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Elasto-plastic analysis of coupled shear walls
Elasto-plastic analysis of coupled shear walls O.
A.
Pekau
Department of Civil Engineering, Concordia University, Montreal, P.Q., Canada VladimirGocevski Canatom lnc., Montreal, P.Q., Canada (Received September 19 79; revised July 1980)
The problem of yielding in the components of coupled shear walls under strong earthquake excitation reduces, for purposes of design, to estimation of ductility requirements. To provide a basic treatment of this problem, a simplified elasto-ptastic analysis of coupled shear walls subjected to lateral loading is suggested. It is based on an approximate distribution of the coupling shear for post-elastic behaviour. A continuous lamina replaces the coupling beams, and hinges are permitted in the walls as well as in the beams. Four examples are included to demonstrate the application of the proposed analysis and to assess the validity of the approximations involved. Results confirm the proposed analysis to within good accuracy. Introduction A theory for the elastic analysis of coupled shear walls subjected to lateral loading was first proposed by Chitty. 1 Three basic techniques have evolved since then: (1), continuum approach; 1-~ (2), frame analogy;4, s and (3), finite element method. 6, 7 While it has many limitations, the continuum idealization has proved sufficiently accurate to allow study of the basic behaviour of coupled systems. The work by Beck 2 applied the technique for various types of static lateral loading, with subsequent extensive studies by Rosman a and others. 8' 9 For wind loading on tall structures, such elastic analysis is generally adequate, since the criterion governing design is usually the deflection. However, only in exceptional cases will it be possible to resist earthquake-generated forces within the elastic range. For strong seismic excitation, the walls as well as the connecting elements (slabs or beams) may experience extensive yielding, and nonlinear behaviour results. Ultimate analysis employing the above continuous lamina theory was introduced by Winokur and Gltick 1° w h o assumed a complete collapse mechanism. Extensive work by Paulay n, 12 resulted in an approximate elastoplastic stagewise procedure in which a fixed sequence of partial failure mechanisms involving the walls and beams is assumed prior to total collapse. Finally, Gldck's formulation 9 for the elasto-plastic behaviour of the coupling continuum, with walls assumed elastic, provides lateral load capacity based on the rotational yield limitation in the connecting beams. 0141-0296/81/020087-09/$02.00 © 1981 IPC Business Press
This paper presents a simplified elasto-plastic procedure which attempts to examine the complete history of response for a coupled shear wall loaded laterally up to overall collapse. The elastic continuum method is modified to allow plastic action in the coupling beams as well as in the two walls. Figure i shows the structure and assumed loading pattern. The latter consists of upper triangular loading together with a concentrated top force, thus modelling the seismic design forces of present building codes. Numerical examples are provided to demonstrate the application of the present method and to compare it with other available techniques.
Elastic b e h a v i o u r The coupling beams formed by the vertically arranged openings in the structure of Figure 1 are replaced, for purposes of analysis, by the continuous connection of Figure 2a. The differential equation 2,11 expressing equilibrium and compatibility conditions is: d2T(~) -
-
d~ 2
Mo(~ ) -
a 2 T(~)
= -7
-
-
H
(1)
where ~ = x/H; H = height of the structure; T(~) = axial force in the walls; Mo(~) = cantilever moment produced by the external loads; and ~, 7 are parameters that depend on the geometric properties of the shear wall system and are defined in the Appendix. For tile combination of
Eng. Struct., 1981, Vol. 3, April 87
Elasto-plastic analysis o f coupled shear wails." O. A. Pekau and V. Gocevski
pw
i
where l = distance between wall centroidal lines; s = clear span of connecting beams; E = Young's modulus lbr concrete; 10 = (11 + 12), the sum of moments of inertia of the walls; and Ai, A 2 = cross-sectional areas of walls 1 and 2, respectively. After integration, equation (5) reduces to:
i
f~ E3 F-q
0(~) WeK1 , 2
E~ EE3 E3 C3
rl_4
3
12
..... 1 /3 (cosha - cosha~) + - (sinho~
1
- 2 (~2 _
A~ [--7 I~ [--']
~.
sinbad)
2~+1)}
(6)
where parameters K 1 and K z are defined in the Appendix. For the assumptions 2, 3 implicit in equation (1), the general expression for elastic wall rotation 0~(~) is:
A2 I~
v--q v-q [Z] V--] v--q y--q v--q
1
H
1
Employing equation (7) in the following integration for lateral displacement:
1 8(~) = H } Ow(~) d~ Figure I
(8)
Coupled shear wall subjected to lateral load
results in the expression for elastic wall deflection: triangular and point loads of Figure
~3 f Mo(~) = W,Ht£ 3 + p~]
1, Mo(~) is given by:
~(~)= Weg 3 {K4 [ ~460 ~11 +-p
(2) /3
where We = resultant of the distributed load for elastic behaviour; and p = ratio of the point load to the distributed load. Solving equation (1) for the appropriate boundary conditions leads to the following solution for axial force:
+-
5\3--~2__p~)]
2
(3)
in which/3 is a parameter defined in the Appendix. The coupling shear force q(~) in the lamina is obtained by differentiating the above expression for T(~); this yields:
2 (~2 _ 2~ - p )
- a sinha~
1
]
(4)
By considering wall elastic rotation and axial deformation, it is easily shown that laminar rotation 0(~), defined in Figure 3, is obtained from: 1
1
/4/
O(~):sE~o { f Mo(~)d~ - l f T(~)d~ } H(I+
88
1
1
Eng. Struct., 981, Vol. 3, April
~~
4
12
'¢ P( ~-- ~)]
60
2
1 - ~) cosha - - (sinha - sinha~)
(1 - ~ ) sinha - - ( c o s h a
__+~2_~
2~w~r1/3 sinha~ - cosha~ - ~ + 1 T(~) = ~ -
27We [ q(~) = ~Ha /3a cosha~
3
3
- cosha~)
(9)
Elasto-plastic analysis
Simplifying assumptions Loading beyond the yield strength qu of the coupling continuum will create a plastic region in the central portion of the curve for shear distribution. To simplify the analysis, the variation of coupling shear is approximated by the solid curve of Figure 2b. Boundary coordinates ~ and ~, corresponding to associated elastic load We, define the plastic region. In the upper and lower zones where beams remain elastic, it is assumed that the distribution of coupling shear follows the same pattern as for elastic behaviour, but with magnitude based on the reduced load W corresponding to the elasto-plastic state o f the structure. These assumptions produce a stepped transition o f q at the boundary points, which will be shown to be relatively unimportant to the solution for displacements and forces within the structure generally. For inelastic load W, the solutions for axial force and bending moments, as well as for rotations and deflections, must be developed separately for the three zones in the' above approximation for the nonlinear distribution of the laminar shear.
Elasto-plastic analysis of coupled shear walls: O. A. Pekau and V. Gocevski a
~
/
=!
force and coupling shear distribution leads to ~5(~)expressed in the concise form:
b
HZl ~(~)=6e(~)+-~o(DI-D2-D3-D4
~8 c~O
~x
h
~N
~=x
\
H
\
\ \
x \
\ /
while equation (5) yields the coupling beam end rotation:
/
0(~) =I4,'
L.
sEIoL4 /////
"/,
Figure 2 Idealization of coupled shear wall: (a) coupling beams replaced by continuous lamina; (b) assumed elasto-plastic shear distribution. (-- -- --), elastic; ( ), inelastic
Derivation o f response equations In the upper zone (0 ~< ~ ~< r/) the coupling beams behave elastically and equations (3) and (4) are assumed to express, for load W, axial force and laminar shear, respectively. Thus, the coupling effect represented by IT(~) is known and M(~), the bending moment on an arbitrary cross-section through both walls, is:
M(~) = W
(1-~2)
]
1 2 _ ~.2 + ~(~+ ~21] + C2_ T~.(I _ ~.)}
HZl ~(~) = 6e(~) + - - (06 - D 7 - D 8 - D 9 +Ds)
Elo
a4
(16)
- 3 + p~
for which terms Ds to D 9 are listed in the Appendix. In the lower zone (~"~< ~ ~< 1) the coupling beams behave elastically; thus, equation (4) approximates the distribution of laminar shear q (~). Allowing for the plastic action of the middle zone, it follows that axial force T(~) is given by:
s i n h ~ - cosha~ - ~ + 1
where T~ = T(~) from equation (3).
