. From P. J. Pike and B. J. Vickery,
@ Decx Slruc'iure
(enclosed
@
+
)
Dritting Pockcges ( 2 )
@ P- ronts
5B.O'-
u(r6t
CT. CMT
@ Crews Quorters (2)
Subslructure
Well Conduclors
Elv. O.O
(14.1.6)
where F and M are the mean force and the moment of interest, p is the air
density, U(16) is the mean wind speed at 16 ft above the water surface, and
the reference area AR and length l,a were chosen as 1 ft2 and I ft, respectively.
in ll4-231are represented in Fig. 14.I.9,
which also shows the notations for the respective aerodynamic coelficients. The
The force and moments obtained
Deck
r56'x 156'x 57
Flore Boom
to'
rJ-
moments characterized by the coemcients CMD and CMT were taken with
respect to a distance of 6.2 in. (62 ft full-scale) below the still water level.
The measured values of the aerodynamic coefficients are represented in Fig.
14. l. l0 for several platform configurations. The configuration for the base case
was the same as in Fig. 14. 1.8, except that the deck structure was not enclosed.
Additional results in Il4-231 show that the effect of enclosing the deck is
ncgligible for practical purposes, as is the effect of the well conductors. Rernoving the flare boom results in torsional moment reductions, but has negligible effects otherwise. It is shown in|4-231that drag forces and drag moments
based on wind tunnel measurements are smaller by about 30% and 2O%, respectively, than the calculated values based on tl4-71.
To check the extent to which the results depend upon the laboratory facility
bcing used, the same structurc was subscqucntly tcsted indcpendently in a
.-J
Skid Bcse
67'x32'x8'
lro
dction of
Boom
I
eo'
I
t+2'
I
I
different wind tunnel ll4-2-5 1 ltt tnosl c:rscrs ol' signilicrrncc lnrrn ir rlcsign
vicwpoint the results obltrirrt:tl irr ll,1 2.5 1 wt'n' liugr'r llurrr tlrosc ol'll4 211 hy
anlounts that did not cxcct'rl lO to \Oi)l,
I
Nole: Helidecks Resl on
Top of Crews
Quortcrs
Itl(llJlll,l
14.
l.tt. (ittycrl lowcl-Platlirrlrr:
(b)
(a) siilc t'lt'v:tltott, (/') l)l;trr ll4
rl)illr'rerrtt'slrt'lwt'ettttsttllsol;r,'ro,l\tt.rtttt,
25
1
lirtilitrr's;ttt':tlso
ttrtlt'rl itt Srr't
I
(r
nr.l.nrrrrrl.,rrrrrlrril'1lrr'Il'r'n'l{lll\
utrltll(t.11
1.1
F.
U
u
O
-.
()
-
O
;, IrYNnMt(. Wtt.tt) | lllclli
()N (;()Ml'l lnNl ()l l:;l l()l l
:;llill(:l([il
:;
497
BASI CAS[
L w/0 fasl otRRlcK
w0 B0rll oEnnlcl(s
'
w/0
DBtU-t]{G EOUIPiIEM
zDECl(COilFlGURATloll
N
r
L
u
a
r:
U
O
O
L
C
o
F
z
U
I
a
=
I'IIND DIBECTION
b)
0
l,,llND DIRECTI0N
(c)
SIASE CASE
EAST OBREK
WO BOIH DERRrcT\S
() WO
I
-
z
o
W0 oRlUJilG EoUlPillEl{I
2
FIGURE f4.1.f0. (Continued)
DECK GoilHGURATtoil
U
/ ,**-tt-
/P
I
I
u
a
./ *F*t^:
14.2
:.--t","--- \:
- ' ,\V''i
U
E
a
DYNAMIC WIND EFFECTS ON COMPLIANT OFFSHORE
STRUCTURES
Compliant offshore platforms are designed to experience significant motions
TM,T{SVEBSE MOMEIIIT (CMT)
under load. An important advantage of compliance is that the forces of inertia
Irssociated with these motions contribute to counteracting the external loads.
ln the case of large offshore structures installed in deep water, compliance
has the additional advantage of making it possible to design platforms with
vcry low natural frequencies in the surge, sway, and yaw degrees of freedom*
( c. g. , I /30 Hz to I / 150 Hz, depending upon type of platform and water depth) .
Wavc motions have narrow spectra centered about relatively high frequencies
(c.g., trom lll5 Hz for extreme events to about I Hz for service conditions).
'l'lrus, aside from possiblc second-order effects, compliant platforms generally
r
IJ
t
:
)
tkr not cxhibit any dynarnic amplilication of the wave-induced response.
t.lnlikc wave motions, winrl spcccl lluctuations in the atmospheric boundary
lrrycr arc charactcrizccl l'ry brorrtl bltrttl spcc(ra (scc Fig. 2.3.4). For this rcason
I'IIND DIRECTION
(b)
14. 1.10. Wintl tLrnncl tcst rcsults. From P..1. l)ikc rrntl ll..l . Vickcry, "A
Wirrrl 'l'trrrrrcl Irrvcstigalion ol'l-uacls antl Prcssurc on a'l'ypic:rl ( irryt'rl lirwcr-Oll,shon'
l,'l(Jlllll,)
ll,ilrt'irt tttolions ilr lltc lorrliitrrrlur;tl. tr;rr:vrt:r', rrttrl vr'tlir':rl tlirct'liolt ltlt' lt'lt'rtt'tl l() its .rrrrli(,
llt\',;rtttl ,/tcilr,, tcspcclivcly. Arrl',rl;u rlolroli rr .r lr;rrtv('rs(', lurrlr,iltrtlirr;tl, :ttttl ltotizottlltl Pl;tttr'
rrtt ttlt'rtul lo lts toll. pilt lr, rttr,l \/it\\. lr".l! ' lr\' lY
l'l:rllilnt,"l'irPt:r'()'l'('42llll, I'nxcttling.s,O.ll.shrtrt'll'r'ltttolrt,t:\'('t,nl('tt'ntt'.llorrslorr.
l'\. Miry l()1{2. ('opyriglrl l()tl2 ( )ll.slrorc 'l'ccltrtology ( 'ortlt'tt'rrt r'
496
rti&.
498
()t
Il;l t()nt liutt,(itt,lu
l;l l' l)YNnMl(: Wl[]]) I lll(;11; ()N O()Ml'l lnNl ()l l:;l l()l tl l;llll,(;ll,ltl
I
il ltrrs ltccrt sllr(ctl irr lltc litcr-ulurc llrrrt wintl incluccrl rlynanric iurrlllilir'ir(ion
cllccts orr cotttpliilnt structurcs arc signiliclril ll4-23,ll4-261. A rnorc gt-liu:(lc(l
ol'thc cl]'ects of wind gustincss was prcscntetlinl14-271as pafl rll'
of thc response to environmental loads of the North Sca Hutton
tcnsion lcg platform (Fig. 14.2.1, see also tl4-281). According to ll4-271:
"Wirrd gusts are typically broad-banded and may contain energy which could
cxcitc surgc motions at the natural period. These would be controlled by surgc
tlrrrrrping. 'l-hcorctical and experimental research is required to clarify the iml)()r'liulcc ol' this mattcr. "
irssossrfronl
ilrr cvaluation
l;
499
Irrve:sligirliorrs irrlo llrt'bchirvior ol'(otlsi()tl lcg pl:rtlirrrrrs utrclor wind loads
rcJrortotl in ll4 291 trrrtl ll4-.101 wcro basccl ott lhc assulnption that the response
to wind is dcscribccl by a systcrn with proportional damping, the damping ratio
bcing ol'the rtrclcr of 5%. Howcver, it was shown in [14-31] that for structures
cornparablc to the Hutton platform, the effective hydrodynamic damping is
considerably stronger, and that the wind induced dynamic amplification for
krw-f'requency motions are for this reason negligible. Section 14.2.1 describes
thc approach used in t14-311 to estimate the response of a tension leg platform
to wind in the presence of current and waves, and a simple method for estilnating the order of magnitude of the damping inherent in the hydrodynamic
loads.
14.2.1 Turbulent Wind Effects on Tension Leg Platform Surge
Under the assumption that the extemal loads are parallel to one of the sides of
the platform shown in Fig. 14.2.1 , the equation of surge motion can be written
AS
Mt
:
(14.2.1a)
F,(t)
where
F,(i) = F,(t) + Fh(t) + R(t)
(14.2.tb)
In Eq. 14.2.1b, F,(t), FhG), and R(/) denote the wind force, the hydrodynamic force, and the restoring force, respectively. Not included in Eq.
14.2.lb is the damping force due to internal friction within the structure, which
corresponds to a damping ratio of the order of l% and is negligible compared
to the damping forces associated with hydrodynamic effects.
Wind Loads. Like the hydrodynamic loads, wind loads consist of a component
flow separation, and an inertial component associated with the relative
fluid-body accelerations. However, it can be verified that the inertial component
is about two orders of magnitude smaller than the component due to flow
clue to
scparation, and can therefore be neglected in practical applications.
To estimate the wind-induced drag force it is assumed that the elemental
rlrag fbrce per unit of area projected on a plane P normal to the mean wind
spccd can be written as
p(y.:.
l'f (Jllfll'l 14.2-l- Sclrcttntic vicw ol'thc Hutkrn tcnsion lcg plrtlirrrrr. Irnrrn N. llllis.
.l ll.'l'clkrw. li. Atttlctsott, urttl A. L. Wrxxlhcatl. "lltrllorr'l'l ,l'Vcsscl Slnrclrulrl
('rrttlillttr':tlrrrrt rtlttl l)trsig,lt liclrturcs," l'}apcr ()'l'(' 442J, l'tttt t,t'rlirr.r;.t. O llillrrc 'l'ct.lr
llrlol'y ('ottli'rt'trt't', l lottslolt, 'l'X, Miry l()t12. ('opyl'iglrl l()l.i-! ( )llslrort' 'll't lrrrology
('orrlt'tt'lrt r'.
11
-
!p,,C,,(t, z)Iu(y, z,
t)
- i?)f
(14.2.2)
whc:rc /r,, is thc air tlcrtsily. { ),( r'. l) is thc pressurc cogfiicient at clcvatiolr l:
Irrrtl lxrrizolr[:tl crxrtrlitut(t' t'ttt lltc plrtttc P, / is thc titttc,,r is thc surgc tlis
plirt't.r1etr(, lhc tl<lt tlt'trolt:, rlrll, r, rrlr:rliorr witlr tcspcct trl litrrc, ltlttl tr(.y, :, /)
is lltt' wirrtl spccrl rrpwirrrl ol llr,' :,lnr(lur(' irr tlrc tlirct'liott ol-lltc tttclttt witttl.
500
t
otIr;il()ril riilr(,(;ilililli
'f'lrc spccrl u(.y, ?., l), can bc cxprcssr:tl lts a stttrr ol'lhc ntcan spc:ccl
tlrc llrrctuirting spccd u'(y, 2., t)'.
u(y, z, t)
:
U(z)
*
u'(y, z, t)
{/(l)
unrl
wlrt'rt'
..1,,
Io tlrt'
rrrcrrn
:
(14.2.3t
s7,;,.,(n)
dz
J^,,rrr.z. 1dy
(14.2.4)
is thc projection of above-water part of the platform on a plane normal
wlnd speed.
'f 'lrt'
nrerrn speeds can be modeled by the logarithmic law (Sect. 2.2.3). Thc
sl)r'( llr ol thc longitudinal velocity fluctuations can be modeled by Eqs. 2.3.25.*
'l'lrt' t'r'oss spcctra of the longitudinal velocity fluctuations are modeled by Eq.
'l'hc cllcct of longitudinal separation should also in principle be taken
irrto lrecotrnt . However, it follows from information presented in [2-89] that
tlris cllct't is ncgligible as far as fluctuating aerodynamic loads on offshore
l.l.lO.
slru('tlrrcs irrc concerned.
lt cirrr hc verified that the mean square values of u' and i and the mean
vrrlrrc ol'thc product u'* are small compared to the square of U. It then follows
l'nrrrr llr1s. 14.2.2, 14.2.3, and 14.2.4 that the mean drag load can be written
SL",(.y,,
'' #Tu\^"'"''
z)u2(z) dY dz
- ,,,\n"Cn',
z)u(z)u'(2, t) dy dz
.t. l.rl).
nrttgc
dy1 dv; tl?.1 tl:.;
S"
(n)
:
as
(t4.2.9)
lrsr^"r;rt
of the process
uLqU)
:
?
"'"rt
cos(Zrn1
-t
$,)
(14.2.10)
In Eq. 14.2.10 the phase angle S; is generated by random sampling from
a
uniform distribution in the interval 0 < fi < 2tr.
Let the spectrum of the force F',c,,,(t), defined as
:
(r4.2.1t)
p oC oAoU(2")uLq(t)
be denoted by Sp"o,,(n). ClearlY
s"*,,I,) =
(14.2.6)
(14.2.1)
rNolt' lh:rl lor lhc licrlucncy r - 0 Bqs. 2.3.25 yicld a spectral onlirurlc ,S(0) prrrgrrlional to tlrc
irrlt'1ir:rl lrrrllrlcncc scllc /,), in accordancc with lundarncntal princilllcs (st'c Iitq. 2.1 l9). On llrt^
ollrt'r lr:rrrtl. li1. 2.-1.2.1 (c.9,., quotc(l in ll4-231) yiclds S(0) O, urrtl i1 tlrcn'lon'rrult'rcstirrrirtt's
lltt'spt'tlrrl otrlirtirlt's itt llrc
z)
(r4.2.5)
:rrrtl l;,, is thcr clcvation of the aerodynamic center of the above-water pan of the
lrlrrtlirrrrr. linrrn lic;s. 14.2.2 to 14.2.5 it follows that the fluctuating part of the
wirrrl rlrrrlq lorrtl (hut would act on the platform at rest (i.e., with i : 0) is
I'i,.,(r)
2.1,
From Su,"o(n) it is possible to generate by Monte Carlo simulation realizations
whcrc lhc overall aerodynamic drag coefficient is
:
:
s,.*o(n)
F 1c,,,(t)
= ip"C/"U'(2")
lz,
The spectrum Sn,(n) can be estimated numerically by assuming Cr(li, z)
c,(i :1,2). An equivalent wind fluctuation spectrum can then be defined
lts
F,(i)
501
: ,i,\^, Jr,, ,r,,.u,' ;1)( ),( v.,' ::.,){/(::')l/(;:,)
(l.r.l.ri)
x
I
:;
'l'ltt'liottltt't
wlrcrc l[c sgbst'r'i1tt l rcl'crs to llre llrct tlrir( tlrt'pl:rllirlrrr is irl lr's( .
translilnrr ol' (hc auloc<lvariattcc litttc(irtlt ol' /"i ,(l) yie ltls
'l'hc totul wind-induced drag force is
F,(r)
()N (i()Ml'l IANI ()l l:ill()l ll :;llltl('ll'lll
l4:' l)YNnMl(i wlNl) I lll(;l:;
ol lypicirl nalttral lictlttt'tttit's lot torrrplrurl slrrrclrrrcs (liig.
(t4.2.12)
sr,.,(n)
Thus aio(l) can be viewed as an equivalent wind speed fluctuation that is
perfectly'coherent over the area Ao and whose effect upon the structure at rest
is the same as that of the actual fluctuating wind field.
The total wind load acting on the platform can thus be expressed as
F,(t)
:
ip,CA,lU(2,) + u'.r(t)
-
*(t)12
(14.2.t3)
Numerical calculations have shown that if the difference between the elevation of the helideck (or the top of the crew quarters) and the underside of
the lower deck in a typical drilling and production platform is less than about
two-thirds to three-quartors ol'tlre-witlth ol'thc rnain clcck, thc tcrm C](z' in lit1. 14.2.|t.
z2)2 of p,q.2.3.30 can bc ncglct'tctl wlrt'rr ev:rlrurling tlrc ilt(c:grlrl
litcttlIol'ltlXrtt(
lltlrrr(',by.lt
llrt'ttlr:rt('issrrritllel.
Thisisaconsequenccol'the
1.5. 'l'hc appnxinruligrr illtt.tt'ttl rrt ttclr,lct lirrp, ( "(.11 l:,)' is sliP,lrlly t'6tr
scrvltivc l-nlln a slntt'trrllrl cnllln('('r lrll lt()lttl ol vir'w ltltorrlllr trrsll'.ltillt';rlrtIV
()l l:;l t()ilt liiltt,(;tUtil
A/llJl)
I
IIrI(;t:i
()N (;()Mt't tnNt ()t t:il
l()nt tiillt,(:tt,llt ti
503
tllrrrrpirrg cocllicicrrl . ll wlrs rrssrrrrctl lirl converrit'rrct' irr ll4 .lll tlrirt lltt' wirvt'
rrrotiorr is rrronoclrronlrtic', lrt:rrcc lhc:rbscttcc rtl sct'otttl ortlcl tllilt lirtccs irt
l;.q. 14.2.ltt ll4-311. l( was lssunrcd in lrtklilion lhlr( /J O, sincc llrc I'lrrlilr(ion
tlarrrping at low licqucncics is ncgligiblc ll4-121.
'l'he total wave-induced exciting lirrcc arrtl tltc sut'gc-irtltlctl tnrrss cutt bc
cstimated numerically on the basis of p<ltcntial thcory. Altcrn:rlive:ly, thcsc lwo
Lcrms may be assumed to be given by the incrLia c()rllp()ncnl ol'lltc Morisott
equation:
lil(lllltl,l
I
r
14.2.2. Integration domain
t)
A= p.Xlv4(c-ut.l
and
lrrrslirrrrrrl iorr ol' variables.
