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Tier buildings with shear cores, bracing, and setbacks
Computers & Structures, Vol. 1, pp. 57-83. Pergamon Press 1971. printed in Great Britain
TIER BUILDINGS
WITH SHEAR CORES, BRACING, AND SETBACKS
WILLIAM WEAVER, JR. and GREGGE. BRANDOW Department of Civil Engineering, Stanford University, Stanford, California, U.S.A.
and THOMAS A. MANNING, JR. Department of Engineering Science, Louisiana State University, Baton Rouge, Louisiana, U.S.A. Abstract--The theory of bending and torsion in thin-walled members of open cross section is applied to shear cores included within the analytical model of the tier building. Representative types of bracing arrangements are also considered, and their stiffnesses augment those of the beams, columns, and shear walls. Changes in floor plan, referred to as “setbacks”, further complicate the analysis; but a relatively simple routine allows automatic handling of this characteristic. Computer programs named STATLER and DYNATIER include all of these features within the linear static and dynamic analyses of three-dimensional multi-story buildings. Results for a twenty-story structure demonstrate the significance of the interaction between a shear core and the skeletal framing. The effect of warping restraints upon the stresses in the shear core is large at the base but diminishes rapidly with height. The normal stress at the base associated with the bimoment is of the same order of magnitude as that due to the combination of biaxial bending and axial force.
NOTATION The following symbols are used in this paper: A
A, AB
Aw ii Cl, G, G
DB D, DW E F
smgy G
47 Iy J K L
cross-sectional area; action in the i sense ; vector of beam actions; vector of shear wall actions; sectorial property of shear wall ; bimoment ; arbitrary constants; vector of beam displacements ; sectorial property of shear wall; vector of shear wall displacements; modulus of elasticity; structural reference point; shear flexibility factors ; elastic shear modulus; second moments of area; torsion constant; warping constant ; length ; 57
58
WILLIAM WEAVER, JR. GREGG E. BRAND~W and THOMAS A. MANNING,JR.
moment; length of middle line of shear core: concentrated force : distributed longitudinal loading; rotation matrix for wall ; radial distance to tangent plane ; beam stiffness matrices; bracing stiffness matrices; member stiffness matrix for brace: shear wall stiffness matrices; torque ; translational transformation matrices; shear wall thickness; axial displacement in wall due to warping; x coordinate of a row of bracing; y coordinate of a row of bracing; location of shear center of wall (local); structure reference axes; coordinates of shear center (global) ; shear wall member axes; rotation angle for shear wall axes; angle between x axis and outward normal ; angle between axis of brace and horizontal; direction cosines: angle of twist; normal stress due to warping restraint. INTRODUCTION SHEAR cores,
bracing, and setbacks often appear in multi-story buildings as complicating factors in the design and analysis of such structures. Figure 1 shows a tier building that includes all of these features in a rather irregular pattern. Planar shear walls present no great challenge to the analyst, but three-dimensional shear cores such as those enclosing elevators, stairways, and utilities are more difficult to handle. Their plan dimensions are large relative to column sizes, and the effects of shear strain and warping due to torsion can be significant. The purpose of this investigation is to incorporate three-dimensional shear wall configurations, bracing between floors, and setbacks at various framing leveis into the analytical model for the static and dynamic analysis of tier buildings. Toward this end, the theory of bending and torsion in thin-walled members of open cross section is utilized for the shear cores. The interaction between frames and shear walls in multi-story buildings has been discussed extensively in the literature, but most authors have limited their investigations to two-dimensional systems. Coull and Smith [7] presented a rather comprehensive summary of this topic, emphasizing techniques for distributing lateral loads between planar shear walls and structural framing, Clough et al. [4-61 automated the linear and nonlinear analyses of such frames for both static and dynamic loads, treating the shear walls as deep columns with flexural, axial and shearing deformations.
Tier Buildings with Shear Cores, Bracing, and Setbacks
59
X-BEAM GRID WORK
Y-BEAM
/Y-BRACING
z
X-BRACING
I
k
SHEAR WALL
Y
X FIG. 1. Tier building.
The finite-element method fl8] for analyzing shear walls within frameworks offers the important advantage of high numerical accuracy when many elements are used, as demonstrated by McLeod [13], Girijavallabhan [IO], and others. Oakberg and Weaver [16] used this approach and compared results with those obtained by the less accurate method of Clough et nl. This work tended to verify the suitability of the deep-column model for tall shear walls in two-dimensional building frames. Several investigators have proposed the analysis of coupled shear walls by replacing rows of spandrel beams with elastic continua. Beck [Z] and Rosman [19] developed the method in two dimensions, and Michaet [15], Rosman [20], and Gluck [I I] extended it to three-dimensional applications. In the latter studies, torsional characteristics of the shear cores played a major role. Jenkins and Harrison [12] compared theoretical and ex~rimental results for a threedimensional plexiglass model of coupled shear cores. They found a lack of correlation attributed to neglecting the effect of nonuniform torsion due to warping restraints. Experimental evidence from existing buildings 18, 231 also indicates the need for more accurate analytical procedures if realistic values for natural periods of tall buildingsa~ to be obtained. A three-dimensional analytical model was introduced by Weaver et al. [24, 251 and applied to the linear static and dynamic analysis of tier buildings. The model involves open rectangular framing and floor diaphragms assumed to be rigid in their own planes, with no restrictions of symmetry for either the structure or the loading. The effects of ail joint displacements and structural rigidities were included, but shear walls were not considered.
60
WurAh% WEAVER, JK., GREGG E. BRAND~W and THOMASA. MANNIN~~,JR.
Possibilities for subdividing three-dimensionai shear cores into finite elements were discarded as impractical because of the large number of nodal displacements introduced. Furthermore, the verification of the deep-column model in two dimensions [16] generated confidence for extending the concept to three dimensions. Thus, the primary objective of this paper is to present a suitable formuiation for threedimensional shear cores and to incorporate it into the analysis of tier buildings. For additional versatility, the possibilities of bracing and setbacks are also included. The pattern of shear walls and bracing is assumed to be constant for ail levels between setbacks, but changes in their stiffnesses are permitted at every level. Connections are assumed to be rigid, and the building is taken to be constructed entirely of linearly-elastic materials. Displacements of joints are assumed to be small in comparison with the overall dimensions of the structure.
SHEAR CORES Torsion in thin-w&led members The theory of torsion in thin-walled members is available in the literature; therefore, only the concepts and equations required in this study are reiterated herein. The primary source of this material is Timoshenko [21], hut Vlasov [22] and Oden [17] have also published contributions. Consider the thin-walled member of open cross section (see Figs. 2a and b), which is assumed to be prismatic and subjected to both bending and twisting actions. The x and J axes shown in the figures are principal centroidal axes, and the wall is indicated with variable thickness t in Fig. 2b. The thickness is assumed to be small compared to the length m of the middle line of the cross section. In the development which follows it will be assumed that the transverse forces are applied to the shear-center axis, so that the bending and torsional effects can be analyzed independently. The location of the shear center 0 with respect to the centroid C is given by the expressions:
In equation (1) x and J’ are the coordinates of an arbitrary point on the middle line of the wall, s is the distance to the point measured along the middle line from one extremity of the cross section,I, and 1, are the familiar second moments of area about the x and y axes. respectively, and a is defined by:
(2) The dimension rc is the perpendicular distance from the centroid to the tangent to the wail at the point x, y (see Fig. 2b). This distance is taken to be positive if a tangential force acting in the direction of increasing s produces a positive moment about the centroidal axis. If one or more cross sections of a thin-walled member are constrained against warping or if the torque varies along the length, the warping is not constant; and longitudinal stresses are developed internally. The rate of twist also varies along the length of the member, distinguishing this condition of “nonuniform” torsion from that of “pure” torsion, (in which the torque is constant and all sections are free to warp).
Tier Buildings with Shear Cores, Bracing, and Setbacks
61
Y
Shear- center
axis
(a)
(b)
FIG.2. Thin-walled member with open cross section.
In the case of nonuniform torsion, the total torque Tatting at any section will be resisted partially by shearing stresses of the kind developed in pure torsion and partially by shearing stresses resulting from the nonuniform axial deformations. The first part of the torque, denoted by T,, is given by: Tl = GJqY
(3)
where G is the elastic shear modulus, C/Iis the angle of twist, 4’ (the prime denotes differentiation with respect to z) is the rate of twist, and J is the torsion constant, defined by:
s
(4)
T2 = - EK@”
(5)
m 3
J= The second portion of the torque is:
t ds. 03
in which E is Young’s modulus and K is the warping constant, defined by: K =
a(DS)2tds.
s0
(6)
62
In equation
WILLIAM WEAVER, JR., GREGG E. BRANDQWand THOMASA. MANNING.JK.
(6) the symbol
D, represents:
(7)
The identifier ye is similar to the previously-defined rc, but the origin is now taken to be the shear center instead of the centroid. The total resisting torque is determined by combining equations (3) and (5), resulting in: EKcjY - GJ@ = - T.
(8)
When the notation (9) is introduced
The general
into equation
solution
(8), it becomes:
of this equation
is:
where C,, C2 and C, are arbitrary constants which must be determined from the conditions at the ends of the member. The axial displacement u’,, defining the warping of a cross section, is determined from \I’, =
Furthermore,
the longitudinal
D,gY.
stress produced o,= Ez=
due to restrained
(12) warping
ED,@’ .
is defined
by:
(13)
Since Q, results only from torque on the cross section, the resultant normal force and bending moment due to these stresses must be zero. To facilitate the analysis of partial warping restraints, Vlasov [22] introduced a stress resultant called the bimoment B, which is statically zero. It is defined as the product of 6, and D,tds integrated over the cross section. When equation (13) is substituted for the longitudinal stress, the bimoment is found to be related to the twist by: B=
The integral (6); thus,
in the above equation
“‘~,Dstds=E#’ f 0
‘“(D,)‘tds. s cl
is recognized
as the warping
B= EK&‘.