T(~) = T o + Hqu(~ - 7 ) + T~ - T: 271 [~
(15)
where the notation for axial force at boundary points is: To = T(~) for ~ = 77, and T~. = T(~) for ~ --- ~', both from equation (3). The corresponding lateral displacement of the walls is obtained from the same considerations that led to equation (13); thus, in concise form:
(10)
Since both walls have the same deflection through the entire height, this moment is assigned to individual walls according to their moments of inertia. Substituting T($) from (3) and Mo(~) from (2) gives:
+-----+ 12 3
Hl 2 sEI° { Tn(1 - ~) + Hq u [(~ - r/)(1 - ~) + ~"-
qu
M(~) =Mo(~) - IT(~)
(14)
T(~) = T o + H q u ( ~ - r l )
/
(13/
where 6e(~) is the elastic component of displacement given by equation (9) and where terms D 1 to D s have definitions listed in the Appendix. In the middle zone (r/~< ~ ~< ~') the coupling beams have plastic hinges at both ends and axial force becomes:
I I
I
_'H
+Ds)
(17)
~2 (~3 2'3--~2
(11)
p~j)]}
Equation (5) defines the rotation of coupling beams and, for T(~) of (3) and M(~) above, reduces to: 0(~)=W
sEl o
+-----+ 12 3
S
(1-~2
1
- cosha~) - - ( s i n h ~ - sinbad) O/
0~2 L
,,
+~--~ --2(n
--~
3,}+ C l + C 21
I
(12)
for which the terms C1 and Cz appear in expanded form in the Appendix. Lateral deflection 6(~) is defined by equation (8). Separate consideration of the effects of external load, axial
s l
Figure 3 Laminar rotation resulting from bending and axial deformation of walls
Eng. Struct., 1981, Vol. 3, April
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Elasto-plastic analysis of coupled shear walls: O. A. Pekau and V. Gocevski
1
The corresponding laminar rotation, obtained using equation (5), is: 0(~)=W--
+-- +--+-(1-~ 12 3 2
sEIo H2l 1 gel o
O~2 [ 1
2)
+2(1
~2) + ; 0 - ~ 2 - 2 ~ )
-Tr(1-O (24)
and:
2~,wt t3 (cosha - coshaO
EIo
1
- - (sinha - sinha~) --+
12
(l--~ 2
3
+-(1+~ 2 2
- Tr(1
(18)
Finally, expressed in a form similar to equation (13), lateral displacement is:
H2l 6(~) = 6e(~) + F~oo [ r r - T~ - H q u ( f ×(1-~)(~
~3
- a- (sinh~ - sinh~() + 2 [4 + .......... 12 3
[ T n ( l - ~) + Hq u ( f - r/) (1 - ~)
2 [_4
~4
r/)]
f)
(19)
In the case of relatively stiff (and strong) walls, plastification of coupling beams may extend to the top of the structure (special case 1). Hence, only two zones remain: a plastic upper zone and an elastic lower zone. The laminar shear in the plastic upper zone is constant at qu, whereas equation (4) represents the assumed shear distribution in the lower elastic zone. Axial force at any section in the upper plastic zone is:
T(~) =Hqu~
(20)
while in the elastic lower zone:
T(~) =Hquf + T~ - Tr
(21)
With axial force known, response equations are formulated following previous procedures. This results in laminar rotation for the upper zone given by:
H21[ 1 ~ 4 0(~) = W sEio 4 + - 12
The equations for the preceding case, where laminar plastification has spread to the top of the structure, apply only if both walls are still elastic. To continue the analysis after a plastic hinge occurs at the base of one of the walls before yielding of the topmost coupling beams (special case 2) it is assumed that this wall continues to participate as part of the system by resisting additional axial force only. At the base of the wall, rotation of the hinge is accompanied by increasing axial force and an appropriate decrease in moment resistance, according to the moment-axial force interaction relationship for the cross-section. With wall 1 yielding, the moment assigned to wall 2 at its base is given by: M 2 = (M~= 1 -Mul), where M~= 1 is the moment obtained from equation (10) and Mm is the current plastic hinge capacity at the base of wall 1. This procedure allows increase of axial force in the wall until the entire connection continuum has yielded or a hinge forms in the second wall, whichever occurs first. Two wall hinges are assumed to constitute overall failure even though in certain cases a few of the topmost beams may remain unyielded. The equations developed previously are employed in the calculation of force and displacement response of the structure.
External load The approximate inelastic shear distribution must be in equilibrium with the applied external loading. The external load W for any elasto-plastic state of the structure (all cases except special case 1) follows from equation (12) after setting 0 = Oy at ~ = r/, where 0y is the laminar yield rotation. Thus: W = -G
--~3+P(1--~2 3 2
(26a)
with:
///2
- - {Hqu [~(1 - ~) + ~"- ~ - ~-z sEIo + ~(f2 + ~2)1 + C2 - T~(1 - f)}
F=H 2
-
H21
G =-
[3 + ~4 - 4r/3 + 6p(1 - r72)]
1210 (23)
where t e r m s D m and Dal are defined in the Appendix. In the lower zone, laminar rotation and wall displacement are obtained from equations (18) and (19), respectively, for r/= 0. Thus:
~3 p
q
0(~)= w sEIo 4+ 12 - - 3+z (1- ~2 HI s [ 23,W ~E~otHqu~(1-~)+ ~ - {~ (cosha - cosha~)
90
Eng. Struct., 1981, V o l . 3, A p r i l
~(~2-rTZ)]qu+sEOy (26b)
I
[12
H21[ 1 ~ 4
[(~" 7?)
Al
(22)
The corresponding displacement reduces to:
H21 6(~)=~)e(~)+~o(D6-D 7 DlO + D l l )
+--+
1
- ? t ( 1 - ~)1
1][
f
(26c)
where T,7 = Tn/W; T~ = T~/W; and T(~) = T(~)/W. A similar equation holds for special case 2, obtained using equation (22) and ~ = ~'. The total external load is then W(1 +p) for the combination of distributed and point loads of Figure 1.
Elasto-plastic analysis of coupled shear walls: O. A. Pekau and V. Gocevski
Computational procedure It is convenient to compute the load-displacement history 13 by proceeding in increments of the associated elastic load We O.e. assuming completely elastic behaviour). Once first yield has occurred in the connecting continuum, every increase in this load results in new values for boundary coordinates r/and ~', which are obtained from equation (4) for the condition that q(~) = qu. With n and ~ known, displacements and forces follow readily from the preceding formulations.
Comparative examples The nonlinear response to lateral loading of a coupled shear wall structure can easily be followed through stages of incremental loading employing the method proposed in this paper. Four numerical examples will be presented. Example 1 demonstrates the application of the method, whereas examples 2, 3 and 4 evaluate its performance by comparing numerical results with the results from other methods. Example 1: nonlinear behaviour o f a typical 20-storey structure The structure consists of the 'shear core' of a 20-storey building that is subjected to the lateral loading of Figure 1 with p = 0.13. Figure 4 shows a cross-section through the structure, while Table 1 contains a summary of the corresponding stiffness and strength properties. For this structure, the value of the plastic coupling shear capacity qu provided by the beams is 438 kN/m. The effects of cracking of concrete are introduced by the following stiffness reductions: (a) 70% loss in beam stiffness as a consequence of diagonal and flexural cracking, and (b) 30% loss in the axial and 50% loss in the flexural stiffness of wall 1 because of cracking due to axial tension and bending moment, respectively.
356mm-- 1 E
356 mm
oE
w
!
!
L-- 1830mm" " 3660.mm 3660 mm Figure 4 Wall section of 20-storey structure (examples 1 and 2). Beam depth, 9 1 0 ram; storey height h, 2 6 7 0 ram; total height H , 53 3 0 0 m m
Table I Section properties for 20-storey example structure (Figure 4) AI (m2)
As (m2)
/1 (m4)
/2 (m4)
Ib
Notes
( m 4 x l 0 -a) 1.935
1.935
2.616
2.616
2.170
Uncracked sections
1.354
1.935
1.308
2.616
0.652
Beams and wall 1 cracked
Note: (1) ultimate capacity in bending: wall 1,Mul = 5.48 x 104 kN.m; wall 2,Mu2 = 2.14 x 104 kN-m; (2) ultimate axial capacity: wall 1 in tension, Tu = 2.57 x 10*kN; wall 2 in compression, Cu = 6.87 xl04kN
0
2
I I 4 6 Ductility factor, #b
l 8
El
I 10
12
1.0 08 - 06
.O
O4 O 0 .-s02 I
I I I I 100 200 Displacement, 5Too (mm)
I 300
I
b 400
Figure 5 Nonlinear behaviour of a typical 20-storey structure: (a) distribution of ductility factors for connecting beams; (b) loaddisplacement diagram
Considering progressive stages of loading the analysis, with results depicted in Figure 5, proceeds as follows: (1) The onset of yielding in the lamina occurs at ~ = 0.73. Curve 1 of Figure 5a shows the distribution of beam rotational ductility factor expressed as/a b = 0/0y, where Oy is the yield rotation. Equation (4) determines the magnitudes of the yield load at this stage for q(~) = qu and ~ = 0.73. The load and corresponding deflection are: W/Wu = 0.60, where Wu is the load at overall failure; 6top ---H/460. (2) Further load increase causes other coupling beams to enter the plastic range and yielding in either wall or of the topmost beam terminates this loading stage. In the present structure 30% of all beams remain elastic when wall 1 begins to yield at W/Wu = 0.87 and 6top = H/326. (3) This stage describes the behaviour from yielding of wall 1 up to formation of plastic hinges in the topmost coupling beam. The latter occurs when W/Wu = 0.96 and 6top = 11/193. Thereafter, wall 1 is ineffective in carrying additional load. (4) Beyond load point 3 of Figure5b, wall 2 acts as a simple cantilever and carries all increase in applied load. Failure takes place when a hinge is formed at the base of wall 2. This occurs when Wu = 4,592 kN; concentrated top force = 596 kN; and ~top = 345 mm. The presence of a plastic hinge at the base of wall 2 completes the kinematic failure mechanism, with wall 1 and all coupling beams already yielding. The ratio of top displacement at the ultimate to first yield load (i.e. points 4 and 1, respectively, in Figure 5b) in this case is 3.0 and corresponds to the displacement ductility factor commonly employed in seismic studies. Curve 4 of Figure 5a shows, however, that the critical coupling beam located at about mid-height of the structure undergoes a much greater
Eng. Struct., 1981, V o l . 3, A p r i l
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Elasto-plastic analysis of coupled shear walls: O. A. Pekau and V. Gocevski
degree of plastic action, as indicated by/% = 9.95. It is common practice to use a ductility factor of 4 in earthquake resistant design of ductile reinforced concrete structures. For the present structure, this implies a top displacement of 453 mm and a ductility requirement for the critical coupling beams of more than 20. The latter value emphasizes the need for the careful attention that must be paid to the ductility in coupling beams whenever coupled shear walls are to resist seismic forces.
