Frame
Equiualent
MethodAppliedto
GonqeteSheatwalls
uildings t ha t i n c o rp o ra te
concreteshearwallsas structural elementsto resist both
vertical and lateral loads are
commonplace. Shearwall and coupled shearwall structures have been
found to be economicalup to the 30
to 4 0 s t or y r ange , a n d s h e a rwall,/frame structures have shown
their effectivenessup to 50 stories.l
Th e c alc ulat ion o f s tre s s e sa n d
deflection in a simple shearwall req u i res only r udim e n ta ry b e n d i n g
theory. Often however, one or more
columns of door and window openings create two or more shearwalls
coupled together at each floor. UnIike the simple shearwall,the analysis of coupled shearwalls is by no
me a ns t r iv ial. F ina l l y , th e s h e a rwall,/frame system adds yet another
degreeof complexity.
There are four main methods for
analyzingcoupled shearwalls:
Scale model testing - Typically
u se d in r es ear c h f ac i l i ti e s to v a l i date or confirm proposed theories,
it is not normally employed by design offices due to resource, time,
and cost restrictions.
L a mi na M et hod - Al s o re fe rre d
to a s t he c ont inuo u s m e d i u m
method, it replaces the individual
coupling beams between shearwalls
with a continuous, uniform, homog e n e ousm edium , re fe rre d to a s a
lamina. It assumesthat the point of
co u n t er f lex ur e oc cu rs i n th e mi d sp a n of t he c ouplin g b e a m s , th a t
the walls deflect equally when subjected to horizontal loads, and that
the walls resist the loads in proporNovember1991
tion to their stiffness. The method
takes into considerationthe contrib u ti o n m ade to the shearw al l s by
the bending and shearin connecting
beams. It is a hand method that in
its pure form is tedious, but graphical methods proposed by researchers assist in removing the drudgery
fro m th e method.2' 3H ow ever, i t i s
l i mi te d t o rel ati vel y hi gh shearwalls, with constant floor heights
and uniform openings.
F i n i te El ement Method (FE M) This method partitions a complex
elementinto smaller componentsof
a fi n i te si ze and number. The geometry of these finite elements are
simpler than the boundaries of the
overall element. Usually the analys i s i s b a s ed on assumed di spl ace,me n t fu n cti ons. B ecause of the
n u m b e r of cal cul ati ons requi red,
e v e n fo r si mpl e el ements, thi s
method is limited to computer applications. Even so, with large complex elements,idealized into small,
numerous finite elements, computation time can be significant. It is
gaining wider use, and may be the
most appropriate method of analysis for some complex problems.
FrameMethod(EFM)
Equivalent
Also referred to as the wide column
a n a l o g y , i t repl aces the coupl ed
shearwall components with an idealized frame structure that behaves
i d e n ti c a l l y to the shearw al l . Thi s
idealized structure is resolved using
matrix analysis techniques. A first
order linear elastic analysis is performed. Although possibleto carry
o u t a m a tri x anal ysi s by hand, i t
qui ckl y becomes ti me consum ing
and compl ex as the si ze of t he
structure increases.Like the FEM,
analyzing a structure in matrix form
is ideally suited to the digital computer. The matrix method of analysi s i s l ess computati on i nt ensive
than the FEM, and consequently is
of great interest to practicing engineers. Its adaptability and flexibili ty have made i t popul ar in engineering offices.
The equi val ent frame met hod
provides a good balance of effectiveness,efficiency, and easeof use.
The application of the EFM to concrete shearw al l s i s by no means a
new topic. But this paper attempts
to assembleinformation that has either been scatteredin a number of
sources or typically has been covered as an asi de. C onsi dering t he
number of structures that r ely on
shearwalls to resist part or all of the
l ateral l oads i mposed on t hem , a
more unified presentationof the information is warranted.
Background
The two main proceduresof matrix
analysis are the flexibility and the
stiffness methods. The flexibility (or
force) method is a generalization of
the Maxwell-Mohr method, developed by J.C Maxwell in 1864, and
O. C. Mohr a decadelater. By writing compatibility equations in terms
of fl exi bi l i ty coeffi ci ents and selected redundants, statically indeterminate structures are analyzed.
