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Simplified analysis of coupled shear walls of variable cross-section
I Sfb Ab4 UDC 624.075
Build. Sci. Vol. 5, pp. 11-20. Pergamon Press 1970. Printed in Great Britain
Simplified Analysis of Coupled Shear Walls of Variable Cross-Section AVRAHAM PISANTY* ELIAHU E. TRAUM t
A simplified analysis of a symmetrical coupled shear wall of variable crosssection is presented. The solution matches the boundary conditions of the upper and lower part of the wall at the plane of contiguity, at which an abrupt change in cross-section occurs. This yields a series of linear simultaneous equations for the determination of the constants of integration with which all internal forces and displacements are readily obtained. The solution is illustrated by a numerical example. showing the relative vertical displacement of the two upper wall segments at the junction where the cross-section changes was most likely responsible for the error in establishing their basic equation. Since it appears that this error might significantly effect the results of the analysis, and, incidentally, might explain the discrepancy between the analytical and experimental values in Coull and Pouri's paper, it is of interest to first clarify it.
INTRODUCTION AN EVER increasing proliferation of publications on the analysis of coupled shear walls is certainly attesting to the interest of the profession in this subject as well as to the prominence of such walls as economical structural spinesin multi-story buildings. Indeed, one rightly hesitates to add even further to the numerous papers that already have been presented in recent technical literature on the analysis of shear walls. However, when scrutinizing the subject material somewhat more closely it is found that the suggested methods of analysis of coupled shear walls with variable cross-section are few and still leave much to be desired. Moreover, notwithstanding their considerable complexity, some are outright erroneous. Shear walls of uniform cross-section for the full height of the building have been treated extensively by Chitty[1], Beck[2], Rosman[3], and others. The case of shear walls with an abrupt change in crosssection has been considered by Rosman[4] in an approximate solution and more exactly by Traum [5]. A more recent paper by Coull and Pouri[6] takes exception with Traum's method of analysis and presents what is claimed to be an improved and simplified analysis of a coupled shear wall with an abrupt change in cross-sectional properties at some level. However, while basically using in their solution the same approach as taken by Traum, namely establishing compatibility of the upper and lower parts of the wall along their planes of contiguity, Coull and Pouri seem to have erred in setting up their basic differential equation for the determination of the internal shearing forces in the upper part of the wall. The absence of a sketch
6H
z.
Fig. 1.
Figure 1 illustrates the deformed shape of the coupled shear wall, treated by Coull and Pouri, at the junction between its upper and lower parts. The method of analysis follows the pattern of the common approach by which the coupled shear wall is cut into two segments along the center line of the connecting beams, assumed as plane of contraflexure. Compatibility is then established along this line by letting the relative displacement of the two cut segments vanish. The basic differential equation combines all factors affecting this relative displacement and sets them equal to zero. One of these factors, relating to the upper part of the wall, is the relative displacement at its base
* Lecturer, Department of Civil Engineering, Technion, Israel Institute of Technology, Haifa, Israel. t Professor of Construction, Graduate School of Design, Harvard University. 11
Arraham Pisanty and Eliahu E. Traum
12
due to axial forces acting on the lower part. Combining the effect of this axial deformation with the rotation OH of the plane of junction, we find that the relative displacement of the connecting lamellae at the plane ofcontraflexure is
A,, = 5H-IIOH.
(1)
Rather than considering this relative displacement, between points C and D in the plane of the section (figure 1), as was done correctly for all other factors contributing to it, Coull and Pouri had erroneously taken the distance between A and B, the axes of the upper wall segments. They have thus introduced into the compatibility equation the distance 6H-(I~ -lz)OH (2) which is considerably at variance with that of equation (1). Depending on the relative width of the upper and lower wall portions this could lead in many cases to quite erroneous results of the analysis. The present paper suggests a different approach to the problem by treating the entire wall as a continuous element. It aims to reduce the complexity, and its inherent likelihood for errors, in previous solutions by establishing separate differential equations for each wall portion of constant cross-section. These are then linked together by the boundary conditions which yield a set of simultaneous linear equations for the determination of all constants of integration.
This solution adopts the basic assumptions made in all the references cited above, which replace the connecting beams by continuous rigid lamellae that conceptually transform the real structure into its substitute form, shown in figure 2b. Again, as before, the structure is cut along its plane of symmetry, which coincides with the plane through the points of contraflexure of all connecting beams. Assuming now that each wall segment is independently deformed, thus causing a relative displacement between pairs of lamellae at any level, compatibility is stipulated by letting this displacement vanish. Referring to figure 3, it is noted that
(a)
(b)
(c)
(d)
Fig. 3. Relative vertical displacements of the pierced elastic media at contraflexure points.
METHOD
OF SOLUTION
The object of the solution is to determine all internal forces and deformations of a coupled shear wall with an abrupt change of cross-section, as shown in figure 2. A symmetrical wall is treated, the physical dimensions of which are evident from the figure.
this relative displacement at the section along the points of contraflexure is the combined effect of four contributions, namely: (a) Bending of the wall segments (figure 3a): dl'
6j(x) = a - = . d.r
ii~ 2 "2,~ ~
One might wish to add to this the effect of shear deformation of the wall segment; this, however, may be neglected in all practical cases where the segments are relatively tall and slender. This effect is, therefore, not included here.
d
2
,'il 2 -~
d,
II - -- li~
[
(b) and (c) Bending and shear deformation of the connecting lamellae (figure 3b and 3c):
1
-q
I -4
k L
I
-I " -I
It~l ~'b' I
I I
[,'
I ~'
• L]
m
"i #
.~
!
_[ : I :± I : r
62(x) =
T •
(3)
l
Fig. 2(a) Real structure, (b) Substitute structure.
hc 3
t,c ~
ll2Elh +G--~h*| q(x)
(4)
The effective value A* of the equivalent crosssection of the beams for shear deformation has been disputed by several authors and treated in
Simplified Analysis of Coupled Shear Walls of Variable Cross-Section
13
various ways. There is no intention here to further dispute it and A* = (1/1.2)A b will be adopted for the purposes of the analysis.
The moment at any section x of each wall segment for 0 < x < b is given by
(d) Displacement due to axial deformation of the wall segments (figure 3d) :
MEI(x) = Mo(x) + ½al S q1(2) d2 + ½a2 S q2(2) d2
x
H
2f[ q 1
63(x) =-E--A
(2) d2 dr/.
(5)
0
b
H
x
b
(10) where Mo(x) is the moment due to external loading in one segment of the shear wall, as defined by equation (18) below. Expressing the moment in terms of curvature as
Now the condition of compatibility along the plane of the section is satisfied by:
EI1 d2y~ -
ME,,
dx 2
Z6(x) = O.