(14)
constant
defined in equation (15)
Tier Buildingswith Shear Cores, Bracing,and Setbacks
63
Shear wall stiflnesses A shear wall element is defined to be that portion of a shear core bounded above and below by two adjacent framing levels in a tier building. The stiffness properties of such an element may be evaluated as if it were a deep column, including the effects of shear strain and nonuniform torsion. For this purpose, the shear-center axis serves as the primary local reference for actions and displacements. In addition, the bimoment B and the rate of twist 4’ are introduced as a generalized action and corresponding displacement at each end of the element. Figure 3 depicts the principal centroidal axes and a parallel set of shear-center axes of a restrained wall element, along with seven types of nodal displacements at each end. Singleheaded arrows in the figure denote translations; rotations are represented by doubleheaded vectors; and the rate of twist at each end is indicated by a triple-headed arrow. The latter entities carry sequence numbers (4 and 8) following other nodal displacements that are not directly associated with the rigid-body motions of the floors (1,2, 3 and 5,6, 7). Partitioning the shear wall stiffness matrix into the following submatrices facilitates the transfer of terms to the stiffness matrix for the whole structure:
SW=
S wa,
0
S Wa,b
S Wa.c
S Wo,
S Wb,
(I
S Wb, b
S Wb, c
S Wb, d
S WC,a
S WC, b
S WC,c
S WC, d
S Wd, a
S Wd, b
S Wd,
S Wd,
c
Shear-center
axis
Fro. 3. Restrainedwall element.
d
d
64
WILLIAM
WEAVER,
JR.,
GREGG
E.
and
BRA&DOW
A. MANNING,JR.
THOMAS
In this matrix the subscript a denotes displacements of types 1-4; the 0 denotes displacements of types 5-8; the c stands for displacements of types 9-l I ; and the rl represents displacements of types 12-14. Thus, the matrix S, is of size 14 x 14, consisting of submatrices of sizes 4 x 4, 4 x 3, 3 x 4, and 3 x 3. Expressions for the shear wall stiffnesses are obtained by inducing unit displacements of each possible type, one at a time, at each end of the wall element. Nonzero terms below the main diagonal of S, determined by this procedure are given in Table 1 for submatrices are related to these as follows : S wu,IITSW+ SWC+’ and SW., only. The other submatrices S Wn,cr=s Wh,h SWC,c,= -SwJ,fl=Swc,lJ= S wc,c= -S TABLE
~
It’d, c =
-SWl,,h (17)
SWYI,d.
1. Shear wall stiffnesses.
~~~
S
_(4+g,Yx+ w’3L- (l+g,JL
S w2,1=
vz g ‘” 1,
AE XoYq-
-
S w3,1=
(3) Unit z,, translation
at .i
(1) Unit .Y,,,rotation
yo----
AE
s w3,3
=, L
sw5,3=-sw1..3
AE
-
at j
L
S Wh,
3=
-
s
3 =
-S
W7.
S w7,
3
w3.3
(4) Unit rate of twist at j 7-
7 W6,
L
1 =
-’
swz,
SWd, 4 =T[cL
1
cosh(cL) - sinh(cL)]
S w7,*=-sw3.1
sM/8,4=
S
S wt1,4(2) Unit y, rotation S
S W&2’ S
AE
.oL
-
GS[c0sl1(cL)
-
(5) Unit x,, translation S
at j
12Ef, w9Y9=(1 +gx)L3
L -s
WI,
2
AE (2-g3~~,_y2 (l+g,)L do L
w6p2=
S W7,2= S
y2
AE =x,-
-
sinh(cL)]
+_
at j
_V+gWy+ w2*2- (l+g,)L
S W3.2
-
-s
w3,
(6) Unit y,,, translation
at j
12E1,
S wto. 10=(,+g,)L”
2
6EI, W9y2=(1+gX)L2
(7) Unit twist at ) S W11, 11= cG.i sinh(cL)
1I
Tier Buildingswith Shear Cores, Bracing,and Setbacks
65
The symbols L, A, and E in Table 1 are the length, cross-sectional area, and modulus of elasticity of the wall element. The identifiers g, and gY denote shear flexibility factors, as given by: 12E1, (18) 9.X- 12ElY ” = GA,L2
GA.&’
where A, is the area effective in shear in the x direction, and A, is that for the y direction. Shear wall stiffnesses other than those associated with twist of the wall were determined by utilizing space-frame member stiffnesses relative to centroidal axes, which are readily available in the literature [9]. Standard axis-translation techniques were used to transform stiffnesses from centroidal reference axes to shear-center axes. Axial and flexural terms are coupled for shear-center axes, as is indicated by expressions containing x0 and y, in Table 1. Torsional stiffnesses in Table 1 were generated using the theory of thin-walled members. As an example, consider a unit twist of the j end of the element (displacement type 11) while all other displacements are zero. Application of the conditions 4,= o = 0 and @, = o = 0 to equation (11) yields: cl=
cf=$.
-cz
(19
The constant C, may be evaluated [14] in terms of the bimoment BzsL associated with the restraint condition r$‘, XL= 0. C2=
1
B,=,+%nh(cL) 1 GJ cGJ cosh(cL)
(20)
in which
Substituting equations (19)-(21) into equation (11) and setting #,= L equal to unity produces the following expression for the restraint torque at the j end of the member:
sW1I, ,,=T=cGJsinh(cL)
(22)
in which GJ
G.J=
(23)
2 - 2 cosh(cL) + CL sinh(cL) ’ The bimoment at the.j end may be seen from equation (21) to be: Swy4,11
=BzzL=
-GJ[cosh(cL)-11.
Similarly, the torque and bimoment at the k end of the member are found to be:
s w14,ll=-%ll,
11
S W8,
ll=sW4,
All other restraint actions in column 11 of SW are zero.
11.
(25)
66
WILLIAMWEAVER,JR., GRECS E. BKANU~Wand THOMASA. MANNING,JR.
Warping actions and displucements
Expressions are required for the statically equivalent actions at the shear center associated with the Ioad cases shown in Figs. 4a, b and, c. Those due to the concentrated force in Fig. 4a are: A ,==Pv,
A,=
-P.Y,
A,=P
/f,=PD,.
(26)
The last expression is the bimoment determined from equation (14), which shows that the longitudinal force produces torsion in the member if the quantity D, is nonzero at the point of application. Consider next the distributed Iongitudina~ loading qS indicated in Fig. 4b, which may be due to local floor loads. The statically equivalent actions at the shear center can be computed by the following integrations: $2
Al=
J
- q,x& c13’ x2 A, = f q,D,ds . .?I
J
PI
Fro. 4. Actions applied to end cross section. (a) Concentrated force. (b) Distributed force. (c) Concentrated moment.
(27)
Tier Buildingswith Shear Cores, Bra&r&and Setbacks
67
The concentrated moment shown in Fig. 4c acts in the positive direction of the outward normal to the plane tangent to the middle surface at the point of application. The bimoment developed on the section is determined by replacing the moment with a couple consisting of two longitudinal forces of magnitude P a distance As apart. B= -PD,+P(D,$
in which P=M/As. equation (7)] :
AD,)
(28)
In the limiting case when As+ds+O,
the bimoment becomes [see
If the moment M is not in the direction of the normal to the tangent plane, only the normal component of M contributes to the bimoment. Displacements at a typical point due to warping of the section are also of interest, and they consist of two types. The longitudinal translation corresponding to the force in Fig. 4a is given by equation (12), while the rotation corresponding to the moment in Fig. 4c is:
dwt -=- d4 # , =: - r& . ds
00)
ds
This rotation is taken to be positive when its vector representation direction of the outward normal.
points in the positive
Shear wall transformations For purposes of evaluating the contributions of a shear wali to the overall structure stiffness matrix, the shear wall stiffness matrix S, must be transformed from local wall axes to structural reference axes. If necessary, the first step may consist of a rotation through the angle o( (about the z, axis) from principal wall axes to a set parallel to the structural reference axes. S Ws= R$S,Rw. The rotation transformation
(39
matrix R, in equation (31) is of the foflowing form:
-4 Rw=
0
0
0
0
RI
0
0
o
0
R2
0
0
0
R2 I
0
(32)
in which f
R1=
sina
0
-sinor 0
cosdl
0
0
1
0
0
0
cos a
(33)
68
WILLIAMWEAVER,JR., GREGG E. BRAND~Wand THOMASA. MANNING,JR
and cos a R,=
sinci
-sina
cos
0
0
1
0
a
0
.
(34
1
-
Portions of the wall stiffness matrix associated with the rigid-body motions of the floors must be transformed to the structural reference axes at each framing level. S,, = T;SwsTw * The translational operator
(33
T’ in equation (35) takes the form: 1
0
oz T,=
0
0
0
0
0
0
0
0 0
TWJ
(36)
T Wk
0
in which Z signifies a 4 x 4 identity matrix, and
Twj=
1
0
- YSCj
0
1
x SCj
0
0
1
(37)
Identifiers Xscj and yscj in equation (37) represent the x and y coordinates of the shear center of the wall relative to structural axes at level j. The definition of submatrix Twk is obtained from equation (37) by replacingj with k. As given by equation (35), the shear wall stiffness matrix is in a form suitable for direct augmentation of the overall structure stiffness matrix. Beam transformations
Beams framing into shear walls increase the wall stiffnesses because the ends of such beams are constrained to move with the walls at the points of connection. Geometric transformations between displacements, actions, and stiffnesses at the end of a beam and those at the shear center of a wall prove to be useful in several phases of the analysis. Figure 5 shows the cross section of a shear wall at a framing level with the axes at the shear center taken parallel to structural reference axes. It is assumed that a beam frames into the wall at point, j. Generalized displacements of the shear wall are indicated by labeled arrows at point 0, whereas the three arrows at point j denote the x and y rotations and the z translation at the end of the beam. The symbol /? in the figure represents the angle between
Tier Buildingswith Shear Cores, Bracing,and Setbacks
69
FIG.5. Shear wall displacements. the x axis and the outward normal at point j. Displacements at j may be computed from those at 0 by the following operation:
Da,= TBj’,j
(38)
in which D,,=
@,,,
Dzt2,
1
0
0
10
4~1
Dw,=
WV,,
4~2,
. . .,
4~)
-r,cos/3
0
0
0
-rosin/I
0
0
0I .