The probl em w i th thi s me t hod is
that the choice of redundantsis not
unique, and an inappropriate one
65
Equivalent
Frame
Outline of coupled
shear wall (deflected
shape undsr lateral load)
i *'+4 P
'-Lr
T
b 1
Lrl
Fi g. 1 - Coupleds h e a rw a lsl u b j e c tto h o ri z o n talloadi ng
wit h equiv alent
f r a me s u p e ri mp o s e d .u
can lead to significantly increased
calculationrequirements.
In the stiffness (or displacement)
method, no input beyond defining
the structural model is required to
car r y out t he an a l y s i s . T h e u n known quantities are the joint displacements,and the number of unkn owns is equal to th e d e g re e o f
ki nem at ic indet e rm i n a c y o f th e
structure. The directnessof this app ro ac h m ak es it s u p e ri o r to th e
flexibility method for applicability
to computer analysis.
The principle of superposition is
fundamental to the application of
the stiffness and flexibility methods
of analysis.For this principle to apply, the structure must be linearly
e l a s t ic , whic h m e a n s th a t i t m u s t
satisfy the following three requirements:
o T he m at er ial o f th e s tru c tu re
must be elastic, having a linear relationship between stress and strain
(Hooke's law).
r The displacements of the structure are small. so that calculations
involving the overall dimensions of
th e s t r uc t ur e c an b e b a s e d o n i ts
original dimensions.,
o No interaction exists between axial and flexural effects in the memb e r s ( t he ef f ec t th a t a x i a l fo rc e s
Fi g.2 - C oupl edshearw aland
l
concretef ramewit h
equi val ent
f rame superi mposed
have on members, when combined
with even small deflections is commonly referred to as the P-A effect,
which is nonlinear. This topic will
be coveredlater in the article, along
with other considerations).
It i s not necessaryto know the
specifics of the theory behind matri x methods of frame anal ysi s to
a p p l y i t, but a practi ci ng engi neer
must be aware, at least conceptually, of the underlying principles. It
i s o n l y i n thi s w ay that the engi neer
can have a better understanding of
th e ty pes of structures to w hi ch
such analysesare applicable, as well
as the method's limitations.
generally those for the corresponding wall sections,sincethe structure
i s assumed to behave i n a linear
elastic fashion. Note however that
for squat w al l s (l ength is gr eat er
than 2 ti mes hei ght) shear def lection predominates;for slenderwalls
(hei ght i s greater Ihan 2 t im es
l ength) bendi ng defl ecti o ngover ns;
and for walls in between, a combinati on of shear and ben ding controls the deflection.aThe combined
deflection for a cantilever fixed at
the support, subject to a uniformly
distributed load is:
wLa
Aaror-
gg1
Equivalent
FrameMethod
Equivalent
Frame
In applying matrix analysis to concrete shearwalls,the coupled shearwalls are replaced with a centerline
frame that displays the same behavior as the elementsbeing modelled.
T h i s m odel i s referred to as an
equivalent frame. Fig. I shows the
essenceof the approach, where the
center lines of walls and connecting
b e a m s form the members of the
frame.
SectionProperties
The sectional properties of the columns in the equivalent frame are
0.6wL2
T
(L^o,n",,)
GA
(L,n"o,)
*
where
lr : uniformly distributed load
Z : length
E : Young's modulus of elasticity
G : shear modulus
1 : moment of inertia
A:
area
The combi ncd defl ect ion f or a
cantilever fixed at the support, subj ect to a concentrated load at it s
free end is:o''
1
-tot
PL3
--
I
3EI
(L,o^",,)
I.2PL
cA
* (L,*",)
ConcreteInternational
Pecknold,
Allen
andDarvall
Carpenter,
MehrainandAalami
Brotchie,
square
Brotchie,round
II
t-
- It
.. .
no|o
flexible
MehrainandAalami
AllenandDarvall
KhanandSbarounis
squale
round
SquarePanel
SquareColumn
Poisson's
Ration= 0
Forusewih programusing
zerosizejoinb
- ExrapolatedCurve
0
Fin
iv |Y.
f
' o-
_
6 ' h3^/l2
where
6 : thickness of connecting beam
/ro : depth of connecting beam
D : width of opening
1, : reduced moment of inertia of
connecting beam.
