Microcomputer analysis of torsionally coupled multistorey buildings—I. Shear beam model

Compurers & Sfrucrur~s Vol.32,No. 5.pp.1175-1182, 1989

00457949/89 S3.00+0.00 Q 1989MsxwcllPerg~mcnMacmilh plc

PrintedinGreatBritain.

MICROCOMPUTER ANALYSIS OF TORSIONALLY COUPLED MULTISTOREY BUILDINGS-I. SHEAR BEAM MODEL DAVID P. THAMBIRAIXAM

and

H. MAX IRVINE

School of Civil Engineering, University of New South Wales, P.O. Box 1, Kensington, N.S.W. 2033, Australia (Received 1 June 1988)

Ahstra&--A simplifiedanalysis of torsionally coupled multistorey buildings,suitable for use with microcomputers, is presented in this paper. Interaction between the lateral and torsional components of the response is automatically included in the analysis. Floors are assumed to be rigid diaphragms each with three degrees of freedom. Potential and kinetic energies of the building are calculated and the equations of motion are derived using Lagrange’s equations. Interaction in the vertical direction is limited to a floor above and a tloor below, as befits a shear beam model. Thus the model is close-coupled. A numerical example and a discussion on the use of the equations of motion are given. The analysis technique pmsented may be used purely for static response or the calculation of natural frequencies and associated modes of vibration, or finally, for a full time history analysis should deterministic forcing be known or available.

1. ~TRODU~ION

A multistorey building that is torsionally balanced has, at each floor level, coincident centres of mass and stiffness which lie on a common vertical axis. However, the conditions necessary for torsional balance are so restrictive as to never actually occur [I]. Hence, all buildings are torsionally unbalanced to some extent and display interaction between the lateral and torsional components of their response. Analysis of tall buildings has received extensive coverage in the literature and references are available for treating to~io~lly coupled buildings. Extensive work pertaining to the response of torsionally coupled buildings, especially single storey buildings, to various forms of loading, particularly earthquake loading, has been carried out by Hutchinson and his coworkers [Z-4]. Some researchers have focussed their attention on providing simplified procedures for analysing torsionally coupled buildings. But most of these methods seem to depend on either hand calculations or elaborate computer programmes [5-g]. Practising engineers may not find these simplified methods convenient for u,se in their design office. There are also many general purpose programmes which are capable of doing almost eve~thing. However, these programmes require large computational effort and time, and hence may not be attractive for those with restricted computer facilities, budgets, and time. Herein a simple method for analysing torsionally coupled buildings is presented. This matrix analysis can be conveni~tly pro~ammed on a microcomputer and may be useful to practising engineers in their design office. The method is an extension to that for a single storey building [Q]. CAS 32/s-P

The present study can be outlined as follows. In Section 2, the analytical model is described and the assumptions used in the analysis are given. Potential and kinetic energies of the system are calculated in the next section which highlights the contribution made by this study. Lagrange’s equations are used to derive the stiffness and mass matrices in Section 4 where the equations of motion are presented in a manner suitable for programming. A simple example is treated in Section 5 to illustrate the procedure. Salient points applicable in the use of the equations of motion are discussed in the final section which also concludes the paper. 2. ANALYTICALMODEL

Consider the asymmetric building shown in Fig. 1. In order to be quite general, a set-back at an intermediate level is also introduced. Vertical members such as r go up to the set-back level, while vertical members such as s continue right through the entire height of the building. z,(i = 1 + n) denotes the intersections with the floor levels of a common vertical axis originating at T, which is taken as the origin of co-ordinates x-y. Thus the position P of a vertical member at any floor level is referenced by the x-y co-ordinates as shown in Fig. 2. The degrees of freedom at each floor level pertain to the displacements at T,as shown in Fig. 3. The following assumptions are made in the analysis. (i) Floors are treated as rigid diaper which possess three degrees of freedom per floor, viz, u, u, and 6, with respect to (the reference point)

1175

Ti.