D,
0
0
0
(39
and
Ti#j=
Y
0
1
-X
Terms in equation (40) result from superposition of warping displacements [see equations (I 2) and (3011upon plane-section displacements. Similarly, actions at the shear center 0 may be computed from those applied at point j (see Fig. 6) by the operation: A wj =
TijA,
(41)
in which A,,=
{Am,
A~23
Ad
A,,=
b%v~,
44~29
. . .v &v).
In this case the relationships are based upon equations (26) and (29).
(42)
70
WILLIAMWEAVER,JR., GREGGE. BRANDOWand THOMASA. MANN~G, JR.
FIG. 6. Shear wall actions.
If a beam frames into a wail at its j end only, the 6 x 6 beam stiffness matrix may be transformed to shear-wall coordinates by the operation:
SBW in which Tg=
=
T%I~B
[ 1. 2
;
(43)
(44)
The transformation matrix TB is of size 6 x 10 and includes an identity matrix of size 3 x 3. If both ends of the beam frame into the same shear wall or different shear walls, matrix Z’, takes the form: TB=
[
‘osi T” Bk1
(45)
where subscript k refers to the k end, and the matrix is of size 6 x 14.
The structure shown in Fig. ‘7simulates a reinforced concrete shear core isolated from a twenty-story building. Floor beams of size 16 WF 40 stiffen the core at each framing level and are assumed to be encased in fireproofing concrete with 2 in. of cover all round. Beam properties were computed on the basis of an equivalent steel section (modular ratio= lo), and the thickness of the wall is taken to be 1 in. of equivalent steel When the unstiffened core is loaded by a torque of 1440 kip-in. applied at the free end, it twists as shown by the plot labeled “shear core only” in Fig. 8. Also shown on the figure p the effect of partial warping restraints provided by the stiffening beams. The curve labeled “stiffened shear core” indicates a reduction of the twist at all levels by approximately 35 per cent. In particular, the twist at the top level was reduced from 0*02089 radians
Tier Buildingswith Shear Cores, Bracing,and Setbacks
FIG.7. Isolated shear core structure. to 0.01346 radians, Thus, even a modest amount of floor framing may have a considerable effect on the torsional stiffness of a shear core of open cross section.
E.wnple 2 : Twenty-story building Figure 9 shows a typical framing plan for a hypothetical twenty-story building with the shear core of the previous example positioned to emphasize the unsymme~~ character of the structure. Beam and column sizes vary over the height of the building, as indicated in Fig. 10, but the shear core thickness is taken to be constant. All framing members are assumed to be encased in fireproofing concrete with 2 in. of cover. This building was analyzed statically and dyna~~al~y with and without the core, and some of the results obtained are given below. The static analysis program named STATIER was used to analyze the building for a lateral pressure of 20 psf acting in the positive y direction, in combination with a uniform vertical loading of 100 psf at all levels. ~ispla~men~ for all stories in terms of the rigidbody motions of the reference point F at each level appear in Fig. 11. The maximum joint displacement occurred at point (4,l) at the top level, which translated 0.610 in. in the x direction and 6.628 in. in the y direction. Table 2 gives the bimoment and the y-shear in the core at various levels as well as the percentage of the total shear carried by the wall. The bimoment in the tabIe is that at the lower end of the shear wall segment associated with
72
WILLIAM WEAVER, JR., GREGG E. BKAND~W and THOMAS A. MANNISH, JR
FIG. 8. Twist of shear core due to torque.
FIG. 9. Typical framing plan for 20-story building.
Tier Buildings with Shear Cores, Bracing, and Setbacks
20x - 14WF30 y - 16WF50 z8WF31
IQIa17-
x - 14WF34 y - 18WF 50 z - lOWF60
16?514-
x - 16WF36 y - 18WF55 z -12WF85
1312ll-
x - 16WF40 y - 2lWF55 z - 12 WF106
IOQ8-
x - 16WF45 y - 21WF62 z - 14 WF127
76-
x - 16WF50 y - 2?WF68 z - 14WF142
54-
x - 18WF50 y - 24WF68 z - 14WF158
32-
x - 10WF55 y - 24WF76 z - 14 WF176
l-
X
o-
FIG. 10. Member sizes for 20-story building.
TABLE2. Bimoment and shear in core
Bioment x 10-S (kips-in.9 (2)
Shear (kips) (3)
Percent of total shear (4)
3
-8-340
239.13
71 *o
3
- 1 a303
147.19
48.5
5
-0.866
108.72
40.5
7
-0.327
83.75
36.0
9
-0.595
73-16
37.0
12
-0.152
39.09
26.5 28.0 15-5
Story 0)
1.5
0.064
26.74
18
0.375
6.68
73
74
WIILIAM WEAVER, JH., and GHEC,GE. BHAN~OWTHOMASA. MANNING, JR,
FIG.
II. Floor displacements for building with shear core.
the particular story. It is seen that the bimon~ent decreases rapidly with height, and at the fifth story it is approxinlately 10 per cent of the vafue at the base of the structure. Above this level the bimoment fluctuates considerably but is not very large. Figure 12a depicts the distribution of normal stress in the base of the shear wall due to the combination of biaxial bending, axial force, and the bimoment. In addition, the normal stress for the bimoment only appears in Fig. 12b. These diagrams show that the bimoment contributes approximately 35 per cent of the maximum normal stress at the base. At higher levels in the structure this percentage is much smaller due to the rapid decay of the bimoment. When this building was analyzed without the shear core (which was replaced by structural framing), the s and J’ translations of the point F at the top level increased by 55 and 46 per cent, respectively. The torsional stiffness of the structure was only slightly reduced, however, as indicated by a 7.3 per cent increase in the rotation at the top level. Dynamic analyses of this building without the shear core were reported in f25]; so in the present study the program named DYNATIER was used to re-analyze the structure with the shear core included. As might be expected, the frequency of the fundamental mode of vibration increased by 27 per cent (from O-318 to O-403 cps). Within the first ten
Tier Buildings with Shear Cores, Bracing, and Setbacks
75
(a)
6.00
(b)
FIG. 12. Normal stress in shear wall at base. (a) Resultant normal stress. (b) Normal stress due to restrained warping.
the largest increase in frequency (133 per cent) occurred in the seventh mode. Furthermore, the mode shapes were altered considerably by the presence of the shear core. The dynamic load condition applied to the building with shear core in this study corresponds to load case 5 (ground motion in the y direction) of [25]. Comparison of the displacement-time histories for story number 20 in both structures indicates that the maximum translation (in the y direction at point 4,l) was reduced from 15.2 in. to 11-l in. by the addition of the shear core. On the other hand, the maximum translation in the x direction increased from 2.26 in. to 4.74 in., and the maximum rotation increased from O-010 rad to 0.014 rad. Thus, the response of the two structures to the ground motion differed markedly. In summary, Examples 1 and 2 show that the interaction between shear cores and skeletal framing is of great significance, and neither can be analyzed independently of the other. Furthermore, any attempts at approximating this complicated interaction should be checked against the approach described herein. modes,
BRACING AND SETBACKS Representative types of bracing Figure 13 illustrates four representative types of bracing selected for this study. Types 1 and 2 each consist of one pinned-end member, while types 3 and 4 are each composed of two braces of equal cross-sectional areas. The members in type 4 are not connected at the crossing point.
76
WILLIAMWEAVER,JR., GREGGE. BRANDOWand THOMASA. MANNING,JR.
The member stiffness matrix S, for a typical pinned-end brace may be handled as if it were a member of a plane truss, as follows:
(46)
where LAt and A, denote the length and cross-sectional area of the member, respectively. The direction cosines of the member axis in terms of the angle y (see Type 1 in Fig. 13) are
This stiffness matrix is sufficient to describe bracing types 1 and 2, and type 4 may be considered to be the sum of these two.
Tier Buildings with Shear Cores, Bracing, and Setbacks
77
Bracing type 3 is slightly more difficult to handle because it does not conform to the pattern of joints in a tier building. The joint at the midspan of the beam involves two additional local displacements, as shown in Fig. 14. Displacements A, and A, are eliminated from the set of ten local displacements before the contributions to the overall stiffness and load matrices are assessed. For this purpose, action-displacement relationships for the beam and braces shown in Fig. 14 are written in partitioned form. (48) Solve for D,., in equation (48).
DA=S;;(AA-SSABDB).
(49)
Substituting equation (49) into equation (48) produces:
S&DB=A;
(50)
in which SL=S,,
- Si7.4S24B
A;=A,-s,J~A,.
(51)
(52)
Then the beam with the two braces attached may be treated as a subassemblage having an 8 x 8 stiffness matrix S&, and an 8 x 1 action matrix A;. It should be noted that the beam also has torsional stiffnesses that are uncoupled from the other types. After the displacements of type B are obtained, those of type A may be calculated using equation (49).
L---xor
y
FIG. 14. Displacements
for bracing type 3.
Bracing transformations Geometric transformation of bracing matrices from local to global reference axes are analogous to those for columns and shear walls. Thus,
S;F=T;BS;BT,,.
(53)
78
WILLIAM WEAVER, JK., GREGG E. BRANDOWand THOMAS A. MANNING, JR
* in equation (53) takes various forms, depending upon the The translational operator TtlB type of bracing. For example, Figs. 15a and b show the local and global displacements nvolved with transforming bracing of type 4 that is parallel to either the .Vor the 3%axis.
j/ f..._“j
!‘I
-‘F_i
I
’
,’
Y+-
I ,
/ ! * -.-i’._..._._-._A
_/’
/
7
DF~
j
/’
/ /’ -v’
FIG.