16
12
k
>o
Example 2: comparison with frame analogy
o
In order to justify use of the approximate elasto-plastic shear distribution of Figure 2b and at the same time to examine the magnitude of the errors resulting from this assumption, an extensive investigation of the internal forces at various stages of loading up to overall collapse has been conducted. Both the simplified analysis discussed here as well as a solution based on the wide-column frame analogy are employed. The latter is accomplished by use of a version of the inelastic plane frame computer program described in Reference 14. The 20-storey structure of example 1, with properties defined by Figures 1, 4 and Table 1, is adopted for this study. The results are presented in Figures 6 to 10. A comparison of the load-displacement diagrams is presented in Figure 6. Good agreement is observed for all stages of loading including final collapse. It is worth noting that the portion of the displacement curve beyond point B involves modelling of the interaction diagrams for the cross-sections of both wails. The approach taken has been summarized as special case 2 ; however, a detailed discussion of this aspect of the problem is beyond the scope of this paper. Figures 7 to 10 compare the proposed analysis with the frame analogy in terms of the internal forces in the beams and walls. The load level presented here (W = 3700 kN) corresponds to a partially plastic state of the structure in which approximately 60% of the connecting continuum has yielded. The beam shear force for the proposed analysis (Figure 7) is obtained by integrating the coupling shear between mid-heights of adjacent storeys, whereas the curves for wall bending moments (Figure 9) represent average values at floor levels in the case of the frame solution. The data indicate that the variation of shear force in beams (Figure 7), axial force (Figure 8), and bending moments in walls (Figure 9) agree closely for the proposed
bl
2°F~
8
4b I i
0
500
10OO O 500 1000 15oo Shear force in b e a m s , ( k N ) Figure 7 Proposed analysis versus frame analogy: distribution of shear force in beams. (a}, elastic state, [4t = 2760 kN. (b}, Partially plastic state, W = 3700 kN. ( ), proposed analysis; (------), frame analogy 20
X
16
_
~7
12
o Y_ 8
0
I
0
I
_.
I
I
II
I
__
I
4
8 t2 16 20 24 Axial force in walls , ( k N × l O 3) Figure 8 Proposed analysis versus frame analogy: distribution of axial force in walls (W = 3700 kN). ( }, proposed analysis; ( - - - ) , frame analogy
b 16
5
4
-12 £y
o
3
×
o EL 8
z v -
2
o, 1
I
O
50
I
I
I
100 150 200 Displacement , 6rap (ram)
J
250
Figure 6 Proposed analysis versus frame analogy: load-displacement diagram. A , first yield; B, wall 1 yields; C, wall 2 yields. ( - - ) , proposed analysis; (-- -- --), frame analogy
92
Eng. Struct., 1981, Vol. 3, April
300
. . . . I I I 8 16 24 16 O Bending moments in w a l l , ( k N - m ) Figure 9 Proposed analysis versus frame analogy: distribution of bending moments in walls (14/= 3700 kN). (a), Wall 1 ; (b), wall 2. ( - - ) , proposed analysis; (------), frame analogy O
8
Elasto-plastic analysis o f coupled shear walls: O. A. Pekau and V. Gocevski
analysis and the frame analogy. However, the variation of beam ductibility factor lzb (Figure 10) requires additional comment. With the definition ~ub = O/Oy,the proposed analysis predicts values within good accuracy when the structure is entirely elastic, as well as for the nonlinear portion when beams have yielded over a part of the structure. As Figure lOb indicates, good accuracy is achieved over the entire structure if the beam ductibility factor is defined as the ratio q/qu for the elastic state and as 0/0y for the yielded portion of the structure. Table 2 provides a summary of maximum values for selected response parameters at the critical stages of loading. Errors remain within readily acceptable levels (i.e. less than 5%), thereby confirming the usefulness of the simplified analysis proposed herein.
Example 3: comparison with GEick's solution The 18-storey shear core described in Reference 9 serves for this comparison, where the present analysis is compared with Glfick's method. The results are summarized in Table 3 and Figures 11 and 12. The tabulated values compare the internal forces in terms of the axial force at the base of the walls (T~=l), the top displacement 5top, and the maximum coupling beam rotational ductility factor/a b . The corresponding load-displacement diagram is shown in Figure 11. Gl~ick's method terminates with yielding of wall 1 and the agreement up to this stage of behaviour is seen to be good.
Table 3 W (kN)
Comparison with GI6ck's solution (example 3) T~=I(kN )
8to p (mm)
max./~b
Notes First yield
3 559
19 4 8 0 (19480)*
72.1 (72.1)
1.00 (1.00)
3 946
21 470 (21,720)
82.0 80.8
1.47 (1.39)
4 177
22 430 (22 570)
89.1 (87.9)
1.92 (1.92)
4 435
23 430 (23,530)
97.8 96.0
2.52 (2.45)
4 559
23 900 (23 750)
102.1 (100.8)
2.84 (2.86)
3 5594 559
1.2%
1.9%
Wall 1 yields
5.8%
Maximum difference
* Values in brackets are for GliJck's method of Reference 9 (case C)
6 5 o
C
-~4 v
3 a
16-
\\
\
o
2
b
1
~
~ = q/qu ' ~ <1 ~b~ 1
12
I
0
I
I
I
I
40
80 120 160 200 Displacement , 6To p ( mm ) Figure t l Proposed analysis versus GILick's solution: loaddisplacement diagram. A, elastic; B, first yield; C, wall 1 yields. ( ), proposed analysis; (-- -- --), Gl~ck's solution
E 0
b
4
0
I 1
A¢~'t
I 2
I
[
1
2
Ductility f a c t o r , p,a Figure 10 Proposed analysis versus frame analogy: distribution o f beam ductility factors (W = 3700 kN). ( - - ) , proposed analysis; ( - - - - - - ) , frame analogy
O;= k~n
~ 0.4 c
o o
~ Q6
Table 2 Comparison with frame analogy (example 2) W (kN)
T~=! (kN)
~top (mm)
max-/~ b
Notes
3 168
16 456 (16 652)*
74.3 (76.9)
1.00 (1.00)
First yield
3 983
19 679 (20 230)
189.4 (192.7)
2.99 (2.97)
Wall 1 yields
4 593
21 551 (22 037)
252.3 (256.2)
5.59 (5.61)
Wall 2 yields
3 1684 593
2.7%
3.4%
3.8%
* Values in brackets refer to computer frame analysis
Maximum difference
08 m
10~,.,,~1 -~0 0 200 4 0 0 6 0 0 Coupling shear, q (~) (kN/m)
I
5
10 15 20 Axial f o r c e , T ( ~ ) (kN×lO 3)
25
Figure 12 Proposed analysis versus Gl~ck's solution: (a) distribution of coupling shear; (b) distribution of axial force in walls (W = 4435 kN). ( ), proposed analysis; (-- -- --), Gli3ck's solution
Eng. Struct., 1981, Vol. 3, April
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Elasto-plastic analysis o f coupled shear walls: O. A. Pekau and V. Gocevski
Since Gl(ick's solution is more accurate over this range of behaviour, it is worthwhile to examine in further detail the magnitude o f the error involved in the analysis proposed herein. Thus, Figure 12a compares the assumed distribution of coupling shear q with the more accurate shear flow of Gl(ick's solution, for a level of loading where approximately 60% of the coupling continuum has yielded. The corresponding variation of axial force in the walls is presented in Figure 12b. Except for the expected local difference in the magnitude o f q at the elastic to plastic transition levels, the agreement for both the coupling shear and the axial force in the walls is seen to be good. Other internal forces, as well as the resulting displacements, within the structure follow from these parameters; hence, the magnitude o f errors resulting from the approximate distribution o f q employed in the analysis proposed herein will be acceptably small. Example 4: comparison with Paulay's method A final comparison is now made, this time using the results for the 18-storey structure reported by Paulay n in order to demonstrate the accuracy of the present analysis for stages o f behaviour when all coupling beams yield prior to formation of a plastic hinge at the base of wall 1. Table 4 and Figure 13 show that the displacements and internal forces agree closely for the two methods. It should be noted that, whereas Paulay's procedure presupposes yielding o f the topmost connecting beam prior to formation of a plastic hinge in wall 1, tile analysis proposed herein is not confined to a fixed sequence of yielding in the structure.