And
Ib
where
4 : moment of inertia of connecting beam
d : depth of connectingbeam
u : Poisson'sratio
T h e a d di ti onal hori zontal secti o n s b e t w een the frame col umns
and the connecting beams are stiff
ended elements that rotate but do
not bend. Theoretically, they should
have infinite areas and moments of
inertia. Programs exist that do allow for end sections of beams to be
infinitely rigid, but for many matri x a n a lysi s programs, extremel y
large section properties can create
errors or large inaccuracies in the
results. If however, small inaccuracies are acceptable, then perfect rigidity is not required. The followi n g c a n be used to determi ne the
properties of the stiff ended beam
element(Fig. 1):'
-___L_
l+ 2. 8( h ./b )' z
rl
| + 2.4(d/b)3 (l + u)
November1991
0.10
0.15
c/g
0.20
0.25
i nto
of severali nvesti gati ons
Fi g.4 - C ompari son
effecti vesl abw i dth.n
F f f e c t i v c s l a h wid th .s
where P : concentrated load
The foregoing is provided to allow the engineer to consider shear
deformation for walls with small
height to depth ratios, where a reduced moment of inertia may be in
order.
Shear def lec t ion mu s t a l s o b e
co n s ider ed t o pr op e rl y m o d e l th e
behavior of the beams connecting
the shearwalls. As can be seen from
Fi g . 1, t he c onnec ti n g b e a m s c a n
undergo relatively large deformations (especially those in the upper
portion of the frame). The equival e n t f r am e beam s h a v e th e s a m e
area as the connecting beams, but
the moments of inertia must incorp o rat e a s hear defl e c ti o n c o rre c tion. The following formulas have
been proposed:6't
0.05
A"/Ar :
I"/\:
100 (e/f)
100(e/f)3 + 300(e/f)'1
+ 300(e/f)
where
e : length of stiff ended section
f : half length of connecting beam
A" : atea,of stiff ended section
1 " : moment of i nerti a of sti ff
ended section
Ar : ajea of connecting beam
1r : moment of inertia of connecting beam
Loadi ng
To model the effect of horizontal
loads on the equivalent frame, the
exterior loading on the shearwall is
replaced by an equivalent concentrated load applied at the joints (intersection of columns and beams) of
the frame.
Analysis
W i th the confi gurati on of t he
equi val ent frame establ i shed, t he
section properties determined, and
the l oadi ng prepared, the equivalent frame is ready for analysis using one of the various matrix analysis programs available.
Otherconsiderations
The foregoing establishesa method
for analyzing coupled shearwalls.
B ut, i t i s unl i kel y that a b uilding
will have just one coupled shearwall
acti ng i n i sol ati on. More r ealist ically, buildings will possessnumerous shearwalls of differing configurations, shearwalls acting together
w i th el evator,/stai r cores, shear w al l s,/frames, or a comb inat ion
thereof. These gi ve ri se to ot her
factors that require consideration,
including the connecting elements,
i n ter ac t ion wit h fra m e s , s u p p o rt
conditions, symmetry, torsion, and
varying opening widths.
Co n nec t ing elem e n ts
Where the connecting elements in
coupled shearwalls are well defined
beams, the section properties of the
beams are as discussedpreviously.
But in many residential buildings
and hotels, the flat concrete floor
slabs are the connecting elements.
The interaction that occurs between
coupled shearwalls depends on the
bending stiffness of the connecting
elernents. In the case of flat slabs.
th e dept h of t he c o n n e c ti n g e l e ments is merely the floor slab thickness, but the width is not quite as
obvious.
A n as s es s m e n t
made by
Schwaighof er on a s y s te m c o mprised of two 21.3 ft (6.50 m) wide
by 194.4 tt (59.25 m) high concrete
sh ear walls s pac e d 5 .3 ft (1 .6 3 m)
apart, and connected by 8 in. (203
mm ) t hic k c onc re te fl o o r s l a b s
showed little difference in shearwall
interaction when using a slab width
o f 2r . 3 f t ( 6. 50 m ) a n d 1 0 .7 ft (3 .2 5
m) respectively.6For larger shearwall spacing, the same may not be
true. Except for very high and low
slab stiffness values, the stiffness of
the system is greatly dependent on
the slab stiffness.TThus, when flat
slabs are the connecting elements
between coupled shearwalls, it is up
to the engineer to consider the system on a case by case basis, review
the literature, and decide on an appropriate effective slab width.