(6)
With the basic compatibility equation thus established, the structure shown in figure 2 is subdivided into two zones, for each of which loading and cross-sections are constant over their respective height. Separate differential equations, in terms of the shear force per unit length ql(x) and qz(x), are then set up for zones (1) and (2) respectively. Finally, the boundary conditions will directly yield a set of simultaneous equations for the determination of all constants of integration. Z O N E (1)--0 < x < b
Introducing into the expressions for the relative displacement of the lamellae at the point of contraflexure, equations (3)-(5), the corresponding values for the lower part of the wall, indicated by the suffix l, equation (6) becomes:
dYl--F hlc3
we can differentiate equation (10) to yield, considering equation (11): d3yt
d d dx Me,(x) = - ~x Mo(x)
EIx dx 3 -
+½alql(x). dMo(x) a~ - a , d ~ - ~ - +--~- q l(X) -
I
hlC311 12Ib~
12EIo, ] dZql(x) + 211 qa(x) = 0 1 + c2GA'-==-~b ~ dx 2
which becomes after dividing by ½a~ H
2ffq
(13)
2
hlc311 [6 ~ [ ]
12EIbt1 d2ql(x) 1 + c2GA*,J
1 [lq_
He L.
=
-5"7--, .4[11
aiA1J ql(x)
2 dMo(x ) a, H2 dx (14)
x b
E~ 1
(12)
Introducing equation (12) into equation (9) we get:
hlC "] z .
al-d-~xl 112Eib +-d-~,Jq~t x)
(11)
1(2) d2 dr/
0 )1
2xQ2 EA1
- 0.
(7)
The following abreviations will be defined:
The last term of equation (7), which is additional to the expressions for displacements derived above, expresses the relative displacement of the lower portion of the wall caused by the axial reaction Q2 from the upper portion. Q2 is simply the summation of the shear forces acting on zone (2), namely:
c~ - 6a~HZIb' hlc311
(15a)
fl~ = 1+ c2GAb, j
(15b)
I ~ =
411 ] 1+ a~A1J
(15c)
H
Q2 = ~ q2(x) dx. b
(8)
~1 - ~r~
After differentiating twice and multiplying equation (7) by EI1, we get:
alEI1 d3yl dx 3
hae311I 12ib '
12EIb' ] d2ql(x) 1 5- c2GA.z J d x 2 2Ix
+ ~ ql(x) = O.
With the constants of equations (15) introduced into equation (14) the differential equation for the lower part of the wall is obtained: dZq,(x)
(9)
(150
dx 2
(~)2
ql(x) =
2(i.t..__L,~2dMo(x) al \~IH] dx (16)
14
A vraham Pisanty and Eliahu E. Traum
The solution of this linear differential equation is: ql{x) = q~'(x) + Cle "'(~'m + C2e-''cv'lt~. (17) The particular solution, qP(x), is obtained from equation (16), after introducing into it the moment Mo(x), which is caused by the external loading on one segment of the wall. Assuming that the applied loading p is uniformly distributed over the entire height of the wall, this external moment is on one segment :
, Mo(x) = - ~pH
,
dx
a')' 2- 1 -
(19)
q-CleU'C':/u~q-C2e-U'(~'/H~. (20)
It now remains to determine the constants of integration in equation (20); this, however, will be linked to the upper portions of the wall. A similar expression for the shearing force qz(x), acting there, must first be developed.
EA~
q;(2) d2 dr/- ~
d,~d ~.
Q2
0q hb
(23)
It should be noted again that the suffix 2 and 1 relate to elements in zones (2) and (1) respectively. After differentiating twice and multiplying by El> equation (23) becomes:
a2Ei2d3y2
h2c312~1
L
(22)
011
This now allows us to set up the compatibility condition for zone (2), which becomes when
12Elb2~d2qz(x)
+
dx2
2b12 dq2(x) _ 0.
212 + ~ 2 qz(X)
A~
(24)
dx
The moment at any section x of each wall segment in zone (2) is:
H M~(x) = Mo(x)+
q2(2) d2.
(25)
x Following now the same procedure as above, we get d
d
dx [Me~(x)] = - dx M°(x)
02
+ -7 q2(x).
(26)
Introducing equation (26) into equation (24), we have" -a2
dMo(x) d-----~" +
qz(x)
h2c312 I
lZElb2"]dzq2(x) + --~q,(x) 2/2 +
bb
2;~ql(k) EA 1
EA2
(21)
where Q2 was defined by equation (8) as the vertical reaction of zone (2) upon zone (1). The additional relative vertical displacement of the lower wall segments due to the action of the shear force ql(x) in zone (1) is:
65 =
x II
EI2 dx 3 -
The compatibility condition of equation (6) for the upper part of the wall will now include not only the deformations, expressed by equations (3)-(5), but also the effect of deformation of the lower wall, zone (1). The relative displacement of the lamellae at their point of contraflexure due to the vertical deformation of the wall segments in zone (1) is: 2b 64 = -- - Q2
h2c
LI2EI< + GA~,_] q2(x)
a2 ~
day2
< x < H
ZONE(2)--b
]12 (-'3
EA
with which the final expression for q~(x) in zone (1) becomes:
ql(x) = alY12 1--
dy2- ~
, .
This then gives the particular solution
q':(x)- a,y2
introducing into equation (6), equations (3)-(5t. (21) and (22):
--
2bi2 dqz(x) AI
dx
dx
- 0,
(27)
which after division by ½a~H2 becomes:
h2c312 [ 12EIb2qdZq2(x) 4blz dq2(x) 6a~H2I~2~l + c2GA*2.J dx 2 + a~H2~ dx 1 [ 4I z -~ - H---5I_l+ a~2A2jqz(x ) =
2
dMo(x)
azH2
dx (28)
Simplified Analysis of Coupled Shear Walls of Variable Cross-Section Similarly to the solution above, the following abreviations are defined:
6a2H2Ib~ ~ = ' h2c312
(29a)
12EIb21 f12 = 1-F c2GA~2] =
412 1 ]
1+
15
The final expression for the shear forces in zone (2) therefore becomes: pH[ q2(x) = a~7~
x 72v4]+C3e-A(X/H)+C4e-B(X/H)
1 . . .H.
(34)
(29b)
(29c) DETERMINATION OF T H E CONSTANTS
2 2 ~2~2
OF INTEGRATION
(29d)
(29e)
The following boundary conditions will serve to determine the four constants of integration C a to C4.