F" =
s()). N()linlt, thcn, that for any arbitrary function O,
I,, J,
*,, Y,
*
Y,l) dY,
(tY2:
*
I' *u,,,
t) dt
(14.2.14)
(l;ig. 14.2.2), and assuming Co(li, zi) = Co, U(zi) : U(2"), and Su,(n) =
,t,(:,,, rr1, (i - I ,2), it follows after some algebra from Eqs. 14.2.8,2.3.30,
rrrxl 14.2.9 that
S,."q(ru)
=
(r4.2.15)
Su(zo, n)J(n)
, n) is the spectrum of longitudinal velocity fluctuations at elevation
J(n) is a reduction factor accounting for the imperfect coherence among
llrc llucl.uating wind pressures at different points of the platform, given by the
wlrcro
5',,(2,,
:,, , and
cxprcssion
.r(n)
r';
lrr
:
-3 [-*0,-E) + (r -
;)
texp(-r)
- r]]
n
: c.b' u(2")
-
(14
Hydrodynamic Loads. The total hydrodynamic load
F1, can
o"
F,,
* F,,- Ax - I].r
(l4.2.lti)
wlrcrc /'), is thc (ollrl hytlr<ltlytltttric viscotts lirlcc, /,,. is llrt'lotrrl w:rvc irrtlrrccrl
r'xt'itirrg lirrcc, y' is llrt'srrrgc:trltlt:tl rrrirss, lrrrtl /J is llrt'srrr1't.wlrvr'r'lrrli:rliorr
t
u11
- rt*)
(14.2.20)
(14.2.21)
/2r\2
:;I (r"i
(t4.2.22)
ll4-34, p. 1571. The total hydrodynamic viscous load may be described by the
viscous component of Morison's equation
F,,
[rr,:
tDi
where 11 is the wave height and k,, is the wave number given by
be written in thc
lirrrrr
.
2z-r\
rH ^"'cos\k.Xt
r.- /.
uii:Te
-T)
lirl. 14.2.1'7, b is thc width of main deck.
Equation 14.2.15 can be used in
Irt'rr trl lirls. 14.2.213 antl 14.2.9 forthe Monte Carlo simulation of the equivrrlcnt vclot'ity lluctuations aiu(t) (see F,q. 14.2.10) needed in the expression of
tlrt'lolrrl wintl kxrcl acting on thc platform, Fu(t).
l*
ll4-34, p. 3l], where p, is the water density, vu is the elemental volume of
the submerged structure, C-u is the surge inertia coefficient corresponding to
v,r, X is the horizontal distance from some arbitrary origin to the center of V;
along the direction parallel to surge motion, ui and u;1 are the current velocity
and horizontal particle velocity due to wave motion, respectively, at center of
v,,. Equations 14.2.19 and 14.2.20 may be employed if for the component
being considered the ratio of diameter to wave length, DIL < 0.2 ll4-34, p.
2831. Since forT* = 15 sec, L: gTz*l2T = 350 mll4-34, p.2831, where
7", is the wave period and g is the acceleration of gravity, it follows that for
members of typical tension leg platform structures, for which D < 2O m or
so, the use of Eqs. 14 .2.19 and 14.2.20 is acceptable if three-dimensional flow
effects are not taken into account. The wave motion can be described by deep
water linear theory, so
2 t6)
(t4.2.17)
p.llr,,r.
(14.2.19)
:
0'5P,,'
LL
{'1,A,,,1.t',
* ttii *l[ui +
u,,
- *l
(14.2.23)
whcrc ,4r,,, is arca ol'clcrut'n(rrl volrrrut' v,, grnricclcrl <ln lr planc rtolrturl (o (lrc
r!ir-ccliorr ol'lho ctrrrcrrl , irrul ( ,7,, rs llrc tlr;r1', tocllit'it'rtl r'olrt'sporttlirtp. lo ,'1,,,,.
ll tltc lclirlivc rnoliorr,rl 1111' lrrxly wrllr rr's|t't'l to tlrc lltritl is lr:rt'tttorur'. lltt'
*
504
()t
Iriilolit r;ilrt,(;t(,til
l;
505
(lnrg an(l incrliu crrcllicicnts itt Mrtrisort's c:(lulriion can bc clctcrntirtrrl ort lltc
brrsis ol'cxpcrirrrcntal results as lunctions ol'local oscillaklry Rcynolrls rrrrrrrbcr.
(11, , 2TD2l(il.t), Keulegan-Calpenter number, K : VTt lD, an<l rclalivr: b<xly
srrrlircc nrughness, where D is the diameter of the body, a is tho kincnratic
viscosity, V and T1 are the amplitude and period of the relative fluid-brxly
vclocity. Howcver, actual relative fluid-body motions are not harmonic. I'his
intrirtlrrccs rrnccftaintics in the determination of the drag and inertia coeflicients
t'vcrr il t'xlrclirnontal information for harmonic relative motions were availablc
rr t('nns ol (11,, rrncl K. Unfortunately, such information is not available tbr thc
:rrrrrll A rrrrrrrbcrs (of the order of 2) and large Reynolds numbers (of the ordcr
ol l0(') ol inlcrcst in tension leg platform design. Forthis reason calculations
:lrrrrrltl lrt't':rllicd out for various sets of values Ca, C., and investigations
:.lrorrltl lrt't'orrtlucted into the sensitivity of the results to changes in these values.
Restoring Force. The surge-restoring force in a tension leg platform is supplit'rl lry (hc horizontal projection of the total tension force in the tethers. Most
ol lhis lirr-cc is the result of pretensioning, which is achieved by ballasting the
lloirting platlirrrn, tying it by means of the tethers to the foundations at the sea
lkxrr'. llrcrr dcballasting it. The tension forces in the tethers should exceed the
t'onrprcssion fbrces induced by pitching and rolling moments due to extreme
lrxrtls.
FIGURE 14.2.3. Notations.
to the hydrodynamic viscous load F,, (Eqs. 14.2.1, 14.2.18, and 14.2.23). For
this reason it is appropriate to solve Eq. 14.2.1 in the time domain.
The nominal natural period in surge is
l.lrrtlcr thc assumption that the tethers are straight at all times, the restoring
lirn't: c:rr.r bc written as
R(/)__(r+srth+Lh
(xll)'l
a
(r4.2.2s)
wlrt'n' (',11 is thcr tkrwnclraw coefficient, equal to the weight of water displaced
;r; tlrt'rlr':rlt is incrcusctl by a unit length [4-32] (Fig. 14.2.3).
lrr n'lrlity. lryrllotlynumic and inertia forces cause the tethers to deform transvt'rst'ly 'l'lrc rrrrglc bctwccn the horizontal and the tangent to the tether axis at
tlrt'plrrtlirrrrr hccl can thcrcfore differ significantly from the values correspondirrp to lhc crrsr: ol'a straight tether. Nevertheless, owing to the relatively small
rolt'ol llrc rcs(oring lirrcc in thc dynamics of typical tension lcg platfbrms, thc
t'lli'cl ol srrclr tlill'crcrrccs on thc motion of thc platfirrrns appL)irrs to hc ncgligiblc
lirr' prircticll purp()scs
ll4-36, l4-37,
(ry)"
(14.2.26)
(14.2.24)
wlrcrc 7'is the initial pretensioning force, AZis the incremental tension due to
strrgc rrrotion, /, is the nominal length of the tethers at r : 0, A/, is the
incrcrrrcnlal lcngth, and
rtlr
T C'*[l tl/I
l, 1 N,,: ,,,*
Tn:2r
l4-381.
Surge Response. 'l'hc strrgc rcsl)onsc is olrtiritrctl by solvrrrl, lrt1. 1.1 .J.1. 'l'ltis
r't;tlrliorr is norrlirrt'ru', llrt' sl nrngt'st corrllilrrrliort lo llrc rrotrlrrrt';uity lrf i111' 1111q'
where M"6 is the coefficient of the term in -t and ft the coefficient of the term
inx in Eq. 14.2.1. From Eqs. 14.2.1, 14.2.18, and 14.2.24, it follows that
:2trltu + 'qlt^f"
"lTl
T.-
(14.2.26a)
A calculated time history of the surge response is represented in Fig. 14.2.4
for a platform with the geometrical configuration of Fig.
14.2.5, under the following assumptions: platform mass M : 34.3 x 106 kg;
total initial tension in legs T: 1.56 x 105 kN; Morison equation coefficients
C-i;: 1.8,x Ca,,: 0.6, wave height and period H :25 m and 7. : 15 s,
respectively; current speed varying from 1.4 m/sec at the mean water level to
0.15 m/s at 550 m depth; aerodynamic parameter C,1": 4320 m2 elevation
of aerodynamic center 2,, : 5O m; atmospheric boundary-layer flow parameters
r:0.002, P :6.0, Li,: 180 m,J, :0.01,f,:0.2, Ct: 16 (scc Eqs.
2.2.23,2.3.2,2.3.4,2.3-25, lLl,tl 14.2.17); and mean wincl spccd [./(:,,) - 45
m/s. lt is shown in ll4-3 ll llurt thc contrihutions ol'(hc rucrrtt wintl itttrl ol (ltt'
wincl lluctuations lo thc llelrk r'('sl)onsc ol'l"ig. 14.2.4 ut-c rtltottt 1ll')/,' lnd l)'/,'.
as a function of time
gl lriA, l,l .t \. rl lollows lrolrr llrcsc irsstuttltliotts itlrl ltottt lirlr l.l .t l(),rrrrl
l(Xl s
l,'l.,l.l(rr llt:rl lltt' tuttttittttl tt;tlttl;tl ltlrlttr'ttr 1' tr /,,
'1'l16r'llre pllrtlirrtrt
506
()t
I1iil()ilt tiiltu(;l(,1il
I'l :'
lAlJl {}l l:'ll()l
Wllllrllll(.1:,()N(.{)Ml'l
ll
"lllll{lll'lll
':
ltll7
sUt!t,(' wits:tltt'tttlltt'rl irr llrt' li(i'l:rllll(' (tll lltt' lr:r:ltr-' t'l lltt'
ol'strlgt'rturliott l('l)l('s('lll:j:l ltttt':tt :;vsl('llr wllll:l
ctlrrirliort
Irsstrrrrllli6rr llrlrt tltc
(crltt
lry it ttotttitl;tl tlirtrrpilrl' llrlto' ( Ilrt't'llt't (
cllltrltclcrizul
viscotr,s tlirrlrping
.l'this tcrrn is p6stulatccl to bc cclrrivllcrtt lo tlrc tl:rrrrpitrl', t'llt't'l ol lltc lrytlrrr
tlynamic viscous lirrcc.
Such an approach is acccptahlc il tlrc or.tlcl ol rrutg.triltttlt' ol lltt' ttortttlt:tl
damping ratiit is consistent wi(h thc hyclnrtlynalrtic'bclt:tvior.ol tltt'syslcllr.
rltlios
Calculations are now presentccl that illustratc lrow such lttltttittitl tlltttt;'riltg
0l'trtrltttlClrl wirul orr
50
E40
U
(5
can be estimated.
It is assumed that aio(r)(see Eq. 14-2.10) is given by the harmonic function
E.
(=_/)
llYll/\Ml{
r
35
uLqQ)
0
500
1000
1500
2000
TIME [s)
uLot cos
2r
(r4.2.27)
,t
where I is the natural period in surge, and the system under consideration
The ampliis linear with mass M + A, natural period 7,, arid damping ratio f'
force
harmonic
of
a
tude of the contribution to the surge response
relation
the
given
by
p,C.A,U(2")uJq r cos Ztrnt is denoted bY r,*u*, and is
p,CuAoUko\u',,
FIGURE 14.2.4, Calculated time history of surge response [14-31].
J!
rr:spcctively. It can be verified that this conclusion is equivalent to stating that
wincl-induced resonant amplification effects are negligible in the cases invesrigarcd in u4-311. Sensitivity studies showed that the results were affected
insignificantly by uncertainties with respect to the actual values of the atmosphcric boundary-layer fl ow parameters.
:
!max
tnt +
,cnnlT))
11l
-
(14.2.28)
1
tnf lrl: +
12ynT,)2\t/2
equal
The nominal damping ratio f is estimated from the condition that x,*,* be
cos(2tl
poCoAoU(Z)u'",
force
the
of
l
to the contribution to the surge response
Tn)t, as obtained by solving- Eqs. 14-2.1' By substituting
14.2.28, it follows that
.
;:6
Nominal Damping Ratio of Pseudo-Linear System Representing the Re-
sponse to Wind. It was indicated previously that the estimation of the effect
jp"C,AuIJ{z)uio
1
l/f for n in Eq'
(14.2.29)
Calculations were carried out in 114-311 for the platform shown in Fig'
14.2.5 (with tether lengths In = 600 m) and for a similar platform with tether
lengths l, = l5O -, .,iittg the mechanical, hydrodynamic' aerodynamic' and
14.2.29
arm"osph;ric boundary-lay"i flo* parameters listed previously. Equation
ln
m
and
600
/,
with
platforms
= l5O
:
for
the
O'2
=
0.5 andf yieldei I
rn, respectively.
:
and Cmj: 1'8 on which these results were based
with large diameters, such as those depicted
members
rnay not be realisiic fbr
calculations were carried out in [14-3ll
rcason
lhis
For
in nig. 14.2.5 |4-34|.
(,1,
ol'
vlrltrc
scts
. C,,,,. Thc calculated nominal clartlplilr a-numbcr of possihlc
l-rc sullicicntly largc t<t prcvclt( tltt'
tri
lirruttl
clrst.s
in
trll
irrg rati.1s f wcic
The- values C,1,,
0rprh;
- 600r
Itl(llllll,l 14.2.5. (it'otnt'lly ol ;tl:tllotnr
o
6
,raa,,rr"n"" gl- s(nlng wirrtl irrtlut t'rl tlyrutrrtit' :rrrrplilicalirln cllccls. Ilrtwt'vt't '
litr sorttc vlrlttr:s tll' (',1,,, ("ll('tllilli()lls irr wlrit'lr {ltt' :tssttttlctl ('tll'rcllls wottltl lrt'
l.wt:r ll''t 1111;5;1:61'lt,i f f l r'orrl,l lr':,rrll rn rcrlut't'tl ttolltitt:tl tl:rlltpirrll l;tlios lol
t't'rlrtilt clirrrit(olo1t,it'ltl t'otttllltott:' ll('i 'ltr"('rvttltl w;tvt'lt"sls viol;tlt'llrt'ltt'ytroltl:;
i
I
()t I
r;il()1il :illlt,(ilt,lrl
lt l llt N(it :;
l;
lln(l Kculcguu-Carpcntcr nuntbcrs, tltcy crtttltot pnrvitlc it rrsclirl ilrtlit'rrtitltt ol'
tlrc c:ll'cctivc darnping lirr thc pn)totypc. 'l'his, in addition to thc ahscrrcc ol'
lcliablc tlrag data fbr large Reynolds numbcrs and small Keulegan-Clilrpclllol'
Irrunbcrs, is a continuing cause of uncertainty in the assessment of dynarlric
t.llr:cts incluccd by wind acting alone or, in the case of a nonlinear analysis, irr
c()n.ir.ulc(ion with wave-induced slow drift.
REFERENCES
ll I
l.l
.l
1.1
]
ltl
4
l,,l
.5
M. lr. l)avics. L. R. Cole, and P. G. G. O'Neill, The Nature of Air Fktws
rtvtr oll.rlutrc Pluforms, NMI Rl4 (OT-R-7726), National Maritime Institutc,
lit'lllrirrrr, U.K., June 1977.
M . lr. l)avies , Wind Tunnel Modelling of the Local Atmospheric Environment
ttl oll.slntre Plaforms, NMI R58 (OT-R-7935), National Maritime Institute,
lrclllrarrt, U.K., May 1979.
1.,. ll. Lirtleburg, "wind Tunnel Testing Techniques for offshore Gas/oil
I'r.otluction Platforms," Paper OTC 4125 Proceedings, offshore Technology
('tnt.lt'rcncc, Houston, TX, 1981.
llult's .for Buitding and classing Mobile offshore Drilling unirs, American
llurcau of Shipping, New York, 1980.
I?ulcs for the Consffuction and ClassiJication of Mobile Offshore Units, Det
norskc Veritas, Oslo, 1981.
14
6
of Offshore Structures,
1977 (Reprint with corrections
the Design, Constuction, and Inspection
for
Rulcs
Appendix B, Loads, Det norske Veritas, oslo,
t919).
l4-1
1,.:l
lt
Rcquirements
for
Verifuing the Structural Integrity of OCS Plaforms, Appen-
rliccs, United States Geological Survey, OCS Platform Verification Division,
l{cston, VA, 1979.
AIrl Rccommended Practice for Planning, Desig,ning, and Constructing Fixed
()l.l.sltorc Platforms, API RP 24, American Petroleum Institute, Washington,
t)(
1981
.
l.l () I) ".l . Norton and C. V. WollT, "Mobile Offshore Platform Wind Loads,"
l';rlx'r'()'l'C 4123, Proceedings, Offshore Technology Conference, Houston,
'l
x. l()t31.
l.l lo ll. lltxrrrsllrr, "Analysis
I
I II
l.l l.)
I,l I \
of Full-Scale Wind Forces on a Semisubmersible Platlrrnrr (lsirtlt ()pct':tlor's Data," J. Petroleum Technol. (1980), 171-776l' .l . l,rrrrslirrtl. Mutsurt,rncnts of the Wind Forces and Measurements of an Oil
I'r,t,lrtr'tittrr .lttt'ktt Structura in Tow-Out Mode, NMI R30 (OT-R-7801)' Nalrorr:rl Mrrli(ittlc Ittstittltc, Fcltham, U.K., Jan. 1978.
('. li. ('owtlrcy, 'l'itrrt-Avtrul4cd Aerodynamic Forces and Moments on a Mtilcl
ol tt Ilrrrt' I.tg,gctl Crnt'rctc Production Plaform, NMI R35 (OT-R-7808)'
Nrrliorlrl Mrrritirnc Institutc, Fcltham, U.K., June 1982.