Interaction
with frames
Many contemporary buildings employ a combination concrete frame
a n d s hear wall ( F i g . 2 ). Si n c e th e
stiffness of concrete frames is sensi ti v e t o t he s la b s ti ffn e s s , th e
shearwall/frame combination will
a l so dis play a s e n s i ti v i ty to s l a b
stiffness.
Many studies have been carried
out to determine the most appropriate values for slab stiffnesses,and
d i ffer ent m odels h a v e b e e n p ro posed. One of the more well known
w a s dev eloped by Kh a n a n d Sb a rounis, who in 1964proposed effective widths of slabs varying from 20
to 60 percent of the slab width, dep e n ding on t he g e o me try o f th e
frame (Fig. 3)8.
However, in 1983 Vanderbilt and
Corley compared the results of sev68
eral investigations with respect to
s l a b wi dths. Thi s compari son
s h o w e d effecti ve w i dths varyi ng
from 20 percent of the slab width to
greater than the full slab width (Fig.
4 )' g . In 1988 C ano and K l i ngner
compared different analysis procedure for two way slabs, which included the effective width method
(Kh a n and S barouni s), the A C I
equivalent frame method, and the
e x te n ded equi val ent col umn and
slab methods (Vanderbilt).r0The results from the various analyseswere
c o mp ared to test resul ts from a
small scale multistory flat plate test
specimen prepared by the National
ResearchCouncil of Canada.
F o r l ateral dri ft. the effecti ve
w i d th method provi ded the most
a c c u ra te resul ts. w i th the other
methods giving similar results, but
over estimating the drift. Thus, like
the effective slab width in coupled
shearwalls. the determination of an
effective slab width for frames is
l e ft to the di screti on of the enei neer.
Supportconditions
All two dimensional matrix analysis
programs allow at least three types
of supports. A roller support is restrained against movement in just
o n e p rinci pal axi s. A pi nned support is restrained against movement
in both principal axes. A fixed support is restrained against movement
in both principal axes, and also resists rotation. Many frame analysis
p ro g ra ms check for stabi l i ty onl y
superficially, so it is worth noting
that for a two dimensional analysis,
a minimum of three restraint components are required for static equil i b ri u m. Thi s can be made up of
one fixed support, one pinned supp o rt a n d one rol l er support, or
three roller supports not located on
the same parallel lines or arc.r' For
th re e d i mensi onal frames. an i nspection of the model may prove to
be the best way to determine stability, and can be accomplished by ass e s s i n g w hether the frame can
tra n s l a te al ong a pl ane, or rotate
about an axis in an unrestrained
m a n n e r.l l
U s u al l y, non yi el di ng supports
are assumedat the founding level of
vertical elements of the equivalent
frame, but conditions arise where
v e rti c a l and rotati onal fl exi bi l i ty
m a y b e requi red.' C omputer programs exist that provide the ability
to model inclined roller supports,
spri ng supports, and suppor t s r estrained partially against rotation.
In the event that an engineer does
not have accessto such a program,
these support conditions can be reasonabl y approxi mated.rl An inclined roller can be modeled with a
short, stiff, pin-ended member inclined at the required angle (Fig. 5).
Spring supports (used to simulate
specific soil conditions for instance)
can be model ed w i th p in ended
members whose properties approximate the spring constant required
(Fig. 6). Partial rotational restraints
(somewhere between a fixed and a
pi nned condi ti on) can be r epr esented with an element perpendicuIar to the actual member to be partially restrained, with properties approximating the degree of restraint
(Fie. 7).
Symmetry
The engineer can take advantage of
buildings that possessa symmetric
structural layout by apportioning
the lateral forces to all shearwall
and frame bents in a single operation (Fig. 8). The frames are idealized using EFM and linked at each
fl oor w i th beams hi nged at t heir
ends. The link beams should theoretically be made infinitely stiff. But
as discussedpreviously, this presents a probl em w i th som e m at r ix
anal ysi s programs, so th e beam s
shoul d be made suffi ci ent ly st if f
such that their axial deformations
are negligible. This is based on the
assumption that the floor slabs act
as rigid diaphragms.
