Analysis of an open spandrel arch including superstructure
by William Lorell
A THESIS Submitted to the Graduate Committee In partial fulfillment of the requirements for the
degree of Master of Science In Civil Engineering
Montana State University
© Copyright by William Lorell (1950)
Abstract:
A procedure for the analysis of an open spandrel arch including the effects of the superstructure is
developed by neons of a special adaptation of the method of moment distribution and the virtual Trork
equations. The arch is considered cut at the center line, and the action of one half of the structure upon
the other is replaced by six unknown forces (one horizontal force, one vertical force and one moment
applied at both deck and arch at the center section). All joints of the remaining arch structure are then
held against translation by sets of restraining forces. The moments resulting from separate application
of the real loads and each one of the six above mentioned unknown forces are determined. Additional
joint moments are introduced, obtained from the solution of so-called equilibrium equations, which
remove all restraints and restore equilibrium of external and internal forces. The joint moments,
expressed in terms of actual loads and the six unknowns, are then utilized to determine rotation and
deflections in horizontal and vertical directions of both deck and arch at the center section. The fact that
rotations and deflections at that section must be identical for the two halves of the structure, provides
six equations, permitting solution for the six unknowns. A model arch, tested by the University of
Illinois experiment Station in 1928, is analyzed by this procedure, and a comparison between
calculated and measured stresses is made. Close agreement is obtained, indicating decided superiority
of this approach over the conventional one of neglecting the effects of interaction between deck and
arch rib. AHATJSi s o: AS Ol /; SP A * ' ARCH
IHCLCDim
SDpmrHDCTims
by
OTLIAU LCRgLL
A THESIS
Submitted to thi* Graduate C ctm lttee
In
p a rtia l. fuJLfillnaint o f the r»qnlrea«ttt»
fo r th e degree of
!Saster o f Sclertoe In C iv il Engineering
at
!Jontana S ta te College
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ABSTRACT
A procedure fo r th e a n a ly sis of an open spandrel arch including the
e ffe c ts o:L th e su p erstru ctu re i s developed by neons of a s^x-cial adapta­
tio n o f th e r t ; cd of memeni d is trib u tio n
nation##
The arch i s considered c u t a t th e c e n ter lin e , and the a c tio n o f one h a lf
of th e stru c tu re upon th e o th e r i s replaced b r s ix unknown fo rc e s (one
h o rizo n tal fo rc e , one v e r tic a l force and one .0 it applied a t bot. deck
and arc,i a t th e cen ter s e c tio n ). A ll jo in ts of th e remaining arch s t n
.
lam ents re s u ltin g fron sep arate ap p lica tio n o f the re a l loads and each one
o f the s ix above ientioned unknown fo rces are determined. A liitio n a l jo in t
ncsnents are introduced, obtained from th e so lu tio n of so -c a lle d equilibrium
eq u atio n s, which remove a l l r e s t r a i n ts and re s to re eru llib riu m o f esrternal
and in te r n a l fo rc e s , The jo in t lam ents, expressed in te rn s o f a ctu a l loads
and th e s ix unknowns, are then u tiliz e d to determine ro ta tio n and deflec­
tio n s in h o rizo n tal and v e r tic a l d ire c tio n s of both deck and arch a t the
c e n te r s'eotion* Th
th a t ro ta tio n s and d e fle c tio n s a t t h a t section
must be id e n tic a l f o r the two halves of th e s tru c tu re , provides six equa­
tio n s , p erm ittin g s o lu tio n lev the s ix unknown#♦ A ; -Odcl arc' , te s te d by
th e U niversity o f I l l i n o i s R p e rin e n t S ta tio n in 1920, i s analyzed by t h i s
procedure, and a comparison between c a lc u la te d and measured s tre s s e s i s
made. Close agreement i s obtained, in d ic a tin g decided s u p e rio rity of th is
approach over th e conventional one of n eg lectin g th e e ffe c ts of in te ra c tio n
b e t r,n dec: and ric h r i b .
-
6-
EfTRODUCTION
The conventional an aly sis of an open spandrel arch bridge as a r ib
w ithout r e s t r a i n t due to the su p e rstru c tu re i s c le a rly incom plete.
Ihe
argument th a t bridges designed by a procedure neglecting deck p a rtic ip a ­
tio n have ayuorenti
given good serv ice i s no t c aside* vd to be a pa-'- tic u —
l a r l y v a lid reason fo r neglectin g tn is c ffe c o .
;irxcc .it i s q u ite obvious
tlia t th e su p e rstru ctu re tends to s t i f f e n the rib* i t i s not p a rtic u la rly
astounding th a t r ib s designed on a b a s is neglecting t n i s s tiffe n in g e iie c t
have stood up w ell under design lo a d s.
A comparison between th e weights
of open spandrel arch s tru c tu re s designed, by conventional means (neglect­
ing the e ffe c ts of th e su p e rstru c tu re ) on one hand, and those designed
w ith due co n sid eratio n to in te ra c tio n on th e o th e r, wo ■?.1 cone c lo se r to
the p o in t.
In c o n tra st to the cu stc ary American approach, Duro ean designs of
open spandrel arches g en erally take in to consideration the in te ra c tio n of
r ib and su p e rstru c tu re * Perhaps the most outstanding r e s u lt of t h i s
o f ana- s is i s th e 160‘-span Grandfe; Viaduct in Sw itzerland, a stru c tu re
.
b rid g e s, however, i s mainly based on model t e s t s , and involves no d e fin ite
mat .c r itic a l met od. “
Current l i t e r a t u r e on th e problem of arch bridge a n a ly sis generally
r e s t r i c t s i t s e l f to s ta tin g th e f a c t of in te ra c tio n and i t s most apparent
e f f e c ts , but sk ip s l ig h t l y over p o ssib le mathematical e o la tio n s .