('. li. ('owtlrcy.'l'irtrr'Atrrugul At'ruxl.ynutnit'Flrtcs ttrul Mttnrnls rtn tt Mtxlcl
tll tt'l'ltrrr'l,t',r4gul ('tntt'rt'tt' Itnxlutliott l'lttl.lrtrnr, NMI It.l(r (O'l'It-7ltol3),
()112
N:rliotutl Mltritirrrt' ltrslilttlc. licl{llltltt' I J. K.' .lrrlle l
l':l
l4 l]. L. Millcr
lrrul
M. ll. l)rrvics, lilirtrl
509
ltxttlirr,r4 ttrt ()llsltort'Slru<'lurcs-A
ol Wittrl 'litttrrtl ,\rrttlir,s, NMI l{l-16 (()'l'R-tJ22-5), National Maritime
lnslittrtc, licltlrarrr, .K., Scpt. l9tt2.
l4-15 lr. l).jcrrcgaarcl ancl S. Volschou, "Wind Ovcrturning Effect on a SemisubmerSttttrttrttry
l..J
siblc," Papcr OTC 3063, Pntcccdings, Offshore Technology Conference,
Houston, TX, May 1978.
14-16 E. Bjcrrcgaard and E. Sorenscn, "Wind Overtuming Effects Obtained from
Wind Tunnel Tests with Various Submersible Models," OTC Paper 4124,
Proceedings, Offshore Technology Conference, Houston, TX, May 1981.
14-11 J. H. Ribbe and J. C. Brusse, "Simulation of the AiriWater Interface for Wind
of Floating Structures," Proceedings Fourth U.S. National
Conference, Wind Engineering Research,B. J.Hartz (ed.), Department of Civil
Engineering, University of Washington, Seattle, July 1981.
l4-18 J. M. Macha and D. F. Reid, Semisubmersible Wind I'oads and Wind Effects,
Paper no. 3, Annual Meeting, New York, Nov. 1984. The Society of Naval
Architects and Marine Engineers, New York, 1984.
14-19 R. F. W. Gould, The Prediction of the Wind Loading on a Wind Tunnel Model
of an Offshore Drilling Unit Based on the SEDCO 700 Design, NMI Rl8 (OTR-1729), National Maritime Institute, Feltham, U.K., July 1979.
14-20 C. F. Cowdrey and R. F. W. Gould, Time-Averaged Aerodynamic Forces and
Moments on a National Model of a Semisubmersible Offshore Rig, NMI R26
(OT-R-7748), National Maritime Institute, Feltham, U.K., Sept. 1982.
Tunnel Testing
Measurements of Aerodynamic Forces and Moments on a Model of a Semisubmersible Olfshore Rig, NMI R34 (OT-R-7807),
14-21 P. J. Ponsford, Wind-Tunnel
National Maritime Institute, Feltham, U.K., June 1982.
14-22 A. W. Troesch, R. W. Van Gunst, and S. Lee, "Wind Loads on a 1:115
Model of a Semisubmersible," Marine Technol., 20 (July 1983),283-289.
14-23 P. J. Pike and B. J. Vickery, "A Wind Tunnel Investigation of Loads and
Pressure on a Typical Guyed Tower Offshore Platform," Paper OTC 4288,
Proceedings, Offshore Technology Conference, Houston, TX, May 1982.
14-24 M. S. Glasscock and L. D. Finn, "Design of a Guyed Tower for 1000 ft of
Water in the Gulf of Mexico," J. Struct. Eng. ll0 (1984), 1083-1098.
14-25 T. A. Morreale, P. Gergely, and M. Grigoriu, Wind Tunnel Study of Wind
Loading on a Compliant Olfshore Plaform, NBS-GCR-84-465, National Bureau of Standards, Washington, DC, December 1983.
14-26 J. R. Smith and R. S. Taylor, "The Development of Articulated Buoyant
Cofumn Systems as an Aid to Economic Offshore Production," Proceedings,
European Offshore Petroleum Conference Exhibition, London, Oct. 1980, pp.
545 551.
14-27 J. A. Mercier, S..l . LcvL:rc(lc, antl A. [,. Bliault, "Evaluation o1'Hutton T[,P
Response to Envirorrrrcntirl l,()r(1s." I'irpcr'O1-C 4429, Procccrlirgs, Ofl,shorc
Tcchnology Ctttrli'tctrtr', Ilottslott.'l X. Mlry l()t32.
14-28 N. tillis, J. H.'l't'llow. l; Aruk'r.';orr. lntl A. 1.. Wtxxllrt'ltl, "lltrltolr'l'l,l'
Vcsscl Strlrclrrrrl ('otrlrl'ur;rltorr :rrtrl I )r':,i1'tt lir':tltttt's," lt:rlx'r ( ) l'(' 'l'l.l'1. I'r,t
t't'ulitt.qs, Ollslron' 'lt'r'lrttnlol't, ('rltrlr'tt'rt( r'. lltttlsl()tt. 'l \. Mlry lt)li,l
.
142() A. Klrrcerrr lrnrl ('
l);rll,rrr.
lllrr:rrrrt
l,llr'rlr, ol Wltrl otr'li'lt:'iott
l.r'1'l'llrl
510
()t
t:;il()nt :;tlrt,(;l(,lll 1;
lirrrrrs," I'apcr O'l'C 422(). I'nn't'r'tlirt.q.s, ()ll.slutn"l?clrtutltt,q.v
('rtrr.li'tr'rrr't',
Houslon, 'l'X, May 1982.
l4 l0 lJ. J. Vickery, "Wind Loads on Compliant Offshore Structurcs," l'nrtulitt,g.s'
l4-3
I
l4-32
1.1
.ll
Ocean Structural Dynamic Symposium, Oregon State University, Dcpitrtnlcttl
I'Civil Engineering, Corvallis, Oregon, Sept. 1982, pp. 632-648.
E. Simiu and S. D. Leigh, "Turbulent Wind and Tension Leg Plattbrm Surgc"'
J. Struct. Eng., ll0 (1984), 785-802.
N. Salvesen et al., "Computations of NonlinearSurge Motions of Tension Lcg
Platfbms," Paper OTC 4394, Proceedings, Offihore Technology ConJbrenct.
Houston, TX, May 1982.
J. A. Pinkster and G. Van Oortmerssen, "Computation of First- and SecondOrder Forces On Oscillating Bodies in Regular Waves," Proceeding,s, Seutrul
lnternational Conference on Ship Hydrodynamics, Univ. of California, Berkc-
ley, 1917.
T. Sarpkaya and M. Isaacson, Mechanics of Wave Forces on Offshore Structures, Yan Nostrand Reinhold, New York, 1981.
14-35 A. G. Davenport and E. C. Hambly, "Turbulent Wind Loading and Dynamic
l,J .14
ofJackup Platform," Paper OTC 4824, Proceedings, Offshore Techrutlogy Conference, Houston, TX, May 1984.
14-36 E. R. Jefferys and M. H. Patel, "On the Dynamics of Taut Mooring Systems,"
CHAPTER 15
WIND.INDUCED DISCOMFORT IN
AND AROUND BUILDINGS
Response
Eng. Strucr., 4 (1982),37-43.
14-3"7 E,. Simiu, A. Carasso, and C. E. Smith, "Tether Deformation and Tension
Leg Platform Surge," J. Struct. Eng., ll0 (1984), 1419-1422.
l4 38 E. Simiu and A. Carasso, "Interdependence between Dynamic Surge Motions
of Platform and Tethers for a Deep Water TLP," Proceedings, Fourth Inter'
national Conference on Behavior ofOffshore Structures (BOSS), pp.557-562,
I
5 July 1985, Delft, The Netherlands.
14-39 R. L. Wardlaw, P. H. Laurich, and G. R. Mogridge, "Modelling of Dynamic
Loads in Wave Basin Tests of the Semisubmersible Drilling Platform Ocean
1.1
40
l,l
,ll
Ranger," Proceedings, Interncttional Conference on Flow-lnduced Vibrations,
Bowness-on-Windermere, England, May 12-14, 1987 .
.l . M. Macha, "Modeling Wind Loads on Mobile Offshore Structures-A Sumrrrrry of Wind Tunnel Results," in Wind Elfects on Compliant Offihore Plat.litrns, C. E. Smith and E. Simiu (eds.), American Society of Civil Engineers,
Ncw York, 1986.
lr. Iljerrcgaarcl and S. Hansen, "Wind Effects on Semisubmersibles and Other
liloirling ()lllshorc Structures," in Wind Effects on Compliant Offshore Plat
litrrns, (. li. Srnith and E. Simiu (eds.), American Society of Civil Engineers,
New Yor.k. 19t36.
It is required that structures subjected to wind loads be sufficiently
strong to
has
viewpoint.
Recent
experience
safety
perform adequately from a structural
must
take
the
designer
also
shown that in the case of tall, flexible buildings,
into account wind-related serviceability requirements. The latter may be formulated, in general tems, as follows: structures should be so designed that
their wind-induced motions will not cause unacceptable discomfort to the building occupants.
Wind-induced discomfort is also of concern in the altogether different context of the serviceability of outdoor areas within a built environment. Certain
building and open space configurations may give rise to relatively intense local
wind flows. It is the designer's task to ascertain in the planning stage the
possible existence of zones in which such flows would cause unacceptable
discomfort to users of the outdoor areas of concern. Appropriate design decisions must be made to eliminate such zones if they exist.
The notion of unacceptable discomfort, which is seen to play a central role
in the statement of serviceability requirements, may be defined as follows. In
any given design situation various degrees of wind-induced discomfort may be
expected to occur with certain frequencies that depend upon the degree of
discomfort, the featurcs ol'thc clcsign and the wind climate at the location in
qucstion. The discomlirrt is uttircccptlrhlc il'any of these frcquencies is .iuclgcd
to bc too high. Statcrrrt:rrls s;x't'ilyirrg nurxinrtrrn acccptahlc tncan l't'ct;ucnt'ics
<ll'<lccurrcncc firr virriorrs rlt'1'.rt't"s ol rlist'orrrlirl ruo ktrowtr:ts cotttlirll ttilr'r'i:r.
ln crlrnlilrt critcritr tlt'vt'loP1'11 lor rrsr' irr rlt'sigrr il is irrtPlrrctit';rl lo Itli'r'
cxplicitly to rlcgrccs ol tlisr'orrrl,rrl ;r: r;rrt lr ltlrllrt'r', ttlt'tt'rtt'r' is ttt:trk' lo ;t
suilirltlt: l)itf:illc:(L:r, vluiotr- v;rlrr':,,r1 s,lrt, lt:ttt':tssor'irtlt'rl willt v:rrl()tts (l('l't('('s
ol'tlist'orrrlirrl. ln llrc t'lr:u'ol \\'irrrl rrrlu(,'il 111111111111' ltlrliotts, lltls Pltl;tlttt'lt't ti
5l
I
512
wtNt) tNt)t,ot
t) t)[;(;()Ml
()t
tt tN nfil]
nt t(
)t
||) I :il llvl(.1 nllll llv {)l lnl I llt,ll I)lN(ill llNl)l ll llll
tNt) ttltll l)lN(iti
tlrr: lruiltlirUl ilccclcralion- ln critct-ia pr:tlrrinirtg to thc scrviccability ol'pc:tlcslrirtrr
rrrcirs, (hc paranrctcr cnrployed is an appnrpriatc mcasurc rtl'thc wintl s;lccrl
noar lhc gnrund at the location of concern. Clearly, to develop comfirrt c:ritcria,
i{ is rcquircd that parameter values be established that correspond to various
tlcgrccs <ll'human discomfort. Furthermore it is necessary that to various clcgrccs ol'discomtirft-or, equivalently, to the parameter values that correspond
Io thcnr thcrc bc assigned maximum acceptable probabilities of occurrencc.
Vcrilying thc compliance of a design with requirements set forth in a given
st't ol cornlil( critcria involves two steps. First, an estimate must be obtaincd
ol tlrc wirrrl vckrciLics under the action of which the parameter of concem will
r'xct'ctl tlrt' vrrlrrcs specified by the comfort criteria (these values may be referred
Io irs t'r'ilir'irl). Socond, the frequencies of occurrence of these velocities must
lrt't'stirrur(cd on the basis of appropriate wind climatological information. lf
llrt' ln't1rrt'rrcics t.hus estimated are lower than maximum acceptable frequencies
spccilictl by thc comfort criteria, then the design is regarded as adequate from
ir st'r'v it:cirbil ity viewpoint.
l{clcvrurt c:ornputational fluid dynamics methods are discussed in [4-89].
l{c:rsorrirhlc qualitative results have been obtained in some instances, but no
t[:lirritivc validations appear to be available. The development of comfort crirclia lirr thc dcsign of tall buildings and questions related to their practical use
irrc tliscusscd in Sect. 15.1. Comfort criteria forthe design of pedestrian areas
rrntl rclatcd design information are dealt with in Sects. 15.2 through 15.4.
n(.ll()N ()l wlt'll)
513
llcsul(s 6l cxprl.irrrcrrls ltirrrctl :rt r:stlrblislrirtg pt't't't'grliott llttt'slrolrlr lol pt'
rirxlic rrrrtlions ol' 0.(Xr7 llz to 0.2 Flz Irtvr: ltt'r.'tt t't'1rot1t'tl ilr l l5 Jl. 'lltr'
cxpcrirncnts, carricd out ort Il2 sLrb.ic:cts itt tttol iott sitttttlltlots l('l)r( \('llljrll\/('
of an ofiicc environmcnt, wcrc clcsigrrctl to litkt' ittto lteeottttl lltt' tltllttt'ttt t' ttlrott
perception thresholds of bocly oricrrla(iott, llrtly tturvr:lttt'rtl, lrotly lxtslttlc, ;ttttl
the extent to which the motion is anticipatctl by (lrt: srtbict'ts ol lltc t'xpt'lttttt'ttls
The perception thresholds as reportcd 1'>y 50')(, tll'lltc: sttll.ic('ls w('le lirtttttl trr
be approximately 1 % g,0.9% g, and O.6% g lirr frcqucncics ttl' v ibrat ion ol 0.(Xr7
Hz, 0. I Hz, and 0.2 Hz, respectively. It is noted that within this l'rcquoncy
range-the perception thresholds decrease as the frequencies increase. Additional experimental results are used in [15-2] as a basis for a tentative relation
between the horizontal acceleration of a floor and the percentage of the individuals on that floor for whom the acceleration will be perceptible'
Studies of human response to vibrations of a motion simulator have also
been reported in [15-3] and [15-4] for frequencies in the range 0.1 Hz to 1 Hz.
Average perception thresholds were found to vary from about o.6%s for fre-
quencies of 0.1 Hz to about 03%S for frequencies of 0.25 Hz. Motions were
distinctly perceptible and the subjects were annoyed while working at their
desks ilthe accelerations exceeded 1.2%g. Beyond accelerations of 4%g, the
perceptions were described as strong and the subjects experienced difficulties
in *ulking. The motions were described as extremely annoying or intolerable
beyond accelerations of the order of 5%g to 6%9. Similar results have been
in [15-5].
A study presented in [15-6] and [15-7] is based on observations of human
reported
15.1
SERVICEABILITY OF TALL BUILDINGS UNDER THE ACTION
OF WIND
15.1.1
Human Response to Wind-lnduced Vibrations
Stuilics ol'human response to mechanical vibrations have been conducted within
thc: lust two decades mainly by the aerospace industry. Because the frequencies
ol'vibrir(iorr of interest in aerospace applications are relatively high (usually I
llz to .15 Hz), the usefulness of these studies to the structural engineer is
11t'nclrrlly lirrritccl. Ncvcrtheless, results obtained for high frequencies have been
r'xtrrrpol:rltril irr
ll5 ll to frequencies
lower than 1 Hz, with the following
t'on1'slxrnrlt:rrcc bcirrg prlposed between various degrees of user discomfort and
llrt' ;rt't't'lt't'rrliorts cltttsing them:
Acceleration (in
I)cgrcc of
l)iscomfbrt
percentages of the
acceleration of gravity g)
lrnpcrccptiblc
Pcrccptiblc
Annoying
Vcry Annoying
<)ns
)nga!'t,g
Irrlolt:t':rblt:
ll%,s
response to actual rather than simulated wind-induced accelerations. The investigation covered the behavior during a storn of two buildings and of their
occupants. Estimates of the rms value of the top floor accelerations during the
storms were based on response measurements for one of the buildings, and on
wind speed measurements and wind tunnel testing for the second. These estimates represented averages (1) in time over the periods of highest storm intensity (20 min to 30 min)* and (2) in space over the entire area of the floor-the
to account for wind-induced torsional momotions see also t15-321 .) The rms values
(For
torsional
of
the effect
tions.
first, and 0.5%g for the second of the two
for
the
were
0.2%S
thus obtained
occupants then revealed that about 35% of
building
with
buildings. Interviews
first building experienced motion sickness
in
the
floors
higher
the persons on the
the
second building the reported percentage
For
storm.
the
symptoms during
creaking noises that occur during the
that
in
is
noted
45%.
It
was about
[15-7]
space averaging being performed
building motion may increase significantly the feeling of discomfort and should
therefbre be minimized by pnrper structural detailing.
Rcsults of surveys contlrrc{ctl among occupants of tall buildings in .Iapan arc
rcportcd in ll5-81.
5'/,,,c,
5%,g l5'n,7i
) l5'x,,q
'r'l inrt'lrvcr':rl'.t's wt're ttlso
cllt'tlt'rl .r't't lottltt'l 1x'rirxls I l5
(rl.
514
wlNl)
tNt )1,(.t
t) t)[i(,()Mt
()l
il
tt,t At.]t r
/\l()t,Nr)
nl
*
,l t)tN(i:i
rrrt r,t ilvl(.1 nlill llYol
lnl I lll lll l)lN(i:itllllrl ll llll nr llrrl l{rl wllll r
fil.'i
15-1-2 Comfort Criteria
('oltrlittl critcria should in principlc bc basctl on an cxtcnsivc ktr<lwlctlgc ol'the
llLrgrcc ttt which building users are prepared to accept discomlort ass<lciirlcrl
wi(lr wirrd-incluccd accelerations. However, at present such knowlcdgc is sculcc.