With the constants of equations (29) introduced into equation (28), the differential equation for the upper part of the wall is obtained:
(1) No rotation at the fixed base of the wall, i.e. for x = 0, (dy/dx) = 0 which becomes from equation (7)
4bI 2 v22 - a2A 1H
[q,(X)]x=O =0 d2qz(x) dx 2 =
v~{~_32)Zdq2(x) (~2)2 + n\?2] dx ~ q2(x)
_._2_2( . 2
~dMo(x). a 2 ~k]~2HJ dx
(30)
(2) At the plane of contiguity between the different zones of the wall, at which the crosssection is abruptly changed, namely at x = b, the continuity of the internal shear force is maintained :*
The homogeneous solution of equation (30) is:
qh(x) = C3e-a(x/n) +C4 e-B(x/n)
[ql(X)]x=b = [q2(x)]x=b (31)
where A = 2[ 1-~(1
+4/z2~1 -~2z]J
(32a)
B=~-[I+ /(i+4/~'~I
(32b)
m-
2 2 V2//2
?,~
(32c)
To obtain the particular solution of the differential equation (30) we have to introduce into it the value of the external moment M o. It was assumed above that the applied horizontal loading p is uniformly distributed over the full height of the wall. This by no means is an essential assumption for lhis solution and could have just as easily been considered as different loading on each zone of the wall. However, with uniform loading p, the external moment is still expressed by equation (18) above, which when introduced into equation (30) yields the particular solution:
pH[
q~(x) = a2~' 2
1
x
H
~l
y2J"
(33)
(35)
(36)
(3) At the same section, x = b, continuity stipulates that the slope of the deflection curve of the upper wall equals that of the lower wall:
dx Jx=b L dx ax=,~
(37)
Extracting (dyl/dx) and (dy2/dx) from equations (7) and (23) respectively and differentiating once, equation (37) becomes:
I
dq~(x) 1 = RFdq2(x) 1 dx Jx=b L dx Jx=b
(38)
* This boundary condition requires some qualifications. It is not a priori clear that the internal shear force does not abruptly change at the plane of change in cross-section. However, recognizing the fact that this shear force is taken as acting in a continuousconnecting medium, such an internal force cannot change abruptly, as well known by St. Venant's principle. The effect of different section properties is felt only at some distance from the plane of change and there it is properly accounted for in all the expressions developed above. This fact has been brought out also by other authors who had offered a different solution to the subject of this paper [6,5]. In light of this, the boundary condition of equation (36) is believedto be valid.
16
Avraham Pisanty and Eliahu E. Traum
where
with
h=c ]
h=c 3
.V
o L12EI~ + CA~'J R = _! "~ V h'c~ /,,o ]
(39)
[)2E1h, + GAy,J
= -tf
;I
and
x= -2H !}H 2 2H Vk~(g~) = aff~ 2s(1--~k)+--/~ ~(C 1e - " ' ~ (52)
+C2e-"':'~sinh(t~ts).
(4) At the top of the wall, for x = H, the moment vanishes. Thus, d 2yz]
"d~-J~=. = 0
(40)
which yields from the derivative of equation (23): h2c312
dqz(x)
2bl2
~-d;-
+ ~
q~(x) ~=~ = 0 .
(4i)
These four boundary conditions now yield a set of linear simultaneous equations for the determination of the four constants of integration C~ to C4: C~ +
C2
= K
e~'~rC~ +e-"~rCi-e-arc3-e-BrC, + L(I - r - N)]
=
vd¢~) -
2~[1 - ¢~-
2C4H _ 8~, 2 C 3 H e - a ; " sinh(As) + - - - - ~ e sinh(Bs), A (53) and finally, for the shear force in the top beam:
+
V~2
r/ - - -h- = -//h- -
-
PHLaz?~ \ 8 H
(45)
~/1~)2
I---1 I
J t % c_) c->
v~
I
(48)
S =
[(J]
(49)
r-
b H"
(50)
l+/t~ v2
1 "
I
]
I I
1
't
I
With Ct to C a determined, q~(x) and q2(x) are computed from equations (20) and (34) respectively. Integrating q(x) over the height of one story, we obtain the shear force Vk(x ) in the connecting beam at floor level x k by xk+h/2
j"
q(x)dx.
This will give in zone (1), where
0_<~_<_r
(51)
(D
1 8
1
BEAMS
k:
7
T H E S H E A R F O R C E IN T H E C O N N E C T I N G
xk-h]2
(54)
5(t/m!
(47)
N - 72
Vk(x) =
.
A 21-story coupled shear wall with an abrupt change in cross-section at the l 1th story is analyzed (see figure 4). The wall is subjected to a uniformly
(46)
pH L = a2722
]
NUMERICAL EXAMPLE
(44)
/SH =
)
+ C 4 ~H e- B(eBr - 1 )
4
In the expressions above some additional abreviations were used, namely: K
H l v~2 \ + C3A e - a(ea'-- I ) 2 72
[-K(I -r) (43)
t~ ~e"'~C 1 - - 111e - "'~C2 + A R e - Arc 3 2i- B R e - B r c = K-RL B e - ~Ca + A e - ~ C , = SL.
(42)
In zone (2), for 1 < ~ < I, equation (51) will yield with equation (34):
~)
E:
1
I--] I I I 1 I 1 ""'~A1"(4~ Fig. 4. Shear wall analysed in numerical example (dimensions in metres).
Simplified Analysis o f Coupled Shear Walls o f Variable Cross-Section d i s t r i b u t e d h o r i z o n t a l l o a d i n g p f o r its full height. A l l d i m e n s i o n s a r e g i v e n in m e t r i c units.
For
17
0 < ~ < 0.5238 q 1(~) = p[4.20525(1 - ~) + 0.230643 e 2"2 a 3o~] - 4 . 4 3 5 8 9 3 e -2"23a°¢]
h = 3.00 m
a n d f o r 0.5238 < ~ < 1
H = 63.00 m
q2(¢) = p[5.64858(0.9712 - 3) + 0-060732 e 3'1339~
d~ = 8.40 m
- 8.950221 e -
d2 = 5.40 m
3"4449~].
T h e s h e a r f o r c e s in t h e c o n n e c t i n g b e a m s at e a c h f l o o r level are g i v e n b y :
c --- 4.80 m a l = 13.20 m
For
a2 = 10.20 m
0 =< ~ = < 0.5238
Vk1(~) --~ fiH[0"200240(1
t = 0.50 m (wall thickness)
- ~) + 0"010965 e 2.2 330~
--0"210888 e -2"2330~]
G = 1/2E a n d f o r 0.5238 < ~ < 1
b = 33.00 m
Vk2(¢) = p H [ 0 " 2 6 8 9 9 0 ( 0 " 9 7 1 2 - - 0
h b = 0.60 m
+ 0"002889 e 3 . 1 3 3 9 : _ 0.425622 e -
r = (b/H) = 0.5238 A 1 = 4.20m 2
1: = 24.6960 m 4
A z = 2-70 m 2
12 = 6.5610 m 4
Ag" = 0.25 m 2
1~ = 0.0090 m 4
1/",2
= 0"022633 pH.