For long narrow buildings, or for
buildings whose lateral load resisting elements are almost as stiff as
the floor slab diaphragm, then the
distribution of horizontal forces is
not the same as that derived assumi ng a ri gi d di aphragm.T The engineer should be aware of conditions
where the validity of the rigid diaphragm assumptionshould be questioned in order to not be misled by
invalid results.
Even when the structural layout is
not symmetri c, i t may b e t hat a
gi ven frame w i thi n the st r uct ur e
displays symmetry. Engineers can
take advantage of favorable geometry to significantly reduce the size
of the frame to be analyzed.
In two dimensional matrix analysis, if the line of symmetry is coinci dent w i th a col umn l ine ( even
ConcreteInternational
Structure
being
supported
Structurebeing
supported
Member with section
propenies to simulate
beingmodeled.
Verticalpin endedmemb€r
to simuldeelasticsoil
supportconditions.
Sff ing
constentis AE/L.Adjust
A&Ltoapprdximate
soilbehavior.
Short, stiff (large area)
pin endedmemberslqped
at angle to simulate
inclinodroller
FRAMING PLAN
Clockwise from above left:
Fi g. 5 -
M odel f o r i n c l i n e d ro l l e r.l l
Fi g. 6 -
M odel f o r s p ri n g s u p p o rts ."
Fi g. 8 - Lat er a l fo rc e d i s tri b u ti o n b e tw e e n el ement s ( f or s y m m e tri c p l a n l a y o u ts ,d i s tri b u ti o nof
l at er al loads c an b e a c h i e v e d i n a s i n g l e o p erati o n by link ing el e me n ts .)
Fi g. 7 -
M odel f o r s p ri n g s u p p o rts ."
Shd, stifi (lilge a@)
pin ended mmbss
lir*ing fril6
at
e@h floor.
Structurebeing
supported
W2 E
Horizor$almernber
to simulateoartialrestraint.
Springconstantis 3El/1. Adjust
| & L to modeldegreeof resEaint.
number of bays), then only one half
of the frame needs to be analyzed,
with one half of the lateral forces
a p p l i ed ( F ig. 9) . T h e s e c ti o n a l
p ro p e r t ies of t he c o l u m n o n th e
symmetry line are likewise reduced
by one half. Finally, the column on
the line of symmetry is restrained
from axial deformation with roller
supports introduced at the column
joints at each floor. This is to simulate the cantilever bending action
of the frame where. for horizontal
loads. the neutral axis is assumedto
pass through the line of symmetry.
If t he line of s y m m e try p a s s e s
th ro u gh t he c ent er o f a b a y (o d d
number of bays), again one half of
the frame is analyzed, with one half
o f th e hor iz ont al fo rc e s a p p l i e d
(Fig. 10). To simulate the neutral
axis of the frame subjected to horizontal loads, roller supports are introduced where the beams intersect
November1991
the line of symmetry at each floor.
In three dimensional matrix analy s i s , p l anes of symmetry can be
u s e d to al l ow the anal ysi s of one
half (for one plane of symmetry), or
one quarter (for two planes of symm e try ) o f the three di mensi onal
frame. The procedure is similar to
the two dimensional case. but additi o n a l s upport el ements must be
p l a c e d s o that the hal f or quarter
frame cannot rotate when subjected
to horizontal loads (Fig. 1l).
Although strictly speaking not an
item covered by symmetry, lumping
te c h n i q u es represent another
me th o d b y w hi ch the engi neercan
reduce the size of the model to be
analyzed.TWith any computer analysis, the intent is to model the behavior of the structure being analyzed. It may be possible to achieve
this, within reasonablelimits, with
a lumped model as well as the full
(unlumped) model. In this process,
the area and location of the vertical
elements in the lumped model are
the same as in the unlumped model.
If n (two or more) typical floors
in the frame are lumped into one
floor, the moment of inertia, and
the area of the horizontal member
i n the l umped model shou ld be n
times that of the unlumped model.