A sirnplc comfort criterion has been proposed in [5-9], bclicvcd by its
rrrr(hors to hcjustificd by the results of [5-2]. This criterion, which lin-rits thc:
irvc:rirgc nrrrnhcr ol'occurrences of 1%g accelerations at the top occupied lkxrr
Itr :rt rnost 12 pcr ycar, has been applied to the design of the World Tradc
('t'tt(t't'11.5-91. ln ll5-61 an attempt is presented to develop comfort criteriu orr
tlrc brrsis ( I ) ol'r-ccorclccl objcctions by building users to the recurrence of windrrrrlrrt't'rl lrrriltling vihrations and (2) of estimates by owners or developers ol'
llrt' possibk' t't'ottoruic repercussions of user dissatisfaction with the building
pt'r'lrrrr;urt't'. Iinrrrr interviews with building occupants who had experiencc(l
nr()tr(fns wilh lur rrns value of the top floor accelerations of about 0.5%g, I
rv:rs cslirrurlctl lhat about 2% of the people in the top one-third of a building
worrltl olr jer'l l() rnore than one occunence of such motions in six years. Intervit'ws witlr brrilding owners and developers suggested, on the other hand, that
rcrrlrrl
orsrrlcsof o{Iicespacewouldnotbeaffectedsignificantly
if atmost2%
ol' thc occupants in the top one-third of the building found the sway objectionlblc. On thc basis of those findings, it is suggested in [5-6] that the following
tlcsigrr critcrion appears to be reasonable: "The retum periods, for storms
crrrrsirrg iln rrns horizontal acceleration at the building top which exceeds 0.5%5,
shall rxrt be less than six years. The rms shall represent an average over the
20-rrrirr period of highest storm intensity and be spatially averaged over the
builcling floor." This criterion is presented in [15-6] as tentative and in possible
ncccl of ad.justment as additional information becomes available.
15.1.3 Relation between Wind Velocities and Building
Accelerations
A lirst
stc:p in vcrifying the compliance of a design with requirements set forth
irr t'orrrlirrl crilcria consists in the estimation, for each possible direction, of the
wintl spt't'rls lhirt would induce the acceleration levels of interest. Wind tunnel
Ir'sl rt'srrlls rrrry bc uscd to obtain plots of speed versus direction for the wind
vt'kx ilit's llurl ilttlucc critical building accelerations (that is, accelerations equal
t. tlursr'sPt't'ilitrtl by thc comfirrt criteria). An example of such a plot is shown
rrr l;i1' 15.1 I. (Notc that the methods of Chapter 8 can be applied in this
t'orrlt'xl.) Spr:ctls corrcsponding to points outside the curve of Fig. 15.l.l will
irrrlrrt'c irccc:lcnrtions such that-if a criterion of the type proposed in [15-6] is
rrst'tl o ) o {', whcrc o is the spatially averaged rms value of the top floor
;rt't'clt'lrlions ancl o,' is thc critical value of o specificd by thc cornf<lrt critcria
(c.g., irr f l-5-(rl, lt" : O.5ol,g). For cstirnatcs ol'huiklirrg irct'clcrltions, scc
rrlso ('lrirplcrr ().
FIGUR.E
l5.l.l.
Wind speeds inducing critical building accelerations.
15.1.4 Frequencies of Occurrence of Winds lnducing Critical
Accelerations
The second step in verifying the adequacy of a design from a serviceability
viewpoint is to estimate the frequency of occurrence of accelerations o higher
than the critical value o* specified by the comfort criteria. As shown in
[5-6], it is reasonable to define this frequency as the mean number per year
Ns(o > o*) of storns causing accelerations o > o*.It is acceptable, in
practice, to approximate Ns(o > o*) by the number of days per year Np(o >
o*) during which the maximum wind speeds exceed the values corresponding
to the curve of Fig. 15.1.1. It may be argued that, for office buildings, high
speeds occurring at night should not be counted in estimating the mean frequcncy Np. However, in view of the many uncertainties inherent in the design
firr building serviceability, such refinements do not appear to be warranted even
lhough they might reduce Np by a factor of the order of two.
'l'hc number of clays pcr ycar Nzr(o > o*) during which wind vclocities
c:xccctl ccrlain spccifictl virlrrt's (that is, the valucs dcfinccl hy thc curvc <ll'Fig.
l-5.1"l) cirn hc ohtirirtctl n';rrlily l.nrrrr l.ocal Climatological I)ttrr (1.('l)) she:cts
516
wtNt) tNt)(,(;t
t"
t) t)ll;c()Ml ()nl lN nul l nll()llNl) llt,ll l)lN(ili
lilr thc wcathcr slatitln closcst trl lltc klt'irliorl itt clucstitltr (scc Sccts. -1' l' ]'4'
ancl 8.3.1). 'Ihe LCD contain daily rccords ol'thc fastcst-ruilc or pcrtk gtrsl
spocds ancl of the corresponding wincl directions. To usc thc inlilrrrr:rtiotr obtiiincd 1'rom the LCD in conjunction with Fig. 15. 1 . l, proper adjustmcnts ttrust
bc rnacle to account for anemometer elevation, roughness of terrain, and av-
oraging of the wind speed with respect to time, as shown in Sect. 3.1.
ihe estimated mean yearly frequency Np(o > o*) must be compared with
the maximum acceptable annual frequency of occurrence of accelerations o >
ox specified by the comfort criteria. Let this frequency be denoted by N,a(o >
a*) ie.g., the value of N7(o > d*) proposed in [15-6] is 1/6 peryear)' If N2
< Nr,1he design is regarded as adequate from a serviceability viewpoint.
15.2
COMFORT CRITERIA FOR PEDESTRIAN AREAS WITHIN A
BUILT ENVIRONMENT
(.()Ml ()l ll (,lil llllln
is
_not
new
,ll llNVlll(}NMI
Nl
5lI
and thc wind spccds causing thcrn ancl (2) that maximum acceptable frequencies
ol'occurrence be specified fbr thesc wind speeds. The present section is devoted
to a brief discussion of these two requirements.
15.2.1 Wind Speeds and Pedestrian Discomfort
Let V denote the mean wind speed measured at approximately 2 m above
ground and averaged over 10 min to t hr. Observations of wind effects on
people and calculations involving the rate of working against the wind suggest
that the following degrees
of discomfort are induced by various
speeds Z
t5-111:
(see
Fig.15.2.1andp'188).However,inrecentyearsnewtypesofbuildingand
op""n rpu"" configurations have evolved. These may exhibit under certain unfavorable conditions zones of intense surface winds causing unacceptable dis-
ll n:;Wl llllNAlll
t'orrrlirrl lo rrst'rs ol ;x'rlt'sl r;ur iurits.'l'ypit':rllyr'suclr t'ottligttt':tliotts ittvolvc
tall ltuiltlirrgs lisirrg wcll :rlxrvt'llrt'srrrnrrrrttlittg lrrrill eltvilorttttcttt atttl atli:rccnt
lo opcn riplccr; sut'lr rrs l)lirzirsi ()r'rrrrrlls. As irtrlicrrtctl prcviously, t<t dclinc thc
nolion ol'urracccptablc tliseorrrlirrl (luan(i(rrtivcly it is rcquircd (l) that a cttrrcspondcncc bc cstablishctl bc(wccn vlrrious dcgrccs of'pcdestrian disc<lnrlirfl
ll5-10,
The problem of wind-induced discomfort in pedestrian areas
l{'l I l'l lrl :,llllnNnl
:
:
V:
mls
m/s
mls
V
5
tr/
10
2O
onset of discomfort
definitely unpleasant
dangerous
A more detailed description of effects of winds of various intensities (as defined
by the classical Beaufort scale) is presented in Table 15.2.1t15-101. Tentative
information on comfort of strolling pedestrians under various sun exposure,
ambient temperature, clothing, and wind speed conditions is provided in
t1s- l 21.
Experiments reported in [15-13] and [15-14] suggest that pedestrian comfort
is a function not only of the mean speed /, but of wind gustiness as well. It
is therefore reasonable, in principle, to study wind effects on people in terms
of an effective wind speed Z" defined as follows:
I
V':Vll+k-l
I
d2t/2
v
|
(15.2.t)
I
where V is the mean speed, ,121/2 i" the rms of longitudinal velocity fluctuations, and k is a constant reflecting the degree to which the effects of the
fluctuations are significant. According to the results of[15-13] and [15-14], an
appropriate value for this constant is ft = 3.0. However, other investigators
use the value k : I 5 ll.5-l-51 or k : 1.0 tl5-161. According to [15-14] wind
tunncl cxperiments anrl obscrvrtlirlns of pcdestrian performance suggest the
lirllowing crlrrcsp<lnclcttt't'betwt't:rt spcotls Z'' (with k : 3.0) and various degrccs ol' discorllil11.
,l:ull,lr,h,i\
Lll'l.l,
ll"'lrr.
J' l! \
F.I(;URFI 15.2. l. Thc Gust. Lithograph by Marlct, collccliort ol llrc llibliothi:c1trc tlc
la Villo tlc l'itt'is (pltoto llogcr Viollct, I'irris)'
rllrrl rtol cxr'lrrsivr'ly; st'r' I l'r l(rl
'l'Alll
t
,l,l
15.2.1. SumrnarY ol'Wintl lilli'cls
lirlt
Dcscription ol'
ttlrt:t
Wind
llcrrrr
N tt
ll,:' (l()Ml {)lll (:lll llllln l()l I I'l l)l :;tltlnN
wlNt)tNl)(,ol t) t)t1ic()Ml ()l ll lN nNl r nll(,llNl) llt,ll l)lN(;l;
518
o
I
J
t
ll5-l([
Spced (m/s)
Less than 0.4
Calnr
Description ol' Wintl lrllccts
No noticeablc wind
0.4-1.5
Mrxlct':tlc brccze
5.5-7.9
No noticeable wind
Wind felt on face
Wind extends light flag
Hair is disturbed
Clothing flaps
Wind raises dust, dry soil,
8.0-10.7
Hair disarranged
Force of wind felt on bodY
3.4,5.4
and loose paper
lire slr btccze
Drifting snow becomes
airbome
Limit of agreeable wind on
Strong breeze
10.8-13.8
land
Umbrellas used with
difficulty
Hair blown straight
Difficulty to walk steadily
Wind noise on ears
unpleasant
Moderate gale
13.9-11 .r
Fresh gale
17.2-20.7
Strong gale
20.8-24.4
Windbome snow above head
height (blizzard)
Inconvenience felt when
walking
Generally impedes progress
Great difficulty with balance
in
V":
V":
V"V', :
6 m/s
9 m/s
15 m/s
20 m/s
ll Ati wt lillt'l A ilUil I til\/iltrrt.tMt ilt
gusts
People blown over bY gusts
onset of discomfort
performance affected
control of walking affected
dangerous
of pedestrian perfolrnance in a large wind tunnel
building, conducted in Japan on over 2000
high-rise
ol'
a
birsc
Irntl :rt tlic
of the following proposed criteria:
dcvelopment
the
lctl
to
lravc
pctlcstriirns,
Srrltsetlrrcttl obscrvations
Ir.5-141.
Thc ability ol'pcdcslriarts to ittlittsl l() slrl)nll wirrtls is lrllt't'letl lrtlvt'r'st'ly il
the exposure to such wincls is lclirlivcly stttklt'n, lrs is llrc t'lrsc in zolrr's willr
flows that are highly nonunilonn irr s1.xrcc. lt is thcrclirrc rxrtctl in ll5 l.rll tlrirt
if the mean speed varies t>y 7O%, ovor a distaucc ol' lcss thirrr 2 rrr or so, lltc
effects of wind on people are more severe than suggcstod abovc.
Measurements of wind drag on people are reported in [5-291.
15.2.2 Comfort Criteria
Comfort criteria were previously defined as statements specifying maximum
of occurrence for various degrees of discomfort. The
following simple criterion based on extensive experience with the study of
ground level wind effects in built environments is suggested in [15-11]. Complaints about wind conditions are not likely to arise if, in pedestrian areas,
winds with mean speeds V > 5 m/s are estimated to occur less than 1O% of
the time. Complaints might arise if such speeds are estimated to occur between
lO% and2O% of the time. Estimated frequencies higher than20% coffespond
broadly to situations where in existing shopping centers remedial action had to
be taken to reduce wind speeds.
More detailed comfort criteria reflecting individual opinions on acceptable
frequencies of occurrence of various wind speeds have been proposed in
[5-151, t15-18], and [15-19]. An example of such criteria is given in Table
rs.2.2l15-181.
The first criterion in Table 15.2.2 is roughly equivalent to the criterion
previously quoted of [5-11]. The limiting gust speed of 25 m/s corresponds
to winds that could knock a frail person to the ground 115-191. Otherwise, as
indicated in [15-18], the values of Table 15.2.2 are subjective and have been
arrived at in the absence of reliable data.
acceptable frequencies
TABLE 15.2.2. Comfort Criteria for Various Pedestrian Areas
Area
Description
Criterion
Limiting Wind
Frequency of
Speed
Occurrence
Plazas and Parks
Occasional gusts to
about 6 m/s
Walkways and other areas Occasional gusts to
subject to pedestrian
about 12 m/s
l0% of the time or
pcrfbrmance not all'cctctl
All of
pcr'lirnnant:t: rrll'cctcxl
pcrlilrlnanc:o scrirrtrsly lrlli't.lctl
All
porlilrtttltttcc vct'y st'tiottsly lrllct'tcrl
abovc
ol-abovcr
I
or 2 times per month or
about 50 h/yr
Occasional gusts l() Alxrrrl .5 lr/yr
ahout 20 rn/s
( )t'r'irsiortal gusls l() Less llrirrr I lr/yr
:rlxrrrl lJ5 rrr/s
about
1000 h/yr
access
/, < -5 rrr/s
5 rrr/s ( /1 < l0 rn/s
l() rrr/s < V| < 1.5 Irr/s
l5 rrr/s .- /r
5lq
whe:rcr /r is tlrt' wirrtl spt:t'tl trvcttrgctl ovt'r' 1 s ll5 I /1. A:, rrolt.rl rrr I l'r I /1.
tltcsc crilcrirt ittc cc;uivlrlLrnt Io rtr rturrgitlrlly trlrt't',sr'v('r('llr;rrr llto:;r'rrl
_
Light airs
l.ight brccze
(lcntlc breeze
1.6-3.3
nt
52O
r,,
wtNl) tNI)t,ot l) l)llio()Ml ()lll lN nNl ) /\ll()t,Nl) ltl,ll l)lN(il;
r
.'()Nr :;
()t ilt(ilt
:i(,1il
A(;t
wtNt
)lt wt ililN A Iil.Jil I t
NViltoNMLNt 521
As slurwl in Scct. l-5.4, thc culcrrllrlccl licrlucncy ol'()ccttrrL:llcc: ol witttl
sltcccls in pcdcstrian arcas dcpends vcry stK)ngly up<ln thc cstilnation pnrcctlttrc
bcingusctl. ltisnotedthatthecomfbrtcritcriaof [5-lll-andsirnilarcrilct'il
srrggcstccl by other authors-are applicable only if the wind speed I'rcqucne ics
iuc cslilnatccl by the simplified procedure of Sect. 15.4. These critcria arc rro
krngcr applicablc if the detailed procedure of Sect. 15.4 is used.
ln thc abscncc ol established criteria, decisions regarding the acceptability
ol'corrrlirr1 conclitions in a pedestrian area are left, in practice, to the judgmcnl
ol'llrr: sitr: ()wncrs ll-5-201.
15.3
ZONES OF HIGH SURFACE WINDS WITHIN
A
BUILT
ENVIRONMENT
15.3.1 Wind Flow near Tall Buildings
As rlrtctl irr ll-5-l ll, high wind speeds occurring
at pedestrian level around tall
llriltlings arc in gcneral associated with the following types of flow:
l.
2.
l'I(;URE 15.3.1. Wind flow in front of a tall building (wind blowing from left to
right.). *
Vortcx flows that develop nearthe ground, as shown in Fig' l5'3'1'
I)csccnding air flows passing around windward corners, as shown in Fig.
15.3.2.
flows through ground floor openings connecting the windward to the
lccward side of a building (Fig. 1 5. 3.2) or cross-flows from the windward
sidc of one building to the leeward side of a neighboring building.
3. Air
'l'hc flow visualization in Figs. 15.3.1 and 15.3.2 was obtained by injecting
srrurkc in the airstream. It is seen that the flow pattems in the immediate vicinity
ol' thc windward face are consistent with the pressure distributions shown on
tlrc winclward face in Fig. 4.6.7b (i.e., the air flows from zones of high to
zoncs ol' krw pressures). Part of the air deflected downward by the building
liy'rrrs ir voncx (Fig. 15.3.1) and thus sweeps the ground in a reverse flow (area
around
..1 , rrlrrkerl "vorlcx flow" in Fig. 15.3.3). Another part is accelerated
near
jets
ground
that sweep the
tlrt. lruiklilg c()ntcrs (Fig. 15.3.2) and forms
If
an
15.3.3).
Fig.
tlrt. lrrrrIirr1i sitlt:s (irrcas B, marked "corner streams" in
the
or
near
present
at
()lx.nlnll r'orrrcctirrg tlrc winclward to the leeward side is
from the zone of rela1ir,,rrrr,l lcvcl, put-l ol'fhc dcsccnding air will be sucked
tively higlr prcssurcs <ln thc windward side into the zone of relatively low
will thus
1r,"rir,.", (suctions) on the leeward side (Fig. 15.3.2). A through-flow
caused
have
type
of
this
swccp the area C shown in Fig. 15.3.3. Through-flows
in
Cambridgc,
Building
serious discomfort to users of the MIT Earth Sciences
Massachusetts, a structure about 20 stories high [15-21.|. Cross-flows bctwccn
pairs of buildings are caused by similar pressure differcnccs, its slrown in Fig.
15.3.4.
The pattern of thc surfircc wincl flow within 11 5llg 1ls:l.tctttls itt tt t'olttltlt:x wlty
lfl(;URE 15.3.2. Wind lkrw
lirrrr lcli to right).
rlrillrrr.es
1.5..1.