T h e c o m p u t e d v a l u e s o f q ( O a n d Vk(O a r e listed in T a b l e 1. T h e v a l u e s f o r ~ = 0.5238 (at t h e l l t h floor) are listed twice, as o b t a i n e d b y t h e e q u a t i o n s for the upper and lower zone respectively. In a d d i t i o n , t h e v a l u e s o f M r a n d M a a r e g i v e n at each floor which represents the moment beneath
All d e t a i l e d c o m p u t a t i o n s are o m i t t e d h e r e a n d o n l y t h e final e x p r e s s i o n s f o r q(~) a r e g i v e n :
Table 1 Story
21 20 19 18 17 16 15 14 13 12 11 II I0 9 8 7 6 5 4 3 2 1
0
1.00000 0.95238 0.90476 0"85714 0.80952 0.76190 0.71429 0"66667 0-61905 0-57143 0.52380 0"52380 0.47619 0.42857 0.38095 0.33333 0.28571 0.23810 0.19048 0.14286 0.09524 0"04762 0.00000
3.4449~]
-p
pH
0'946388 0'971136 1.013488 1"068435 1.130570 1"194971 1"256670 1.309292 1'350917 1'372137 1'367790 1'368072 1'338868 1.300017 1"248605 1"181621 1"096602 0'989957 0.857962 0"697433 0-504069 0"272977 0.00(0~
0"022633 0.046206 0"048237 0"050866 0"053836 0-056916 0"059866 0'062385 0"064381 0"065407 0.065216 0"065190 0'063818 0.061986 0-059557 0-056387 0-052359 0-047305 0'041045 0"033429 0"024251 0.013280 0.000000
M~ (~) pH 2
-0.001265 -0.003305 -0.004376 - 0'004524 -0.003780 - 0.002154 + 0.000370 +0.003824 + 0.008249 +0.013728 +0.006163 +0.012516 +0.020195 + 0.029262 +0.039796 +0'051881 +0.065633 +0.081174 + 0.098648 +0-118216 +0.140068
Mg (~ pH 2 -0.001832 - 0-005006 -0.007210 -0.008493 -0'008882 -0-008388 0-007000 - 0.004680 --0-001388 + 0.002954 -
-- 0'005739 --0.000523 + 0-006028 +0.013956 +0'023357 + 0"034311 + 0"046925 +0-061333 + 0"077672 +0.096107 +0'116825
18
Avraham Pisanty and Eliahu E. Traum
the connecting beam (including the action of the shear force at that beam) and above it respectively. For the k-th story (at Ck): a M[(~k) = Mo(~k)-- y --
~
Vi(~.)
i=k+l
\ v,(¢)
MI(~k) = Mo(~k)-- a
\
i=k
Figure 5 shows the distribution of q(O over the height of the wall as well as the total shear force in every connecting beam. ]n figure 6 the moment distribution on the wall segment is plotted. The computations have been carried out to a high degree of accuracy for illustrative purpose, which, however, is not essentim with manual calculations. Nevertheless it is recommended to compute fairly accurately the values of q(~) and Vk(~) at the plane of change in cross-section, which will serve to check the correctness of the solution.
1
/ /
j/"
/
~q
<
/
/
/ C f
S
CONCLUSIONS
002 !r
Compared with previous solutions, which treat each wall portion separately and match them at their junctions, the method of analysis here is
!).I
'"
, ,'
:
(a)
Fig. 5. (a) Distribution of q(~), (b) Distribution of Vk(O.
-
010
008
: ~
(b)
1/I
(}!;>
00:.
¢
Story
0.12,
/
00.6
00-,
002
Fig. 6. Moment distribution in a wall segment.
('~
.L
00~
-
Simplified Analysis o f Coupled Shear Walls of Variable Cross-Section believed to considerably reduce and simplify the numerical work. It follows the simple pattern of establishing for each part of the wall its compatibility equation, each following the same straightforward format, and then setting up the boundary conditions. Those link all constants of integration directly together as the unknowns in a set of simultaneous linear equations. The generality of the method put forward here, obviously transcends the application that served to illustrate it. (a) To apply the solution to a coupled shear wall with several abrupt changes in cross-section, two additional boundary conditions are used at each plane of change, similarly to equations (37) and (38). This will yield, together with the other boundary conditions, all necessary simultaneous equations for the determination of all constants of integration. (b) The "abrupt change" implied change of cross-section of the wall or of the connecting beam, the story height, the loading or any other discontinuity. (c) Any distribution of horizontal loading may be considered. It will only affect the value of the external moment, Mo(x ), which is reflected in a different particular solution of the differential equation.
a
distance between the centroids of the wall segments
b
height of the lower zone
e E
clear span of the connecting beams modulus of elasticity of concrete
G
shear modulus of concrete
H
height of shear wall
h
story height
hb
height of connecting beam
I
moment of inertia of wall segment
Ib
moment of inertia of connecting beam
ME(I)
total moment on the wall segment at
MF.(~)
moment in wall element beneath connecting beam at
M~(~)
T
moment in wall element above connecting beam at
M0(~ )
external moment in one wall segment due to p at
b
horizontal loading per unit height of shear wall
q(~)
shear force per unit length in the substitute system
r
= (b/H)
s
= (h/2H)
Vk(~)
shear force in connecting beam at
NOTATION The suffix 1 and 2 pertain to the elements of zone (1) and (2) respectively. A
cross-sectional area of one wall segment
Ab
cross-sectional area of the connecting beam
Ai
equivalent cross-sectional area of the connecting beam for shear deformation
19
= (x/tO
other constants are defined in the text.
REFERENCES
1. L. CHITTY, On the cantilever composed of a series of parallel beams interconnected by cross members. Philosophical Mag. (Lond.) Vol., 685 (1947). 2. H. BECK, Contribution to the analysis of coupled shear walls. Proc. Am. Concr. Inst. 59, 1055 (1962). 3. R. ROSMAN, Beitrag zur statischen Berechnung waagrecht belasteter Querwfinde bei Hochbauten (I) (II). Bauingenieur, 35, 133 (1960), 37, 24 (1962). 4. R. ROSMAN, Beitrag zur statischen Berechnung waagrecht belasteter Querw/inde bei Hochbauten (III). Der Bauingenieur, 37, 303 (1962). 5. E. TRAUM,Multistory pierced shear walls of variable cross-section, Symposium on Tall Buildings held at Southampton 1966. Pergamon Press, Oxford, 1967. 6. A. COULL, and R. D. PORt, Analysis of coupled shear walls of variable cross section. Building Sci. 2, 313 (1968). 7. A. PISANTY,Coupled shear walls in multistory buildings, M.Sc. Thesis, Faculty of Civil Engineering, Technion--Israel Institute of Technology, Haifa, February 1965.
20
Avraham Pisanty and Eliahu E. Traum
On pr6sente une analyse simplifi6e d'un mur de cisaillement accoupl6 symm6trique de section variable. La solution est en accord avec les conditions limitrophes de la pattie sup6rieure et de la partie inf6rieure du tour au plan de contiguit6, auxquelles un changement soudain de la section a lieu. Une s6rie d'6quations lin6aires simultan6es est produite pour la d6termination des constantes d'int6gration avec lesque[les routes les forces et d6placements internes sont ais6ment obtenus. La solution est illustr6e d'un exemple num6rique. Eine vereinfachte Analyse einer symmetrisch gekuppelten Absteifwand mit ver~inderlichem Querschnitt wird dargestellt. Die L6sung kommt den Grenzbedingungen des oberen und unteren Teils der Wand an der Angrenzebene gleich, bei welcher ein plt~tzlicher Wechsel des Querschnitts erfolgt. Daraus ergeben sich eine Reihe linearer Simultangleichungen zur Ermittlung der Intergrierungskonstanten, mit welchen alle inneren Kr~ifte und Verschiebungen leicht erhalten werden ktinnen. Die I_~Ssungwird mit zahlenm~issigem Beispiel illustriert.