Conclusions A simplified technique is presented for the approximate elasto-plastic analysis o f coupled shear walls under the effects of applied lateral loads. Comparisons of numerical results with existing methods indicate close agreement over the applicable ranges o f behaviour. A detailed analysis of the sequence of lateral collapse behaviour for a typical 20-storey shear core is discussed. It is noted that connecting beam ductility requirements, as expressed by the required rotational yield capacity, can become excessively high for magnitudes of the structural ductility factor normally assumed in aseismic design of ductile reinforced concrete structures. It should be emphasized that the merits of the proposed analysis are demonstrated by comparison with other analytical procedures, whereas experimental verification is obviously the future requirement for the present as well as the other available analytical techniques. Table 4 Comparison with Paulay's method (example 4) W (kN)
T~=I(kN )
f r o p (mm)
3 100
20 680 (20 680)*
81.8 {81.8)
4 786
28 090 (27 790)
5 351 5 636 3 100-5 636
Notes
i o 04to
0
I
28 090 (27 790)
233.2 (238.8)
Wall 1 yields
28 090 (27 790)
296.4 (299.7)
Wall 2 yields
AI,A2 dl, d2 E H h I1,12
Eng. S t r u c t . , ! J81, V o l , 3, A p r i l
I
l 400
1 Chitty, L. 'On the cantilever composed of a series of parallel beams inter-connected by cross-bars', Phil. Mag., 1947, 38 2 Beck, H. 'Contribution to the analysis of coupled shear walls', J. Amer. Concrete lnst., 1969, 59 (8), 1055 3 Rosman, R. 'Approximate analysis of shear walls subject to lateral loads',J. Amer. Concrete lnst., 1964, 61 (6), 717 4 Kumarapillai, K. N. and Coull, A. 'Elastoplastic analysis of coupled shear walls', J. Struct. Div., ASCE, 1976, 102, ST9, 1845 5 MacLeod, I. A. "Lateral stiffness of shear wails with openings', In 'Tall buildings' (eds Coull, A. and Stafford Smith, B.), Pergamon Press, NY, 1967 6 Chiyyarath, V. 'Analysis of sheax walls with openings', J. Struct. Div., ASCE, 1969, 95, ST10, 2093 7 MacLeod, I. A. 'New rectangular finite element for shear wall analysis', J. Struct. Div., ASCE, 1969, 90, ST3,309 8 Coull, A. and Choudhury, R. J. 'Stresses and deflections in coupled shear walls',& Arner. Concrete lnst., 1967, 64 (2), 65 9 Gl/~ck,I. 'Elasto-plastic analysis of coupled shear walls', J. Struct. Div., ASCE, 1973, 99, ST8, 1743 10 Winokur, A. and Glfick, J. 'Ultimate strength analysis of coupled shear waits', J. Amer. Concrete Inst., 1968,65 (12), 1029 11 Paulay, T. 'An elasto-plastic analysis of coupled shear walls', J. A mer. Concrete Inst., 1970, 67 (11), 915 12 Paulay, T. 'Coupling beams of reinforced concrete shear walls', J. Struct. Div., ASCE, 1971,97, ST3, 843 13 Gocevski, V. 'Analysis and behaviour of coupled nonlinear shear walls', M.Eng. Thesis, Concordia University, Montreal, 1977 14 Kanaan, A. E. and Powell, G. H, 'General purpose computer program for inelastic dynamic analysis of plane structures', Rep. EERC 73-6, Earthquake Engineering Research Centre, University of California, Berkeley, 1973
Uppermost beam yields
Maximum difference
I I I 200 300 Displacement, 5Top (mm)
References
162.6 (165.1)
2.3%
I
displacement diagram. A, first yield; B, topmost beam yields; C, wall 1 yields; D, wall 2 yields. ( ), proposed analysis; (o), Paulay's method
Nomenclature
1.1%
I
100
Figure 13 Proposed analysis versus Paulay's method: load-
First yield
* Values in brackets are for Paulay's method of Reference 11
94
~oP
lo lb
cross-section areas of walls 1 and 2, respectively widths o f walls 1 and 2, respectively modulus of elasticity of concrete overall structure height storey height cross-section moments of inertia for walls 1 and 2, respectively sum of moments of inertia of walls = 11 +12 coupling beam moment of inertia
Elasto-plastic analysis of coupled shear walls: O. A. Pekau and V. Gocevski
Mo(~)
total cantilever moment produced by external lateral load bending moment on an arbitrary cross-section through both walls distributed shear in lamina ultimate coupling shear capacity of connecting beams clear span of coupling beams axial force generated in walls values of T(~) from equation (3) at 7, ~, and ~', respectively triangular component of lateral load for inelastic behaviour triangular component of lateral load for elastic behaviour triangular component of ultimate lateral load parameter defined by equation (27) parameter defined by equation (28) lateral displacement elastic component of displacement given by equation (9) displacement at top of structure plastic zone boundary coordinate parameter defined by equation (29) distance between wall centroidal lines coupling beam rotational ductility factor point load parameter laminar rotation at support wall rotation non-dimensional coordinate plastic zone boundary coordinate
M(~) q(~) qu S
T(~)
W We W.
a(f)
(~e(~) (~top 7 7 l /ab P
o(~)
Ow(~)
+--
+-----+-(1-~'2
2
12
3
+ 2 (1 + ~.2 _ 2~')} D, = ~
1 0l
- - (sinha~ - sinha~
(1 + 1 +12] 12IbH2 3 = tanha +
1
1
1 2
(A.1)
(A.3)
_14 (~.4 _ 74 ) _ ~ (~.3 _ 73)1 } D 2 = -~- ~j
(cosha~" - coshe7)
_ 1,~(sinbad" - s i n h , ~ ) - 1 8 2 - n 2) + ~"- n
a22 [11(~4--7 4) 1 3(~" 3 -- 73)- 2(.~-2 72)]} (A.11) D3 = Tn [1(~.2 _ 7 2) _ ~(~.--7)]
)
(A.12)
½(7 + ~)(~-2 _ 72 ) + 7~(~"- 7)] (A.13)
~) (A.14)
[~"cosha~" - ~ coslloe,
- - (sinha~" - sinha~) -
sinha~"
- ~ sinha~ - -, (cosha~" - cosha~
O~ + 1_
,]
a2
Dv=~ 1
--(sinha~" - sinha~) -
A 2]
1
(~-2_ ~ 2 ) + t + ~
2
27H2l
(A.6)
E I o a4 a2
(A.7)
2"1l 2
2
(A.16) (A.17)
D8 = Tn [1~2 + ~2) _ ~.]
DO = Hq u [~(~-3 _ ~3) _ 1(7 -- ~j)(~-2 _/j2) + r/~(~ -- ~)]
The following terms are expressed in concise form in the text: C~ = T n ( 1 - 7 )
D6 = 7 { ~
1 3
(A.5) . /o +12(1+1 \A 1
K4 -
(A.10)
(A.4)
I-Io~4
Ha 4
a 2 [1
-72)-2 i7 ( s-75)
0~
K2 =
K3 -
]
Ds = [Tt - Tn - Hqu(~ -7)I ( I -~) ( ~ -
(A.2)
2a cosha
K1- 27hl2 Ii + ! I°+ il EIoa4S 12 [ 1 1
27l
l[ 0~
- - ~ sinha~
2--a2(l+p)
12IblH a 7 = hs3i°
I
)]
- 7 sinha-rl - - (coshoe~"- coshar~) a
Additional equations The parameters used in the text are:
(A.9)
{~ [~" cosha~" - ~ cosha7
D4 =Hqu [~(~-3_ 73) _
Appendix
2
1
+Hqu(~-
- Tt(1 -- ~')
7)[~(S" - 7 ) + (1 -~')]
(A.8)
hm =aqu [~(~-3 _ ~3) + ~(~-2 _ ~2)]
(A.18) (A.19)
Dn= [Tt-Hqu~](1-~) ( 7 -
(A.20)
~)
Eng. Struct., 1981, Vol. 3, April
95
A.