IS. Related floor and bracing displacements.
For bracing of types I or 2 parallel to the .Yaxis, the transformation form :
Ti?li=
-YCR
0
0
0
0
0
0
0
1
0
0
0
0
1
I
0
0
I
0 0
matrix is of the
(54)
Tier Buildingswith Shear Cores, Bracing,and Setbacks
79
Similarly, for bracing of type 3 parallel to the x axis, the matrix takes the form:
TBB=
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
- YCR
-YCR 0
The operators for bracing types 1, 2 and 3 parallel to the y axis are of the same forms as equations (54) and (55), but with - YCR replaced by XCR. A separate operator is not required for bracing type 4, because each brace in that configuration may be transformed individually. Changes in joor plan
In order to handle setbacks that may occur in multi-story buildings (see Fig. l), some automatic technique for altering the floor plan at any level must be introduced into the analytical procedure. During the forward-elimination phase, the structural stiffness and load matrices are condensed story-by-story from top to bottom. When the information for a given story is assembled, the numbering of the degrees of freedom is established for that level and the next level below, assuming that the latter has the same layout as the former. If the floor below contains additional bays, the residual stiffness and action matrices must be expanded in order to accommodate the new degrees of freedom. This expansion may be accomplished by the insertion of rows and columns of zeros corresponding to each new degree of freedom encountered. Subsequently, additional stiffness and action quantities from the story below will create nonzero terms in those positions. During the backward-substitution phase of the analysis, the joint-displacement matrix must be contracted wherever a setback occurs. This contraction may be accomplished by removing rows corresponding to non-existent joints (see Fig. 1) and shifting other rows upward in the matrix. Example 3. Four-story building
A hypothetical four-story building with shear cores, bracing, and setbacks was contrived to illustrate these features and was analyzed both statically and dynamically. The framing plan for the upper two stories is one bay by one bay and for the lower two stories is four bays by three bays, as shown in Fig. 16. Four shear walls occur in the lower portion of the building, and the elevation views in Fig. 17 include four braced frames. Not shown is frame E, which is a simple two-story, three-bay frame without bracing or walls. Member properties for this structure are listed in Table 3.
80
WJLLIAM WEAVER, JK., GREGG E. BRA&DOW and THOMAS A. MANNING,
-_
..~._ -
1st B2n3
-
.
LEVEL
Jn.
. _.._
f-l?AMitG
P...ANS
Fra. 16. Framing plans for I-story building.
TABLE
_
3. Member properties for example 3.
__ ~. Beams
Columns ‘4 (in.*) (4)
f (in.4) (7)
Story (1)
I J (in.4) (in.“) (3) (2)
3 and 4
2,ooo
300
25
200
200
75
I and 2
4,000
500
50
400
400
150
fz b/ (in.4) (in.4) (5) (6)
Bracing: Type 3 A=5 in.2 Type 4 A=6 in.2 Shear wall thickness: 1 in. equivalent steel E=30,000 ksi G=12,000 ksi
Static vertical loads on the building include the uniformly-distributed beam loadings indicated by the numbers in Fig. 16, as well as two IO-kip forces acting downward at points (B,2) and (B,3) at the second level. Lateral loads for each of the upper two stories consist
Tier Buildings with Shear Cores, Bracing, and Setbacks
81
of 14 kips in both the .Yand y directions at their centers. Similarly, a 160-kip lateral load is applied in both directions at the center of each of the lower two stories. The story displacements resulting from this condition of loading are given in Table 4.
--x
TABLE4.
Stoty displacements for example 3 due to static loads.
x-Translation (in.) (2)
y-Translation (in.) (3)
z-Rotation (rad.) (4)
4
4.036 x IO-2
3.501 x10-2
1.014x IO-5
3
2.999 x 10-2
2.416 x 10-Z
1.010 x 10-5
2
1.215x10-2
5.758 x
10-3
1.033 x 10-s
1
7.179 x 10-3
3.195 x10-3
6.021 x IO-6
story (1)
82
WILLIAM WEAVEK, JR., GREGG E. BRANDOWalld THornAs
A. MANNING, JR.
For purposes of dynamic analysis, a uniformly distributed dead was assumed to exist at all levels of the building. The frequency of the of vibration for this structure was found to be 5.576 cps, and computer shape are exhibited in Fig. 18. No other analytical results are included of space, but further details are given in [3].
weight of 200 psf fundamental mode plots of the mode herein due to lack
MODE SHRPE NO. 1 FREQUENCY IS 5.576 C.P.S. X-TRRNS .
Y-TRRNS.
FIG. 18.
CONCLUSIONS The analytical model for tier buildings has been expanded to include three-dimensional shear cores and bracing between floors, as well as the typical beam and column framing. The floor framing plans may vary from one level to another, provided that the plan for the level below merely involves the addition of bays to the plan above. Programs STATlER and DYNATIER for the static and dynamic analyses of tier buildings were considerably upgraded in this study. These programs have been documented in [3] and [14], which may be obtained from the first co-author upon request. FORTRAN coding is also available on magnetic tapes for a nominal charge to cover the cost of handling. Additional features could be incorporated into the programs without great difficulties. These include additional shear wall configurations and bracing types, shear strains in beams and columns, nonprismatic members, finite joint sizes, and flexible connections. The desirability of adding such features must be weighed against the additional programming tasks and the increased computer run times. Further developments for the tier building model lie in the areas of automated design [l], nonlinear analysis, and three-dimensional soil-structure interaction.
Tier Buildings with Shear Cores, Bracing, and Setbacks
83
REFERENCES [l] V. L. AOAXAR, Automated design of tier buildings. Department of Civil Engineering, Stanford University, Tech. Report No. 124 (December 1969). [2] H. BECK, Contribution to the analysis of coupled shear walls. J. Am. Concrete Inst. 59, 1055-1070 (1962). [3] G. E. BRANDOW,Computer programs for statics and dynamics of tier buildings. Department of Civil Engineering, Stanford University, Tech. Report No. 129 (June 1970). [4] R. W. CLOUGH,I. P. KING and E. L. WILSON,Structural analysis of multistory buildings. J. Structural Division, ASCE 90, No. ST3, Proc. Paper 3925, 19-34(1964). [S] R. W. CLOUGH, K. L. BENUSKAand E. L. WILSON,Inelastic earthquake response of tall buildings. Proceedings, 3rd World Conference on Earthquake Engineering, Wellington, New Zealand (1965). [6] R. W. CLOUGH and K. L. BENUSKA,Nonlinear earthquake behavior of tall buildings. J. Engng. Mechanics Division, ASCE 93, No. EM3, Proc. Paper 5292, 129-146 (June 1967). [7] A. COULL and B. S. SMITH, Analysis of shear wall structures. Tail Buildings, 139-155. Pergamon Press, Oxford (1967). [8] R. CRAWFORDand H. S. WARD, Determination of the natural periods of buildings. Bull. Seismological Sot. Am. 54, 1743-1756 (1964). [9] J. M. GERE and W. WEAVER,JR., Analysis of Framed Structures. D. Van Nostrand, Princeton, New Jersey (1965). 1101 C. V. GWAVALLABHAN,Analysis of shear walls with openings. J. Structural Division, ASCE 95, No. STlO, Proc. Paper 6824, 2093-2103 (1969). [l l] J. GLUCK, Lateral load analysis of asymmetric multistory structures. J. Structural Division, ASCE 96, No. ST2, F’roc. Paper 7089, 317-333 (1970). [12] W. M. JENKINSand T. HARRISON,Analysis of tall buildings with shear walls under bending and torsion. Tall Buildings, 413444. Pergamon Press, Oxford (1967). [13] I. A. MCLEOD, New rectangular finite element for shear wall analysis. J. Structural Division, ASCE, 95, No. ST3, Proc. Paper 6464, 399409 (1969). [14] T. A. MANNING,JR., The analysis of tier buildings with shear walls. Department of Civil Engineering, Stanford University, Tech. Report No. 128 (April 1970). [ 151 D. MICHAEL,Torsional coupling of core walls in tall buildings. The Structural Engineer 47(2), 67-71 (1969). D61 R. G. OAKBERGand W. WEAVER,JR., Analysis of frames with shear walls by finite elements. Proceedings, Symposium on Application of Finite Element Methods in Civil Engineering, Vanderbilt University 567-607 (November 1969). r171J. T. ODEN, Mechanics of Elastic Structures, Chapter 7. McGraw-Hill, New York (1967). J. S. PRZEMIENIECKI, Theory of Matrix Structural Analysis. McGraw-Hill, New York (1968). ;:i;R. ROSMAN,Approximate analysis of shear walls subject to lateral loads. J. Am. Concrete Inst. 61, 717-732 (1964). PO1 R. ROSMAN,Torsion of perforated concrete shafts. J. Structural Division, ASCE 95, No. ST5, Proc. Paper 6578, 991-1010 (1969). Theory of bending, torsion and buckling of thin-walled members of open cross PI S. P. TIMOSHENKO, section. J. Franklin Inst. 239, March, April, May, 201-219, 249-268, 343-361 (1945). PI V. Z. VLASOV,Thin- Walled Elastic Beams, 2nd ed., translated from the Russian Tonkostennye Uprigie Sterzhni by the Israel Program for Scientific Translations for the N.S.F. and Dept. of Commerce, U.S.A., Office of Technical Services, Washington, D.C. (1961). [231 H. S. WARD and R. CRAWFORD,Wind-induced vibrations and building modes. B14ll.Seismological SOC. Am. 56, 793-813 (1966). ~241W. WEAVER,JR. and M. F. NELSON,Three-dimensional analysis of tier buildings. J. Structural Division, ASCE 92, No. ST6, Proc. Paper 5019, 385-404 (1966). 1251W. WEAVER,JR., M. F. NELSONand T. A. MANNING, JR., Dynamics of tier buildings. J. Engng Mechanics Division, ASCE 94, No. EM6, Proc. Paper 6293, 1455-1474 (1968). (Received 3 February 1971)
TIER BUILDINGS
WITH SHEAR CORES, BRACING, AND SETBACKS
WILLIAM WEAVER, JR. and GREGGE. BRANDOW Department of Civil Engineering, Stanford University, Stanford, California, U.S.A.
and THOMAS A. MANNING, JR. Department of Engineering Science, Louisiana State University, Baton Rouge, Louisiana, U.S.A. Abstract--The theory of bending and torsion in thin-walled members of open cross section is applied to shear cores included within the analytical model of the tier building. Representative types of bracing arrangements are also considered, and their stiffnesses augment those of the beams, columns, and shear walls. Changes in floor plan, referred to as “setbacks”, further complicate the analysis; but a relatively simple routine allows automatic handling of this characteristic. Computer programs named STATLER and DYNATIER include all of these features within the linear static and dynamic analyses of three-dimensional multi-story buildings. Results for a twenty-story structure demonstrate the significance of the interaction between a shear core and the skeletal framing. The effect of warping restraints upon the stresses in the shear core is large at the base but diminishes rapidly with height. The normal stress at the base associated with the bimoment is of the same order of magnitude as that due to the combination of biaxial bending and axial force.