-T L

I

Gi

Y,

zi Fig, 2. Co-ordinate system.

I of

(bl

ELEVATION

PLAN

Fig. 1. Analytical model of building: (a) elevation, (b) plan.

An alternative choice for the degrees of freedom will be the centroidal displacements, rr,,, v,, 8, at each floor level leading to a somewhat different analysis. In this case, the mass matrix will be diagonal but the derivation of the stiffness matrix will be more complicated. However, the authors prefer to choose the degrees of freedom with respect to q which leads to considerable simplification in the stiffness matrix derivation with marginal increase of effort in computing the mass matrix which will not be diagonal now. It is recognised that not all buildings are covered by the above assumptions. In buildings where there are heavy service cores, assumption (iii) above will not hold completely and an alternative procedure will be necessary in the analysis. A simplified procedure for treating such buildings is currently being pursued, which is not si~~~~y more detailed than the present procedure. In any event, modelling by the shear beam approach has been the traditional starting point for most simplified 3-D analysis procedures and much of the info~~tion in respect of the coupling action is relevant in a relative sense. 3.

(ii) Kinetic energies of vertical resisting members (columns, shear walls, etc.) are neglected. However, we can smudge this by adding, say, 10% to each floor inertia. Alternatively, for buildings with heavy cores, we can add in point masses when working out the moment of inertia for floors. (iii) All vertical members are assumed to be restrained at the floor levels, i.e., a shear building model is considered. (iv) Principal axes of all vertical members are along the x-y directions. This is a reasonable assumption as it is uncommon to have vertical members with axes in arbitrary directions. 69 The system is close coupled, i.e., effects at any ftoor fevel are not propagated beyond the levels just below and above that floor level.

PO=WiAL AND KIh’ETtC ENERGIJS

3.1. Potential energy V Consider a vertical member such as r between the floor levels i and (i - 1). Since the rotations at the floor levels are assumed to be zero, the potential energy of this member is given by

Fig. 3. Degreesof freedom.

Microcomputer analysis of torsionally coupled multistorey buildings

or

where

kf., 00 kj+= 1

and

0

k;j

0

[ 0

0

kfi

*

K,, = [

(2)

0

k fi

-w:_~KT~w,~] (10)

-W:k;iW,i_I (3)

In the above equations, E is Young’s modulus, I the second moment of area about the axis denoted by the subscript, L the length of the member between floor levels, G the shear modulus and J the polar moment of inertia about the vertical axis through the centroid of the member. Shear effects are brought in by the parameter 4 given by (4)

where A, is the effective shear area and the subscripts for 4 [eqn (3)] pertain to the axis about which bending takes places. If shear effects are to be neglected, then 4 will be set to zero. The degrees of freedom in the analysis am the displacements wTiand wzi_1for the levels i and (i - 1), respectively. With reference to Fig. 2, these are related to the end displacements of the vertical member r by

(9)

Vr,=t[wf;~iWli+Wf;-I~iW,i-l

in which Ki=

4 = 12EI/L2GA,

- k!,y, kfix,

The total potential energy of all the vertical members between i and (i + 1) levels is therefore

+ f$,)L3,

k;, = GJ/L.

0

kf = kjiy; + k;ix; + kfi.

vi=C + f&)LJ, kZi= 12HJl

-kfiYr krixr k? 1. (8) kfi

In the above matrix

The elements of /c,~ denote the principal stiffness coefficients of the member and are given by Ghali [lo] kii = 12EZJl

1177

c K*. r-1

(11)

Equation (10) above holds for the potential energy of vertical members between any two floor levels. If there are any set-backs, as in the present case, then the summation over i will vary above the set-back level(s). The total potential energy of the structure is obtained by summing up the contributions Vi for i ranging from 1 to the number of floors (say n) in the form V= i i-l

V -‘i

w;K,,~~~+wfi_,ll;,w,~_~

‘-2i*1 -

WZX;,Wri-

I- wfi- IK,w~,* (12)

In the above equation, wb will be taken as zero since the building is assumed to be restrained at its base. Equation (12) will be used to obtain the stiffness matrix for the building by using Lagrange’s equation. 3.2. Kinetic energy

w,t-

1

=

Wd-

(3

I

10-Y, c, =0 1

where the connectivity

matrix C, is given by

1

[ 00

Since all loading, Using potential

X, .