In s p ite
o f a most thorough search, th e author has succeeded in lo c a tin g only one
I.
E la stic Arch Brld c s , by HcCollough and Thayer, John Wiley and Sons,
I n c ., I!. Y ., 1931# Pl . 313.
-
7-
o u tlin e of on exact so lu tio n , ^ -which, hov,'-ver^imrolves no le s s than 6k
equations -wit!
unknowns.
I t i s f a i r l y c le a r th a t a method of t h i s type
1Erould add l i t t l e toTr r<; th e prim ary purpose of th is th e s is j
to fin d a
workable, com paratively sh o rt approach to th e analysis o f open spandrel
Lng the o f f r e ts of th e su p e rstru c tu re .
For the above men­
tio n ed reaso n s, referen ces to e x is tin g books and u b licatio n s w ill be extr e n e l
scarce throughout t h i s th e s is .
The netiiod subs- npiently to be developed i s used in th e f i n a l p a rt o f
tills th e s is to analyze m athem atically a model arch which had been experi­
m entally te s te d to f a ilu r e a t the IM v e rs ity of I l li n o i s Experiment S ta tio n
i n 1923.'' Since s tre s s e s fo r various loadings had been recorded during
these t e s t s , a comparison of a n a ly tic and experim ental r e s u lts i s ' o ss ib le .
,/hereas s tr e s s e s c alc u la te d by th e e la s tic theory neglecting deck p a r t i c i ­
p a tio n show dev iatio n s up to 1$0# from observed s tre s s e s , much clo ser agree­
ment i s obtained by th e method of an aly sis developed in t h i s t h e s is .
Ho
m istaken conclusions as to th e accuracy o f t h i s method, however, should be
drawn*
The u n c e rta in tie s involved in th e determ ination of q u a n titie s such
cs th e modulus o f e l a s t i c i t y of concrete, th e moment of I n e r t i a of re in ­
forced concrete se c tio n s, e t c . , make c lo s e r agreement than, say $0$ purely
a c c id e n ta l.
I t has been demonstrated conclusively by the U niversity of I l li n o i s
Experiment S ta tio n i n th e above mentioned experiments th a t th e decks of
th e arch
o d d s te s te d incre ased th e stre n g th of th e stru ctu re s to IsC '*
Any co n sid eratio n f o r economy o f design cannot overlook a fa c to r of th a t
2.
3.
Ib id , pp. 309-310.
"
Laboratory Tests of Reinforced Concrete Arches b ith Decks, B u lle tin
-- , JhTv " d :t " b f "dl/hhoxs S y crim cnt 3 tc £ Io h /'Urliaria,' 'H i . , 1929.
. .a iltudo.
The author th e re fo re b eliev e s t h a t sny end «0.1 a ttc jp ta to
dem lop a ; roeodurc fo r tlse a m ly a is o f in te g r a l Cznh o tio n are
j LatificxU
-9 -
A r CM'tr. v a i l be
f;-
«'"■
^
veil, a
. - ': "
.vVtiLcrrvX'. fo r th e cm dyilo of -Hie o -OroaiicntiL type
0:1
i e one c
.,n
in H g . I .
s . c.-Lcc:, _c i a i l L e . ..c..
The i f f sorts of in c o . ...i, . .cL
u.
H g. I
The a n a ly sis v a il involve an exfceneion of the tlzardy Grooo : icthod of
a
ont < -sti-ibution, ocrM aed w ith th e p rin o irlo s o f v i r tu a l vrarte.
In G eneral, t e d loei aeioo v d li bo I i .ite d to on arcli of th e physical
cluc.-acteristica o f th e ay-... .iet r i e e l 7-oaael iech Bnal^vatxi i n the I . .it p a it
oi v ..a to. ice ,
iUi oxtensJon t ; any cliicg? h c of arch, ’,cccvi -> osn be
3
,
s^ a q s uttitpo ptto TPttBd H O ^ rim o c%x
ptn» •eytpof 6nop,MA ■:?•/:, jo
uoTtvpmm.iq. %T- -:«% %ou «Mp ojt\pooo,td
" O- <*> eatMaj ustoTttfuti
«*4) euoi^-t vdo #$%.*%&*
» 1 0 0 »?^
^trtg, pguuoc^i ®q ^ens: %r
jo <0
0
aoj
000
•(gqm
pt?c? *ep% %TBrt^oy air% jk>j
trp poq.nqp^eB> pwo punoj ate soswoj tiBcxucfun
XTS Otrr jo SjOoo m » j puo ep 0 %%oa% oq-j
umj
nu nin sca e%aemni pus poeqy?
atpr *sataqgiaat snoptsm mu jo #o#tedoJd ojjtraia aqq. 3ur|ti^te%op
Z "3%
®
0
0
0
*3 *3m Uj tssoup o i eCjTaox Tpn-Joa oqj -;tm so a io j
WjtitCfGn Xjo osGTjj sjjp& popoox *-••
aoXTV' - - 0
Jonptrpiwa
‘soaxoj u
tcf
~uu x js j o scktoj u j pMsauKim s j uoqjo oqj uo <xmjoru$je oq j jo j%sq ouo
JD U
O
TJO
C 0% *0Ui%JOJUOO nqj j* jno c? %' 3% %
%tzioqs ypuu or&
(Q lirm o HOIJATOG
-O I-
•
jo in ts WUxchj
ft r Having been imposed on the stru c tu re during moment d is ­
tr ib u tio n , must suboenuently be elim in ated .