NOTATION The following symbols are used in this paper: A
A, AB
Aw ii Cl, G, G
DB D, DW E F
smgy G
47 Iy J K L
cross-sectional area; action in the i sense ; vector of beam actions; vector of shear wall actions; sectorial property of shear wall ; bimoment ; arbitrary constants; vector of beam displacements ; sectorial property of shear wall; vector of shear wall displacements; modulus of elasticity; structural reference point; shear flexibility factors ; elastic shear modulus; second moments of area; torsion constant; warping constant ; length ; 57
58
WILLIAM WEAVER, JR. GREGG E. BRAND~W and THOMAS A. MANNING,JR.
moment; length of middle line of shear core: concentrated force : distributed longitudinal loading; rotation matrix for wall ; radial distance to tangent plane ; beam stiffness matrices; bracing stiffness matrices; member stiffness matrix for brace: shear wall stiffness matrices; torque ; translational transformation matrices; shear wall thickness; axial displacement in wall due to warping; x coordinate of a row of bracing; y coordinate of a row of bracing; location of shear center of wall (local); structure reference axes; coordinates of shear center (global) ; shear wall member axes; rotation angle for shear wall axes; angle between x axis and outward normal ; angle between axis of brace and horizontal; direction cosines: angle of twist; normal stress due to warping restraint. INTRODUCTION SHEAR cores,
bracing, and setbacks often appear in multi-story buildings as complicating factors in the design and analysis of such structures. Figure 1 shows a tier building that includes all of these features in a rather irregular pattern. Planar shear walls present no great challenge to the analyst, but three-dimensional shear cores such as those enclosing elevators, stairways, and utilities are more difficult to handle. Their plan dimensions are large relative to column sizes, and the effects of shear strain and warping due to torsion can be significant. The purpose of this investigation is to incorporate three-dimensional shear wall configurations, bracing between floors, and setbacks at various framing leveis into the analytical model for the static and dynamic analysis of tier buildings. Toward this end, the theory of bending and torsion in thin-walled members of open cross section is utilized for the shear cores. The interaction between frames and shear walls in multi-story buildings has been discussed extensively in the literature, but most authors have limited their investigations to two-dimensional systems. Coull and Smith [7] presented a rather comprehensive summary of this topic, emphasizing techniques for distributing lateral loads between planar shear walls and structural framing, Clough et al. [4-61 automated the linear and nonlinear analyses of such frames for both static and dynamic loads, treating the shear walls as deep columns with flexural, axial and shearing deformations.
Tier Buildings with Shear Cores, Bracing, and Setbacks
59
X-BEAM GRID WORK
Y-BEAM
/Y-BRACING
z
X-BRACING
I
k
SHEAR WALL
Y
X FIG. 1. Tier building.
The finite-element method fl8] for analyzing shear walls within frameworks offers the important advantage of high numerical accuracy when many elements are used, as demonstrated by McLeod [13], Girijavallabhan [IO], and others. Oakberg and Weaver [16] used this approach and compared results with those obtained by the less accurate method of Clough et nl. This work tended to verify the suitability of the deep-column model for tall shear walls in two-dimensional building frames. Several investigators have proposed the analysis of coupled shear walls by replacing rows of spandrel beams with elastic continua. Beck [Z] and Rosman [19] developed the method in two dimensions, and Michaet [15], Rosman [20], and Gluck [I I] extended it to three-dimensional applications. In the latter studies, torsional characteristics of the shear cores played a major role. Jenkins and Harrison [12] compared theoretical and ex~rimental results for a threedimensional plexiglass model of coupled shear cores. They found a lack of correlation attributed to neglecting the effect of nonuniform torsion due to warping restraints. Experimental evidence from existing buildings 18, 231 also indicates the need for more accurate analytical procedures if realistic values for natural periods of tall buildingsa~ to be obtained. A three-dimensional analytical model was introduced by Weaver et al. [24, 251 and applied to the linear static and dynamic analysis of tier buildings. The model involves open rectangular framing and floor diaphragms assumed to be rigid in their own planes, with no restrictions of symmetry for either the structure or the loading. The effects of ail joint displacements and structural rigidities were included, but shear walls were not considered.
60
WurAh% WEAVER, JK., GREGG E. BRAND~W and THOMASA. MANNIN~~,JR.
Possibilities for subdividing three-dimensionai shear cores into finite elements were discarded as impractical because of the large number of nodal displacements introduced. Furthermore, the verification of the deep-column model in two dimensions [16] generated confidence for extending the concept to three dimensions. Thus, the primary objective of this paper is to present a suitable formuiation for threedimensional shear cores and to incorporate it into the analysis of tier buildings. For additional versatility, the possibilities of bracing and setbacks are also included. The pattern of shear walls and bracing is assumed to be constant for ail levels between setbacks, but changes in their stiffnesses are permitted at every level. Connections are assumed to be rigid, and the building is taken to be constructed entirely of linearly-elastic materials. Displacements of joints are assumed to be small in comparison with the overall dimensions of the structure.
SHEAR CORES Torsion in thin-w&led members The theory of torsion in thin-walled members is available in the literature; therefore, only the concepts and equations required in this study are reiterated herein. The primary source of this material is Timoshenko [21], hut Vlasov [22] and Oden [17] have also published contributions. Consider the thin-walled member of open cross section (see Figs. 2a and b), which is assumed to be prismatic and subjected to both bending and twisting actions. The x and J axes shown in the figures are principal centroidal axes, and the wall is indicated with variable thickness t in Fig. 2b. The thickness is assumed to be small compared to the length m of the middle line of the cross section. In the development which follows it will be assumed that the transverse forces are applied to the shear-center axis, so that the bending and torsional effects can be analyzed independently. The location of the shear center 0 with respect to the centroid C is given by the expressions:
In equation (1) x and J’ are the coordinates of an arbitrary point on the middle line of the wall, s is the distance to the point measured along the middle line from one extremity of the cross section,I, and 1, are the familiar second moments of area about the x and y axes. respectively, and a is defined by:
(2) The dimension rc is the perpendicular distance from the centroid to the tangent to the wail at the point x, y (see Fig. 2b). This distance is taken to be positive if a tangential force acting in the direction of increasing s produces a positive moment about the centroidal axis. If one or more cross sections of a thin-walled member are constrained against warping or if the torque varies along the length, the warping is not constant; and longitudinal stresses are developed internally. The rate of twist also varies along the length of the member, distinguishing this condition of “nonuniform” torsion from that of “pure” torsion, (in which the torque is constant and all sections are free to warp).
Tier Buildings with Shear Cores, Bracing, and Setbacks
61
Y
Shear- center
axis
(a)
(b)
FIG.2. Thin-walled member with open cross section.
In the case of nonuniform torsion, the total torque Tatting at any section will be resisted partially by shearing stresses of the kind developed in pure torsion and partially by shearing stresses resulting from the nonuniform axial deformations. The first part of the torque, denoted by T,, is given by: Tl = GJqY
(3)
where G is the elastic shear modulus, C/Iis the angle of twist, 4’ (the prime denotes differentiation with respect to z) is the rate of twist, and J is the torsion constant, defined by:
s
(4)
T2 = - EK@”
(5)
m 3
J= The second portion of the torque is:
t ds. 03
in which E is Young’s modulus and K is the warping constant, defined by: K =
a(DS)2tds.
s0
(6)
62
In equation
WILLIAM WEAVER, JR., GREGG E. BRANDQWand THOMASA. MANNING.JK.
(6) the symbol
D, represents:
(7)
The identifier ye is similar to the previously-defined rc, but the origin is now taken to be the shear center instead of the centroid. The total resisting torque is determined by combining equations (3) and (5), resulting in: EKcjY - GJ@ = - T.
(8)
When the notation (9) is introduced
The general
into equation
solution
(8), it becomes:
of this equation
is:
where C,, C2 and C, are arbitrary constants which must be determined from the conditions at the ends of the member. The axial displacement u’,, defining the warping of a cross section, is determined from \I’, =
Furthermore,
the longitudinal
D,gY.
stress produced o,= Ez=
due to restrained
(12) warping
ED,@’ .
is defined
by:
(13)
Since Q, results only from torque on the cross section, the resultant normal force and bending moment due to these stresses must be zero. To facilitate the analysis of partial warping restraints, Vlasov [22] introduced a stress resultant called the bimoment B, which is statically zero. It is defined as the product of 6, and D,tds integrated over the cross section. When equation (13) is substituted for the longitudinal stress, the bimoment is found to be related to the twist by: B=
The integral (6); thus,
in the above equation
“‘~,Dstds=E#’ f 0
‘“(D,)‘tds. s cl
is recognized
as the warping
B= EK&‘.