The hei ght of the col umn in t he
l umped model w oul d be m ade n
times that of the unlumped model.
Fi nal l y, the moment of i ner t ia of
the col umn i n the l umped m odel
w oul d be n' ti mes the mom ent of
i nerti a of the unl umped colum n.
The technique could be quite useful
w hen carryi ng out a prel im inar y
matrix analysis of a frame in a tall
bui l di ng. W hen usi ng th e t echnique, the engineer should use both
care and caution to ensure that the
correct modellins of the structural
Rollersupponsrestraining
verticalmovementareadded
at eachfloor.
Symlnerry
Lil
?
(
3)
t5 :
4;
i
I
F
Ab3,4=Ab2,3 Ab4,F=Abr,zi
lbE;4=lbz;3 i h4,5= ht,z i
Abz,i
Abl,2
|be,3
lr l
Varyingopeningwidths
Ni
oi
<t s
ili-
*i
N
o
<5
il-
r!
<3
<: 3
!i1$
I
Lr,z
ffi""'
i
Lz,s
=
Le,+
I
=
I L+,s'"""'/:.
/------
lllzg
ll
I
Lt,z
Onlyhalfof frame
is analyzed
I
Ac, Ab= areaolcolumn&beam
lc, lb = momentoi inertiaot column& beam
L = tengttr
I
t i th c o l u mn .'
S y m m et r yl i n ec o i n c i d e nw
Fl g.9 -
Roller suppons restraining
vertical movement are add6d
at each floor.
.)a D
4
YY
wt2E'
<i E
EI
fl i T
tl
<: - 9
E
L}!\a
Fi g.10 -
Onlyhalfof frame
is analyzed
Ls,+=
t,
Lt,z
I
Ac, Ab = areaot column& beam
lc, lb = momentot inertiaof column& beam
L = lengrh
l
S y m m e tryl i n eth ro u g hc e n te ro f b a y .'
b e hav ior is not c o m p ro m i s e d fo r
the sake of reduced computations.
Torsion
Th e ef f ec t s of t o rs i o n o n l a te ra l
l o ad r es is t ing ele me n ts c a n re s u l t
from either an asymmetrical structural layout, unbalanced wind loading, or eccentrically applied earthq u ak e loads . W h e re s h e a rw a l l o r
shearwall/frame structures are subiect to eccentric loads the distribu-
It i s possi bl e to use the EFM t o
model coupled shearwalls with nonuniform openings simply by shifti ng the centerl i ne of the wall elements (Fi g. l 2). H ow ever , t his ignores the rotation of the walls at the
offsets, resulting in deflections that
are sl i ghtl y l i beral , but accur at e
enough to predi ct the over all behavi or of the bui l di ng.' Since t he
behavi or i n the zone of t he wall
offset may not be correct ly pr edicted using the EFM, the finite element method should be considered
for a detai l ed assessm entof t he
stressdistributions in the zone.
P-AEffect
As statedearlier,thisis thetermfor
Abi,4=,4bl,z:
lbir,4;lhi,t :
l
I
S tructures w i th Torsi on" and includes a worked out example of its
application.
tion of lateral forces to the various
elements is non-trivial. If a threedimensional, matrix-analysis comp u te r p rogram i s avai l abl e, the
analysis can be greatly facilitated.
In the event that the engineer does
n o t h a ve accessto a three di mensional frame analysis program, it
can be done either by hand or with
th e h e l p of a spreadsheet. MacL e o d ' 2 di scusses i n detai l the
"Component Stiffness Method for
the effect that axial forces have on
members that undergo even small
deflections. To accurately take this
effect into consideration, a second
order non-linear analysis including
the effect of sway is required. However, the ACI reinforced concrete
code gi ves provi si ons for t he approximate evaluation of this effect,
through the use of a moment magnification factor (ACI 318-89, secti on 10.11).' 3 N evertheless, t he
commentary of the A C I does endorse the use of second order frame
anal yses to di rectl y i nclude t he
sway, or P-A, effect.