I t is
o ssib le to e s ta b lis h a
s e rie s o_ equations, henceforth to be c a lle d equilibrium equations, whose
so lu tio n vti.ll y ie ld c o rre c tio n moments, which, when ap H ed to th e various
j o in t s , w ill e s ta b lis h equilibrium between external, and SatenuO. tm
A lt, v t. d s o p eratio n , the oonpletely balanced jo in t moments w ill be exoressed in terms of one constant (due to th e a c tu a l loads) and s ix unknowns (due
to th e a ctio n of one h a lf of th
s tru c tu re upon tin oth er h a lf ) .
% 13enns
of the v i r tu a l work equations, the h o riz o n ta l and v e rtic a l d e fle ctio n s end
the r o ta tio n of both deck and arch a t th e cen ter can new be form ulated in
terms of th e s ix unknowns.
Since the d e fle ctio n s of the l e f t h a lf of the
stru c tu re must be id e n tic a l w ith those of the r ig h t h a lf a t t h i s p o in t, s ix
equations can be s e t ur. perm ittin g so lu tio n f o r th e s ix unknowns.
n o ta tio n :
The follow ing general ru le s concerning n o tatio n w ill be established*
q u a n titie s re fe rrin g to th e l e f t h a lf of th e arch vti.ll be denoted by c a p ita l
l e t t e r s , wH e q u a n titie s re fe rrin g to th e r ig h t h a lf of th e arch w ill be de­
noted by lo v e r case l e t t e r s .
A ll q u a n titie s re fe rrin g to th e arch r i b , l o f t
or r i g n t, w all be denoted by a prime.
Deck and arch, re ente
i l l be numbered
consecutively by s u b s c rip ts , ranging from % and VJ1 (n^ and rd-,) a t jo in ts
0 and O ', respective!1 , to Hg and
.
*q (qq an-’
*q) a t
o in ts C and c* (Ti .
.' columns w ill be denoted by a subscript c ,
fo llo w :r by a s c icnd su b sc rip t giving I c a t on.
This system of n o tatio n i s
i l l u s t r a t e d below*
-
deck uo .cnt, l e f t h a lf , a t jo in t 2, member 2-3.
-
12 -
EI*^ = crch MCTBWttt , r ig h t
#
;.g21 “ oolunn a m e n t, l e f t h d f ,
-
co lu n no;.sent, r i g ' t
join t 2 », ncnber l « - 2 «.
t jo in t 2 '.
S lf , a t jo in t h,
h*^
= h o rizo n tal force in ax-ch-scctd.-vn 0 « -l« , r i g l t
If.
HI*
s hospteeotal fe m e in a rc h -se ctio n l«-2», l e f t h alf*
= v e r t i c a l fo rce in deck-section 2-3, l e f t h a lf.
v*t
a v e r tic a l fo rce in a rc h -sc c ti .n 31- », r i f t ' .a lf.
L
= h o rizo n tal len g th of a l l deck and arch sectio n s (assumed con­
s ta n t throughout t h i s discussion)*
Lq0
» len g th of column 2-2*.
X
= mar.cnt of i n e r t i a of a l l deck sectio n s (assumed constant
throughout t h i s d isc u ssio n ),
I
- rac. icnt o f i n e r t i a of a l l column sectio n s (assumed constant
throughout t h is d isc u ssio n .)
c
E
-
modulus of e l a s t i c i t y (assumed constant throughout th is d is­
cussion) .
N otations o th er than th e ones given above Tfill be defined where they
are f i r s t u see.
Sign Coitventi on:
The so le sig n convention employed c o n siste n tly throughout t h i s th e s is
w ill be t h a t c hunter-clockwise moments (a c tio n of jo in t on member) w ill be
considered r o s itiv e .
I t w ill prove n. ivanhageous not to e s ta b lis h the cazae
sign convention as to p o s itiv e d ire c tio n o f fo rces and d e fle c tio n s fo r both
halves o f th e s tr u c tu r e .
Conventions as to th e sign o f those q u a n titie s
Tdll be e sta b lish e d where th ey are f i r s t needed.
Determination of E la s tic P ro p e rtie s:
The determ ination of th e e la s tic p ro p e rtie s of the v a rio u s arch sec­
tio n s between columns .oscs a s p e c ia l problem.
For any such arch sectio n
-1 3 -
A-B, i t vdl
bo wees»G*y t c fin d aew n e ltis tia ornstimfcsi
OV T
A to 3 and fy.
k one a t Bi
B to A;
5) the dt*.' o tio a a tiffftc s e f
I & 2) the
cc^ ty-
3 & ) t o r e t a t l a C tiffaooe a t
and C- & 7) the- rUoexWod
Ce uei . b; inti'C-.-aativ.-a of la^-isoatal end v e r tic a l foi-e^s.
The h i r s t th ro e
o f tiKi,< I :. a , since they ore n o t self-c:.:; lirr&tez-- •» a h t i l be lofincsd as f o l I GSiBt
■ . . . »' -■
:
-- ,
.
“A
A
end A of a stractuh‘a l no. b< r , and bL i s th e roealtonb
it a t t .: f t :■■
'
■ ....
t
B« ! i f
c
'-on*
AB
•',
s t - Iy euBT.orted end of a given -xr&er w ith the oth -r
Cttl f i x I,
This q u a n tity i s net d tc o b tain S^aenfc d ia »
trib w tio n frete-'; .
:
Ho' t
:i:
.
f lo a t:o n a t the f r e e end o f a given member w ith th e
ot WP Qd fix e d ,
rtov.t..om
Di ....