Build. Sci. Vol. 5, pp. 11-20. Pergamon Press 1970. Printed in Great Britain
Simplified Analysis of Coupled Shear Walls of Variable Cross-Section AVRAHAM PISANTY* ELIAHU E. TRAUM t
A simplified analysis of a symmetrical coupled shear wall of variable crosssection is presented. The solution matches the boundary conditions of the upper and lower part of the wall at the plane of contiguity, at which an abrupt change in cross-section occurs. This yields a series of linear simultaneous equations for the determination of the constants of integration with which all internal forces and displacements are readily obtained. The solution is illustrated by a numerical example. showing the relative vertical displacement of the two upper wall segments at the junction where the cross-section changes was most likely responsible for the error in establishing their basic equation. Since it appears that this error might significantly effect the results of the analysis, and, incidentally, might explain the discrepancy between the analytical and experimental values in Coull and Pouri's paper, it is of interest to first clarify it.
INTRODUCTION AN EVER increasing proliferation of publications on the analysis of coupled shear walls is certainly attesting to the interest of the profession in this subject as well as to the prominence of such walls as economical structural spinesin multi-story buildings. Indeed, one rightly hesitates to add even further to the numerous papers that already have been presented in recent technical literature on the analysis of shear walls. However, when scrutinizing the subject material somewhat more closely it is found that the suggested methods of analysis of coupled shear walls with variable cross-section are few and still leave much to be desired. Moreover, notwithstanding their considerable complexity, some are outright erroneous. Shear walls of uniform cross-section for the full height of the building have been treated extensively by Chitty[1], Beck[2], Rosman[3], and others. The case of shear walls with an abrupt change in crosssection has been considered by Rosman[4] in an approximate solution and more exactly by Traum [5]. A more recent paper by Coull and Pouri[6] takes exception with Traum's method of analysis and presents what is claimed to be an improved and simplified analysis of a coupled shear wall with an abrupt change in cross-sectional properties at some level. However, while basically using in their solution the same approach as taken by Traum, namely establishing compatibility of the upper and lower parts of the wall along their planes of contiguity, Coull and Pouri seem to have erred in setting up their basic differential equation for the determination of the internal shearing forces in the upper part of the wall. The absence of a sketch
6H
z.
Fig. 1.
Figure 1 illustrates the deformed shape of the coupled shear wall, treated by Coull and Pouri, at the junction between its upper and lower parts. The method of analysis follows the pattern of the common approach by which the coupled shear wall is cut into two segments along the center line of the connecting beams, assumed as plane of contraflexure. Compatibility is then established along this line by letting the relative displacement of the two cut segments vanish. The basic differential equation combines all factors affecting this relative displacement and sets them equal to zero. One of these factors, relating to the upper part of the wall, is the relative displacement at its base
* Lecturer, Department of Civil Engineering, Technion, Israel Institute of Technology, Haifa, Israel. t Professor of Construction, Graduate School of Design, Harvard University. 11
Arraham Pisanty and Eliahu E. Traum
12
due to axial forces acting on the lower part. Combining the effect of this axial deformation with the rotation OH of the plane of junction, we find that the relative displacement of the connecting lamellae at the plane ofcontraflexure is
A,, = 5H-IIOH.
(1)
Rather than considering this relative displacement, between points C and D in the plane of the section (figure 1), as was done correctly for all other factors contributing to it, Coull and Pouri had erroneously taken the distance between A and B, the axes of the upper wall segments. They have thus introduced into the compatibility equation the distance 6H-(I~ -lz)OH (2) which is considerably at variance with that of equation (1). Depending on the relative width of the upper and lower wall portions this could lead in many cases to quite erroneous results of the analysis. The present paper suggests a different approach to the problem by treating the entire wall as a continuous element. It aims to reduce the complexity, and its inherent likelihood for errors, in previous solutions by establishing separate differential equations for each wall portion of constant cross-section. These are then linked together by the boundary conditions which yield a set of simultaneous linear equations for the determination of all constants of integration.
This solution adopts the basic assumptions made in all the references cited above, which replace the connecting beams by continuous rigid lamellae that conceptually transform the real structure into its substitute form, shown in figure 2b. Again, as before, the structure is cut along its plane of symmetry, which coincides with the plane through the points of contraflexure of all connecting beams. Assuming now that each wall segment is independently deformed, thus causing a relative displacement between pairs of lamellae at any level, compatibility is stipulated by letting this displacement vanish. Referring to figure 3, it is noted that
(a)
(b)
(c)
(d)
Fig. 3. Relative vertical displacements of the pierced elastic media at contraflexure points.
METHOD
OF SOLUTION
The object of the solution is to determine all internal forces and deformations of a coupled shear wall with an abrupt change of cross-section, as shown in figure 2. A symmetrical wall is treated, the physical dimensions of which are evident from the figure.
this relative displacement at the section along the points of contraflexure is the combined effect of four contributions, namely: (a) Bending of the wall segments (figure 3a): dl'
6j(x) = a - = . d.r
ii~ 2 "2,~ ~
One might wish to add to this the effect of shear deformation of the wall segment; this, however, may be neglected in all practical cases where the segments are relatively tall and slender. This effect is, therefore, not included here.
d
2
,'il 2 -~
d,
II - -- li~
[
(b) and (c) Bending and shear deformation of the connecting lamellae (figure 3b and 3c):
1
-q
I -4
k L
I
-I " -I
It~l ~'b' I
I I
[,'
I ~'
• L]
m
"i #
.~
!
_[ : I :± I : r
62(x) =
T •
(3)
l
Fig. 2(a) Real structure, (b) Substitute structure.
hc 3
t,c ~
ll2Elh +G--~h*| q(x)
(4)
The effective value A* of the equivalent crosssection of the beams for shear deformation has been disputed by several authors and treated in
Simplified Analysis of Coupled Shear Walls of Variable Cross-Section
13
various ways. There is no intention here to further dispute it and A* = (1/1.2)A b will be adopted for the purposes of the analysis.
The moment at any section x of each wall segment for 0 < x < b is given by
(d) Displacement due to axial deformation of the wall segments (figure 3d) :
MEI(x) = Mo(x) + ½al S q1(2) d2 + ½a2 S q2(2) d2
x
H
2f[ q 1
63(x) =-E--A
(2) d2 dr/.
(5)
0
b
H
x
b
(10) where Mo(x) is the moment due to external loading in one segment of the shear wall, as defined by equation (18) below. Expressing the moment in terms of curvature as
Now the condition of compatibility along the plane of the section is satisfied by:
EI1 d2y~ -
ME,,
dx 2
Z6(x) = O.