Pekau
Department of Civil Engineering, Concordia University, Montreal, P.Q., Canada VladimirGocevski Canatom lnc., Montreal, P.Q., Canada (Received September 19 79; revised July 1980)
The problem of yielding in the components of coupled shear walls under strong earthquake excitation reduces, for purposes of design, to estimation of ductility requirements. To provide a basic treatment of this problem, a simplified elasto-ptastic analysis of coupled shear walls subjected to lateral loading is suggested. It is based on an approximate distribution of the coupling shear for post-elastic behaviour. A continuous lamina replaces the coupling beams, and hinges are permitted in the walls as well as in the beams. Four examples are included to demonstrate the application of the proposed analysis and to assess the validity of the approximations involved. Results confirm the proposed analysis to within good accuracy. Introduction A theory for the elastic analysis of coupled shear walls subjected to lateral loading was first proposed by Chitty. 1 Three basic techniques have evolved since then: (1), continuum approach; 1-~ (2), frame analogy;4, s and (3), finite element method. 6, 7 While it has many limitations, the continuum idealization has proved sufficiently accurate to allow study of the basic behaviour of coupled systems. The work by Beck 2 applied the technique for various types of static lateral loading, with subsequent extensive studies by Rosman a and others. 8' 9 For wind loading on tall structures, such elastic analysis is generally adequate, since the criterion governing design is usually the deflection. However, only in exceptional cases will it be possible to resist earthquake-generated forces within the elastic range. For strong seismic excitation, the walls as well as the connecting elements (slabs or beams) may experience extensive yielding, and nonlinear behaviour results. Ultimate analysis employing the above continuous lamina theory was introduced by Winokur and Gltick 1° w h o assumed a complete collapse mechanism. Extensive work by Paulay n, 12 resulted in an approximate elastoplastic stagewise procedure in which a fixed sequence of partial failure mechanisms involving the walls and beams is assumed prior to total collapse. Finally, Gldck's formulation 9 for the elasto-plastic behaviour of the coupling continuum, with walls assumed elastic, provides lateral load capacity based on the rotational yield limitation in the connecting beams. 0141-0296/81/020087-09/$02.00 © 1981 IPC Business Press
This paper presents a simplified elasto-plastic procedure which attempts to examine the complete history of response for a coupled shear wall loaded laterally up to overall collapse. The elastic continuum method is modified to allow plastic action in the coupling beams as well as in the two walls. Figure i shows the structure and assumed loading pattern. The latter consists of upper triangular loading together with a concentrated top force, thus modelling the seismic design forces of present building codes. Numerical examples are provided to demonstrate the application of the present method and to compare it with other available techniques.
Elastic b e h a v i o u r The coupling beams formed by the vertically arranged openings in the structure of Figure 1 are replaced, for purposes of analysis, by the continuous connection of Figure 2a. The differential equation 2,11 expressing equilibrium and compatibility conditions is: d2T(~) -
-
d~ 2
Mo(~ ) -
a 2 T(~)
= -7
-
-
H
(1)
where ~ = x/H; H = height of the structure; T(~) = axial force in the walls; Mo(~) = cantilever moment produced by the external loads; and ~, 7 are parameters that depend on the geometric properties of the shear wall system and are defined in the Appendix. For tile combination of
Eng. Struct., 1981, Vol. 3, April 87
Elasto-plastic analysis o f coupled shear wails." O. A. Pekau and V. Gocevski
pw
i
where l = distance between wall centroidal lines; s = clear span of connecting beams; E = Young's modulus lbr concrete; 10 = (11 + 12), the sum of moments of inertia of the walls; and Ai, A 2 = cross-sectional areas of walls 1 and 2, respectively. After integration, equation (5) reduces to:
i
f~ E3 F-q
0(~) WeK1 , 2
E~ EE3 E3 C3
rl_4
3
12
..... 1 /3 (cosha - cosha~) + - (sinho~
1
- 2 (~2 _
A~ [--7 I~ [--']
~.
sinbad)
2~+1)}
(6)
where parameters K 1 and K z are defined in the Appendix. For the assumptions 2, 3 implicit in equation (1), the general expression for elastic wall rotation 0~(~) is:
A2 I~
v--q v-q [Z] V--] v--q y--q v--q
1
H
1
Employing equation (7) in the following integration for lateral displacement:
1 8(~) = H } Ow(~) d~ Figure I
(8)
Coupled shear wall subjected to lateral load
results in the expression for elastic wall deflection: triangular and point loads of Figure
~3 f Mo(~) = W,Ht£ 3 + p~]
1, Mo(~) is given by:
~(~)= Weg 3 {K4 [ ~460 ~11 +-p
(2) /3
where We = resultant of the distributed load for elastic behaviour; and p = ratio of the point load to the distributed load. Solving equation (1) for the appropriate boundary conditions leads to the following solution for axial force:
+-
5\3--~2__p~)]
2
(3)
in which/3 is a parameter defined in the Appendix. The coupling shear force q(~) in the lamina is obtained by differentiating the above expression for T(~); this yields:
2 (~2 _ 2~ - p )
- a sinha~
1
]
(4)
By considering wall elastic rotation and axial deformation, it is easily shown that laminar rotation 0(~), defined in Figure 3, is obtained from: 1
1
/4/
O(~):sE~o { f Mo(~)d~ - l f T(~)d~ } H(I+
88
1
1
Eng. Struct., 981, Vol. 3, April
~~
4
12
'¢ P( ~-- ~)]
60
2
1 - ~) cosha - - (sinha - sinha~)
(1 - ~ ) sinha - - ( c o s h a
__+~2_~
2~w~r1/3 sinha~ - cosha~ - ~ + 1 T(~) = ~ -
27We [ q(~) = ~Ha /3a cosha~
3
3
- cosha~)
(9)
Elasto-plastic analysis
Simplifying assumptions Loading beyond the yield strength qu of the coupling continuum will create a plastic region in the central portion of the curve for shear distribution. To simplify the analysis, the variation of coupling shear is approximated by the solid curve of Figure 2b. Boundary coordinates ~ and ~, corresponding to associated elastic load We, define the plastic region. In the upper and lower zones where beams remain elastic, it is assumed that the distribution of coupling shear follows the same pattern as for elastic behaviour, but with magnitude based on the reduced load W corresponding to the elasto-plastic state o f the structure. These assumptions produce a stepped transition o f q at the boundary points, which will be shown to be relatively unimportant to the solution for displacements and forces within the structure generally. For inelastic load W, the solutions for axial force and bending moments, as well as for rotations and deflections, must be developed separately for the three zones in the' above approximation for the nonlinear distribution of the laminar shear.
Elasto-plastic analysis of coupled shear walls: O. A. Pekau and V. Gocevski a
~
/
=!
force and coupling shear distribution leads to ~5(~)expressed in the concise form:
b
HZl ~(~)=6e(~)+-~o(DI-D2-D3-D4
~8 c~O
~x
h
~N
~=x
\
H
\
\ \
x \
\ /
while equation (5) yields the coupling beam end rotation:
/
0(~) =I4,'
L.
sEIoL4 /////
"/,
Figure 2 Idealization of coupled shear wall: (a) coupling beams replaced by continuous lamina; (b) assumed elasto-plastic shear distribution. (-- -- --), elastic; ( ), inelastic
Derivation o f response equations In the upper zone (0 ~< ~ ~< r/) the coupling beams behave elastically and equations (3) and (4) are assumed to express, for load W, axial force and laminar shear, respectively. Thus, the coupling effect represented by IT(~) is known and M(~), the bending moment on an arbitrary cross-section through both walls, is:
M(~) = W
(1-~2)
]
1 2 _ ~.2 + ~(~+ ~21] + C2_ T~.(I _ ~.)}
HZl ~(~) = 6e(~) + - - (06 - D 7 - D 8 - D 9 +Ds)
Elo
a4
(16)
- 3 + p~
for which terms Ds to D 9 are listed in the Appendix. In the lower zone (~"~< ~ ~< 1) the coupling beams behave elastically; thus, equation (4) approximates the distribution of laminar shear q (~). Allowing for the plastic action of the middle zone, it follows that axial force T(~) is given by:
s i n h ~ - cosha~ - ~ + 1
where T~ = T(~) from equation (3).