B etatdon o r th e resain iag . tiU .'tic fa c to rs w ill be as fd lo o ia t
I
J
TI
.
i
J
I n gener C l, th e v e t m is a tio n o f th e rcqudrec' e la s tic
rr-ctors involves
no new a t cda o r ocnee.. t@» and w ill th e re fo re not be d isco sh: d fu rth e r a t
t h i s -O in tw fha 3ft-,; Xe caleulfstioM i n t o nati .aactioa
an aly sis o f wa arch
v&ih deck in th e I s n t p o r t Cf W s tfcf-sia 1Will e lc -^ ly 111u s tr s te th e Kioth,■Sn
Li.
. *
i'A siu trih n tii nt
I t rig gamed th t • 13 - jpeulCAjaft.y .n vMoiKxl e l ".Stic <3ox*str-'Ita for- o o lpc
I raid dc@k
cttl/ nr»
"-rC tx-cn found.
f|*t
c tn c l lo a d s , find each
c m o f t$'t si;: unloriown f c .con, a re new p i: ^ /I c-p rrrf o l
on th e . • :dnii\p
otr-ictvu-o, H^cd-cnal irx .nnts ore determ ined, sad d ll JoinM ere held ec.ainst
ta*r n a trtio ri but rro n i t - .-rtel; T-cn'd.ttrd to r e - to*
Zn othor wv-’s , fl*cv —
end -o < t - m e d is trib u te d hr/ the ln a ic Bsrciy Sress method.
I t id 11 be
a: cm t h a t th e actu?1- lo rd s m ' th e raoribuntcl fcroc- R... a etln g on the dec*
CfiKiss no finod-end inoeuants, and th a t jo in t Boxmtg fo r th ese two cond itio n s
of lead in g v d l be zero.
A ctually, i t w ill only be neecaoury to go thrcmgli
# # # t o obtain JcdLnt moments f o r a l l
Six C- ildit - Uy Of Ic- -Z.nr.
St i a • isuacr-i t h a t a «• c a t of s#y, 1000 u n its i s & Iird to Jcf-a t 3
■ the ' f t h a lf c
t
.
- T arch c h r -n an H ;;♦ . «5 ntei :: I
I t ia -'v oua t:
rc tin
I t" a Jcdnt
the an iio n r Vr and
t I ( f i r , 2) coil t; on be te tc m tm d by r c r e r tlo n , since these
tvxc ria R titie e w ill of -:c f i x vd~ond.
r-
nta esu r.f! b
;> o ut- f i s t z i b f -
. o '"vo\
ilrrl # r
exits V, x 1 /2 sad I! @t JcvOt 3»
3
D
nt o f ID "> m aits e l i e " r t 3* -rnf R.ibsew
ucTML d istrib u te ! w i l l solve th e
p
:ble% o f finding th e ,D lid om enta due
to V,, n, m 3 U: $ ap1 Lb f a t -I*,
A fter ,joint
. , aie -Jue to tb. VarJ ous coiKfiti ns o f Iood n Ihvo th u s
been d e te r laud, i t w ill be tibac n r 4 th a t » -ltfecr co>,$m ncr ? SBtiL shears
..." tc,
< n. r.Ucti leads#
Tiiig Iu etity n e t .ro l, a im s tlse .•<
ti-m prcecdui'c-- wee eaeided - u:t i-if
,joints locked e c Lnat I
it d istx lb u m ic tio n . The
-
15 -
tran a j-atlcn -p rev an tlii; j o in t forces are now exerting t h e i r due influence
ic tu r e in th e s tru c tu re .
t
The next
:: '■vc t ' jo in t ■" - .: /.a:.s .
Jo in t irstr-: i.nts and i q u il i b r im l. u a tlo n s 3
The removal o f th e re s tra in in g fo rce s i s equivalent t o p erm itting de­
f le c tio n s to occur -sliicii th ese r e s tr a in ts had
a lly prevented,
since
i t i s the primary purpose here to determine the e ffe c ts of th ese d e fle c tio n s,
th e a ctu a l magnitude of th e r e s t r a i n ts i s of no p a rtic u la r i n t e r e s t , as long
as the prevented d e fle c tio n s can be imposed on the stru c tu re ; n ' tiie ir e f -
ending to each of th e preccoding nc ent d is trib u tio n s w ill th ere fo re be
c a lc u la te d , th e i r Ioviaticns iro n th e r t r r l r e d values (H f o r h o rizo n tal
load H a t Cf zero fo r a l l o th er loads) vti.ll be noted ( i t i s re a liz e d th a t
t i s , of course, i s a leisure of the rc .t r a i n ts ) , and then d e fle ctio n s are
imposed, -which -will b rin g panel and column shears back to th e require ’ sag—
n itu
s.
In order to solve the l a t e r problem, the p o ssib le mays in which th e
stru c tu re can d e fle c t -rust f i r s t bo c la s s if i e d .
©
©
0
©
I'lG. 3
©
©
-1 6 -
C ::
a ., ,.m ape
stteii w skv,' > in P it> 3, i t i s dbiloae th a t deck
I
ietiU
a i t U re p a t to I ,
-to.# but# n lc c tin g f ;Ioruatlcm due to d ir e c t
s tre ss# tlo r o can be no rol-utive h g eiao n tel diC vlm a cnt bet:-eon the deck
jo in t 5.
: I
Cai t i e oi , r h; nd# thore ocn bo b©t;» %y-Iatiwe h o rizo n tal
I ot '
L vvt
n th
• .iOb jo in t-i,
Plyiclt , I
nd v - r t-
v,<112 Ie E ro r; I
tdve v u rtd c a l . rOVeicnt b e t aeon ocrroopond.:.nc deck end arch Jcdxtte, such a s ,
2-2*,
.-c,# b-.it de-ok i i n t# % !,?# >
>2 C can d e fle c t a," m&tm - xaly
.