(6)
With the basic compatibility equation thus established, the structure shown in figure 2 is subdivided into two zones, for each of which loading and cross-sections are constant over their respective height. Separate differential equations, in terms of the shear force per unit length ql(x) and qz(x), are then set up for zones (1) and (2) respectively. Finally, the boundary conditions will directly yield a set of simultaneous equations for the determination of all constants of integration. Z O N E (1)--0 < x < b
Introducing into the expressions for the relative displacement of the lamellae at the point of contraflexure, equations (3)-(5), the corresponding values for the lower part of the wall, indicated by the suffix l, equation (6) becomes:
dYl--F hlc3
we can differentiate equation (10) to yield, considering equation (11): d3yt
d d dx Me,(x) = - ~x Mo(x)
EIx dx 3 -
+½alql(x). dMo(x) a~ - a , d ~ - ~ - +--~- q l(X) -
I
hlC311 12Ib~
12EIo, ] dZql(x) + 211 qa(x) = 0 1 + c2GA'-==-~b ~ dx 2
which becomes after dividing by ½a~ H
2ffq
(13)
2
hlc311 [6 ~ [ ]
12EIbt1 d2ql(x) 1 + c2GA*,J
1 [lq_
He L.
=
-5"7--, .4[11
aiA1J ql(x)
2 dMo(x ) a, H2 dx (14)
x b
E~ 1
(12)
Introducing equation (12) into equation (9) we get:
hlC "] z .
al-d-~xl 112Eib +-d-~,Jq~t x)
(11)
1(2) d2 dr/
0 )1
2xQ2 EA1
- 0.
(7)
The following abreviations will be defined:
The last term of equation (7), which is additional to the expressions for displacements derived above, expresses the relative displacement of the lower portion of the wall caused by the axial reaction Q2 from the upper portion. Q2 is simply the summation of the shear forces acting on zone (2), namely:
c~ - 6a~HZIb' hlc311
(15a)
fl~ = 1+ c2GAb, j
(15b)
I ~ =
411 ] 1+ a~A1J
(15c)
H
Q2 = ~ q2(x) dx. b
(8)
~1 - ~r~
After differentiating twice and multiplying equation (7) by EI1, we get:
alEI1 d3yl dx 3
hae311I 12ib '
12EIb' ] d2ql(x) 1 5- c2GA.z J d x 2 2Ix
+ ~ ql(x) = O.
With the constants of equations (15) introduced into equation (14) the differential equation for the lower part of the wall is obtained: dZq,(x)
(9)
(150
dx 2
(~)2
ql(x) =
2(i.t..__L,~2dMo(x) al \~IH] dx (16)
14
A vraham Pisanty and Eliahu E. Traum
The solution of this linear differential equation is: ql{x) = q~'(x) + Cle "'(~'m + C2e-''cv'lt~. (17) The particular solution, qP(x), is obtained from equation (16), after introducing into it the moment Mo(x), which is caused by the external loading on one segment of the wall. Assuming that the applied loading p is uniformly distributed over the entire height of the wall, this external moment is on one segment :
, Mo(x) = - ~pH
,
dx
a')' 2- 1 -
(19)
q-CleU'C':/u~q-C2e-U'(~'/H~. (20)
It now remains to determine the constants of integration in equation (20); this, however, will be linked to the upper portions of the wall. A similar expression for the shearing force qz(x), acting there, must first be developed.
EA~
q;(2) d2 dr/- ~
d,~d ~.
Q2
0q hb
(23)
It should be noted again that the suffix 2 and 1 relate to elements in zones (2) and (1) respectively. After differentiating twice and multiplying by El> equation (23) becomes:
a2Ei2d3y2
h2c312~1
L
(22)
011
This now allows us to set up the compatibility condition for zone (2), which becomes when
12Elb2~d2qz(x)
+
dx2
2b12 dq2(x) _ 0.
212 + ~ 2 qz(X)
A~
(24)
dx
The moment at any section x of each wall segment in zone (2) is:
H M~(x) = Mo(x)+
q2(2) d2.
(25)
x Following now the same procedure as above, we get d
d
dx [Me~(x)] = - dx M°(x)
02
+ -7 q2(x).
(26)
Introducing equation (26) into equation (24), we have" -a2
dMo(x) d-----~" +
qz(x)
h2c312 I
lZElb2"]dzq2(x) + --~q,(x) 2/2 +
bb
2;~ql(k) EA 1
EA2
(21)
where Q2 was defined by equation (8) as the vertical reaction of zone (2) upon zone (1). The additional relative vertical displacement of the lower wall segments due to the action of the shear force ql(x) in zone (1) is:
65 =
x II
EI2 dx 3 -
The compatibility condition of equation (6) for the upper part of the wall will now include not only the deformations, expressed by equations (3)-(5), but also the effect of deformation of the lower wall, zone (1). The relative displacement of the lamellae at their point of contraflexure due to the vertical deformation of the wall segments in zone (1) is: 2b 64 = -- - Q2
h2c
LI2EI< + GA~,_] q2(x)
a2 ~
day2
< x < H
ZONE(2)--b
]12 (-'3
EA
with which the final expression for q~(x) in zone (1) becomes:
ql(x) = alY12 1--
dy2- ~
, .
This then gives the particular solution
q':(x)- a,y2
introducing into equation (6), equations (3)-(5t. (21) and (22):
--
2bi2 dqz(x) AI
dx
dx
- 0,
(27)
which after division by ½a~H2 becomes:
h2c312 [ 12EIb2qdZq2(x) 4blz dq2(x) 6a~H2I~2~l + c2GA*2.J dx 2 + a~H2~ dx 1 [ 4I z -~ - H---5I_l+ a~2A2jqz(x ) =
2
dMo(x)
azH2
dx (28)
Simplified Analysis of Coupled Shear Walls of Variable Cross-Section Similarly to the solution above, the following abreviations are defined:
6a2H2Ib~ ~ = ' h2c312
(29a)
12EIb21 f12 = 1-F c2GA~2] =
412 1 ]
1+
15
The final expression for the shear forces in zone (2) therefore becomes: pH[ q2(x) = a~7~
x 72v4]+C3e-A(X/H)+C4e-B(X/H)
1 . . .H.
(34)
(29b)
(29c) DETERMINATION OF T H E CONSTANTS
2 2 ~2~2
OF INTEGRATION
(29d)
(29e)
The following boundary conditions will serve to determine the four constants of integration C a to C4.