(14)
constant
defined in equation (15)
Tier Buildingswith Shear Cores, Bracing,and Setbacks
63
Shear wall stiflnesses A shear wall element is defined to be that portion of a shear core bounded above and below by two adjacent framing levels in a tier building. The stiffness properties of such an element may be evaluated as if it were a deep column, including the effects of shear strain and nonuniform torsion. For this purpose, the shear-center axis serves as the primary local reference for actions and displacements. In addition, the bimoment B and the rate of twist 4’ are introduced as a generalized action and corresponding displacement at each end of the element. Figure 3 depicts the principal centroidal axes and a parallel set of shear-center axes of a restrained wall element, along with seven types of nodal displacements at each end. Singleheaded arrows in the figure denote translations; rotations are represented by doubleheaded vectors; and the rate of twist at each end is indicated by a triple-headed arrow. The latter entities carry sequence numbers (4 and 8) following other nodal displacements that are not directly associated with the rigid-body motions of the floors (1,2, 3 and 5,6, 7). Partitioning the shear wall stiffness matrix into the following submatrices facilitates the transfer of terms to the stiffness matrix for the whole structure:
SW=
S wa,
0
S Wa,b
S Wa.c
S Wo,
S Wb,
(I
S Wb, b
S Wb, c
S Wb, d
S WC,a
S WC, b
S WC,c
S WC, d
S Wd, a
S Wd, b
S Wd,
S Wd,
c
Shear-center
axis
Fro. 3. Restrainedwall element.
d
d
64
WILLIAM
WEAVER,
JR.,
GREGG
E.
and
BRA&DOW
A. MANNING,JR.
THOMAS
In this matrix the subscript a denotes displacements of types 1-4; the 0 denotes displacements of types 5-8; the c stands for displacements of types 9-l I ; and the rl represents displacements of types 12-14. Thus, the matrix S, is of size 14 x 14, consisting of submatrices of sizes 4 x 4, 4 x 3, 3 x 4, and 3 x 3. Expressions for the shear wall stiffnesses are obtained by inducing unit displacements of each possible type, one at a time, at each end of the wall element. Nonzero terms below the main diagonal of S, determined by this procedure are given in Table 1 for submatrices are related to these as follows : S wu,IITSW+ SWC+’ and SW., only. The other submatrices S Wn,cr=s Wh,h SWC,c,= -SwJ,fl=Swc,lJ= S wc,c= -S TABLE
~
It’d, c =
-SWl,,h (17)
SWYI,d.
1. Shear wall stiffnesses.
~~~
S
_(4+g,Yx+ w’3L- (l+g,JL
S w2,1=
vz g ‘” 1,
AE XoYq-
-
S w3,1=
(3) Unit z,, translation
at .i
(1) Unit .Y,,,rotation
yo----
AE
s w3,3
=, L
sw5,3=-sw1..3
AE
-
at j
L
S Wh,
3=
-
s
3 =
-S
W7.
S w7,
3
w3.3
(4) Unit rate of twist at j 7-
7 W6,
L
1 =
-’
swz,
SWd, 4 =T[cL
1
cosh(cL) - sinh(cL)]
S w7,*=-sw3.1
sM/8,4=
S
S wt1,4(2) Unit y, rotation S
S W&2’ S
AE
.oL
-
GS[c0sl1(cL)
-
(5) Unit x,, translation S
at j
12Ef, w9Y9=(1 +gx)L3
L -s
WI,
2
AE (2-g3~~,_y2 (l+g,)L do L
w6p2=
S W7,2= S
y2
AE =x,-
-
sinh(cL)]
+_
at j
_V+gWy+ w2*2- (l+g,)L
S W3.2
-
-s
w3,
(6) Unit y,,, translation
at j
12E1,
S wto. 10=(,+g,)L”
2
6EI, W9y2=(1+gX)L2
(7) Unit twist at ) S W11, 11= cG.i sinh(cL)
1I
Tier Buildingswith Shear Cores, Bracing,and Setbacks
65
The symbols L, A, and E in Table 1 are the length, cross-sectional area, and modulus of elasticity of the wall element. The identifiers g, and gY denote shear flexibility factors, as given by: 12E1, (18) 9.X- 12ElY ” = GA,L2
GA.&’
where A, is the area effective in shear in the x direction, and A, is that for the y direction. Shear wall stiffnesses other than those associated with twist of the wall were determined by utilizing space-frame member stiffnesses relative to centroidal axes, which are readily available in the literature [9]. Standard axis-translation techniques were used to transform stiffnesses from centroidal reference axes to shear-center axes. Axial and flexural terms are coupled for shear-center axes, as is indicated by expressions containing x0 and y, in Table 1. Torsional stiffnesses in Table 1 were generated using the theory of thin-walled members. As an example, consider a unit twist of the j end of the element (displacement type 11) while all other displacements are zero. Application of the conditions 4,= o = 0 and @, = o = 0 to equation (11) yields: cl=
cf=$.
-cz
(19
The constant C, may be evaluated [14] in terms of the bimoment BzsL associated with the restraint condition r$‘, XL= 0. C2=
1
B,=,+%nh(cL) 1 GJ cGJ cosh(cL)
(20)
in which
Substituting equations (19)-(21) into equation (11) and setting #,= L equal to unity produces the following expression for the restraint torque at the j end of the member:
sW1I, ,,=T=cGJsinh(cL)
(22)
in which GJ
G.J=
(23)
2 - 2 cosh(cL) + CL sinh(cL) ’ The bimoment at the.j end may be seen from equation (21) to be: Swy4,11
=BzzL=
-GJ[cosh(cL)-11.
Similarly, the torque and bimoment at the k end of the member are found to be:
s w14,ll=-%ll,
11
S W8,
ll=sW4,
All other restraint actions in column 11 of SW are zero.
11.
(25)
66
WILLIAMWEAVER,JR., GRECS E. BKANU~Wand THOMASA. MANNING,JR.
Warping actions and displucements
Expressions are required for the statically equivalent actions at the shear center associated with the Ioad cases shown in Figs. 4a, b and, c. Those due to the concentrated force in Fig. 4a are: A ,==Pv,
A,=
-P.Y,
A,=P
/f,=PD,.
(26)
The last expression is the bimoment determined from equation (14), which shows that the longitudinal force produces torsion in the member if the quantity D, is nonzero at the point of application. Consider next the distributed Iongitudina~ loading qS indicated in Fig. 4b, which may be due to local floor loads. The statically equivalent actions at the shear center can be computed by the following integrations: $2
Al=
J
- q,x& c13’ x2 A, = f q,D,ds . .?I
J
PI
Fro. 4. Actions applied to end cross section. (a) Concentrated force. (b) Distributed force. (c) Concentrated moment.
(27)
Tier Buildingswith Shear Cores, Bra&r&and Setbacks
67
The concentrated moment shown in Fig. 4c acts in the positive direction of the outward normal to the plane tangent to the middle surface at the point of application. The bimoment developed on the section is determined by replacing the moment with a couple consisting of two longitudinal forces of magnitude P a distance As apart. B= -PD,+P(D,$
in which P=M/As. equation (7)] :
AD,)
(28)
In the limiting case when As+ds+O,
the bimoment becomes [see
If the moment M is not in the direction of the normal to the tangent plane, only the normal component of M contributes to the bimoment. Displacements at a typical point due to warping of the section are also of interest, and they consist of two types. The longitudinal translation corresponding to the force in Fig. 4a is given by equation (12), while the rotation corresponding to the moment in Fig. 4c is:
dwt -=- d4 # , =: - r& . ds
00)
ds
This rotation is taken to be positive when its vector representation direction of the outward normal.
points in the positive
Shear wall transformations For purposes of evaluating the contributions of a shear wali to the overall structure stiffness matrix, the shear wall stiffness matrix S, must be transformed from local wall axes to structural reference axes. If necessary, the first step may consist of a rotation through the angle o( (about the z, axis) from principal wall axes to a set parallel to the structural reference axes. S Ws= R$S,Rw. The rotation transformation
(39
matrix R, in equation (31) is of the foflowing form:
-4 Rw=
0
0
0
0
RI
0
0
o
0
R2
0
0
0
R2 I
0
(32)
in which f
R1=
sina
0
-sinor 0
cosdl
0
0
1
0
0
0
cos a
(33)
68
WILLIAMWEAVER,JR., GREGG E. BRAND~Wand THOMASA. MANNING,JR
and cos a R,=
sinci
-sina
cos
0
0
1
0
a
0
.
(34
1
-
Portions of the wall stiffness matrix associated with the rigid-body motions of the floors must be transformed to the structural reference axes at each framing level. S,, = T;SwsTw * The translational operator
(33
T’ in equation (35) takes the form: 1
0
oz T,=
0
0
0
0
0
0
0
0 0
TWJ
(36)
T Wk
0
in which Z signifies a 4 x 4 identity matrix, and
Twj=
1
0
- YSCj
0
1
x SCj
0
0
1
(37)
Identifiers Xscj and yscj in equation (37) represent the x and y coordinates of the shear center of the wall relative to structural axes at level j. The definition of submatrix Twk is obtained from equation (37) by replacingj with k. As given by equation (35), the shear wall stiffness matrix is in a form suitable for direct augmentation of the overall structure stiffness matrix. Beam transformations
Beams framing into shear walls increase the wall stiffnesses because the ends of such beams are constrained to move with the walls at the points of connection. Geometric transformations between displacements, actions, and stiffnesses at the end of a beam and those at the shear center of a wall prove to be useful in several phases of the analysis. Figure 5 shows the cross section of a shear wall at a framing level with the axes at the shear center taken parallel to structural reference axes. It is assumed that a beam frames into the wall at point, j. Generalized displacements of the shear wall are indicated by labeled arrows at point 0, whereas the three arrows at point j denote the x and y rotations and the z translation at the end of the beam. The symbol /? in the figure represents the angle between
Tier Buildingswith Shear Cores, Bracing,and Setbacks
69
FIG.5. Shear wall displacements. the x axis and the outward normal at point j. Displacements at j may be computed from those at 0 by the following operation:
Da,= TBj’,j
(38)
in which D,,=
@,,,
Dzt2,
1
0
0
10
4~1
Dw,=
WV,,
4~2,
. . .,
4~)
-r,cos/3
0
0
0
-rosin/I
0
0
0I .