T(~) = T o + Hqu(~ - 7 ) + T~ - T: 271 [~
(15)
where the notation for axial force at boundary points is: To = T(~) for ~ = 77, and T~. = T(~) for ~ --- ~', both from equation (3). The corresponding lateral displacement of the walls is obtained from the same considerations that led to equation (13); thus, in concise form:
(10)
Since both walls have the same deflection through the entire height, this moment is assigned to individual walls according to their moments of inertia. Substituting T($) from (3) and Mo(~) from (2) gives:
+-----+ 12 3
Hl 2 sEI° { Tn(1 - ~) + Hq u [(~ - r/)(1 - ~) + ~"-
qu
M(~) =Mo(~) - IT(~)
(14)
T(~) = T o + H q u ( ~ - r l )
/
(13/
where 6e(~) is the elastic component of displacement given by equation (9) and where terms D 1 to D s have definitions listed in the Appendix. In the middle zone (r/~< ~ ~< ~') the coupling beams have plastic hinges at both ends and axial force becomes:
I I
I
_'H
+Ds)
(17)
~2 (~3 2'3--~2
(11)
p~j)]}
Equation (5) defines the rotation of coupling beams and, for T(~) of (3) and M(~) above, reduces to: 0(~)=W
sEl o
+-----+ 12 3
S
(1-~2
1
- cosha~) - - ( s i n h ~ - sinbad) O/
0~2 L
,,
+~--~ --2(n
--~
3,}+ C l + C 21
I
(12)
for which the terms C1 and Cz appear in expanded form in the Appendix. Lateral deflection 6(~) is defined by equation (8). Separate consideration of the effects of external load, axial
s l
Figure 3 Laminar rotation resulting from bending and axial deformation of walls
Eng. Struct., 1981, Vol. 3, April
89
Elasto-plastic analysis of coupled shear walls: O. A. Pekau and V. Gocevski
1
The corresponding laminar rotation, obtained using equation (5), is: 0(~)=W--
+-- +--+-(1-~ 12 3 2
sEIo H2l 1 gel o
O~2 [ 1
2)
+2(1
~2) + ; 0 - ~ 2 - 2 ~ )
-Tr(1-O (24)
and:
2~,wt t3 (cosha - coshaO
EIo
1
- - (sinha - sinha~) --+
12
(l--~ 2
3
+-(1+~ 2 2
- Tr(1
(18)
Finally, expressed in a form similar to equation (13), lateral displacement is:
H2l 6(~) = 6e(~) + F~oo [ r r - T~ - H q u ( f ×(1-~)(~
~3
- a- (sinh~ - sinh~() + 2 [4 + .......... 12 3
[ T n ( l - ~) + Hq u ( f - r/) (1 - ~)
2 [_4
~4
r/)]
f)
(19)
In the case of relatively stiff (and strong) walls, plastification of coupling beams may extend to the top of the structure (special case 1). Hence, only two zones remain: a plastic upper zone and an elastic lower zone. The laminar shear in the plastic upper zone is constant at qu, whereas equation (4) represents the assumed shear distribution in the lower elastic zone. Axial force at any section in the upper plastic zone is:
T(~) =Hqu~
(20)
while in the elastic lower zone:
T(~) =Hquf + T~ - Tr
(21)
With axial force known, response equations are formulated following previous procedures. This results in laminar rotation for the upper zone given by:
H21[ 1 ~ 4 0(~) = W sEio 4 + - 12
The equations for the preceding case, where laminar plastification has spread to the top of the structure, apply only if both walls are still elastic. To continue the analysis after a plastic hinge occurs at the base of one of the walls before yielding of the topmost coupling beams (special case 2) it is assumed that this wall continues to participate as part of the system by resisting additional axial force only. At the base of the wall, rotation of the hinge is accompanied by increasing axial force and an appropriate decrease in moment resistance, according to the moment-axial force interaction relationship for the cross-section. With wall 1 yielding, the moment assigned to wall 2 at its base is given by: M 2 = (M~= 1 -Mul), where M~= 1 is the moment obtained from equation (10) and Mm is the current plastic hinge capacity at the base of wall 1. This procedure allows increase of axial force in the wall until the entire connection continuum has yielded or a hinge forms in the second wall, whichever occurs first. Two wall hinges are assumed to constitute overall failure even though in certain cases a few of the topmost beams may remain unyielded. The equations developed previously are employed in the calculation of force and displacement response of the structure.
External load The approximate inelastic shear distribution must be in equilibrium with the applied external loading. The external load W for any elasto-plastic state of the structure (all cases except special case 1) follows from equation (12) after setting 0 = Oy at ~ = r/, where 0y is the laminar yield rotation. Thus: W = -G
--~3+P(1--~2 3 2
(26a)
with:
///2
- - {Hqu [~(1 - ~) + ~"- ~ - ~-z sEIo + ~(f2 + ~2)1 + C2 - T~(1 - f)}
F=H 2
-
H21
G =-
[3 + ~4 - 4r/3 + 6p(1 - r72)]
1210 (23)
where t e r m s D m and Dal are defined in the Appendix. In the lower zone, laminar rotation and wall displacement are obtained from equations (18) and (19), respectively, for r/= 0. Thus:
~3 p
q
0(~)= w sEIo 4+ 12 - - 3+z (1- ~2 HI s [ 23,W ~E~otHqu~(1-~)+ ~ - {~ (cosha - cosha~)
90
Eng. Struct., 1981, V o l . 3, A p r i l
~(~2-rTZ)]qu+sEOy (26b)
I
[12
H21[ 1 ~ 4
[(~" 7?)
Al
(22)
The corresponding displacement reduces to:
H21 6(~)=~)e(~)+~o(D6-D 7 DlO + D l l )
+--+
1
- ? t ( 1 - ~)1
1][
f
(26c)
where T,7 = Tn/W; T~ = T~/W; and T(~) = T(~)/W. A similar equation holds for special case 2, obtained using equation (22) and ~ = ~'. The total external load is then W(1 +p) for the combination of distributed and point loads of Figure 1.
Elasto-plastic analysis of coupled shear walls: O. A. Pekau and V. Gocevski
Computational procedure It is convenient to compute the load-displacement history 13 by proceeding in increments of the associated elastic load We O.e. assuming completely elastic behaviour). Once first yield has occurred in the connecting continuum, every increase in this load results in new values for boundary coordinates r/and ~', which are obtained from equation (4) for the condition that q(~) = qu. With n and ~ known, displacements and forces follow readily from the preceding formulations.
Comparative examples The nonlinear response to lateral loading of a coupled shear wall structure can easily be followed through stages of incremental loading employing the method proposed in this paper. Four numerical examples will be presented. Example 1 demonstrates the application of the method, whereas examples 2, 3 and 4 evaluate its performance by comparing numerical results with the results from other methods. Example 1: nonlinear behaviour o f a typical 20-storey structure The structure consists of the 'shear core' of a 20-storey building that is subjected to the lateral loading of Figure 1 with p = 0.13. Figure 4 shows a cross-section through the structure, while Table 1 contains a summary of the corresponding stiffness and strength properties. For this structure, the value of the plastic coupling shear capacity qu provided by the beams is 438 kN/m. The effects of cracking of concrete are introduced by the following stiffness reductions: (a) 70% loss in beam stiffness as a consequence of diagonal and flexural cracking, and (b) 30% loss in the axial and 50% loss in the flexural stiffness of wall 1 because of cracking due to axial tension and bending moment, respectively.
356mm-- 1 E
356 mm
oE
w
!
!
L-- 1830mm" " 3660.mm 3660 mm Figure 4 Wall section of 20-storey structure (examples 1 and 2). Beam depth, 9 1 0 ram; storey height h, 2 6 7 0 ram; total height H , 53 3 0 0 m m
Table I Section properties for 20-storey example structure (Figure 4) AI (m2)
As (m2)
/1 (m4)
/2 (m4)
Ib
Notes
( m 4 x l 0 -a) 1.935
1.935
2.616
2.616
2.170
Uncracked sections
1.354
1.935
1.308
2.616
0.652
Beams and wall 1 cracked
Note: (1) ultimate capacity in bending: wall 1,Mul = 5.48 x 104 kN.m; wall 2,Mu2 = 2.14 x 104 kN-m; (2) ultimate axial capacity: wall 1 in tension, Tu = 2.57 x 10*kN; wall 2 in compression, Cu = 6.87 xl04kN
0
2
I I 4 6 Ductility factor, #b
l 8
El
I 10
12
1.0 08 - 06
.O
O4 O 0 .-s02 I
I I I I 100 200 Displacement, 5Too (mm)
I 300
I
b 400
Figure 5 Nonlinear behaviour of a typical 20-storey structure: (a) distribution of ductility factors for connecting beams; (b) loaddisplacement diagram
Considering progressive stages of loading the analysis, with results depicted in Figure 5, proceeds as follows: (1) The onset of yielding in the lamina occurs at ~ = 0.73. Curve 1 of Figure 5a shows the distribution of beam rotational ductility factor expressed as/a b = 0/0y, where Oy is the yield rotation. Equation (4) determines the magnitudes of the yield load at this stage for q(~) = qu and ~ = 0.73. The load and corresponding deflection are: W/Wu = 0.60, where Wu is the load at overall failure; 6top ---H/460. (2) Further load increase causes other coupling beams to enter the plastic range and yielding in either wall or of the topmost beam terminates this loading stage. In the present structure 30% of all beams remain elastic when wall 1 begins to yield at W/Wu = 0.87 and 6top = H/326. (3) This stage describes the behaviour from yielding of wall 1 up to formation of plastic hinges in the topmost coupling beam. The latter occurs when W/Wu = 0.96 and 6top = 11/193. Thereafter, wall 1 is ineffective in carrying additional load. (4) Beyond load point 3 of Figure5b, wall 2 acts as a simple cantilever and carries all increase in applied load. Failure takes place when a hinge is formed at the base of wall 2. This occurs when Wu = 4,592 kN; concentrated top force = 596 kN; and ~top = 345 mm. The presence of a plastic hinge at the base of wall 2 completes the kinematic failure mechanism, with wall 1 and all coupling beams already yielding. The ratio of top displacement at the ultimate to first yield load (i.e. points 4 and 1, respectively, in Figure 5b) in this case is 3.0 and corresponds to the displacement ductility factor commonly employed in seismic studies. Curve 4 of Figure 5a shows, however, that the critical coupling beam located at about mid-height of the structure undergoes a much greater
Eng. Struct., 1981, V o l . 3, A p r i l
91
Elasto-plastic analysis of coupled shear walls: O. A. Pekau and V. Gocevski
degree of plastic action, as indicated by/% = 9.95. It is common practice to use a ductility factor of 4 in earthquake resistant design of ductile reinforced concrete structures. For the present structure, this implies a top displacement of 453 mm and a ductility requirement for the critical coupling beams of more than 20. The latter value emphasizes the need for the careful attention that must be paid to the ductility in coupling beams whenever coupled shear walls are to resist seismic forces.