PiKujCi I P lo o tiono te ll, ',am be , reaJiown in F ig .
d.
The ty. Io a l ; unel loflooticm
, can be oenoidcitd to fee eo-jpooed o f th e p r r td a l deflc o*
t ...m sliom In F igt I , b,c*
vjxI
d.
h
. T-ck one of th ese p a r t i a l d e fle c tio n s b , o, end d vtiQl e x is t to a
o ertcd a e x te n t in every panel o f th e ..-,truotm-c., Wiloii io j u s t enotiier mg*
o f -W iag th a t th e re v&ll fee a-: .o v e r tic a l dofle-otioa of any given p a ir o f
ooirespcW lng Pock and trctii J o in ts w ith re a p e e t to t h e i r ad jacent jo in ts m
th e l e f t ,
io h o risc n ttil d e fle c tio n o f each arch jo in t Y.lth .respect to tiie
-1 7 -
;r e ': <c%nt on tlj® l e f t ,
j '.Tnt
' i t : rtr
;>r- o u tu a l horigonfcal d e fle c tio n s o f o il cteok
Ot to tlic ir o i'ig in e l i.o ritio n .
, th is coo
be esprcssed by assig n in g to each oar of th ese d e fle e tie n s an in te n s ity
footin’ -T Ioh re: Iocv
u t te n a n t.
th ' very va^ae nord. «ac:.3C« used i n th e -rcc
The v e r t i c a l d e fle e tic n o f Jc in ta I c a t I* M tb resp ec t to Jo in ts
0 . n. 'H i s th u s ris lc n c d on '. r t t . a l t / f r o t e r $:*, of jo ln ta 2
Xf
2* w ith
' '■’■ to " (,*! I* lb I , ,-Vifrl 3 : Ki 3* vr. t : resp co t to 2 ,:.-d 2» »c», th e Iivww
le o n ta l <;’ Clf itlf ii o f I* . -ih .«•-■. oot to D* * % of 2* i f . rosrsoct to 1«
1 N of
1 ‘-v.t
' sp- c t to 2 * i f i ,
1 '0 Tc
jc'jd.v
ItT r - .
o\
: hr t..
't
Oi
KTt T' ' m itudl
:... C '.[Tnal
inte —lticn n u irrio d ly .
c
TTcn
It
fin s to
Tic T fcct- of any Ivon one of those
IT H L rl CcfUott.: is on the rersidaal stru c tu re c
C
Vift--^AeJ dtxflv.ctdon o f
c whole c m be leteasaSaed# by
I : O dry; tL t le flo c tio n on th e e th e n d s e unloaded stru c tu r-r, c a lc u la -
. --C C-c:.! Ki
r ta
d io triW tin c.
Thuo, rig . $ ahoro a iu d t v ertic al
dofl notion lnpoacd on Jointo % and I* , rnf the r vn d tin e fixed-end no .imt#
n T: t a , r rc t o
1 C:
'T ■
-if r n f :, vrz-tlc&l m J
V:•• • .; ; , t r> ' ' ,U
rriccnt.i^ h i c r s r f t r r oofr.
—lu —
«a.) UfUdion
4
fix&c/-end Aio/nan^
Ay) DidribuhdZdonKtfjhj ^arIic^ / / boriyonlnl
fctn<d Forc<L5
Condition # I
Fig. 5
*
-1 9 -
S"
i.lo tw .- OLsAix t .
$ SkaLl Iionccfw w be
tcxl Con*
• ■' -i I (u. I c ia o f u n it v a v tic u l (ML#pl6ocamt of jo in ts I ixui I * ) .
ill
fc-reoa ai ^ n rn t in S i l s o n d itio n w ill th e re fo re beer the se te o rl; b I f o l ^vl'- •• t.
Ii--Cution aubociiot*
V^i i s total th e v t . t i e d fo rc e i n v«nel 2
.
... '
3.
r.
.
3
fa
.
2
,
% "^
,
Wkl O f o r o il o f . c r Io W s), and the shear in t.x .t : en el a f te r :tls-
Lribro -n e f f l
t e ..•: . I:,..e
.b-vu i
i.
a cr.- r...l Ir '. .:at lotid m 3 f t m
to be o , then
u eto sj Wlcrv :." to W fXect u n t i l tin vddltdem l sh-: cor St-Q- i s
ixitrc:.; n d in to to e p: iel.
I t fc s been s ta te d t h a t Condition A w ill eod-.-t with on in te n s ity * '
■' t
'b-.fl !."Lvotb-ii ; i t L
L ... 'i
Tio
7
I b t, e tc * , and f in a lly
' L" I * *
... ■ . ;. • f. rtf
LL-L' .
*
Condition W iS it
of c "tl ns I to la ta*.
.-re ue.-;
.
th r
-'J- led i n
...
ok;
i i IL v 'b
f
suexr I .: to
Tn f
'
.•
Iccc
of
if • s) eye oft
a i/„ + b Vtl+ c v,3 + d v,4 +C-Vtsr +f\flto + j vn = 5, - J-,
a
''li *- c 14? * c/ 4.4 * e
3 vJI + h ^iz +
3 hU + bHn.-r
f 4.6 + j 4?
p/ 4* ^ C t$r -I1 ff
r- 9 4?
+ ^ ^ 4 + CL Hts. + f U lu +
~ -Sz- ^z
c
5 ?- sA,
p,
;e t
—20—
SH2./ *
*■ c.