D,
0
0
0
(39
and
Ti#j=
Y
0
1
-X
Terms in equation (40) result from superposition of warping displacements [see equations (I 2) and (3011upon plane-section displacements. Similarly, actions at the shear center 0 may be computed from those applied at point j (see Fig. 6) by the operation: A wj =
TijA,
(41)
in which A,,=
{Am,
A~23
Ad
A,,=
b%v~,
44~29
. . .v &v).
In this case the relationships are based upon equations (26) and (29).
(42)
70
WILLIAMWEAVER,JR., GREGGE. BRANDOWand THOMASA. MANN~G, JR.
FIG. 6. Shear wall actions.
If a beam frames into a wail at its j end only, the 6 x 6 beam stiffness matrix may be transformed to shear-wall coordinates by the operation:
SBW in which Tg=
=
T%I~B
[ 1. 2
;
(43)
(44)
The transformation matrix TB is of size 6 x 10 and includes an identity matrix of size 3 x 3. If both ends of the beam frame into the same shear wall or different shear walls, matrix Z’, takes the form: TB=
[
‘osi T” Bk1
(45)
where subscript k refers to the k end, and the matrix is of size 6 x 14.
The structure shown in Fig. ‘7simulates a reinforced concrete shear core isolated from a twenty-story building. Floor beams of size 16 WF 40 stiffen the core at each framing level and are assumed to be encased in fireproofing concrete with 2 in. of cover all round. Beam properties were computed on the basis of an equivalent steel section (modular ratio= lo), and the thickness of the wall is taken to be 1 in. of equivalent steel When the unstiffened core is loaded by a torque of 1440 kip-in. applied at the free end, it twists as shown by the plot labeled “shear core only” in Fig. 8. Also shown on the figure p the effect of partial warping restraints provided by the stiffening beams. The curve labeled “stiffened shear core” indicates a reduction of the twist at all levels by approximately 35 per cent. In particular, the twist at the top level was reduced from 0*02089 radians
Tier Buildingswith Shear Cores, Bracing,and Setbacks
FIG.7. Isolated shear core structure. to 0.01346 radians, Thus, even a modest amount of floor framing may have a considerable effect on the torsional stiffness of a shear core of open cross section.
E.wnple 2 : Twenty-story building Figure 9 shows a typical framing plan for a hypothetical twenty-story building with the shear core of the previous example positioned to emphasize the unsymme~~ character of the structure. Beam and column sizes vary over the height of the building, as indicated in Fig. 10, but the shear core thickness is taken to be constant. All framing members are assumed to be encased in fireproofing concrete with 2 in. of cover. This building was analyzed statically and dyna~~al~y with and without the core, and some of the results obtained are given below. The static analysis program named STATIER was used to analyze the building for a lateral pressure of 20 psf acting in the positive y direction, in combination with a uniform vertical loading of 100 psf at all levels. ~ispla~men~ for all stories in terms of the rigidbody motions of the reference point F at each level appear in Fig. 11. The maximum joint displacement occurred at point (4,l) at the top level, which translated 0.610 in. in the x direction and 6.628 in. in the y direction. Table 2 gives the bimoment and the y-shear in the core at various levels as well as the percentage of the total shear carried by the wall. The bimoment in the tabIe is that at the lower end of the shear wall segment associated with
72
WILLIAM WEAVER, JR., GREGG E. BKAND~W and THOMAS A. MANNISH, JR
FIG. 8. Twist of shear core due to torque.
FIG. 9. Typical framing plan for 20-story building.
Tier Buildings with Shear Cores, Bracing, and Setbacks
20x - 14WF30 y - 16WF50 z8WF31
IQIa17-
x - 14WF34 y - 18WF 50 z - lOWF60
16?514-
x - 16WF36 y - 18WF55 z -12WF85
1312ll-
x - 16WF40 y - 2lWF55 z - 12 WF106
IOQ8-
x - 16WF45 y - 21WF62 z - 14 WF127
76-
x - 16WF50 y - 2?WF68 z - 14WF142
54-
x - 18WF50 y - 24WF68 z - 14WF158
32-
x - 10WF55 y - 24WF76 z - 14 WF176
l-
X
o-
FIG. 10. Member sizes for 20-story building.
TABLE2. Bimoment and shear in core
Bioment x 10-S (kips-in.9 (2)
Shear (kips) (3)
Percent of total shear (4)
3
-8-340
239.13
71 *o
3
- 1 a303
147.19
48.5
5
-0.866
108.72
40.5
7
-0.327
83.75
36.0
9
-0.595
73-16
37.0
12
-0.152
39.09
26.5 28.0 15-5
Story 0)
1.5
0.064
26.74
18
0.375
6.68
73
74
WIILIAM WEAVER, JH., and GHEC,GE. BHAN~OWTHOMASA. MANNING, JR,
FIG.
II. Floor displacements for building with shear core.
the particular story. It is seen that the bimon~ent decreases rapidly with height, and at the fifth story it is approxinlately 10 per cent of the vafue at the base of the structure. Above this level the bimoment fluctuates considerably but is not very large. Figure 12a depicts the distribution of normal stress in the base of the shear wall due to the combination of biaxial bending, axial force, and the bimoment. In addition, the normal stress for the bimoment only appears in Fig. 12b. These diagrams show that the bimoment contributes approximately 35 per cent of the maximum normal stress at the base. At higher levels in the structure this percentage is much smaller due to the rapid decay of the bimoment. When this building was analyzed without the shear core (which was replaced by structural framing), the s and J’ translations of the point F at the top level increased by 55 and 46 per cent, respectively. The torsional stiffness of the structure was only slightly reduced, however, as indicated by a 7.3 per cent increase in the rotation at the top level. Dynamic analyses of this building without the shear core were reported in f25]; so in the present study the program named DYNATIER was used to re-analyze the structure with the shear core included. As might be expected, the frequency of the fundamental mode of vibration increased by 27 per cent (from O-318 to O-403 cps). Within the first ten
Tier Buildings with Shear Cores, Bracing, and Setbacks
75
(a)
6.00
(b)
FIG. 12. Normal stress in shear wall at base. (a) Resultant normal stress. (b) Normal stress due to restrained warping.
the largest increase in frequency (133 per cent) occurred in the seventh mode. Furthermore, the mode shapes were altered considerably by the presence of the shear core. The dynamic load condition applied to the building with shear core in this study corresponds to load case 5 (ground motion in the y direction) of [25]. Comparison of the displacement-time histories for story number 20 in both structures indicates that the maximum translation (in the y direction at point 4,l) was reduced from 15.2 in. to 11-l in. by the addition of the shear core. On the other hand, the maximum translation in the x direction increased from 2.26 in. to 4.74 in., and the maximum rotation increased from O-010 rad to 0.014 rad. Thus, the response of the two structures to the ground motion differed markedly. In summary, Examples 1 and 2 show that the interaction between shear cores and skeletal framing is of great significance, and neither can be analyzed independently of the other. Furthermore, any attempts at approximating this complicated interaction should be checked against the approach described herein. modes,
BRACING AND SETBACKS Representative types of bracing Figure 13 illustrates four representative types of bracing selected for this study. Types 1 and 2 each consist of one pinned-end member, while types 3 and 4 are each composed of two braces of equal cross-sectional areas. The members in type 4 are not connected at the crossing point.
76
WILLIAMWEAVER,JR., GREGGE. BRANDOWand THOMASA. MANNING,JR.
The member stiffness matrix S, for a typical pinned-end brace may be handled as if it were a member of a plane truss, as follows:
(46)
where LAt and A, denote the length and cross-sectional area of the member, respectively. The direction cosines of the member axis in terms of the angle y (see Type 1 in Fig. 13) are
This stiffness matrix is sufficient to describe bracing types 1 and 2, and type 4 may be considered to be the sum of these two.
Tier Buildings with Shear Cores, Bracing, and Setbacks
77
Bracing type 3 is slightly more difficult to handle because it does not conform to the pattern of joints in a tier building. The joint at the midspan of the beam involves two additional local displacements, as shown in Fig. 14. Displacements A, and A, are eliminated from the set of ten local displacements before the contributions to the overall stiffness and load matrices are assessed. For this purpose, action-displacement relationships for the beam and braces shown in Fig. 14 are written in partitioned form. (48) Solve for D,., in equation (48).
DA=S;;(AA-SSABDB).
(49)
Substituting equation (49) into equation (48) produces:
S&DB=A;
(50)
in which SL=S,,
- Si7.4S24B
A;=A,-s,J~A,.
(51)
(52)
Then the beam with the two braces attached may be treated as a subassemblage having an 8 x 8 stiffness matrix S&, and an 8 x 1 action matrix A;. It should be noted that the beam also has torsional stiffnesses that are uncoupled from the other types. After the displacements of type B are obtained, those of type A may be calculated using equation (49).
L---xor
y
FIG. 14. Displacements
for bracing type 3.
Bracing transformations Geometric transformation of bracing matrices from local to global reference axes are analogous to those for columns and shear walls. Thus,
S;F=T;BS;BT,,.
(53)
78
WILLIAM WEAVER, JK., GREGG E. BRANDOWand THOMAS A. MANNING, JR
* in equation (53) takes various forms, depending upon the The translational operator TtlB type of bracing. For example, Figs. 15a and b show the local and global displacements nvolved with transforming bracing of type 4 that is parallel to either the .Vor the 3%axis.
j/ f..._“j
!‘I
-‘F_i
I
’
,’
Y+-
I ,
/ ! * -.-i’._..._._-._A
_/’
/
7
DF~
j
/’
/ /’ -v’
FIG.
IS. Related floor and bracing displacements.