T = +r,(ti;, + \i&) + f J&.

(6)

1

members are assumed to be vertical prior to C, will be the same at all levels. eqn (5) in eqn (1) and simplifying, the energy V,, now takes the form

Vrr=f[wzK,lw,l+ Wf;-lK&,-I-

WZKd%l-1

- wf;- IK,%l where Kri= CTk,,C,

Kinetic energy of any one floor (say at level i) which is assumed to be rigid and hence possessing three degrees of freedom is given by (13)

In the above expression uc,, cc,, 13,are the displacements of the centroid G, in the x and y directions and the rotation respectively, while m, and JG1denote the mass and the polar moment of inertia about the centroid. The dot above any quantity indicates differentiation with respect to time. As the unknown displacements in the analysis pertain to the point r,, it will be necessary to express the velocities in eqn (13) in terms of those at t,. This is easily accomplished by * -YG,h #Gi = hi

(7)

fiGi = ti,, + XGib,

where 8, is the same at r, and G, due to the rigidity of the floor.

DAVIDP. -TNAM

1178

and H. MAX IRVINE

Using eqn (14) in eqn (13) gives

T,= fm,{zi;, + ti;i - 28,(&y, - &x,)} + f Jl,& (15) where Jfi = JGi + M/(X; + yi)

(16)

is the polar moment of inertia about T,. Kinetic energy T for the whole structure tained by summing up Tiin the form

is ob-

T = c q = fz [q{tif,+ tif,2&1&y~,- i)r,xG,)} + J&l.

Mt&+Kw,= Q,

M=

EQUATIONS OF MOTION

Lagrange’s equations (in the absence conservative forces such as damping) are

(23)

where w: = (u+ vIir 0,) and the mass matrix M and the stiffness matrix K are given by

(17)

Equation (17) will be used to derive the mass matrix of the structure with the aid of Lagrange’s equations. 4.

The above expression indicates that the system is closed coupled. From eqns (18). (20) and (22) the equations of motion for the building are given by

of non-

M, 0 0 M2 0 0 0 0 0 0 0 0 0 0

0 0 M, 0 0 0 0

.

.

. .

. .

ifi :

.

.

.

.

0 0 0

0 0

4

J

(24

(18)

K= where Qk are the generalised forces and qk the general&d coordinates which are the displacements (wr,= ufi, I),~, 0,) at 7, in the present analysis. Since the kinetic energy Tis here not a function of the displacements, the second term in the above equation may be dropped. Using equation (17) the derivatives of T with respect to the velocity components are given by

The stiffness matrix is shown only for a building with four floors, as the pattern is repetitive. Its non-zero elements are given by

-K,,+,... i=l...n. Ki+l.i= Ki,i+,= aT - = m,li,im,8,y,i ati,,

The general&d load vector Q,( = Q,, k = 1 --, n) in equation (23) pertains to lateral (horizontal) loads and couples applied at each point z,. This equivalent system at t, can easily be determined from the actual

aT - = m,tj,,+ rn,8,XG, ah g

=

loads acting at each floor level.

m,(-uki, + ti,,xo,)+ JJ,

(19)