3 H
+ b H Jt
<3H 1
+
t d Hz4 + G H2F + JHtb *■ 9 Ht7 = JHt - SA1
f cM;;
b Ht
+
cHj
■*■ d H j t^
t
dl H 4
+ C H js" +
+ « H g-
J ^ Sb
1^ H j
^ 9^*7
t- J H b
t
jH
7
=
SH
~ -TAj
- sA
Tliia a c t o£ equcM- ns a h a ll be c a lle d th e c y ^ l i b r i i f : eductions fo r
tile stiru ctu re# A fter th< valceg fo: i'. c tu rs a t/a* ugl$ g ht c been detop*
■1; n©d» t
mb
o in t
c easts in Condi tio n -,I os*© s i nly
in Con;'.it.'-on 2 ly, ' , e t c . , and t b ;<
n
j l t i lie d by a , these
t s ■ c t:;en ■
■s> >'mta e x is tin g in the w tieflccto u u nafl*ure*
' to t ... joint,
S tw o z . ca*, ©q>. " ib s lm
.
I t «111 be viot-od t h a t t e l e f t aide of th e cooi iIbrlt-: eoustlons v d ll
al'.ivy
>f .Un too
c -f
I c
are »
<1
t!v :.', t a id es tr ill •’ a-■, a
accord ng t o th e a d d itio n a l ahecra oetsieU#
Aft as* Co lf>ti n of t3ae wwpis; w to t l a
-oint, 0-
I? tel: • b n lto cr ". ■ :iv$t
#
lssgs and too
<t n t ■m m tm r e m i t : nu JTyea the-1 are new c oobincd end : laced
on the aljMcrturc £
liltaricov .^ly*
As a
,-t >oiS' ■ in Ujx a of .. a© 0 i ; t , s t (da
JCsrctc.
-J xn v
t e
s u it,
:. t
. - I :■ n i l
be r: -
t: aotto.-;) I c 0/1Injij ) ms:, abe a d d itio n al
in s $! , V r., U-., I'&, V.,
A a , -I ' .
-
21-
DEFLBffriiOH EQUATIONS
t'> d ir e c t B trc ji ;.u*c
tl
TB OUii bo
I . " ' "i t - c • r..-.
I .)
t
' -U .1
vc.
T h e ^ d o f lc o -
a m to bo em ail, and ere no r l r i£Lw /3 e d i t e d .
' ■ i
I f cos-
Cl::. Cd 1. I t l o u t d i f f i c u lt; .
Doofc D eflections
a)
“epk H dctdon
Tlio ro ta tio n o f th e center o f th e dock ’.Till be
ew of e ll
d o ck -jo in t ro ta tio n s plun tlie ro ta tio n o f the c e n te r p o in t C
w ith r : a c o t to jo in t 3.
QaefWj (.solumn 0*-0 end cede se c tio n 0—1 under th e influence
o f t jc
Ji-Clvri
c CntoHc0, Lcq, end L1, H2, re s p e c tiv e ly .
A rb itra rily
c 003l1v^ On-CCLutiur r o ta tio n . o o ltiv e , tlie r o t .t i o n of jo in t 0 , l e f t span
is :
.
Mco f- Mco1
Mco J c/y
ZfZ<
(Meo' ~ Mco )
-2 2 -
:-::l Hat-1.---, th e ro ta tio n of Jo in t I is*
6 ^ i i J ( ~ i M' z - M1) cL x- - ~ f ( M 1- M i ) t
B:z Gnalr-St rotations Sg and
^ ~ zei ( mi3 - /W4 ) j
#*& th e r e t - ti.cn o f the
For th e l e f t
~
Dtrt : -
0
aret
<9j =
(ry\s - ZVi6)
e n te r m in t fl r d th re sreot to Jo in t 3$
Ai f , the t o t a l r o ta tio n o f th e c e n te r i s th.
( m ^o'- M co) +
(fZVl1-AI2 f-M3 -M 4 +
uoi-e:
-/Vl4 ^
~
)
O of ay:, ' t r y , th e ic c n tic e l ex;TceSaicn can be us d "or th e cc v.
' Cf t c v ■
ft : ?lf.
®/Z =
( hrxCe,' - nxCo) * J fl (m , - W2 r tti, - n<4 +Hxs - M it+
m-f- W1
Unit ntoixxen+
©
r
TX f
7
)
-
23 -
F ig . 7 shows th a t th is expression a lso gives a p o sitiv e sign i f ro­
ta tio n s are c l ocKva.se.
The common fa c to r
I)
i s dropped, and corresponding terms are combined:
Mt o ) - ( / r j c o - Mc o j J ^ J K- m ,
-
M
3
f
- M t) + z (/r ? 7 -fy j7) - j(rr?g -M ff) - O
b)
V e rtical Deck D eflection
The t o t a l v e r tic a l d e fle c tio n a t the center of th e deck w ill be
taken as the sum of the v e r tic a l d e fle ctio n s of the various deck
jo in ts .
Demoting the d e fle c tio n of jo in t i from th e tangent to
the e la s tic curve a t jo in t (i-1 ) by y . ,
the d e fle c tio n a t the
c en ter of the deck (Fig. 8) i s given by:
A y *Jy,+ A yt +Ayj ^Ay4 tjfoo+A Q , +AGt +G3)
F ig. 8
-
"•
("■'
24 -
* ti)*
= JTi Jo
= ^ 1 CZM1- H J
L*
A y 1 = — ^ M 5-M 4 ) .......
e fc
a t th e ooiitei- i s ■
+ YjTl
(M t-M z )+ 3 CMi
- M *) > (M r -M ^ )J J -
= ^ ( 9 . 51* , - &.S M z+ 6.5 M a -S -S M 4, + 3.5 M s -Z .5 M 6 * O S M 7 - O.Z 5 M 8 ) «■
'Jaln1 tiso i c 1:1Qc: toc rea o lcn c: t be uaod f or
r i ■t
<Uti of the
' -•'**IS
Ay
= — — f S . 5 tvi, - f l .5 Wz + 6 . 5"irij - 5 . 5 m * + 3.5 w # - Z .S Wg +O.Snty - O.ZS n ig ) +
T...1.3 c
I E l c ( rrtC o ' ~ rrtCo ) •
x'30:, .i, Lie .
v;:,i [,.vc a ,
-vc " , 1 o r
M^y d e flo o -
t l OG* (c<e Fige ?)