I lhn)ullll
15.
t
I'1.
llriltlirrg llcst'irrt lt Iislirhlislutrcnl
liorrcry ()llitc.
rte:rr
lltc wintlward facc of a tall building (wind blowing
l.r l
llh
-),1.;rrrrl
l\ l.)\torrlrilrrrletl lrylx'r'rrrissiorrol llrt'l)rrr.tlrrr
('olr\'rlllrt, ('orrlrollcl ol llcl llriltrrtrrit M:rjcsly's St;r
522
*
wtNt)tNt)tt(;l l) l)lt;(i()Ml ()l il lN Aul) nl r{,{,Nl } lrt,ll l)lN(i:;
ll'lt
/'()Nl 1,1,1 lll(ill :itllil n(;l WlNl):; Wl llllN n ltlJll I lNVlll()NMl Nl
523
tlrc rclaiiv(: loclr(itlrr, lltc: clitttclnsirttts, tltt: sltitl)t:s, ittttl t't:rlltilt rll tltc ltt'
chiir:c(rrnrl lbl(uros (c.g., glorrrrtl lkxrr opr:rtittgs) ol' tlte: lttriltlirrgs ittvolvctl,
upon thc K)ughncss and thc lopoglirlllriclrl lclrtttl'cs ol'tltt: lcrrltitt:rtrrtttttl lltt:
sitc, and upon the possiblc prcscnco ncul lltr: silc ol'ottt: rlr scvtritl tlrll lrttiltlirrgs.
'fo study the surface wind llow in any givcrr brrilt cttvinrrrrtcttl , it is thctclirrt'
necessary, in general, to conduct wintl turrncl tcsls. Ncvcrtltclcss, its irrtlit'itlt:tl
in [5-ll], experience has shown that inlirnlation hirsctl ott ircrotlyttitlttic sltttlies of the basic reference case represented in Fig. 15.3.3 is uscl'ul lor thc
prediction of surface winds in a wide range of practical situations. Such infbrmation is presented in [15-11] and will be summarized below. Its range of
applicability includes built environments that retain a basic similarity with the
configuration shown in Fig. 15.3.3 and in which the height of the buildings
does not exceed 100 m or so. Detailed information on the wind environment
around single buildings and around groups ofbuildings is presented in [15-30].
rrgxrn
streams
Vortex
f
low
Through-
htr
15.3.2 Wind Speeds at Pedestrian Level in a Basic Reference
Case
[1s-11]
lil(llJltl,l
15.-].-1. Regions of high surface wind speeds around a tall building (after
lr.5 r rl).
Surface winds around models of the tall building shown in Fig. 15.3.3 were
measured in wind tunnel tests conducted at a 1/120 scale. The roughness conditions simulated in the tests were typical of a suburban environment, the mean
wind profile being given, approximately, by a power law with exponent cv :
0.28. The surface winds depend upon the dimensions H, W, L, and h defined
in Fig. 15.3.3 and are expressed in terms of ratios VlVs, where V an:d V, are
mean speeds at pedestrian level and at elevation Il, respectively. In certain
applications it is useful to estimate the ratio VlVs, where Iz0 is the mean speed
at l0 m above ground in open terrain. The ratios VlVs can be obtained as
rl
Main wind direct
fbllows:
V
(rs.3.l)
va vo
vo-VVH
Approximate ratios V1/V1, corresponding to the experimental conditions reported in [15-11] are given in Table 15.3.1 for various heights I1.
In the material that follows, the wind direction is assumed to be normal to
the building face (angle 0 : 0') unless otherwise stated.
Speeds in Vortex Ftow. Vo and V11 denote the maximum mean wind speed
at pedestrian level in zone A of Fig. 15.3.3 and the mean wind speed at
c
o
.F
o
*
!
c
t
'l'Alll,E 15.3.1. Approximatc Ratios
o
II
(nr)
_9
V,,
Itl(,llll{l,l 15.-1.4. ('lrss lkrw lrt'lwt't'rr lwo llrll lrtrrlrlirrl'. (.rltlr ll5
I
ll)
vu
20
0.7.r
30
0.1{2
,10
VrrlV,)
50
[5-lll
60
70
I O,l
1{( )
I
Oli
(x)
t(x)
tlt
lt,t
.r
15.3 loNl
WIND.INDUCED DISCOMFORI tN AND AnOUNt) DUil t)tNGS
524
o.7
o.1
0.6
0.6
0.5
0.5
0.4
o.4
>< 0.3
\< 0.3
o.2
o.2
0.1
L/H =
8> H/h>
O.25
2
L=O.4m
h=0.1
0=o"
L/H =
8> H/h>
1.0
|l=
m
O.4 m
h=O.1
0 =0"
m
1.0
2
0 0.1 0.2 0.3 0.40.50.60.7 0.8
H (m)
h)
0
0.5
525
lt=O.4m
W=O.4m
0.1
0
li ()l lll(lll frulltAct, wtNt)ti wt ililN A iltilt I I NVItoNMfNr
1.5
w/H
o.7
0.5
0.6
o.4
o
o
03
F1
a
$
s
05
s
04
S= 0.2
L/H =
4
0.3
o.2
H=O.4m
L/H =
h=0.1
0 =0"
8> H/h>
0
O.5
2
0_5
0
1.0
Il-O.4m
L=O.4m
2.O
2
0.1
m
W=0.4m
L=O.4m
1
0
0.1 O.2 0.3 O.4 0.s
w/H
0.1
0
> H/h>
o
0
W (m)
1.5
2.O
0.1 0.2 0.3 0.4
h
(c)
1.0
=O"
u)
FIGLIRE 15.3.6. Examples of the variation of vn with individual parameters tl5-111.
w/H
FIGURE 15.3.5. Ratios VnlVo [15-11].
elevation Il, respectively. Approximate ratios VAIYH are given in Fig. 15.3.5
as functions of WlH for various ratios LIH and for the ranges of values Hlh
shown. The height ft corresponded in all the model tests to typical heights of
suburban buildings (7 m to 16 m). It is noted that as the building becomes
more slender (as the ratio WIH becomes lower) the ratio VAIVH decreases.
Typical examples of the variation of Vn with individual variables are shown
in Fig. 15.3.6. If the distance Z between the low-rise and high-rise building is
small, the vortex cannot penetrate effectively between the buildings and Z7 is
small (Fig. 15.3.6h).If I is very large or if h is very small, the vortex that
Ionrrs upwind ol the tall building will be poorly organized and weak; Vn will
thcrclbre be relatively low (Figs. 15.3.6b and 15.3.6d).It h approaches the
value of H, the taller building will in effect be sheltered and the speed Z,a will
thus be low.
It is noted that the ratio
situations.
Speeds
s
0.6
\Q
-
0.5
VllVn is of the order of 0.5 for a range of practical
in Corner Streams. Figure 15.3.7 shows the approximate
0.5
(m)
w/H > o.5
L/H4*
depen-
dence upon Hlh of the ratio VBIVH, where VB and Vn denote the maximum
mean speed at pedestrian level in the zones swept by thc corncr strcams and
the mean speed at elevation //, respectively. A typical cxatttplc ol'thc vlriirtion
4lt67
Ir'l(llfllfrl
il/
h,
1.5..1.7. lldtios V,/V,, I l5-l I l.
il
'I
wtNl) lNl)(,()l D t)ll;coMl ()lll lN ANI ) nl l()ttNt) lll'll l)lN(lt;
526
llr il
:'( )Nl !: ( )l
ilt(
;il : iUnt Aot
wtNt
)ii wl il llN A ilt It I t NVil
)NMt
t(
N
527
t
ti
6
5
5
34
E4
53
L =O.4m
h =0.1 m
0 =0'
r*,
W=0.4m
,t
1
0
0
o
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
o.2 0.3
h
0.4
0.5
Wind direc t ron
t_)
J
(m)
i:I
'.,
4l
H=0.4m
W=0.4m
',:l
h =0m
l,I
0.1
0
o.2 0.3
w
0.4
-90 -45
0.5
0
45
Wind angle (0')
(m)
o.2
6
5
1i^
0.3
E3
,_e
FIGURE 15.3.9. Surface wind speed field in a corner stream
2
m
[5-ll].
1
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
L
(m)
FIGURE 15.3.8. Examples of the variation of
vB
with individual parameters [15-11]'
of /3 with the variables H, L, W, and ir is given in Fig' 15'3'8'
issecntodependweaklyupontheangle0betweenthemeanwinddirection
of the corner
and thc normal to ttre uuilding face. However, the orientation
may depend
ZB
speed
maximum
of
point
of
the
strcarns and, hence, the positioi
wind'
mean
of
the
0
signilicarrtly upon the clirection
of a wide building
lnlirrrrration on thc wind speed field around the corner
:
Fig' 15'3'9' The
in
given
is
:0.4
m)
O'3
L
m,
rrrrxlcl (// - 0.4 n, W
building corner
wintl spcccl clccrcases rather slowly within a distance from the
as in Fig'
defined
Y
is
where
Yl(Dl2),
iatio
The
H.
cqual, apprcrximately, to
j.:.q ana D is the building depth, provides an approximate measure of the
f
for various values
position of the comer streani. Mlasuied values of this ratio
that the points of
of 11and of wl(Dl\) are shown in Fig. 15.3.10. It is seen y:
constant x
Fig. 15.3.10 are fit reasonably well by a curve of the form
:
rt'. p* example, if W :4j m and D : 15 m' then Wl(Dlz\ 6' Yl(Dl2)
The speed Z6
= o.a 6ig.
ol'tho builcling
15.^3.10), and the maximum speed on thc ccn(crlirrc:
: 6 m'
would occur at Y = 0.8 x Dl2
lirr
Itisn<rlocl thattlrc r.lioV1,lV1,is<ll'lhctlrclcrtll'0'()5
sil r url irttts.
illlllll''('ol
pritctic:ltl
Speeds in a Through-Flow. Let Vg and Vo denote the maximum mean wind
spced through a ground floor passageway connecting the windward to the leeward side of a building and the mean wind speed at elevation F1, respectively.
f rigure 15.3.11 shows the approximate dependence of the ratio V7IVH upon the
lrrrameter Hlh as determined in [5-11] by semiempirical formulas and wind
Iunnel measurements. Examples of the variation
r'oO.8
m
ro.4
m
of Vrwith H,
W,
I .n^ nl
I oo.a *
\.0.2.
1
f = constant
X
W2
0
wr\
Itl(lllRl,l l5..l.lll. l;rrrlririt:rl r'rrlvc l/X
vc:rsus l,//X
ll-5-lll.
L, h,
and 0
528
I6,3
WIND.INDUCED DISCOMFORT IN AND AROUND BUILDINGS
ZONES OF HIOH EURFACE WINDS WITHIN
A EUILT
ENVIRONMENT
t2c
I
7
1.4
6
1.3
F
i
€4
1.2
1".0.3m
14 -O.4m
/r -0.1 m
0 -0"
>Q3
2
1.1
I
1.0
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.9
*
!t
H
(m)
(a)
0.8
0.7
8
7
0.6
6
0.5
G
5
5
i4
0.4
I =0.3
=O.4
,r = 0.1
0 =0'
>o3
0.3
H
2
E
m
m
m
1
o.2
w/H
>
o.5
0.1
0.1
H =0.4m
w =0.4m
h =O.1 m
0 =0o
3
2
1
o
O.1<I/H<-
.c)
4
o.2
0
0.3
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
t
t7 (m)
0
(b)
456
(m)
(c)
H/h
FIGURE 15.3.f 1. Ratios V.IV, [15-l
l].
I=0.3m
15.3.3 Wlnd Tunnel and Full-Scale Measurements of Surface Winds:
Case Studies*
Case 1. Oftlce Building (H - 31 m) Spanning a Shopping Center
: 4.4, WIH : 1.6, and LIH =
115-111. A 31-m tall building for which Hlh
*Thc sourcc ol thc material is indicatcd by reference numbers in euch cuse, For ndditional casc
studics, sec ll5-311,
7
H=O.4m
are given in Fig. 15.3.12. It is seen in Fig. I5.3.12b that for WIH < 0.5 the
ratios VglVn are lowerthan in Fig. 15.3.11. Figure 15.3.12e shows forvarious
values of d the range of variation of Z6' with opening width.
Thc graphs of Figs. 15 .3 . 1 1 and 15 .3 .12 are based on measurements in and
neur passageways with sharp-edged entrances. If the edges of the entrance are
nrunded to form a bellmouth shape, the speeds Vg can be reduced with respect
to thosc ol'Figs. 15.3.11 and 15.3.12 by as much as25% or so [15-11].
It is noted that the rat\o Vs lV H is of the order of I .2 for a range of practical
situutions.
w=o.4m
6
0=O"
E
5
Qq
E
\o3
le
2
,l
0.1
FIGURE 15.3.12.
I
rs-1
11.
o.2
0
0.3
-90
0
-45
n (m)
Wind angle (0')
u)
(e)
Examples
0f thc
vari&tion
oi /,. with individual
45
90
parameters
wtNll lNIlr,lcLlJ Dlscc)Ml
olll
lN ANI) nll(xlND BtllLDlN(ls
---+->N
c
trO
t.F
q
9
rn E
;O:L--=
i.9
\-
.-r
o-c
UF
a
I
\q
5.
-.
O\ !9E
C Cr ',tr
/Ec
O
q
trJ
l3
r)F-\O
$Nca
U9UUU-
-O\.a.aO
ca\olr)
o
r)cr.FooOO
a)
oo
q
a)
t^
Sl-v
'-:
ncln
^
co
O\
o\
O\
nnn
nn
OOO
c.l c.l
c.)
I
!o€
\-
Model of a Building in Utrecht, Netherlands [15-11]- A proposed
80-m tall building with width W : 50 m, depth D : 22 m, and for which
Hlh : 8.0, WIH : 0.63, and LIH : 0.5, is shown in plan in Fig. 15.3.14Contours of ratios VlVs, shown in Fig. 15.3.14 for south and lor north winds'
were obtaincd in [-5-lll using wind tunncl data rcporlocl in ll5-221. Moasurcd
rali<ts VnlV11 and VlVllurc about 0.65 (at lhc ccnlor li111r ol'llrc lrttiltlirrg) itntl
2.
n
;6:ii
\ @
O\
': nac'l
\
UF
O\
ioocn 6l
\
Cq C.t \o
\\q
c)
bo
d
r
o
o0
F
coo6.
-:01 'q
n-cl
99999U)a-
Qca!
Ni\O
d a.l c..l
cq:a3,
c.l O t'-
CO\a.i
O cq O
n':-
^t\OO.
99999999U
-:!n nn\
tn-*
o\cai
9VV99!rJUU
-nn
119
OOt"-
V-rco@
r]|.)F-
C-r)\O
co*,- ^'-:
ueu
*F-O
\oicq
-i,^^'
99V999!'9*\n\O
\OOr€
:v1h
.- .- -J
uJlJ
NciO
(\cI\oo
,^.^^'
cir)ca
C\F-oO
r)\o0o
:
aOO
r)\Ocn
oooo-:;
O\-ra)
*Nbo
C)
>
qaa
q)
@
C)
0)
a
q?
m
\oooc!
uuu
o
.o
\nc..!
9UUU_U
v?clq
(-::-=
t\
rn
rn
rc
FJ
Fr
cd
\Oca\O
n1UUU
tr:\9
\
lr)
-.O-
.JC€
c.i
*\o-
cj n09
clqq
999UUaa-i
grxrd.
Case
ca
q \\q
()
c.)
0.85 is shown in plan in Fig. 15.3.13. Full-scale measurements of ground level
speeds (;; at locations i : 1,2, ... ,9 (see Fig. 15.3.13) and of the speeds
236 measured at location l0 at a 36-m elevation above ground were made on
tcn occasions. The results of the ten SetS of measurements are expressed in
Table 15.3.2 in terms of ratios V11y'\6. Also shown in Table 15.3.2 are avcrages of the measured ratios V1iy'V36 for west winds (measurement sets a
through h) and for east winds [measurement sets j and k]. These averages were
28
multiplied by the factor (36/31)0
- 1.04 to yield approximate ratios V()IVH,
whcrc V,,is the mean speed at elevation H : 3l m.
It is notccl that the measured ratios V1i1lV36 vary in certain cases considerably
l'nrrrr rncasurcmcnt to measurement (e.g., V6y'V36: 1.33 and 0.56 for measurcnrcltt scts c and f, respectively). No explanation is offered for these variirlirrns. ljor purposcs ol'comparison, Table 15.3.2 also includes predicted ratios
VAlVy, V4lVrr, anrl V1.lVs based on Figs. 15.3.5,15.3.7, and 15.3.11, respoctivcly. Thc agrccment with the average measured values is seen to be fairly
-ot\o
O\co-
\OO r)
<'O\ O
cln
OO 999U4
cr
,5
.{!\oE
o\ Srot^
H
FIGUR-E 15.3.13. Plan view, case study 1'
6l lr) c.)
OOOO
rrr a-
\o6lo
I
.-l
ca
9
qq\
n
F9
oo
vl
\
\qq
n-$
-q\q
-:oqv? OO+oo
aaO
OOO
co
O
o
cvo
(!
?
J
'-
rl
r.r
=l- 'ar
ll:l
l.rOt'J=
u96
^Q9.;
avts
53t
532
wtNt ) tNl
)1,(
:l l) l)l:;(;( )Ml
( )l
ll lll
n I t( )t
tNt
l', r ,'ot It , ot ilii.i! ',t,ltl n(.1
) lil,ll l)lN( i:;
v/vH = o.15
/o.a5\\
,
Wt
ililIl /\ iil,[ I til\/!ltol.JMt Nt
533
O.(X). tcspt't'ltvt'ly I'rt'tltr'lt'rl nrlios l',l l'1, ;rntl 1,,/1,, lr;r:;t'tl orr Iiigs. 15.-1.-5
irtl(l l-5..1.'/ lue :rl)()u( O.(rO lrrrrl l-(X). n'sllr't lrvt'ly. Ilrc :rgrccrrrcnt l)clwccu
prctliclctl iut(l ln(:asur'(:tl vrrlrrcs is sr't'rr lo lrt' rt'ltsorurlrly grxxl. ll is no(cd,
'Nt
I
0.8
Wtt'lt):
Itowcvcr, thal thc vorlcx llow is irsynun('lrt;rl rrrrrl t'orrlirins rcgions in which
llrc rati<rs VlV11 arc as high its 0.1{
\
\
\ \\\
----==-!--====-a_===-=_=
.r_)
Wind direction
)..