16
12
k
>o
Example 2: comparison with frame analogy
o
In order to justify use of the approximate elasto-plastic shear distribution of Figure 2b and at the same time to examine the magnitude of the errors resulting from this assumption, an extensive investigation of the internal forces at various stages of loading up to overall collapse has been conducted. Both the simplified analysis discussed here as well as a solution based on the wide-column frame analogy are employed. The latter is accomplished by use of a version of the inelastic plane frame computer program described in Reference 14. The 20-storey structure of example 1, with properties defined by Figures 1, 4 and Table 1, is adopted for this study. The results are presented in Figures 6 to 10. A comparison of the load-displacement diagrams is presented in Figure 6. Good agreement is observed for all stages of loading including final collapse. It is worth noting that the portion of the displacement curve beyond point B involves modelling of the interaction diagrams for the cross-sections of both wails. The approach taken has been summarized as special case 2 ; however, a detailed discussion of this aspect of the problem is beyond the scope of this paper. Figures 7 to 10 compare the proposed analysis with the frame analogy in terms of the internal forces in the beams and walls. The load level presented here (W = 3700 kN) corresponds to a partially plastic state of the structure in which approximately 60% of the connecting continuum has yielded. The beam shear force for the proposed analysis (Figure 7) is obtained by integrating the coupling shear between mid-heights of adjacent storeys, whereas the curves for wall bending moments (Figure 9) represent average values at floor levels in the case of the frame solution. The data indicate that the variation of shear force in beams (Figure 7), axial force (Figure 8), and bending moments in walls (Figure 9) agree closely for the proposed
bl
2°F~
8
4b I i
0
500
10OO O 500 1000 15oo Shear force in b e a m s , ( k N ) Figure 7 Proposed analysis versus frame analogy: distribution of shear force in beams. (a}, elastic state, [4t = 2760 kN. (b}, Partially plastic state, W = 3700 kN. ( ), proposed analysis; (------), frame analogy 20
X
16
_
~7
12
o Y_ 8
0
I
0
I
_.
I
I
II
I
__
I
4
8 t2 16 20 24 Axial force in walls , ( k N × l O 3) Figure 8 Proposed analysis versus frame analogy: distribution of axial force in walls (W = 3700 kN). ( }, proposed analysis; ( - - - ) , frame analogy
b 16
5
4
-12 £y
o
3
×
o EL 8
z v -
2
o, 1
I
O
50
I
I
I
100 150 200 Displacement , 6rap (ram)
J
250
Figure 6 Proposed analysis versus frame analogy: load-displacement diagram. A , first yield; B, wall 1 yields; C, wall 2 yields. ( - - ) , proposed analysis; (-- -- --), frame analogy
92
Eng. Struct., 1981, Vol. 3, April
300
. . . . I I I 8 16 24 16 O Bending moments in w a l l , ( k N - m ) Figure 9 Proposed analysis versus frame analogy: distribution of bending moments in walls (14/= 3700 kN). (a), Wall 1 ; (b), wall 2. ( - - ) , proposed analysis; (------), frame analogy O
8
Elasto-plastic analysis o f coupled shear walls: O. A. Pekau and V. Gocevski
analysis and the frame analogy. However, the variation of beam ductibility factor lzb (Figure 10) requires additional comment. With the definition ~ub = O/Oy,the proposed analysis predicts values within good accuracy when the structure is entirely elastic, as well as for the nonlinear portion when beams have yielded over a part of the structure. As Figure lOb indicates, good accuracy is achieved over the entire structure if the beam ductibility factor is defined as the ratio q/qu for the elastic state and as 0/0y for the yielded portion of the structure. Table 2 provides a summary of maximum values for selected response parameters at the critical stages of loading. Errors remain within readily acceptable levels (i.e. less than 5%), thereby confirming the usefulness of the simplified analysis proposed herein.
Example 3: comparison with GEick's solution The 18-storey shear core described in Reference 9 serves for this comparison, where the present analysis is compared with Glfick's method. The results are summarized in Table 3 and Figures 11 and 12. The tabulated values compare the internal forces in terms of the axial force at the base of the walls (T~=l), the top displacement 5top, and the maximum coupling beam rotational ductility factor/a b . The corresponding load-displacement diagram is shown in Figure 11. Gl~ick's method terminates with yielding of wall 1 and the agreement up to this stage of behaviour is seen to be good.
Table 3 W (kN)
Comparison with GI6ck's solution (example 3) T~=I(kN )
8to p (mm)
max./~b
Notes First yield
3 559
19 4 8 0 (19480)*
72.1 (72.1)
1.00 (1.00)
3 946
21 470 (21,720)
82.0 80.8
1.47 (1.39)
4 177
22 430 (22 570)
89.1 (87.9)
1.92 (1.92)
4 435
23 430 (23,530)
97.8 96.0
2.52 (2.45)
4 559
23 900 (23 750)
102.1 (100.8)
2.84 (2.86)
3 5594 559
1.2%
1.9%
Wall 1 yields
5.8%
Maximum difference
* Values in brackets are for GliJck's method of Reference 9 (case C)
6 5 o
C
-~4 v
3 a
16-
\\
\
o
2
b
1
~
~ = q/qu ' ~ <1 ~b~ 1
12
I
0
I
I
I
I
40
80 120 160 200 Displacement , 6To p ( mm ) Figure t l Proposed analysis versus GILick's solution: loaddisplacement diagram. A, elastic; B, first yield; C, wall 1 yields. ( ), proposed analysis; (-- -- --), Gl~ck's solution
E 0
b
4
0
I 1
A¢~'t
I 2
I
[
1
2
Ductility f a c t o r , p,a Figure 10 Proposed analysis versus frame analogy: distribution o f beam ductility factors (W = 3700 kN). ( - - ) , proposed analysis; ( - - - - - - ) , frame analogy
O;= k~n
~ 0.4 c
o o
~ Q6
Table 2 Comparison with frame analogy (example 2) W (kN)
T~=! (kN)
~top (mm)
max-/~ b
Notes
3 168
16 456 (16 652)*
74.3 (76.9)
1.00 (1.00)
First yield
3 983
19 679 (20 230)
189.4 (192.7)
2.99 (2.97)
Wall 1 yields
4 593
21 551 (22 037)
252.3 (256.2)
5.59 (5.61)
Wall 2 yields
3 1684 593
2.7%
3.4%
3.8%
* Values in brackets refer to computer frame analysis
Maximum difference
08 m
10~,.,,~1 -~0 0 200 4 0 0 6 0 0 Coupling shear, q (~) (kN/m)
I
5
10 15 20 Axial f o r c e , T ( ~ ) (kN×lO 3)
25
Figure 12 Proposed analysis versus Gl~ck's solution: (a) distribution of coupling shear; (b) distribution of axial force in walls (W = 4435 kN). ( ), proposed analysis; (-- -- --), Gli3ck's solution
Eng. Struct., 1981, Vol. 3, April
93
Elasto-plastic analysis o f coupled shear walls: O. A. Pekau and V. Gocevski
Since Gl(ick's solution is more accurate over this range of behaviour, it is worthwhile to examine in further detail the magnitude o f the error involved in the analysis proposed herein. Thus, Figure 12a compares the assumed distribution of coupling shear q with the more accurate shear flow of Gl(ick's solution, for a level of loading where approximately 60% of the coupling continuum has yielded. The corresponding variation of axial force in the walls is presented in Figure 12b. Except for the expected local difference in the magnitude o f q at the elastic to plastic transition levels, the agreement for both the coupling shear and the axial force in the walls is seen to be good. Other internal forces, as well as the resulting displacements, within the structure follow from these parameters; hence, the magnitude o f errors resulting from the approximate distribution o f q employed in the analysis proposed herein will be acceptably small. Example 4: comparison with Paulay's method A final comparison is now made, this time using the results for the 18-storey structure reported by Paulay n in order to demonstrate the accuracy of the present analysis for stages o f behaviour when all coupling beams yield prior to formation of a plastic hinge at the base of wall 1. Table 4 and Figure 13 show that the displacements and internal forces agree closely for the two methods. It should be noted that, whereas Paulay's procedure presupposes yielding o f the topmost connecting beam prior to formation of a plastic hinge in wall 1, tile analysis proposed herein is not confined to a fixed sequence of yielding in the structure.