For bracing of types I or 2 parallel to the .Yaxis, the transformation form :
Ti?li=
-YCR
0
0
0
0
0
0
0
1
0
0
0
0
1
I
0
0
I
0 0
matrix is of the
(54)
Tier Buildingswith Shear Cores, Bracing,and Setbacks
79
Similarly, for bracing of type 3 parallel to the x axis, the matrix takes the form:
TBB=
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
- YCR
-YCR 0
The operators for bracing types 1, 2 and 3 parallel to the y axis are of the same forms as equations (54) and (55), but with - YCR replaced by XCR. A separate operator is not required for bracing type 4, because each brace in that configuration may be transformed individually. Changes in joor plan
In order to handle setbacks that may occur in multi-story buildings (see Fig. l), some automatic technique for altering the floor plan at any level must be introduced into the analytical procedure. During the forward-elimination phase, the structural stiffness and load matrices are condensed story-by-story from top to bottom. When the information for a given story is assembled, the numbering of the degrees of freedom is established for that level and the next level below, assuming that the latter has the same layout as the former. If the floor below contains additional bays, the residual stiffness and action matrices must be expanded in order to accommodate the new degrees of freedom. This expansion may be accomplished by the insertion of rows and columns of zeros corresponding to each new degree of freedom encountered. Subsequently, additional stiffness and action quantities from the story below will create nonzero terms in those positions. During the backward-substitution phase of the analysis, the joint-displacement matrix must be contracted wherever a setback occurs. This contraction may be accomplished by removing rows corresponding to non-existent joints (see Fig. 1) and shifting other rows upward in the matrix. Example 3. Four-story building
A hypothetical four-story building with shear cores, bracing, and setbacks was contrived to illustrate these features and was analyzed both statically and dynamically. The framing plan for the upper two stories is one bay by one bay and for the lower two stories is four bays by three bays, as shown in Fig. 16. Four shear walls occur in the lower portion of the building, and the elevation views in Fig. 17 include four braced frames. Not shown is frame E, which is a simple two-story, three-bay frame without bracing or walls. Member properties for this structure are listed in Table 3.
80
WJLLIAM WEAVER, JK., GREGG E. BRA&DOW and THOMAS A. MANNING,
-_
..~._ -
1st B2n3
-
.
LEVEL
Jn.
. _.._
f-l?AMitG
P...ANS
Fra. 16. Framing plans for I-story building.
TABLE
_
3. Member properties for example 3.
__ ~. Beams
Columns ‘4 (in.*) (4)
f (in.4) (7)
Story (1)
I J (in.4) (in.“) (3) (2)
3 and 4
2,ooo
300
25
200
200
75
I and 2
4,000
500
50
400
400
150
fz b/ (in.4) (in.4) (5) (6)
Bracing: Type 3 A=5 in.2 Type 4 A=6 in.2 Shear wall thickness: 1 in. equivalent steel E=30,000 ksi G=12,000 ksi
Static vertical loads on the building include the uniformly-distributed beam loadings indicated by the numbers in Fig. 16, as well as two IO-kip forces acting downward at points (B,2) and (B,3) at the second level. Lateral loads for each of the upper two stories consist
Tier Buildings with Shear Cores, Bracing, and Setbacks
81
of 14 kips in both the .Yand y directions at their centers. Similarly, a 160-kip lateral load is applied in both directions at the center of each of the lower two stories. The story displacements resulting from this condition of loading are given in Table 4.
--x
TABLE4.
Stoty displacements for example 3 due to static loads.
x-Translation (in.) (2)
y-Translation (in.) (3)
z-Rotation (rad.) (4)
4
4.036 x IO-2
3.501 x10-2
1.014x IO-5
3
2.999 x 10-2
2.416 x 10-Z
1.010 x 10-5
2
1.215x10-2
5.758 x
10-3
1.033 x 10-s
1
7.179 x 10-3
3.195 x10-3
6.021 x IO-6
story (1)
82
WILLIAM WEAVEK, JR., GREGG E. BRANDOWalld THornAs
A. MANNING, JR.
For purposes of dynamic analysis, a uniformly distributed dead was assumed to exist at all levels of the building. The frequency of the of vibration for this structure was found to be 5.576 cps, and computer shape are exhibited in Fig. 18. No other analytical results are included of space, but further details are given in [3].
weight of 200 psf fundamental mode plots of the mode herein due to lack
MODE SHRPE NO. 1 FREQUENCY IS 5.576 C.P.S. X-TRRNS .
Y-TRRNS.
FIG. 18.
CONCLUSIONS The analytical model for tier buildings has been expanded to include three-dimensional shear cores and bracing between floors, as well as the typical beam and column framing. The floor framing plans may vary from one level to another, provided that the plan for the level below merely involves the addition of bays to the plan above. Programs STATlER and DYNATIER for the static and dynamic analyses of tier buildings were considerably upgraded in this study. These programs have been documented in [3] and [14], which may be obtained from the first co-author upon request. FORTRAN coding is also available on magnetic tapes for a nominal charge to cover the cost of handling. Additional features could be incorporated into the programs without great difficulties. These include additional shear wall configurations and bracing types, shear strains in beams and columns, nonprismatic members, finite joint sizes, and flexible connections. The desirability of adding such features must be weighed against the additional programming tasks and the increased computer run times. Further developments for the tier building model lie in the areas of automated design [l], nonlinear analysis, and three-dimensional soil-structure interaction.
Tier Buildings with Shear Cores, Bracing, and Setbacks
83
REFERENCES [l] V. L. AOAXAR, Automated design of tier buildings. Department of Civil Engineering, Stanford University, Tech. Report No. 124 (December 1969). [2] H. BECK, Contribution to the analysis of coupled shear walls. J. Am. Concrete Inst. 59, 1055-1070 (1962). [3] G. E. BRANDOW,Computer programs for statics and dynamics of tier buildings. Department of Civil Engineering, Stanford University, Tech. Report No. 129 (June 1970). [4] R. W. CLOUGH,I. P. KING and E. L. WILSON,Structural analysis of multistory buildings. J. Structural Division, ASCE 90, No. ST3, Proc. Paper 3925, 19-34(1964). [S] R. W. CLOUGH, K. L. BENUSKAand E. L. WILSON,Inelastic earthquake response of tall buildings. Proceedings, 3rd World Conference on Earthquake Engineering, Wellington, New Zealand (1965). [6] R. W. CLOUGH and K. L. BENUSKA,Nonlinear earthquake behavior of tall buildings. J. Engng. Mechanics Division, ASCE 93, No. EM3, Proc. Paper 5292, 129-146 (June 1967). [7] A. COULL and B. S. SMITH, Analysis of shear wall structures. Tail Buildings, 139-155. Pergamon Press, Oxford (1967). [8] R. CRAWFORDand H. S. WARD, Determination of the natural periods of buildings. Bull. Seismological Sot. Am. 54, 1743-1756 (1964). [9] J. M. GERE and W. WEAVER,JR., Analysis of Framed Structures. D. Van Nostrand, Princeton, New Jersey (1965). 1101 C. V. GWAVALLABHAN,Analysis of shear walls with openings. J. Structural Division, ASCE 95, No. STlO, Proc. Paper 6824, 2093-2103 (1969). [l l] J. GLUCK, Lateral load analysis of asymmetric multistory structures. J. Structural Division, ASCE 96, No. ST2, F’roc. Paper 7089, 317-333 (1970). [12] W. M. JENKINSand T. HARRISON,Analysis of tall buildings with shear walls under bending and torsion. Tall Buildings, 413444. Pergamon Press, Oxford (1967). [13] I. A. MCLEOD, New rectangular finite element for shear wall analysis. J. Structural Division, ASCE, 95, No. ST3, Proc. Paper 6464, 399409 (1969). [14] T. A. MANNING,JR., The analysis of tier buildings with shear walls. Department of Civil Engineering, Stanford University, Tech. Report No. 128 (April 1970). [ 151 D. MICHAEL,Torsional coupling of core walls in tall buildings. The Structural Engineer 47(2), 67-71 (1969). D61 R. G. OAKBERGand W. WEAVER,JR., Analysis of frames with shear walls by finite elements. Proceedings, Symposium on Application of Finite Element Methods in Civil Engineering, Vanderbilt University 567-607 (November 1969). r171J. T. ODEN, Mechanics of Elastic Structures, Chapter 7. McGraw-Hill, New York (1967). J. S. PRZEMIENIECKI, Theory of Matrix Structural Analysis. McGraw-Hill, New York (1968). ;:i;R. ROSMAN,Approximate analysis of shear walls subject to lateral loads. J. Am. Concrete Inst. 61, 717-732 (1964). PO1 R. ROSMAN,Torsion of perforated concrete shafts. J. Structural Division, ASCE 95, No. ST5, Proc. Paper 6578, 991-1010 (1969). Theory of bending, torsion and buckling of thin-walled members of open cross PI S. P. TIMOSHENKO, section. J. Franklin Inst. 239, March, April, May, 201-219, 249-268, 343-361 (1945). PI V. Z. VLASOV,Thin- Walled Elastic Beams, 2nd ed., translated from the Russian Tonkostennye Uprigie Sterzhni by the Israel Program for Scientific Translations for the N.S.F. and Dept. of Commerce, U.S.A., Office of Technical Services, Washington, D.C. (1961). [231 H. S. WARD and R. CRAWFORD,Wind-induced vibrations and building modes. B14ll.Seismological SOC. Am. 56, 793-813 (1966). ~241W. WEAVER,JR. and M. F. NELSON,Three-dimensional analysis of tier buildings. J. Structural Division, ASCE 92, No. ST6, Proc. Paper 5019, 385-404 (1966). 1251W. WEAVER,JR., M. F. NELSONand T. A. MANNING, JR., Dynamics of tier buildings. J. Engng Mechanics Division, ASCE 94, No. EM6, Proc. Paper 6293, 1455-1474 (1968). (Received 3 February 1971)