Case 3. Models of Place Desjardins, Montreal 115-231. Figure 15.3.15
slrows a model (l/400 scalc) ol'onc arnong several designs considered for a
tlcvelopment in Place Dcs.jardins, Montreal. The predominant wind direction,
tlctcmined from measurements at the top of a tall building near the site, is
slxrwn in Fig. 15.3.16. Wind tunnel tests were conducted for that direction
only. Surface flow patterns were observed by using thread tufts taped to the
-
\
l0
20
Scale
30
40
50
in meters
Wind direction
FIGURE 15.3.14. Plan vicw. clst:
2
l,'l(,llllll,l l-5.-l.l-5. l)llrtt'l)r':;;;ttrlttt:, tnotlt'l ((()url(:iV ol lltt'N:rliott:rl At'nlurrrticlrl lis
l:rlrlisluttr'ttl. Nlrlioturl ltt'sr';rrt lr ('orrrrr rl ol (':rrlrrl;r)
wtNt)tNt)t,ct t) t)t:ic()MI
534
()t
il tN nNIr
nt
t()t,Nt) tt(,ilt)tN{i:
r ill( ,l tl:uttl n(.1 wlNt): ;wililllt/\nUilI
1.66
I
L
t",1,
(Llil,)
{1 .1 5)
(46.57o)
Qz) ,rta
4.i5
535
llre llnllrt','l'ltt't1rr:rrrlilit's llrirl lue rrol lrt'lwt.t'rr lr;rt.rrllrt.:;t.s t'or.r.csl-lottcl to lnclt
stlrclllLlllts Iltlttlc ilt tlte: itbscltct: ol rr plrjt't lr'rl 'rO slory l()w('r'nclu-lhc s<luthwcsl
?.4.1
46.gyo
tl]vilt()l.lMl Nt
('()nlcr ol'thc tlcvclopluclrt. 'l'o irrvcslil',:tlr. tlrt't.llt.t.( ol tlrc tower upon lltc
strrlhcc winds, tncasulctllcnts w('rr lrlso rnlrtlt' willr (lrc rrrodcl of the towcr irr
lrllrcc. Results of thcso nrcirsllr.t:rn('nls :rn' sltown bctwccn parentheses in liig.
t"r,'t
.ut
5%
r-5.3. t6.
case
2.56
4. commerce court Plaza, Toronto lls-241. A l/400 scale morlc:l
;rnd a plan view
of t-he commerce court project in Toronto are shown in Figs.
l-5.3.17 and 15.3.18, respectively. surface flow patterns obtained by smokc
visualization are shown fbr two wind directions in Figs. 15.3.19 and 15.3.2o
l15-251 . Ratios VlVr, where V and V11 are mean wind speeds at2.7 m an<l
240 m above ground, were obtained from measurements in the wind tunnel
and, after the completion of the structures, on the actual site. The results of
llrc measurements are shown in Fig. 15.3.21 as functions of wind direction fbr
Iocations 1 through 7 (see Fig. 15.3.18). The agreement between wind tunnel
and full-scale values is seen to be generally acceptable, although differences
trl'the order of 3O%,50%, and even more can be noted in certain cases.
31.O%
(2.481
33.1%l
T-
(0
tra
L'
Case 5. Model of the DMA Tower, Paris 115-261. Models of the l2}-m
lall DMA tower and of adjacent projected structures are photographed in Fig.
3.11
41.4./.
(s.38)
134.O%t
k2.66
@
'.oo
5O.3o/"
@
t- t
>,.q
(3.82)
(36.'t%l
FIGURE 15.3.16. Wind speeds and turbulence intensities, place Desjardins [15-23]
(courtcsy
of the National Aeronautical Establishment, National
Research Council
of
Canarla).
nrotlcl surllccs, a w(x)l tuft c;n thc cnd of a hand-held rod, and a liquid mixture
ol'kcnrscrro-chalk (china clay) sprayed over the horizontal surfaces of the model.
As thc wind blows over thc model, the mixture is swept away from high speed
zones and accumulates in zones of stagnating flow. After the evaporation of
the kerosene, the white acc:umulations of chalk indicate zones of low speeds
while areas that are dark represent zones where surface winds are high. wind
speed measurements were made in these latter zones. The numbers givcn in
Fig. 15.3.16 represent ratios of mean wind speeds at the locations sh<lwn t<r
the mean speed at 1.8 m above ground at thc norlhwcs( c()n)cr ol'tlrc: tlcvclopment. The percentagcs of Fig. l-5.3.16 rcprcscnt tullrult'rrt't' irrlcnsilir:s, :yttl
the arrows show lltc: tlirccliorr ol'lhc wirrrl c()nrlx)n('nl llurl w:rs rrrt':rsrrn'tl by
Itl(l(lltl'l l-5.-1.17. ('otttttttttt ('rrttrl [\lrrkl ll'r '.1; (to11111'5y l]orrrul:uy l.:rycl-Wintl
lttttltt'l l.:tllotlrloly, 'l'lrt' lJrrrr,, r'.rlt ol \!r..,tr'rri ( tll.rttot
536
wtNI) tNt)t ,(:t
t) t)l;(;( )Mt ( )t lt tN
n
t,|l
r /\ltoLNt ) iltil t)tN(
i
l|):t .'()t.ll , {)t ilt(,il :il ,1il n(.1 wlNlt):; wt illlll n llUlt I lt{vilt()NMl NI
537
\
lltr
,tl
t
iii! tower
|*:-::
iii s+ stories
FIGURE 15.3.18. Plan view, Commerce Court. After N. Isyumov and A. G. Davenport, "comparison of Full-scale and wind runnel wind speed Measurements in
the Commerce Court Plaza,"
J. Ind. Aerodyn.,
I
(1975),201-212.
l5.3.22 against the background of the actual site. Let v" and v"H denote speeds
dcfined as in Eq. 15.2.1 with k : I and measured at2 m and l2O m above
ground, respectively. Ratios v"lv"H obtained in wind tunnel tests for the southwcsl wind direction are shown in Fig. 15.3.23. It is noted that for this direction
(lrr: highcst winds occur between the two curved buildings located northwest
trl llrc lrrwcr (circled value VnlV"11 : 1.08 in Fig. 15.3.23) rather than in the
inrrrrt'tliirtc vicinity of the tower itself. The increase of the wind speeds by the
t lr;urrrclirrg ol'thc flow between buildings forming an angle in plan is sometimes
rt'lt'rctl t() irs it Vcnluri el1-ect [15-16|.
15.3-4 lmprovement of Surface Wind Conditions
ll rrt r'r:r'tlrirr klcutions suface winds are judged to be too high and thus to cause
rrrr;rt't'ePtrrblc rliscornfirrl
to pedestrians, ways must be sought to imprclve cnvinrttttrcttlitl wincl conditions or otherwise protect pedestrians from unplcasant
witrtl clll'cts. lrr cctlltin cxtrcmc cases it may bc ncccssltry lo tlcsign builclirrgs
ol lowcl' hciglrl or ol-tlill'crcnl configuratitlns than wrrrr.. origirurlly irrtcnrlctl. Il'
possilrle, ()l)clt ltleirs sltottltl ltc so rlcsigrrctl :rs lo pn'vt'rrl 1x.rk'sll'iirrr lllrllic
Wind
-//
I"IGURE 15.3.19. Surface wind flow patem, commerce courr (easr wind)
[15-251
through high wind zones. Also, as suggested in [15-12], handrails should be
provided in potentially dangerous areas. In certain extreme cases it may be
nccessary to enclose windy areas frequently used for pedestrian traffic.
Local improvements of surface wind conditions can be achieved by providing (l) roofs over pedestrian areas and/or (2) solid or porous screens at suitable
locations. studies of sheltering effects due to screens are reported in
[15-271
rrnd [15-28]. However, no general design rules exist to date on the basis of
which sheltering effects could be predicted reliably within a built environment.
Also, as noted in tl5-121, solid screens merely deflect the wind from one
krcation to another so that the consequences of their use must be investigated
carcfully.
A f'ew case studies illustrating rcmcdial measures aimed at reducing pedestriitn level wind speccls ilr(: l)rui(l)tc:cl bclow.*
Case 1. Shopping Center, Croydon, Engtand 11S-ttl. Figurc t5.3.24 is
It vicw l'rot'tt thc wcsl ol :rtt ttllir'r'lruiltlirryi,7.5 rrr lirll,70 rrr witlc, irrrtl Ill rrr
tlccp:ttlitlittittg tt slrrt;lpinl'. (('rt( r /'r rrr lorrli A plrss:rll('wity l2 r1 Irilglr:r1tl \. /
r'lltcstttttr't'olllrctrr:rlcri:rlirrrrrlr,,ri,rll,q
1,l,rrrr,i rrrrrrrlr.r.,trrt.;rrlrr.:rst.
wtNt)tNl)l ,(;l l) l)l:;(;()Ml ()l ll lN nNI) nl l()llNl) ll(lll l)lN(i:;
o
c
co
tC=
;;
o5
lL
t.
F-
-t!
.o-a)
I
Za
o
91 iru
E:-*
,9c
!ts
<tT
6-
s 9R
a\
ht 5 -:
Fh
tsr-
1
t
\
^Ci
qJ>
t-9€
-:.;
o
c
^q
.9
o
o
J
b.
oL^
o
o
,a
>\>1
-
E E€
Q
J:6ru tr0)I
\)a
o-1.s
6
o\
\Ol
l'l
J
.
!.:-\
tri
-a'
9F
*r7
(JvN
a t=
FIGURE 15.3.20. Suriace wind flow pattern, Commerce Court (southwest wind)
tl
@ >95
!
aY\J
5-251.
;
m high connects the shopping center on the west side of the building to the
strect on the east side (Fig. 15.3.25). The shopping center was designed and
huilt without the curved roof over the shopping mall that can be seen in Fig.
15.3.24. Alicr the completion of the building complex, it became apparent that
rcnrcrliul rncasurcs wcre necessary to reduce wind speeds in the passageway
irrrtl in llrc shopping mall. The ground level wind flow was investigated in the
wirrtl tunncl, Iirst for thc complex as initially built (i.e., with the mall not
covcrcd) and thcn with various arrangements of roofs over the mall and of
screens within the passageway. Ratios VlVo measrtred in the wind tunnel (lz
and V, are the mean speeds at 1.8 m and15 m above ground, respectively)
are shown in Fig. 15.3.25 in three cases. For the complex as first built, the
highest values of the ratio VlVswere 0.68 in the vortex flow zone and l.0l in
the through-flow zone. The provision of a full roof over the mall but ol no
screens within the passageway reduced considerably pedestrian level spcctls
caused by west winds. However, with east winds, thc lklw wlts lritppctl ttntlcr
the roof ancl the wind spccds within thc rlall wcrc. lir lltis rcrtsott, higlr; rrs
d'
o
-H H"H
rPE
9 Ftr
'i:tr .)
>-YX
al\
-.
.-c
a{v tr
n)
rn*E F!
o
JY
c
.9
,^!2=
"')Oa
aYtsO
:
F
o
J
J
6lr>,<
o
=s
o
539
540
wtNt)tNl)U(;t t) l)t:i(i()Ml
(,1
il
ll\1
nl']lr nl r()l lNl) ltl lll lrlN(,:;
*
l', L'illll'.
0.35o
mcnt, Etablissement de Nantes).
Wt
ililN n tJlilt I tNVilt()NMl Nl
541
0.34 o
o:'
o:r
I,'IGURE 15.3.22. DMA Tower (courtesy Centre Scientifique et Technique du BAti-
o.44o0.96r0.89o.0.87 o 7j-l
0.74. ffi,..x o.zs
Fig. 15.3.25, the speeds were also high at the east entrance of the
passageway. A solid roof close to the tall building followed by a partial roof
D.M.A.
Tower
shown in
over the rest of the mall, and a screen obstructing 75% of the passageway area
rcsultcd in a significant reduction of surface winds, as shown in Fig. 15.3.25.
It is notcd that to protect the mall from strong vortex flows caused by west
wincls, thc solid roof had to extend for at least 18 m from the building face.
'l'hc solu(ion actually applied consisted of providing (l) a full roof over the
cntirc rrurll (l;ig" l-5.3.24) and (2) screens with75% blockage in the passagewiry.'l'his solulion proved elTective in ensuring a comfortable wind environ-
r)t iltriil ,t,ilt n(I WtNl)t;
0.43 o
.0.61
0.63.
.
0.62
FIGURE f5.3.23. Surface wind speeds near the DMA Tower
115-261.
of Place Desiardins, Montreal115-231. It is seen in Fig.
lirct<rr of about I .61 at rocation 10. However, with the tower not installed,
while the mean speeds were reduced by a factor of almost three at location g,
tlrc rcduction at location l0 was insignificant.
l6 that the ground level winds in the Place Desjardins mall (Fig. 15.3.15)
are relatively high: with the 5O-story tower southeast of the development not
installed, V$/V(o: 3.11 and Vosy'Vo, : 2.96: with the tower in place, V6,l
Vtrl : 3.38 and V11s:)lV(\ : 2.48. Wind tunnel measurements of pedestrian
level wind speeds are also reported in [15-23] for the casc in which thc mall
was covered. With the -5O-story tower in placc, thc cll'cc( ol'r'ovt'rittg thc rrrall
was to rcducc thc rncan wincl spcccls by ir luctor ol'livc:tl ltx':tliott tl rrrrtl l.ry a
3. commerce court Plaza, Toronto lls-ls\. After the completion
thc building corrrplc:x slrowrr in Fig. 15.3.18, conditions were found to be
Prrrtictrlarly annoying ott wintly tlrrys lor pcdcstrians walking from the relatively
Plrrlcctcrl zonc n<lt'(h ol'tlrt' ll st()ly lowcr into thc flow funneled through the
l)ilssllllcwlly 2.1. Wirttl lttttltt'l (t'sls irrrlit'rrtcrl llrlr( thc pnrvision of screens at
tllt'gtrrtllttl lcvt'l lts sltowtr rrr lir1l. l.5.l.l(xr woulrl n.sull :rl l<lcirtigrrs 2, -5, and
IilL:nt.
Case
15.3.
2.
Models
case
,l
542
wlNl) lNl)l l( )l l) l)ll;(
l(
)4,41
(
)ll
I lN
n
l.ll
)
n I
11
)l
,Nl) llllll l)lN( ;:;
l',.1
llll rll ll llr
ll
or ililr
'r r
/\:;nNl
wtNt):
i wt ililt.J n nl,il | I Nvlli()NMt Nt
built
053 057 065 068 065 049
With f ull roof and no screen
o 4s o 24 019 020 025 021
With partial roof and '15y" screen 023 o1j 019 028 023 023
As first
036
02a
019
543
012
032
040
N
As f irst built
026
With full roof and no screen
With partial roof and 75o/o screen
048
007 017 044 052 056 078
045 052 061 067 063 071
o17 011 023 043 047 053
021
10'l
088
059
FIGURE f5.3.25. Model test results, Croydon [15-l
ll.
of the velocity vector with speed Zu.
'l'he frequency of occurrence at the location concemed of wind speeds larger
lhan V, denoted by f', can be written approximately as
and let the angle 0 define the direction
FIGURE 15.3.24. Tall building and shopping center, Croydon [5-11].
n
(r in rcdLrctions of undesirable mean speeds of the order of 40%. However,
whilc cll'cctivc acrodynarnically this solution was rejected for architectural reasons. lr.rstcad, pottcd cvergreens about 3 m high were placed as shown in Fig.
15.3.26b.'l'his rcduced the mean winds by about2O% at location 2, l0% at
Iocation 5, and 33% at location 6.
15.4
FREQUENCIES OF OCCURRENCE OF UNPLEASANT WINDS
WITHIN A BUILT ENVIRONMENT
15.4.1 Detailed Estimation Procedure
LetVn(V,0) denotc thc wintl spccrls at l0 rn 11111;vs grtrrrrul irr o;x'tt lt'n:ritt tlutl
induce petlcslrilrn lcvt'l wintl spct'rls /rrl :r g.ivcrt lot';rtiort itt ;r Itttrll ('nvit()nnr('nl,
f': 2t?
(ls.4.l)
in which fv,o arc the frequencies of occurrence in open terrain of winds with
spccds larger than Vo(V, 0i) and the directions 0; - rln <0<0,+r/n,the
rrngle d; being defined as
0i
2ri
II
(i
1.2....,n)
(ts.4.2)
Irt ltractical applicatiorts lr l(r lxrirrl ('()nrl)irss ts t'onunonly rrsctl so that in Eqs.
I5.4.1 irnrl 15"4.2. tt
l(t
'l'rr rrltlitirt.lI it is lr('('('ss;ry, Iirsl, lo t':,lrrrr:rlt'llrt'v;rlrrt's rl l/,,(l/, //,). Iinrrrr
wilttl t'littlt(okrgit'lrl rllrt;r. i( is llrt'tr l)osstlrlr'lo t':,lrrrr;rlt'llrt'lrt't;ut'rrt'i.'s /li'.
544
wtND tNI)tJCt
t) t)t$(:()Mt ()ilt lN ANtr Alt(lt tNlt null l)tN(ili
It,
I lill {lilFll{'l|!r fil ltl ll,t tA!i^til Wlf.ll[; Wiililti A tilllt I tNVilt()NMt:Nt
545
:;tr\rt
f
lrc r1x'ctl J't(]', ll, ) t;rrr lrt. wnllcn
)'tlv,0i)
:ts
I
Vo\o,) ..
vtvtt(q')
w0)'
(
15.4.3)
llrt' r;rlios Vtl|i)lvil(?i) characterize the site from a micrometeorological standllrr standard nrughness conditions in open terrain, these ratios depend
rrurrr (lrr: clcvation Hand upon the roughness conditions upwind of the site, as
',lrrrwrr irr Sccts. 2.2 and 3.1. The ratios VIV{0;) at a given location are an
;rr'rrxlyrrarnic property of the wind environment and are estimated on the basis
ol wirrrl tunncl tests, as seen in Sect. 15.3 (e.g., Fig. 15.3.21).