With the constants of equations (29) introduced into equation (28), the differential equation for the upper part of the wall is obtained:
(1) No rotation at the fixed base of the wall, i.e. for x = 0, (dy/dx) = 0 which becomes from equation (7)
4bI 2 v22 - a2A 1H
[q,(X)]x=O =0 d2qz(x) dx 2 =
v~{~_32)Zdq2(x) (~2)2 + n\?2] dx ~ q2(x)
_._2_2( . 2
~dMo(x). a 2 ~k]~2HJ dx
(30)
(2) At the plane of contiguity between the different zones of the wall, at which the crosssection is abruptly changed, namely at x = b, the continuity of the internal shear force is maintained :*
The homogeneous solution of equation (30) is:
qh(x) = C3e-a(x/n) +C4 e-B(x/n)
[ql(X)]x=b = [q2(x)]x=b (31)
where A = 2[ 1-~(1
+4/z2~1 -~2z]J
(32a)
B=~-[I+ /(i+4/~'~I
(32b)
m-
2 2 V2//2
?,~
(32c)
To obtain the particular solution of the differential equation (30) we have to introduce into it the value of the external moment M o. It was assumed above that the applied horizontal loading p is uniformly distributed over the full height of the wall. This by no means is an essential assumption for lhis solution and could have just as easily been considered as different loading on each zone of the wall. However, with uniform loading p, the external moment is still expressed by equation (18) above, which when introduced into equation (30) yields the particular solution:
pH[
q~(x) = a2~' 2
1
x
H
~l
y2J"
(33)
(35)
(36)
(3) At the same section, x = b, continuity stipulates that the slope of the deflection curve of the upper wall equals that of the lower wall:
dx Jx=b L dx ax=,~
(37)
Extracting (dyl/dx) and (dy2/dx) from equations (7) and (23) respectively and differentiating once, equation (37) becomes:
I
dq~(x) 1 = RFdq2(x) 1 dx Jx=b L dx Jx=b
(38)
* This boundary condition requires some qualifications. It is not a priori clear that the internal shear force does not abruptly change at the plane of change in cross-section. However, recognizing the fact that this shear force is taken as acting in a continuousconnecting medium, such an internal force cannot change abruptly, as well known by St. Venant's principle. The effect of different section properties is felt only at some distance from the plane of change and there it is properly accounted for in all the expressions developed above. This fact has been brought out also by other authors who had offered a different solution to the subject of this paper [6,5]. In light of this, the boundary condition of equation (36) is believedto be valid.
16
Avraham Pisanty and Eliahu E. Traum
where
with
h=c ]
h=c 3
.V
o L12EI~ + CA~'J R = _! "~ V h'c~ /,,o ]
(39)
[)2E1h, + GAy,J
= -tf
;I
and
x= -2H !}H 2 2H Vk~(g~) = aff~ 2s(1--~k)+--/~ ~(C 1e - " ' ~ (52)
+C2e-"':'~sinh(t~ts).
(4) At the top of the wall, for x = H, the moment vanishes. Thus, d 2yz]
"d~-J~=. = 0
(40)
which yields from the derivative of equation (23): h2c312
dqz(x)
2bl2
~-d;-
+ ~
q~(x) ~=~ = 0 .
(4i)
These four boundary conditions now yield a set of linear simultaneous equations for the determination of the four constants of integration C~ to C4: C~ +
C2
= K
e~'~rC~ +e-"~rCi-e-arc3-e-BrC, + L(I - r - N)]
=
vd¢~) -
2~[1 - ¢~-
2C4H _ 8~, 2 C 3 H e - a ; " sinh(As) + - - - - ~ e sinh(Bs), A (53) and finally, for the shear force in the top beam:
+
V~2
r/ - - -h- = -//h- -
-
PHLaz?~ \ 8 H
(45)
~/1~)2
I---1 I
J t % c_) c->
v~
I
(48)
S =
[(J]
(49)
r-
b H"
(50)
l+/t~ v2
1 "
I
]
I I
1
't
I
With Ct to C a determined, q~(x) and q2(x) are computed from equations (20) and (34) respectively. Integrating q(x) over the height of one story, we obtain the shear force Vk(x ) in the connecting beam at floor level x k by xk+h/2
j"
q(x)dx.
This will give in zone (1), where
0_<~_<_r
(51)
(D
1 8
1
BEAMS
k:
7
T H E S H E A R F O R C E IN T H E C O N N E C T I N G
xk-h]2
(54)
5(t/m!
(47)
N - 72
Vk(x) =
.
A 21-story coupled shear wall with an abrupt change in cross-section at the l 1th story is analyzed (see figure 4). The wall is subjected to a uniformly
(46)
pH L = a2722
]
NUMERICAL EXAMPLE
(44)
/SH =
)
+ C 4 ~H e- B(eBr - 1 )
4
In the expressions above some additional abreviations were used, namely: K
H l v~2 \ + C3A e - a(ea'-- I ) 2 72
[-K(I -r) (43)
t~ ~e"'~C 1 - - 111e - "'~C2 + A R e - Arc 3 2i- B R e - B r c = K-RL B e - ~Ca + A e - ~ C , = SL.
(42)
In zone (2), for 1 < ~ < I, equation (51) will yield with equation (34):
~)
E:
1
I--] I I I 1 I 1 ""'~A1"(4~ Fig. 4. Shear wall analysed in numerical example (dimensions in metres).
Simplified Analysis o f Coupled Shear Walls o f Variable Cross-Section d i s t r i b u t e d h o r i z o n t a l l o a d i n g p f o r its full height. A l l d i m e n s i o n s a r e g i v e n in m e t r i c units.
For
17
0 < ~ < 0.5238 q 1(~) = p[4.20525(1 - ~) + 0.230643 e 2"2 a 3o~] - 4 . 4 3 5 8 9 3 e -2"23a°¢]
h = 3.00 m
a n d f o r 0.5238 < ~ < 1
H = 63.00 m
q2(¢) = p[5.64858(0.9712 - 3) + 0-060732 e 3'1339~
d~ = 8.40 m
- 8.950221 e -
d2 = 5.40 m
3"4449~].
T h e s h e a r f o r c e s in t h e c o n n e c t i n g b e a m s at e a c h f l o o r level are g i v e n b y :
c --- 4.80 m a l = 13.20 m
For
a2 = 10.20 m
0 =< ~ = < 0.5238
Vk1(~) --~ fiH[0"200240(1
t = 0.50 m (wall thickness)
- ~) + 0"010965 e 2.2 330~
--0"210888 e -2"2330~]
G = 1/2E a n d f o r 0.5238 < ~ < 1
b = 33.00 m
Vk2(¢) = p H [ 0 " 2 6 8 9 9 0 ( 0 " 9 7 1 2 - - 0
h b = 0.60 m
+ 0"002889 e 3 . 1 3 3 9 : _ 0.425622 e -
r = (b/H) = 0.5238 A 1 = 4.20m 2
1: = 24.6960 m 4
A z = 2-70 m 2
12 = 6.5610 m 4
Ag" = 0.25 m 2
1~ = 0.0090 m 4
1/",2
= 0"022633 pH.