Conclusions A simplified technique is presented for the approximate elasto-plastic analysis o f coupled shear walls under the effects of applied lateral loads. Comparisons of numerical results with existing methods indicate close agreement over the applicable ranges o f behaviour. A detailed analysis of the sequence of lateral collapse behaviour for a typical 20-storey shear core is discussed. It is noted that connecting beam ductility requirements, as expressed by the required rotational yield capacity, can become excessively high for magnitudes of the structural ductility factor normally assumed in aseismic design of ductile reinforced concrete structures. It should be emphasized that the merits of the proposed analysis are demonstrated by comparison with other analytical procedures, whereas experimental verification is obviously the future requirement for the present as well as the other available analytical techniques. Table 4 Comparison with Paulay's method (example 4) W (kN)
T~=I(kN )
f r o p (mm)
3 100
20 680 (20 680)*
81.8 {81.8)
4 786
28 090 (27 790)
5 351 5 636 3 100-5 636
Notes
i o 04to
0
I
28 090 (27 790)
233.2 (238.8)
Wall 1 yields
28 090 (27 790)
296.4 (299.7)
Wall 2 yields
AI,A2 dl, d2 E H h I1,12
Eng. S t r u c t . , ! J81, V o l , 3, A p r i l
I
l 400
1 Chitty, L. 'On the cantilever composed of a series of parallel beams inter-connected by cross-bars', Phil. Mag., 1947, 38 2 Beck, H. 'Contribution to the analysis of coupled shear walls', J. Amer. Concrete lnst., 1969, 59 (8), 1055 3 Rosman, R. 'Approximate analysis of shear walls subject to lateral loads',J. Amer. Concrete lnst., 1964, 61 (6), 717 4 Kumarapillai, K. N. and Coull, A. 'Elastoplastic analysis of coupled shear walls', J. Struct. Div., ASCE, 1976, 102, ST9, 1845 5 MacLeod, I. A. "Lateral stiffness of shear wails with openings', In 'Tall buildings' (eds Coull, A. and Stafford Smith, B.), Pergamon Press, NY, 1967 6 Chiyyarath, V. 'Analysis of sheax walls with openings', J. Struct. Div., ASCE, 1969, 95, ST10, 2093 7 MacLeod, I. A. 'New rectangular finite element for shear wall analysis', J. Struct. Div., ASCE, 1969, 90, ST3,309 8 Coull, A. and Choudhury, R. J. 'Stresses and deflections in coupled shear walls',& Arner. Concrete lnst., 1967, 64 (2), 65 9 Gl/~ck,I. 'Elasto-plastic analysis of coupled shear walls', J. Struct. Div., ASCE, 1973, 99, ST8, 1743 10 Winokur, A. and Glfick, J. 'Ultimate strength analysis of coupled shear waits', J. Amer. Concrete Inst., 1968,65 (12), 1029 11 Paulay, T. 'An elasto-plastic analysis of coupled shear walls', J. A mer. Concrete Inst., 1970, 67 (11), 915 12 Paulay, T. 'Coupling beams of reinforced concrete shear walls', J. Struct. Div., ASCE, 1971,97, ST3, 843 13 Gocevski, V. 'Analysis and behaviour of coupled nonlinear shear walls', M.Eng. Thesis, Concordia University, Montreal, 1977 14 Kanaan, A. E. and Powell, G. H, 'General purpose computer program for inelastic dynamic analysis of plane structures', Rep. EERC 73-6, Earthquake Engineering Research Centre, University of California, Berkeley, 1973
Uppermost beam yields
Maximum difference
I I I 200 300 Displacement, 5Top (mm)
References
162.6 (165.1)
2.3%
I
displacement diagram. A, first yield; B, topmost beam yields; C, wall 1 yields; D, wall 2 yields. ( ), proposed analysis; (o), Paulay's method
Nomenclature
1.1%
I
100
Figure 13 Proposed analysis versus Paulay's method: load-
First yield
* Values in brackets are for Paulay's method of Reference 11
94
~oP
lo lb
cross-section areas of walls 1 and 2, respectively widths o f walls 1 and 2, respectively modulus of elasticity of concrete overall structure height storey height cross-section moments of inertia for walls 1 and 2, respectively sum of moments of inertia of walls = 11 +12 coupling beam moment of inertia
Elasto-plastic analysis of coupled shear walls: O. A. Pekau and V. Gocevski
Mo(~)
total cantilever moment produced by external lateral load bending moment on an arbitrary cross-section through both walls distributed shear in lamina ultimate coupling shear capacity of connecting beams clear span of coupling beams axial force generated in walls values of T(~) from equation (3) at 7, ~, and ~', respectively triangular component of lateral load for inelastic behaviour triangular component of lateral load for elastic behaviour triangular component of ultimate lateral load parameter defined by equation (27) parameter defined by equation (28) lateral displacement elastic component of displacement given by equation (9) displacement at top of structure plastic zone boundary coordinate parameter defined by equation (29) distance between wall centroidal lines coupling beam rotational ductility factor point load parameter laminar rotation at support wall rotation non-dimensional coordinate plastic zone boundary coordinate
M(~) q(~) qu S
T(~)
W We W.
a(f)
(~e(~) (~top 7 7 l /ab P
o(~)
Ow(~)
+--
+-----+-(1-~'2
2
12
3
+ 2 (1 + ~.2 _ 2~')} D, = ~
1 0l
- - (sinha~ - sinha~
(1 + 1 +12] 12IbH2 3 = tanha +
1
1
1 2
(A.1)
(A.3)
_14 (~.4 _ 74 ) _ ~ (~.3 _ 73)1 } D 2 = -~- ~j
(cosha~" - coshe7)
_ 1,~(sinbad" - s i n h , ~ ) - 1 8 2 - n 2) + ~"- n
a22 [11(~4--7 4) 1 3(~" 3 -- 73)- 2(.~-2 72)]} (A.11) D3 = Tn [1(~.2 _ 7 2) _ ~(~.--7)]
)
(A.12)
½(7 + ~)(~-2 _ 72 ) + 7~(~"- 7)] (A.13)
~) (A.14)
[~"cosha~" - ~ coslloe,
- - (sinha~" - sinha~) -
sinha~"
- ~ sinha~ - -, (cosha~" - cosha~
O~ + 1_
,]
a2
Dv=~ 1
--(sinha~" - sinha~) -
A 2]
1
(~-2_ ~ 2 ) + t + ~
2
27H2l
(A.6)
E I o a4 a2
(A.7)
2"1l 2
2
(A.16) (A.17)
D8 = Tn [1~2 + ~2) _ ~.]
DO = Hq u [~(~-3 _ ~3) _ 1(7 -- ~j)(~-2 _/j2) + r/~(~ -- ~)]
The following terms are expressed in concise form in the text: C~ = T n ( 1 - 7 )
D6 = 7 { ~
1 3
(A.5) . /o +12(1+1 \A 1
K4 -
(A.10)
(A.4)
I-Io~4
Ha 4
a 2 [1
-72)-2 i7 ( s-75)
0~
K2 =
K3 -
]
Ds = [Tt - Tn - Hqu(~ -7)I ( I -~) ( ~ -
(A.2)
2a cosha
K1- 27hl2 Ii + ! I°+ il EIoa4S 12 [ 1 1
27l
l[ 0~
- - ~ sinha~
2--a2(l+p)
12IblH a 7 = hs3i°
I
)]
- 7 sinha-rl - - (coshoe~"- coshar~) a
Additional equations The parameters used in the text are:
(A.9)
{~ [~" cosha~" - ~ cosha7
D4 =Hqu [~(~-3_ 73) _
Appendix
2
1
+Hqu(~-
- Tt(1 -- ~')
7)[~(S" - 7 ) + (1 -~')]
(A.8)
hm =aqu [~(~-3 _ ~3) + ~(~-2 _ ~2)]
(A.18) (A.19)
Dn= [Tt-Hqu~](1-~) ( 7 -
(A.20)
~)
Eng. Struct., 1981, Vol. 3, April
95