A rrsclirl basis for the estimation of frequencies flo is provided by weather
rccords of wind speeds and directions, observed at three-hour intervals
"tirliorr
irrrrl puhlished in monthly Local Climatological Data sheets (see Sect. 3.1).
P,rrr(.
o
a
o
o
a
( '()rrrii(lcr, fbr example,
all the three-hour interval observations in a year (8 obs/
365 days : 292O obs), and assume that 58 out of these observations
rr'plcscnt NNW winds with speeds in excess of 6 m/s. The frequency of ocr'un('nco of such winds can then be estimated as follows:*
rlry X
Trees--->
"o
f?:
(b)
l''l(;uRIt 15.3.26.
Rcrnccriar nlc:lsures at Commerce Court: (a)
screens; (b)
trees. Aftcr
N. Isyumov and A. G. Davenport, "The Ground Level
wind Environment in Buirt,
fp.|1eas," in Proceedings of the Fourth International conference on wna E;ffects .tt
Buildings and structures, London, 1975, cambridge
Univ. press, camlridge, r97(r,
pp.403-422.
#=
(ts.4.4)
2%
It is desirable, in practice, to base frequency estimates on several years of
rlrtir. 'fhis is the case fortwo reasons. First, one yearof data might not reflect
tlrt'wind climate in a representative way. Second, the observations taken at
tlrn'e:-hour intervals are instantaneous values, which are sometimes lower,
:;rrilctimes higher than the mean speeds. The estimation error associated with
:,rrt'h differences is small if the sample size is large.
ln certain applications it may be of interest to estimate frequencies for inrlrvitlual seasons, or for a grouping of seasons (e.g., spring, summer, and fall).
lrr such cases the only data used to estimate wind frequencies are those that
t rvcr the season (or seasons) of interest. It is also noted that winds occurring,
s:r,y, fiom 1l p.m. to 5 a.m. are, in many cases, of little concern from the
r.lrrrrclpoint of pedestrian comfort. In estimating wind frequencies, midnight and
I rr.rn. observations can then be eliminated from the data set.
lrrfbrmation on frequencies of wind speeds at a weather station fvio may be
prrscnted either diagrammatically or in the form illustrated by Table 15.4.1.
An example is now prcscntcd of the calculation of frequencies /2. The
trrlculations are carried out lirr krcation 4 of Fig. 15.3.18 for which the plot
l/1V,, is given in Fig. 1.5.3.21. lt is irssurncd that the ratio VolV, = 1.5 and
tlurl thc wind climatc is tlcrsclibcrl by 'l'rblt: 1.5.4. l. The frequency.fv is soughl
r'l'he sttpcrscripl in lht'nolitliott
/l'tr'prcrrrrl: tlrt's1x'r'rl
l',,
s;xrtttlslttlltcvitlttci Iiltit l(r;rrtttl (orrll;riirrrrvlrrrlrllrt
sl:rr'(irrg lirrnr thc NNW rlircclrorr
(rn' l,r1 ll 'l
I
I
(r rtr/s, whilc llrc srrbst.r'ipl cortr.
irrrp,lr'//isrrrt'trsrrrt'tl
totrttlt'rtlotlwist
lr),1 llll ljl
o
<g
tr
o
q,)
o0
q)
xH
9z
2zE]z
Ott-61
cq-o
tIl
u)
gl
I
/aO
a
O\
,
@u)
qJ
CB
naa
\OOc7)
(.{*O
'
o
q)
a
VI
U)
.+>
ol
q
()
o)
L
tri
B
z
/z
z
tf,
lai
f-l
rla
1E
o
o
c)
o
,
I
.??n
o.t-Ol
..lci$
c.{io
r
r
I
a
nnqF-c..lOO
l.n
nnn\O.TOO
':nna
t
ll
(ls.4.s)
C!F-61
*-Ol
o
of direction (in the example of Table 15.4.1, these data are given in the last
column). It is noted in [15-11] that this simplified procedure, even though not
"exact," provides generally reliable indications on the serviceability of pedestrian areas in a built environment of the type represented in Fig. 15.3.3. It
rA
+o
Nrr
-F
(/)
c.;o
is emphasized, however, that the procedure can only be regarded as useful if
applied in conjunction with the comfort criteria proposed in [15-11] (see Sect.
.h F
u)
n'1
t5.2.2).
\OO
sF
9-:
caO
c^,
NT
,
c.;
the climatological information is concemed, the data needed are the frequencies
of occurrence of all winds with speeds in excess of various values Zs, regardless
*e
!
^lOO
o
coO
6A
I
coio
B
To illustrate the procedure proposed in [15-l l], consider the case of a building complex for which H : 70 rrr, W : 50 m, ,L : 35 m, and h : 10 m.*
From Figs. 15.3.5 and 15.3'7, VAIVH = 0'6 and VBIVH = o'95, whete V1
VB are the highest mean speeds in the vortex and in the corner flow,
^nd
rcspectively. For 11 : 70 m, VHIVyOO) = l.O4 (Table 15'3'1), so
o
I
c.i
z
na
a.t o
Yt =
vo
crt
cl -:
CiiOO
a
7.s
WVA;)
A simplified version of the procedure just presented is suggested in [15-11] for
built environments similar in configuration to the basic reference case (Fig.
15.3.3) dealt with in Sect. 15.3. In this version the aerodynamic information
used, rather than being a function of wind direction (as, e'g., in Fig. 15.3.21),
is limited to the results given in Figs. 15.3.5, 15.3.7, and 15.3. 1 1. The ratios
VnlV1lof mean wind at elevation F/in the built environment to mean wind at
l0 m above ground in open terrain may be taken from Table 15.3.1. As far as
io
q-:
El
O\ (,
(n
r
-ivln
oi
ot4
ra
I
0909.1
q
\\\!
:
'f'hc calculations are given in Table 15.4.2.
odd
r
r
c.{io
09 09
5 m/sec. Equation 15.4.3 can then
15.4.2 Simplified Estimation Procedure
-a r!
a'>
6
tr tr tro
co\ooo-
nnnn
d
Q
+
z
BS
c\l
.E
rc
o
t-\
xH rci*Ltr
ra
\a \o \o \o
o.oz
(15.4.6a)
h=t.m
(1s.4.6b)
vo
t\
t
F
546
Vo6.0i)
n
I
q\vl-:
\oF
(h \Oc.lOO
a
\oO
tr.l
cn '.r
r!
?.lc'l
6D
a
+
-z
2zr!
0)
9H
cdo
547
IJJ
-Oc.l
c.: ol o
-,
2zz
>
t NVlll(lNMl Nl
r
I
2zr!
6.1 .,r
:ll !'; lrl llNl'l IAUANT WlNl)S Wl tlllN A llt,ltl
bc wli(lcrn ls
d; o
s.l
,
-0OO.l
co-Ol
li
liqJ
3
t-- a.l $
c.;-id
+
az
lr.1
IJ.]
d
orono
\o c.l
I'll
lirr' ;rcrkrstriiut lcvcl wirrtls with spccds V
2z
r)
'l
V
o
t::
** lo-ri.
l-(
l\\
la
lt
-^.5
\u:
i
9
e
ru*
\
'
I
\
F.i t
'l'hc frequencies of wincls /2 ) 5 m/s and VB ) 5 m/s are now sought,
assuming that the wincl clittlttc is clcscribed by Table 15.4.1. It follows from
liq. 15.4.6a that, in onlcr lltitl V1 > 5 ,nls, Vo > 510.63 = 8 m/s. From
'l'ahlc 15.4.1, the I'rcqtrrrttt'y ol srrclr witttls is 5o/o. However, to speeds Z6 )
tlirr
lhcsc notalions, see
lrig, l5 t
I
548
wtNt ) tNl
|,
)ll(;l l) l)li;(l()Ml ()l ll lN nl ll ) Alt()tll
54q
. ('. lt. llrrrrt. li (' llrultorr.;rrrrl .l (' Mtttttlorrl, 'llr, l'.llr'rl" nl Wtrttl ott
Itr.olllt'. Nt.w ('rrlt'rur lllrst'tl orr Wnttl l'ntrrr'l l't1x'tttttr'ttlr," llrtrl,l Ittttt,'tt
5 rrr/s, wlticlt ltt't: sct:tl irt'l':rlrlt'
51r/s lltcrc corrcsl.rotttls sllculs (, - 5/l
()ccttr
tittlc.
o|(hc
30%,
itb()ut
15.4.1 trt
'l'lrr: cornlirrl critcrion proposccl in [15-lll and prcscntod in Scc{. 15.2'2
s(:rtt:s tllrt lrcas in which wind speeds in excess of 5 m/scc occur rngrc lhan
)ll'/t, ol'rhc tirnc are generally unsatisfactory from a pedestrian comlirrt point
ol' vicw. 'l'hcrclirrc, according to this criterion, the wind conditions of thc
l5 l,l
lirrcg<ling cxittnplc arc unacceptable.
l5-16 J. Candctlcr, "Wind linvir'onrrrerrt Anrurttl lltriklings: Atrtrxlyttitlttit'('ott
cepts," in Pnx'culirtl4s ttl'tltc ["ourlh Inttnrutitnul (1nl|rrttct' tnt Witul l',llt't t,t
on Buildings und Slru(:ture.r, London, 1975, Cambridge Univ. Prcss, Catn-
REFERENCES
l.\ I
l5
4
l5
-5
K. ('hang, "Human Response to Motions in Talt Buildings," .1. Struct.
/)ir',, ASCIE, 98, No. 5T6 (June 19733), 1259-12'12l'. W, ('hcn and L. E. Robertson, "Human Perception Thresholds of Horizontal
Mrrriort." J. Struct. Div., ASCE, 97, No. ST8 (Aug' 1972),1681-1695'
M. Yurnada and T. Goto, Criteria for Motions in Tall Buildings, College of
lirrginccring, Hosei University , Koganei, Tokyo, Japan, 1975'
',l'. (ioto, "Human Perception and Tolerance of Motion," Monograph of Coun'
cil on Tall Builctings and Urban Habitat, Vol' PC (1981)' 817-849'
lr. R. Khan and R. A. Parmelee, "Service Criteria for Tall Buildings for Wind
Loading, in Proceedings ofthe Third International conference on wind Effects
on Buiidings ancJ Structures, Tokyo, 1971, Saikon' Tokyo, 1972' pp' 401401.
l5-6
R. J. Hansen, J. W. Reed, and E. H. Vanmarcke, "Human Response to windInduced Motion," J. Sffuct. Div., ASCE, 98, No' ST7 (July 1973), 15891605.
15
1
J. W. Reed, WinrJ-lnduced Motion and Human Discomfort in Tall Buildings'
Research Report No. R7l-42, Department of civil Engineering, MIT, Cambridge, 1971.
l5-ti
T. Goto, "studies of wind-Induced Motion of Tall Buildings Based on occupants Reaction," J. Wind Eng. Ind. Aerodyn', 13 (1983)' 241-252'
L. Ircld, ..superstructure for 1350 ft. world Trade center," Civ. Eng., ASCE,
41, (r (Junc l97l),66-70.
A. I). l)crrwunlcn, "Acccptable wind Speeds in Towns," Build. sci.,8,3
l5
g
15
l0
15
ll A l). l)crrwrrnlcn rrntl A. F. E. Wise, Wind Environment
(Scl)l
.
l()7.1), 259-261
.
around Buildings'
Establishmcnt Report, Department of the Environment.
lirrikling l{cscarclr
llrriltling ltcscarch tjstablishment, Her Majesty's Stationery Olfice, London,
t915.
15-12 T. V. Lawson and A. D. Penwarden, "The Effects of wind on People in the
Vicinity of Buildings," in Proceedings of the Fourth International ConJerence
l5-13
I, t l()/{r;. I Jl{
l-5-15 N. lsytrtrrov:lrrtl A (i l):tvt'ttlxrrl. " llrc (itottltrl It'vll wttrrl l''ltt'tlrtlltttr'ttl tti
Iltrilt.trp At-clts," it'r I'tt'tt'ttlirt,qs r'l lltt l\trttllt ltttt'tttrtltttttrtl < t'ttlt'tI ttt t tttt
Wind Iillcct,s ott Ihtiltlirr.q,t tltl ,\trrtttrttr'.r, LottrLrtt. l()ll. ('lttttlrtttl;',t'llrtrv
I
Prcss, Cltttbritlgt:, l()7(r, pp. '10.| "12J.
bridge, pp.423-432.
l\ I li.
l\ I
.l
on winrJ Effects on Buildings and Structures, London, 1975, Cambridge Univ.
Press, Cambridge, 1976, PP. 605-622.
E. C. Poulton, J. c. R. Hunt, J. C. Mumford, and J. Poulton, "Thc Mcchanical
Disturbance Pro<Iuced by Steady and Gusty Winds ol'Mtxlc:rrttc Strcrrgth: Skillctl
periormanceandScmanticAsscsstncnts," I'.rgrtttttrtit.t,ltt,6(l()75),65
I
673
15-17 S. Murakami and K. Deguchi, "New Criteria for Wind Effects on Pedestrians,"
J. Wind Eng. Ind. Aerodyn., 7 (1981), 289-309.
15-18 L. W. Apperley and B. J. Vickery, "The Prediction and Evaluation of the
Ground Level Wind Environment," in Proceedings of the Fiilh Australasian
Conference on Hydraulics and Fluid Mechanics, University of Canterbury,
Christchurch, New Zcaland, 1974.
Melbourne and P. N. Joubert, "Problems of Wind Flow at the Base of
Tall Buildings ," in Proceedings of the Third Intemational Conference on Wind
Effects on Building and Structures, Tokyo, 1971, Saikon, Tokyo, 1972' pp.
l5-19 W. H.
105-l 14.
15-20 E. Arens and
D. Ballanti, "Outdoor Comfort of
Pedestrians
in Cities," in
Proceedings of the Conference on the Urban Physical Environmenl, 1975, U.S'
Forest Service, American Meteorological Society, and Syracuse University,
Syracuse,
NY
1975.
15-21 M. O'Hare, "Designing with Wind Tunnels," Arch. Forum (April 1968),
60-64.
Windtunnelmetingen aan een model van het Transitorium II van
de Rijksuniversiteit, Lltrecht, Report No. TR72l10L, National Aerospace Lab-
l5-22 R. Poestkoke,
oratory NLR, The Netherlands, 1972.
15-23 N. M. Standen, A Wind Tunnel Study of Wind Condition.t on Scale Models of
Place Desjardins, Montreal, Laboratory Technical Report No. LTR-LA-101,
National Research Council of Canada, National Aeronautical Establishment,
Ottawa, 1972.
15-24 N. Isyumov and A. G. Davenport, "Comparison of Full-Scale and Wind Tunnel Wind Speed Measurements in the Commerce Court Plaza," J. Ind. Aerodyn., 1,2 (Oct. 1975),201-212.
15-25 A. G. Davenport, C. F. P. Bowen, and N. Isyumov, A Study of Wind Effects
on the Commcrct ()turt Pntict:t, Pan II, Wind Environment at Pedestrian
Level, Enginecring Scient'c llcscarch Rcport No. BLWT-3-70, University of
ol lirrginccring Sc:icncc, London, Canada, 1970.
15 26 J. Ganilcrncr, li)trt,l,' ,!, ltr tt,ut l) AI 1., I'rtrtit 2, I)tttt'rrttitttttiotr rlrt <'lrttrttlt tlt'
t,ilt'.t.st'tttt tuti,sitttt.tlt',!tt,',trt1,l,'t,'l,,tti ,!,'ltt l.,ttt-l).M.A., liN n l)YM 75'l('.
('cntor Scicntilirltrt' t't 'li't ltrttrlttr' rltt ll:tlttttt'ttl, N:tttlt's, lilirlrcc, l()75.
l5 27 M. ()'lllrrt. untl lt l, krrrrr;rrri r. "l'( r(,' l)t sillrt:r lo l(t't'I Witul llrtttr llt'ttt;' ;t
Nttislttlt't'." ,4t, ltit Ii,', ( lrrlr l(f{r'}} l i l l ilr
Wcstern On{ario, lrrtt'trlly
l
550
wrNurNr)rJCr.r)
DtscoMfont tN ANI) Anot,Nt)
RtJilDtNGrl
15-28 V. K. Shrirrin, "Wind Comlirrt and Wind Shcltcr," in I'nx'ctdings r2l'tha
Symposium on External Fktws, University o1 Bristol, 1972.
CHAPTER 16
15-29 A. D. Penwarden, P. F. Grigg, and R. Rayment, "Measurements ol' Wind
Drag on People Standing in a Wind Tunnel," Build. Environ., 13 (1978),
75-84.
15-30 W. J. Beranek, "Wind Environment around Single Buildings of
Rectangular
Shape, and Wind Environment around Building Configurations," Heron, 29
(1984), 1-70.
l5-31 F. H. Durgin and A. W. Chock, "Pedestrian Level Winds: A Brief Review,"
J. Struct. Div., ASCE, f08 (1982), 175l-1767.
15-32 A. Tallin and B. Ellingwood, "serviceability Limit States: Wind Induced Vibrations," J. Strucr. Eng., ll0, (Oct. 1984), 2424-2437.
TORNADO EFFECTS
'fornadoes are storns containing the most powerful of all winds (see Sect. 1.3).
However, their probabilities of occunence at any one location are low compared to those of other extreme winds (see Sect. 3.5). It has therefore been
generally considered that the cost of designing structures to withstand tornado
cffects is significantly higher than the expected loss associated with the risk of
a tornado strike (the expected loss being defined as the product of the magnitude
ol the loss by its probability of occurrence). For this reason tornado-resistant
design requirements are not included in current building codes or standards,
Ior example, the Uniform Building Code [6-1], the Southern Building Code
116-21, or the ASCE 7-95 Standard [17-l].