^yL =
rga .5Mt4-'
JOT.)
JJ [p.Sfa,+ At,)- 5.5
+
*.n oct" on ram
iH'
+6.3 Cmi + M 3)-S .5 Cm4 +M 4 ) + 3.3 (ms * M
o.s(nt7+M 7)-o .Z S C m 8 +M 8 )J +
,) -Z .S On 6 * M6 ) +
£cm Co, + M co,) - ( m Co+ McoJj
=0
—25>—
O)
Uv:.':!'
ec i: Cfloctiv-U
TiiQ lto^lsom a l dciloctior. cT tho deck i s . .ensured by tiie .-orisjenUEl
kj
le c t io f
o f the celvr Uis at ti c l e f t and v i lit abut en t,
e-
.6 :
A lJ ^
{
(rZ v g U l v-
BotIi GXfa1Oaaiona are •o s i t iv e if 1Ofleetionn
ao; inn the c
on factor
)y d y •
„
M
• *o th e rifiit#
_
Z ( ^ ca' ~ Mco') ~ C Co ~ M co) = O
IB.)
2)
Arch efieoticsns
a) Arch dotation
Bm c f a m .«i d Ijoint rdtationa jlus the ro ta tio n of t i e center Ow ith respect to
.
-2 6 -
F ig, 9 ohcyrs cz'c' g ctio n C1 - I 1 vndcr the action of
I1 an Mt0* on- horizontal
o o itiv e xnc ents
nd v e r tic a l f croon Ht 1 and V11, v/hlci’, on the
:
i .
ioi- t u
,.I e an*
TiiiD c< ___ :a\,o ocnti r i s
(
y ^ f 5 =j ~
ua-'
,c by (x,
)♦
a t t t- n ciitiv l
o in t
‘ °
*
A rbitr-^ril., chcooint dlcdnsioe rote . t i t o a i t ' v c t
V1'C r , - - H /Cf,-if)
Q>s J
Iacc
K'z,
-
hI v Jf i
- mI
= J f iW K - Kv,-"I)
•:
©
%
11
By anal
’ I ft
‘o r
f*r V f
0
;
'T 1
C
I ,
I
©
-
-
.
■
JgW K-Kv.t JI
t
C^4%4 ~^4*/* ~ Mg).
Al it - n tic i I expression £oi ..
o f t. :t. ri<jkt ;iolf o f th e arofct, t.-c p o s itiv e U irc c ti.,ns of
v. ... c
. OlCC
11
t/ c -. CU ..: •
•
o r ie n ta l and
-
27 -
y
'it^h B eetle# Ol-I* (TSciit) mt;'cr th e entdots o f p o e ltlv e
I
h
v'T
2F
'.'-'e
' t
Xv ‘j:-’ ion
c
"t'
:c
•v'
x?lv '■t-'-v-
-(mZ - M l p f p f i , &
-M i)] J-
T.
(Fs . n
tz
B tl CA*
28 —
—
* 4 ) 4 '4 %
^^
-2 9 -
I t vr’l? nov,- Iy COtwrnIeni tc -r
o
o
tu to s
<9
©
) i e /c .
O
J g
4' =
O
Q
* -
-iiouv
>q a
ex’tdou ox I
r ■. sjiIV4.es t d l i
o;.cad entirely on the ; r/adcal pro -
ti-oi > • ... u i l neecaauTj I.»-U.4-a *or uacir c
cbtalnod
. incd*
efc.
t>'
o el. ,Lie couutvuv... Ic j. t o vro.'
u ta tio n v d il be
is
.no ure J e t c r -
For t i c l e f t span# i i x tot.a£L v e r tic a l J f le c tio n of t
o ra l c e n ter
ic tiler: :Tc ■:
+ Z4p4 - >/4 ^4 -
r4 ,
.
be. used for i ' ’ u-:i
i v e r tic a l d f ' exit.V iv;, b rt i ■&. t
be posi ti v *V- - - - I I- flo e ti-
Jfj
P/ (^+V1
J~
fp }
■ v.
.
+H
’
>
)- r, (**1+M’
z) -t P
lC
v
^4 V
I) -
V i ) - faCk's + ^ i )
' -•':
- r} Am6 *M4 ) + p4
G 4+ V4)
U
tt U
1
z) -V
1C
m
4+M4^t
- ^ ^A4* H4 )
- r4 (m # + M g ) = O
-
c)
30-
H orizontal Arch D eflections
The t o ta l h o rizo n tal d e fle c tio n a t the arch cen ter (Fig. 12) i s
given by the equation:
A t * A t , + A jl +Ati +Ai
h
i OlZ h +OzZ h +Ojhi,
F ig. 12
Tihere
Xi i s th e h o rizo n tal d e fle c tio n of arch jo in t i due to mom­
e n ts, h o rizo n tal and v e r tic a l forces in arch sectio n (i-1 ) - i, and Il
1
- is
the height of arch se ctio n i , or T i- Y (I-I)*
A r b itr a r ily choosing d e fle ctio n s to th e r ig h t p o sitiv e fo r the l e f t
h a lf of the arch:
-3 1 -
D u b stit.Ltlnci
...