If such a program is not available
to the designer, then it is possible to
take the P-A effect into considerati on by modi fyi ng the fir st or der
(EFM) analysis.TThe technique requires the following procedure:
o The horizontal and vertical loading is applied to the structure, and
usi ng the E FM anal ysi s, t he f ir st
order di spl acements (Ai) at each
story are obtained.
o Knowing the story displacement
and the accumulated gravity load at
each fl oor the addi ti onal st or y
shears caused by the P-A moment
can be cal cul ated. The net st or y
ConcreteInternational
Symmetry
Lrne
\
'/
sh ear at a giv en l e v e l i s th e a l g e braic sum of the story shears from
th e c olum n abov e a n d b e l o w th e
floor.
r The f inal s ec o n d o rd e r d e fl e c tions are calculated from the first
order deflectionsas follows:
\-
Symmetry
Line
Movement in
y direction
restrain€d.
Movernent in
y direction
resrained.
Movement in
z direction
rstrarned.
Y
I
1 - DPLi/Hh
where
EP : cumulative vertical load
Ai : first order story sway
h : story height
H : horizonral shear
o The addit ional s to ry s h e a rs a re
added to the applied loads, and the
structure is re-analyzed (first order
a n aly s is ) , us ing th e n e w l a te ra l
loads.
Thus, a first order (EFM) analysis can be used to give secondorder
d e fl e c t ions , m om e n ts , a n d fo rc e s .
Th e s e c an t hen be u s e d i n th e d e sign of members, without resorting
to the moment magnifier method.
The des igner s h o u l d b e a w a re ,
h o wev er , t hat idea l l y th e fl e x u ra l
stiffness (E1) needs to reflect the
a mount of r einf or c i n g , e x te n t o f
cracking, creep, reduced stiffness
due to axial loads, as well as the inelastic behavior of concrete and the
variation of EI along the length of
a mem ber f r om c ra c k e d to u n cracked regions. In practice, simplifying assumptions are used to compute EI, since it would be impractical to consider the flexural stiffness
o f eac h m em ber o f a h i s h ri s e
building.?
Summary
One of the strengths of the equivalent frame method of analysis is its
a p p l ic abilit y t o a v a ri e ty o f c o n crete shearwall configurations and
shapes. It is unencumbered by the
n e e d f or eit her c o n s ta n t fl o o r
heights or uniform openings (as in
the Lamina Method). Virtually any
type of horizontal and vertical loads
(i n cl uding t em pera tu re i n d u c e d )
ma y be applied t o th e i d e a l i z e d
November
1991
F i g .1 1 -
S ymmetryfor 3D f rame.'
structure. It can also be used in the
a n a l y s i s of shearw al l s i nteracti ng
with frames.
With any method of analysis, it is
important to have a general senseof
how the structure will behave and
the order of magnitude of resultant
reactions. This is particularly true
when using computers to assist in
th e a n a l ysi s of structures. If the
output varies greatly from the exp e c te d sol uti on Lhe engi neer must
determine whether the results contain an error (and if so, why) or if
the structure is behaving in an uhanticipated fashion.
All computer analysis techniques
and programs have basic underlyi n g a s s umpti ons bui l t i nto them.
Further, their output is directly and
strictly related to the input, which in
tu rn i s b ased on further assumptions made by the user. It is up to
the engineer to understand the limita ti o n s of the assumpti ons (prog ra m b ased and user made) and
verify the results. The EFM can assist in this verification process by
allowing the engineer to quickly remo d e l th e structure and test the
outcome of differing assumptions.
In many casesthe validity of comp u te r a nal yses can sti l l be cross
c h e c k e d w i th si mpl i fi ed manual
a n a l y s i s techni ques. The engi neer
must also be sufficiently aware of
the limitations of the EFM to know
Fi g. 12 - C oupl edshearw a llwit h
non-uni form
openi ngs.'
w hen a more detai l ed analyt ical
techniquemay be appropriate.
References
1. "Modern Mul ti -S tory C oncrete B ui l dings," Canadian Portland Cement Associati on, C anada, 1989,p. 3.
2. Rosman, Riko, "Approximate Analysis
of Shearwalls Subjected to Lateral Loads,"
A C I JounN al , P roceedi ngs, V . 6 1, N o. 6,
June 1964, pp.711-133.
3. S chw ai ghofer, J., "Tabl es for the
A nal ysi s of S hearw al l s w i th Tw o V erti c al
Rows of Openings," Publicotion 71-27, University of Toronto Department of Civil Engi neeri ng,N ov. 1971.