However, in designing facilities for which the consequences of failure would
bc exceptionally grave, the effects of a tornado strike must be explicitly taken
into account. Such facilities include nuclear power plants, for which it is required that "structures, systems and components important to safety . . . be
tlcsigned to withstand the effects of natural phenomena such as . . . tornadoes
. . . without loss of capability to perform their safety functions" [16-3]. In the
tJnited States, construction permits or operating licenses for nuclear power
plants are issued or continued only if this requirement is satisfied in a manner
consistent with Regulatory Guides* issued by the U.S. Nuclear Regulatory
Cbmmission (e.g., [6-31 and I l6-41) or otherwise acceptable to the Regulatory
stafl'of that agency. lt is (hc purposc ol'this chapter to describe studies undcrlakcn, as well as dcsigrr clilcliir irrrtl llrlcctlurcs developed, with a vicw to
cnsuring an adcqualc rcsislirttee rtl'ttttclcirr l)()wcr plants t<l tornackr cllbcts.
+'l'hc licgulatirry (ittitlcs rtte n'vir'wrrl
l*'rirxlilirllv, irr nt'crlt'tl. l() itr'('onrrKXlitlc corrrrttt'rtls
lo t'trllcc( nrrw ittlitnttitlirrr or r'rpi:ri.:ttr r. I ifr ,ll
rrttrl
551
552
t()nNnt)()
il
Il tot:;
wltt'l'c K rs it (()ns(iurl
Tornado cllbcts uray bc diviclctl itt(o tltrcc gn)ups:
rigonrttsly
1. Wind pressures,
caused by the direct action upon the structurc
ol'thc air
flow.
2. Pressures
3.
associated with the variation of the atmospheric pressure field
as the tomado moves over the structure (atmospheric pressure changc
effects).
Impactive forces caused by tornado-borne missiles.
To estimate these effects, it is necessary to assume a model of the tornado
wind flow. A model currently accepted fbr use in engineering calculations
consists of a vortex characterized by the following parameters: (l) maximum
rotational wind spccd V^r,* (2) translational speed of the tornado voftex 2,.,
(3) radius <lf'maximum rotational wind speed R^, (4) pressure drop po, and
(5) rate ol' prcssurc drop dp"ldt. (Values of these parameters proposed for the
design of nuclcar power plants in the United States are listed in Sect. 3.5.).
The tomado voftex flow model must then be complemented by assumptions on
the detailed features of the wind flow. Such features are discussed as needed
in the subsequent sections herein. For a survey of recent developments in
engineering practice related to tornado effects, see [6-5].
16.1 WIND
PRESSURES
[16-6], and assumes the following:
1. The wind velocities and, therefore, the wind pressures, do not vary with
'l'hc wintl prosstlrc
tlrcrcol'rnay bc writtcn
yr,,,
,'rprr':,:.rorr, wlrrr lr
l.
rrol
ttsctl irt tlt'sip,rrirrg, s(tttt ttttt's ()r l)iuls :rttl lxrrlrorrs
irs
lr,
t1,(',, I t1y(',,i
1l(r.1.211
whcrc Q, is thc cxtcrnal prcssurc crrcllicicnt, C,i is thc internal pressure coef:
licient, qp is the basic extcrnal pressure,* q, is the basic internal pressure.
Values for the pressure coefficients C, and Cpi zre given, for example, in
ll6-31. The quantities qp and eu may be calculated as follows:
Qr:
CIP^^
:
CYP^
Qv
(16.1.s)
(16.1.6)
*
where
P^o :
(16.1.7)
*PV'-^
ln Eq. 11.16.7, p is the airdensity and Z-u^ is the maximum horizontal wind
If Z-o* is expressed in mph andp-"" inlblft?, |p :
0.00256 lblftzl(mph)z. The quantities Cf and C! are reduction (or size) coeflicients that account for the nonuniformity in space of the tornado wind field.
'l'he size coefficient Clmay be determined from Fig. 16.l.l as a function of
lhc ratio LlR., where L is the horizontal dimension, perpendicular to the wind
tlirection, of the tributary area of the structural element concerned (if the wind
Ioad is distributed among several structural elements, e.g., by a horizontal
rliaphragm, L is the horizontal dimension, perpendicular to the wind direction,
ol'the total area tributary to those elements). The size coefficient Cf may be
rlctermined as follows. If the size and distribution of the openings are relatively
trnifbrm around the periphery of the structure, C! is determined in the same
way as Cf using a value of t equal to the horizontal dimension of the structure
pcrpendicular to the wind direction. If the sizes and distribution of the openings
rrrc not uniform, the following weighted averaging procedure is used:
height above ground.
The tangential wind velocity component is given by the expressions
V,
: ^r V^ (0<r=R.)
K_
V,
:
R'n
r
V,n (R-<r(o)
(16.1.1)
(16.1.2)
l.
3.
ol lttrrlxlltotItlrly Ilrl;
l('('l, is r'ottvt'ttir'rtl in t ;rlr'ttllrl torr,
speed (see Sect. 3.5.1).
A procedure for calculating wind pressures is now described; it is taken from
2.
('()r
whcre V,, is the maximum tangential wind velocity and Ru is the radius
of maximum rotational wind speed.
The total horizontal wind speed is
V:KV,
+The rotational wind spcccl is dcfinccl as thc rcsultanl
componcnts ll6 l3l.
ol'lhc l:rngcrrtiirl
(
r6. l .3)
;rrrtl r:uli:rl wirrtl
vt'kxily
Determine quantity 11lR,,, such that
t't
1i,,,
R,,,
t't I
(r6.
r.r3)
L
rllt't:tttse tto tlislirtt'liorr is ttt;trlr' rtt lltr'. lt,r, , ,lttrr' lrlu'r'r'rt lr;tsir'
Ptt'sstltt's tlscrl rtt lltr' rlt stl'.tr ol
slnt('1ilt('s, rttt (ltt'0nt'lIttttl. ;ilt,1
"l l',ilt, ;ilr,1 l','llr,'r', r'il llr( ollr('t lr;il11, llrr'n()lllll{)il rl, u\('(l rr
I l{r }l lor l)r('ssll('s ()n p:ul!;tttrl lxrtltltt, r', ri}l t rrl'lr'\'r'rl ltr'tr'tl
554
rotlN^rx) tlIloll-;
ilr r wlNt) t,llt lil;(,nl l;
whcrc,rl,, is thc arca ol'opcning itl locirliorr i. (i,, is thcr lirctor (',r ul
l<tcatiorr ri, ancl N is tho nutttbor 9l'gpcrri1gs. ('l'lrr: cgcllir:ic:rrt (1, in liig.
16. 1.2 rcprcscnts nonditncnsionalizcrl winrl prcssures lnrl was calculatccl
using Eqs. l6.l.l, 16.1.2, 16.1.3, anrl l(r.1.7.'lir obtain trig. t6.l.l,
the nondimensionalized prcssurcs ol' tJig. 16.l .2 wcrc intcgratcd bctwccn
the limits 11 and 11 * L, whcrc 11 is givcn by Eq. 16. 1.8, ancl the rcsults
of the integration wcrc thcn nonnalizcd; the coefficient Cl is thus an
approximate measure of the average pressure over the interval L t16-61).
1.0
0.9
F
cs
o.7
0.6
Numerical Example The building of Fig. 16.1.3 is assumed ro be in region
I. The sizes and distribution of the openings (not represented in Fig. 16.1.3)
are assumed to be uniform around the periphery of the structure. The ratio
between area of openings and total wall area is AolA*: O.25.It is assumed
V^u :360 mph (161 m/s), R. : 150 ft (46 m) (see Table 3.5.1). The
pressures on the 100-ft (30.5-m) side walls induced by the wind blowing in
the direction shown in Fig. 16.1.3 are calculated as follows:
0.5
0.4
1.0 1.2 1.4
1
.6
1
.8
2.O
L
ftFIGUR.E 16.1.1. Size coefficient Cf
2.
555
116-61.
Locate plan of structure drawn at appropriate scale within the nondimensionalized pressure profile of Fig. 16.1.2, with the left end of the structure at the coordinate rtlRm.
Determine factor Cn from Fig. 16.1.2 for each exposed opening.
3.
4. Determine Cf; from Eq.
16.1.9
Dl Asicq,
-, _
Ls
r
- -FX
Ll no,
(16.1.9)
pmu*
:0.00256 x
3602
:
330
#
(rr,roo
{)
(Eq. 16.1.7)
(r*uo
{)
(Eq. 16 r.5)
For basic external pressures,
L:20Oft
L 200 :
R_ 150
(61 m)
1.33
: 0.50 (Fig. r6.1.1)
lb
ae : 0.56 x 330 : 185
Cf
-ft'
1.0
0.9
0.8
o.7
0.6
C,I
0.5
0.3
o.2
0.1
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.O 2.2 2.4 2.6 2.8
r
RFIGURE 16.1.2. Cocfticicnt C,/ ll()
61.
4l
3.O
.'9(.'"
/n{'
li'l(illltl,l
16.1..1. $r'lrr.rrrrrtit vit.w ol lrrriltlirrpq
556
t()trNntx)
;
Il t(;tli
Iiol basic inlcrnal prcssLltts,
:
200
_L :
1.33
cf
:
o.so
4u
:
o.56
L
R*
I
ior'
ft
(61 m)
330
:
Lt'i -
lirlr wind
185
q
(tt.o
ft2
I)
(Eq
16 r
: -0.7 x
-0.7 tl6-31
+0.3 ("r*<
185
-
0.3
x
551
0.3, see lro-:l)
185
:
r8s
g
ft'
(*ruo
\
: PVl,
(l
(r.2..11)
lf the structures are completely open, the intemal and external pressures are
cqualized, for practical purposes, instantaneously, so the loading due to atmospheric pressure changes approaches zero. In structures with openings (vented
structures), the intemal pressures change during the tornado passage by an
amountp,(r). Denoting the external atmospheric pressure change by p,(t), the
atmospheric differential pressure that acts on the extemal walls is p"(t) - piG).
A useful model for p"(t) can be obtained by assuming, in Eqs. 16.2.2 and
16.2.3, r : Vot, where 2,, is the translation speed and r is the time. A simpler
model in which the variation of p.(t) with time is given by the graph of Fig.
16.2.1 may also be used [16-6]. The time-varying internal pressures p,(t) may
be estimated by iteration as follows. Assume that the building consists of a
number n of compartments. The air mass in compartment N (where N < n) at
time f +1 is denoted by Wy(\+1) and may be wrirten as
61
pressure,
p*,
i
tornado-'l'hcrclirrc during (lto p:tssirgt'tlte rlillt'rt'rrtt'lrt'lwccrr llrt'lrrtt.r'rr:rl prt.s
surc and the atnrosphcric prcssutc is ctlturl lo 2,,. ll lollows lrorrr llt1s. l(r.l J
and 16.2.3 that thc maxirtrurrr vtrlrrc ol'yr,,, wlriclr (x'(.ut.s ltt /. O, ls
1l',t,tt'^
x
I ( )n I !ll.1r
lrt tlrc ctrsc ol slrrrclrrrcs with rto opt.rrings (tttryt'ttlt,rl \.!ntt.tut(,\ ), llrr' rrr{t.rrr;rl
l)rcssuro tttn:ritts ctlttirl lo (lrr: trlntos;rltt.r'it'pn.ssrrrt. lrt.lort'llrt- lrlr.;:;lr1't'ol llrt.
Pr.t'ssrrrc cocfficients.
Co:
l'lll :,',l ,l tl ( ll^tl( il
I\
m'l
Wy(t1+t)
(Eq. 16.1.a)
: Wp(\) +
[GN(i")(t)
-
G,v,.",,(4)J
Ar
(16.2.4)
where
G1r,,", snd Grq,,",y denote the mass of air flowing into and out of compartN per unit of time, respectively, and Ar is the time increment. The air
mass flow rates G7y can be calculated as functions of the pressures outside and
within the compartment N and of relevant geometrical parameters, including
ment
16.2
ATMOSPHERIC PRESSURE CHANGE LOADING
('onsitlcr thc cyclostrophic wind equation (Sect. 1.3) written
dpo
v?
dr: P;
as
(16.2.1)
wlrt'n'r/,rr,,/r/r'is tlrc ulrnosphcric pressure gradient at radius r from the center
rrl tlrr'torrrirrlo vorlcx. 'lir obtain thc pressure drop po,Eq. 16.2.1 is integrated
lrorrr irrlinily lo r'. ll'thc cxprcssionfor V, given by Eqs. 16. l.l and 16.1.2 is
rrst'tl I l(r (rl:
1,,,1n
-,t;' (, #i)
p,,(r)
- p ;-7
Vi,,
Ri,,
(o
sr-
(R,,,sr(oo)
R,,,)
(t6.2.2)
(t6.2.3\
lll(;lJltl,l 16.2.1. ltleirlizt'rl :tlttto:ltltltt, lrr".',uri'r
A,
lr;rrr1',' v{'r:,n:,
lrn(' lurrt'liorr Il{r
{rl
558
TORNADO EFFECTS
opening sizes, as shown subsequently. The internal pressure in compartment
N at time ti+r,Piu(ti+1), is then written as
p
i.(tj
+, :
N
lry#lo
o,r r,,t
(16.2.s)
(?)
(o
ll
:
1.4 is the ratio of specific heat of air at constant pressure to specific
where k
heat of air at constant volume.
A computer program for calculating loading on vented structures due to
atmospheric pressure changes is briefly described in [16-6]. The program incorporates the following type of model [16-7] for the air mass flow rate:
NO
EE
N
(f,
o
N
o
o
6
lt
G
:
- p)lt''
0.6C,Azf2l,,(pr
(t6.2.6)
@
o)N
OO
dc;
I lt
\o
\o
NO
N.
ci$
0.)
Err
o
(J>
where
. / \2lk
*: [(,1)
ft
k
[1
- tl-
-
(p2lpr1{t'- rtr*
-
Pzlqr
It | -
-lJ"t
| - (A2tAt)2
(AzlA)2(prlpr)''o
(16.2.7)
o
N
N
N
st
I
a)
H
N
N
o
o
o
so)
@
N
o
o
@
FrOto
Err
Err
LO
ciN
N
o.
o>
il
o
o
O)
d)
il
e
(t
o)
sf
El
o.
(J>
ro
(')
o
o
N
rod
N
il
o;
tr
N
t:
c! c')
o.
(J>
o
r
N
(o
(o
(o
(o
ra
ll
il
al
+
\o
r{
/t
Air flow
pattern
f[i
zIIZZ=V
sf
(fJ
ll
o
o
@.
.ro
Err
E>
FIGURS 16.2.2. Illustration of pressure distribution and flow pattern during building
depressurization
[6-6].
6t9
560
r()t tNnt
r
x) I ttt(;t:;
l) ol'tlrc wull bctwctrtt cotttand 2,,42 is thc arca cotltloctirrg cornpartttlcnts I antl 2" (', is ir
nondimensional comprcssibility cocfficient, k: 1.4, p1 is thc prcssurc in cotttpartment l, p2is the pressure in compartment2 (p2 < Pr), and "y1 is thc mass
per unit volume of air in compartment 1. If, in compartments provided with a
blowout panel, the differential pressure exceeds the design pressure for a panel,
a statement in the program transforms the blowout panel area into a wall
opening. In view of the presence of three-dimensional effects not accounted
for by Eq. 16.2.6, the atmospheric differential pressures on extemal walls
obtained by the procedure just described are multiplied by a factor of 1.2
ancl zl 1 is thc arca (rln tho sitlc ol'c()lnl)iu-lrncnl
partmcnts
tl
I
6-61.
An illustration of the pressure distribution and of the flow pattern in a building during clcprcssurization is given in Fig. 16.2.2. An illustration of a structure
dcprcssuriz.ation model with values of geometric parameters required as input
in thc conrputer program, and an example of a corresponding differential pressurc-tirnc history calculated using the program, are shown in Figs. 16.2.3 and
16.2.4, rcspcctively.
I
16.3
l,r,
5ti
I
TORNADO-BORNE MISSILE SPIEDS
'lir
cs(itrtltlc sPcctls irlliritrt:rl lry lrn olrjt't'l nt()vltl, rrrrrlt.r llrr. ;rt.ltorr ol ;rcrorly
rtitltric lorccs intlrrccrl lty lot'llrtkr wilrtls.;r sr.l ol lt:i:illntl)l t()nr, l:. rt'rlurrt.tl
o On thc acnxlynalrric t'lurr':rt'lt'r'islit's ol llrc olr;t.t'l
o On thc dctailctl lctrtrrrcs ol llrt' wilrtl lLrw lickl.
o On the initial positiotr ol tltc olr.jcc( wi(h rcspcct lo llrc grourrtl untl (o lhc
.
tornado centcr, ancl its irritial vcl<rcity.
objects commonly considered as potential missiles in the design of nuclear
power plants are bluff bodies such as wooden planks, steel rods, steel pipes,
utility poles, and automobiles.
The purpose of this section is to review approaches to the tornado-borne
rnissile problem based on
(l)
deterministic modeling, (2) probabilistic modeling
of missile transport as a
involving numerical simulations, and (3) modeling
Markov diffusion process.
Between compartment 3 and outside
16.3.1 Deterministic Modeling of Missile Motions
atmosphere
Between compartments 1 and 3
Equations of Motion and Aerodynamic Modeting. The motion of an object
may be described in general by solving a system of three equations of balance
Note: lnput time history per Fig. 11.2.1
using3R-,//rr=9sec
and po = 432 lb/tt2
Structure depressurization model
shown
in
of momenta and three equations of balance of moments of momenta. In the
case of a bluff body, one major difficulty in writing these six equations is that
Figure 11.2.3
the aerodynamic forcing functions are not known.
It is possible to measure in the wind tunnel aerodynamic forces and moments
acting on a bluff body under static conditions for a sufficient number of positions
of the body with respect to the mean direction of the flow. on the basis of
such measurements, the dependence of the forces and moments on position and
corres