Q
a;; ;
..."
v' '
' b'::~ c l.
<D
... o*
0L
C
'flTf.
4 = « J i-e/
^
H'sI ~ zvfZ+1 + ^ I l - hLsZ ~
:'r r '
be i;
«'u.3o lie
^ ^
t
'
- Hi si - 1^L+S +
- H14 S4. - M rs +4 .
10, I t y t l l bo octm tiin t <m ‘ 1Cnticvn c-—m a l w nay
■“ I- r i "c-a an ’.oriacnta3 deflect* cco, cirl t h a t t ' *g esTreaoion w ill
"-!"v fo r V -n e c f <na to •.bo r"
~
^ l s, -
w Z-A
vL f z -
t,
+ ' L f 3 - ^ 1i S3 -
W >3 +
V j f 4 - h 4 S4
- W ^ 4 .
AX =
n j
^ r
K / ; A , ' - ^
- s s ( A 3 - H3 J - - S ms (Mg -
K) - ^
) + f 4 (^4
- K, 3 ~ J4
^ 4- v j ) +^
H4 ) - 14 (>+ig- M g )
=o
-
32 -
SI r LTrYim ASST FTlOSS
Y..? F
L c ' T: .-mtlomi 3 dLY be n ^ T w ' c ' M U
duce Cm ^ ount o f ca" c a c tio n ; r c r;s c r v 'Ar ' ^
I ^
’iv ec’ stlv-'-S Id v
D efer a t ions t 'ue t
I)
re ­
r rco ' .obi .
The 'tr-’ct'-'C can be
2)
1 - aea.brit t r :
csf!-T a- Viv- 4trite • cl c f seven r e s t r a i n ts
Asmvtrtion I)., ( O r -d - ln co r c ra te d i n the f o r .trV*dr>3 c f d e fle c tio n
^TAfifclons, i s c core on one, general "f er.r Io -ed in t e o n .I'M ' of ei ipec
end c
Lo:: frc o s.
Asa
r iu
I t s v d l d i t y M i l th e re fn -o not- bo rr-nre-'
*
tio n 2 ), v.hie t M i l red—o th e number of Si u ltv oouo
;P ib~
equations to be solved fo r anv given condition of Load ug ?J* ■' -,r vt^n
to fo u r, i s booed on th e ocr^ep t th.-.t creh sootions b e tt e r n lo in ts c: , Lc
tr e a te d as s tr a ig h t
c bom w it out in tro d u c t'o n of c ?f
Yic Vs t i f l o a t ’ - o fo r
i 3 . e - . '0 l l
b rscd on t;ie
akin; any s i IH f iv
<-2TCi’n*
asa -v.pt - - an ait" anal -s i s
re e f t h a t ik e e rro r th e i-b
l u t r L - v i M H be
rigorous m.
I t cru on" ' be sM ' t " !
t:.
f
-to ir f :*o s , s a
'•■d-arus c f C flvaticity of concrete and fo r ta
..j
0
c - tf s c o f
. , :_cvrr
a ' ' v"
, n" d fw
nt of v-v v-e of rM n -
l i ' c l y fx- 'n iro .X 'c crroi% cooM dcrobly
largor than thcae r e su lt!nr- f r r ear: rfcHns I & •
m t i e f b l l 'M n r ,
ro o f in o ffe n d f
l>o VCld a g a in st tr e slat.ion b
Inbs
r t o nuit - *o atasa
t the r e s i d e rroh a tr - o t'M eon
fo u r r e s t r a i n t s , i f arc
t.
:. iu ..tion*
.
s Shown in p o rts a and b*
se ctio n s bet- con
-
33-
Fig. 13
The proof of th is assumption w ill hinge on whether or not arch jo in ts
I ' , 2* and 3 ' can be held ag ain st tra n s la tio n by purely v e rtic a l or purely
h o rizo n tal fo rc e s, o r, in other words, whether or not th ese jo in ts are
capable of d e fle c tio n in one d ire c tio n only.
Proof:
v
Fig. 14
34
The arch sectio n Shovm in Fig. Iii i s acted upon by V, H amd M.
By
v i r tu a l vrork:
Now V i s assumed such th a t A y = O
Solving fo r M in the second eouation and su b stitu tin g in to the f i r s t :
/ 1 ^
4
/4 ;S W
i i 0
M
W
y¥
- A
1M
I f a lin e a r re la tio n sh ip e x is ts between x and y, of th e form y s
/
Hf
V
V zzyyi - /
4
ax,
- J z T a x c/x.
S u b stitu tio n of these fa c to rs in th e above in te g ra l equation v/ill re ­
duce th e c o e ffic ie n ts of V and H to zero.
To evaluate the e rro r introduced, the expression fo r/1 x can also be
w r i t te n : .
-3 5
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A ll these q u a n titie s w ill be
known from the determ ination of the e la s t ic p ro p e rtie s of arch sectio n s.
I t i s th ere fo re p o ssib le to evaluate num erically the c o e ffic ie n ts of V
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At
The equilibrium equations must now be modified in accordance with th e
nreceeding assumptions.
Since th e re w ill be a d e fin ite re la tio n sh ip be—
tween h o rizo n tal and v e r tic a l d e fle ctio n s of arch p o in ts , they can be per­
m itted to occur sim ultaneous!v.
groups shown in F ig. l £ , b and c.
This reduces panel d e fle c tio n s to th e two
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N578_____________ 98579
cop.2 L o r ell. William
Analysis of an open spandrel arch
Including superstructure.________
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