T h e c o m p u t e d v a l u e s o f q ( O a n d Vk(O a r e listed in T a b l e 1. T h e v a l u e s f o r ~ = 0.5238 (at t h e l l t h floor) are listed twice, as o b t a i n e d b y t h e e q u a t i o n s for the upper and lower zone respectively. In a d d i t i o n , t h e v a l u e s o f M r a n d M a a r e g i v e n at each floor which represents the moment beneath
All d e t a i l e d c o m p u t a t i o n s are o m i t t e d h e r e a n d o n l y t h e final e x p r e s s i o n s f o r q(~) a r e g i v e n :
Table 1 Story
21 20 19 18 17 16 15 14 13 12 11 II I0 9 8 7 6 5 4 3 2 1
0
1.00000 0.95238 0.90476 0"85714 0.80952 0.76190 0.71429 0"66667 0-61905 0-57143 0.52380 0"52380 0.47619 0.42857 0.38095 0.33333 0.28571 0.23810 0.19048 0.14286 0.09524 0"04762 0.00000
3.4449~]
-p
pH
0'946388 0'971136 1.013488 1"068435 1.130570 1"194971 1"256670 1.309292 1'350917 1'372137 1'367790 1'368072 1'338868 1.300017 1"248605 1"181621 1"096602 0'989957 0.857962 0"697433 0-504069 0"272977 0.00(0~
0"022633 0.046206 0"048237 0"050866 0"053836 0-056916 0"059866 0'062385 0"064381 0"065407 0.065216 0"065190 0'063818 0.061986 0-059557 0-056387 0-052359 0-047305 0'041045 0"033429 0"024251 0.013280 0.000000
M~ (~) pH 2
-0.001265 -0.003305 -0.004376 - 0'004524 -0.003780 - 0.002154 + 0.000370 +0.003824 + 0.008249 +0.013728 +0.006163 +0.012516 +0.020195 + 0.029262 +0.039796 +0'051881 +0.065633 +0.081174 + 0.098648 +0-118216 +0.140068
Mg (~ pH 2 -0.001832 - 0-005006 -0.007210 -0.008493 -0'008882 -0-008388 0-007000 - 0.004680 --0-001388 + 0.002954 -
-- 0'005739 --0.000523 + 0-006028 +0.013956 +0'023357 + 0"034311 + 0"046925 +0-061333 + 0"077672 +0.096107 +0'116825
18
Avraham Pisanty and Eliahu E. Traum
the connecting beam (including the action of the shear force at that beam) and above it respectively. For the k-th story (at Ck): a M[(~k) = Mo(~k)-- y --
~
Vi(~.)
i=k+l
\ v,(¢)
MI(~k) = Mo(~k)-- a
\
i=k
Figure 5 shows the distribution of q(O over the height of the wall as well as the total shear force in every connecting beam. ]n figure 6 the moment distribution on the wall segment is plotted. The computations have been carried out to a high degree of accuracy for illustrative purpose, which, however, is not essentim with manual calculations. Nevertheless it is recommended to compute fairly accurately the values of q(~) and Vk(~) at the plane of change in cross-section, which will serve to check the correctness of the solution.
1
/ /
j/"
/
~q
<
/
/
/ C f
S
CONCLUSIONS
002 !r
Compared with previous solutions, which treat each wall portion separately and match them at their junctions, the method of analysis here is
!).I
'"
, ,'
:
(a)
Fig. 5. (a) Distribution of q(~), (b) Distribution of Vk(O.
-
010
008
: ~
(b)
1/I
(}!;>
00:.
¢
Story
0.12,
/
00.6
00-,
002
Fig. 6. Moment distribution in a wall segment.
('~
.L
00~
-
Simplified Analysis o f Coupled Shear Walls of Variable Cross-Section believed to considerably reduce and simplify the numerical work. It follows the simple pattern of establishing for each part of the wall its compatibility equation, each following the same straightforward format, and then setting up the boundary conditions. Those link all constants of integration directly together as the unknowns in a set of simultaneous linear equations. The generality of the method put forward here, obviously transcends the application that served to illustrate it. (a) To apply the solution to a coupled shear wall with several abrupt changes in cross-section, two additional boundary conditions are used at each plane of change, similarly to equations (37) and (38). This will yield, together with the other boundary conditions, all necessary simultaneous equations for the determination of all constants of integration. (b) The "abrupt change" implied change of cross-section of the wall or of the connecting beam, the story height, the loading or any other discontinuity. (c) Any distribution of horizontal loading may be considered. It will only affect the value of the external moment, Mo(x ), which is reflected in a different particular solution of the differential equation.
a
distance between the centroids of the wall segments
b
height of the lower zone
e E
clear span of the connecting beams modulus of elasticity of concrete
G
shear modulus of concrete
H
height of shear wall
h
story height
hb
height of connecting beam
I
moment of inertia of wall segment
Ib
moment of inertia of connecting beam
ME(I)
total moment on the wall segment at
MF.(~)
moment in wall element beneath connecting beam at
M~(~)
T
moment in wall element above connecting beam at
M0(~ )
external moment in one wall segment due to p at
b
horizontal loading per unit height of shear wall
q(~)
shear force per unit length in the substitute system
r
= (b/H)
s
= (h/2H)
Vk(~)
shear force in connecting beam at
NOTATION The suffix 1 and 2 pertain to the elements of zone (1) and (2) respectively. A
cross-sectional area of one wall segment
Ab
cross-sectional area of the connecting beam
Ai
equivalent cross-sectional area of the connecting beam for shear deformation
19
= (x/tO
other constants are defined in the text.
REFERENCES
1. L. CHITTY, On the cantilever composed of a series of parallel beams interconnected by cross members. Philosophical Mag. (Lond.) Vol., 685 (1947). 2. H. BECK, Contribution to the analysis of coupled shear walls. Proc. Am. Concr. Inst. 59, 1055 (1962). 3. R. ROSMAN, Beitrag zur statischen Berechnung waagrecht belasteter Querwfinde bei Hochbauten (I) (II). Bauingenieur, 35, 133 (1960), 37, 24 (1962). 4. R. ROSMAN, Beitrag zur statischen Berechnung waagrecht belasteter Querw/inde bei Hochbauten (III). Der Bauingenieur, 37, 303 (1962). 5. E. TRAUM,Multistory pierced shear walls of variable cross-section, Symposium on Tall Buildings held at Southampton 1966. Pergamon Press, Oxford, 1967. 6. A. COULL, and R. D. PORt, Analysis of coupled shear walls of variable cross section. Building Sci. 2, 313 (1968). 7. A. PISANTY,Coupled shear walls in multistory buildings, M.Sc. Thesis, Faculty of Civil Engineering, Technion--Israel Institute of Technology, Haifa, February 1965.
20
Avraham Pisanty and Eliahu E. Traum
On pr6sente une analyse simplifi6e d'un mur de cisaillement accoupl6 symm6trique de section variable. La solution est en accord avec les conditions limitrophes de la pattie sup6rieure et de la partie inf6rieure du tour au plan de contiguit6, auxquelles un changement soudain de la section a lieu. Une s6rie d'6quations lin6aires simultan6es est produite pour la d6termination des constantes d'int6gration avec lesque[les routes les forces et d6placements internes sont ais6ment obtenus. La solution est illustr6e d'un exemple num6rique. Eine vereinfachte Analyse einer symmetrisch gekuppelten Absteifwand mit ver~inderlichem Querschnitt wird dargestellt. Die L6sung kommt den Grenzbedingungen des oberen und unteren Teils der Wand an der Angrenzebene gleich, bei welcher ein plt~tzlicher Wechsel des Querschnitts erfolgt. Daraus ergeben sich eine Reihe linearer Simultangleichungen zur Ermittlung der Intergrierungskonstanten, mit welchen alle inneren Kr~ifte und Verschiebungen leicht erhalten werden ktinnen. Die I_~Ssungwird mit zahlenm~issigem Beispiel illustriert.