4. G l a n v i l l e , J o h n I ., a n d Ha tzin iko la s,
Michael A., "Engineered Masonry Design,"
Winston House Enterprises, Winnipeg, Canada, 1 9 8 9 ,p . 1 6 0 .
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S t ruc t u r e s , " S t r u c t u r a l Co n cr e te Sym p o '
s irm, U n i v e r s i t y o f T o r o n to De p a r tm e n t o f
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1971 ,p p . 1 1 8 - 1 4 5 .
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90, S T 3 , J u n e 1 9 6 4 ,p p .2 8 5 - 3 3 5 .
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33-43.
10. C a n o , M a r y T h er e sa , a n d Klin g n e r ,
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Analysis and Design," Building Structural
72
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Ch a r le s G. S al mon, edi tors, John Wi l ey &
So n s, Ne w Y ork, 1987,pp.530-541.
1 2 . M a c l eod. Iai n A .. "S hearw al l -Frame
In te r a ctio n: A D esi gn A \d," E ngi neeri ng
Bulletin, Portland Cement Association, Skokie , 1 9 7 0 , 17 pp.
1 3 . ACI C ommi ttee 318, "B ui l di ng C ode
Requirements for Reinforced Concrete (ACI
3 1 8 - 8 9 )a nd C ommentary - (A C I 318R -89),"
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p p . 1 1 7 - 129.
1 4 . Co ul l , A l exander, and C houdhury, J.
R., "str e sses and D efl ecti ons i n C oupl ed
Shearwalls," ACI JounNa,l, Proceedings, Y.
6 4 , No . 2 , Feb. 1967, pp.65-72.
1 5 . Co ul l , A l exander, and C houdhurY ,
J.R., "Anal ysi s of C oupl ed S hearw al l s,"
ACI Jo u nN a.l , P roceedi ngs, V . 64, N o. 9,
Se p t. 1 9 6 7,pp. 587-593.
1 6 . Co ul l , A ., and E l H ag, A . A ., "E ffective Coupling of Shearwalls by Floor Slabs,"
ACI JounN,lt-, Proceedings, V. 72, No. 8,
Au g . 1 9 7 5, pp.429-431.
1 7 . Du chesne,D . P . J., and H umar, J' L',
"En g in e e ri ng S oftw are - a C onsul tant's
Perspective," Canadian Journal of Civil Eng in e e r in g ,V .18, A pr. 1991,pp. 303-311.
1 8 . F a lk, H ow ard, "Mi crocomputer S oftware for Concrete Structural Design," Concrete Internationol: Design & Construction,
V. 7 , No . 6, June 1985, pp.49-56.
1 9 . Kh an, A .H ., and S tafford S mi th, B .,
"simplified Method of Analysis for Deflec-
ti ons and S tresses i n Wal l -Frame S truc Ives." Building ond Environment, Y. ll,
N o. 1, 1976,pp.69-78.
20. K ong, F.K ., et al , E di tors, H andbook
of Structural Concrete, Pitman Books Limi ted, London, 1983,pp. 3l -1 to 3' 7-44.
21. S chw ai ghofer, Joseph, and C ol l i ns ,
Michael P., "Experimental Study of the Behavi or of R ei nforced C oncrete C oupl i ng
S l abs," A C I JounN ,ql -,P roceedi ngs ,V . 74,
N o. 3, Mar. 1977, pp. 123'127.
22. Weaver, Wi l l i am Jr., and Gere, J ames
M., "Matri x A nal ysi s of Fram ed S truc tures," 3rd E di ti on, V an N ostr and R ei nhol d, N ew Y ork, 1990.
Received and reviewed under Institute publication policies.
A C I member A ngel o
Mattacchi one i s P resi dentof P rosumE nninoorinn
llrl
2
structuralconsul ti ng fi rm i n N orth
Y ork, Ontari o. H e
has been acti ve i n
the desi gnof numerousstru ct ur esin
ti mber, sl ructural steel , and r einconcr et e.
forcedand post{ ensi oned
Authorizedreprintfrom: Novemberl99l lssue of ACI ConcreteInternattonal