5. NUMERICAL. EXAMPLE

I

or

where

[

A simple example of a torsionally coupled building with three storeys is treated to illustrate the procedure. The elevation and plan of the building are shown in Fig. 4, together with some data. Young’s modulus E and Poisson’s ratio v for the concrete (building) are taken as 30 GPa and 0.20, respectively. All columns are assumed to be 350 x 350 mm in size. Natural frequencies of vibration and the associated mode shapes will be first obtained and then the static response of the building to a uniform wind pressure of 0.7 kN/m* will be determined. It will be necessary to form the structure stiffness matrix K, the mass

-wGi 4xGi1 (21) mi 0

Ml=

(26)

0 m,

- m9, miXGl

Jr,

is the mass matrix for the ith floor. From equation (12), the derivatives of V with respect to w,, are obtained as

matrix M and the load vector Q, (due to the wind

Microcomputer analysis of torsionally coupled multistorey buildings

1179

A3

2.5 kPa

SLAB

c 3m

3m

SLAB

T

‘I

/ 3m I’ Ao la 1

co

Bo ELEVATION

3n6=18m

lb1

PLAN

Fig. 4. Asymmetric building: (a) elevation, (b) plan.

loading). Using eqns (8), (9), (11) and (25) the stiffness matrix K can he determined. As 4 is very small for the columns, it will be set to zero. From equations (21) and (24) the mass matrix M can be determined. Matrices K and M can he computed very conveniently by programming the appropriate equations. However, in a simple example such as this, they could also he (easily) hand calculated. For the data pertaining to this problem the stiffness matrix K is given by

K= lo6

k;,+II;, -4, 0

-42 k;2+K3

-&

0 -43

K,

1

with 266.72

0 266.72

- 2400.48

2400.48

- 2460.48 2400.48 67,213.44

(27)

Pa) 1

0

-900.18

150.03 900.18

900.18 18003.6

150.03 Kr2= [

0 -900.18 66.68

G=

1 (2W 1* (284

DAVID P. T~~~BIRATNAMand H. MAX IRVINE

1180

[

0

0 -200.04

- 200.04

66.68

200.04

200.04

2400.48

shapes are presented in Table 1. Deflected shapes of the vertical line rO,2,. 7*, r3 ire shown in Fig. 5 for the various modes. The wind pressure (of 0.7 kN/m*) is assumed to act in they direction and the wind load on each tributary area is calculated. These loads are then equally distributed to the slabs above and below the tributary areas to act at the mid points. Finally the load vector Q, is obtained from the equivalent load (forcecouple) system at rIr r2 and zg. This is given by

Units used in the above matrices are N, N/m and N . m. The mass matrix M given by

Qz = 10’(0,31.5,245.7,0,

18.9,94.5,0,6.3,

(29) with 0 1.62

- 14.58

14.58

Mz= [

0.72 0 -4.32

1

- 14.58

1.62 0

M,= [

14.58 349.92

(304

-4.32

0 0.72

(W

4.32 69.120 I

4.32

-0.27 0.27 2.16 I

(6) (7) (8) (9)

The units in Q, are N and N +m for the forces and moments, respectively. Using the stiffness matrix Kin eqn (27) and the above load vector Qf the static deflections of r were obtained and are presented in Table 2 (in millimetres and radians). Using the transformation matrix C in equation (6) with (x,=y,=6m) the deflections of ‘A’(A,,A, and A,) can be obtained. The deflected shapes of the vertical lines rO, rI, 22, ‘Fj and &,A,,A*,A3 are given in Fig. 6. Torsional coupling in the building response is evident from the results presented for this example. 6. DISCUS!SION AND CONCLUSION

Natural frequencies (cycles/set): 4.42 4.59 6.72 9.27 9.67

(31)

(3Oc)

Units used in the above matrices are kg, kg * m and kg. m2. Using eqn (23), with Q, = 0, the natural frequencies and the associated mode shapes are calculated. The first nine natural frequencies (in cycles per second) are as below.

(1) (2) (3) (4) (5)

18.9).

11.33 14.68 15.67 21.23.

The mode shapes pertain to deflections in the z-p and z-q planes which are planes of symmetry. With U, V denoting the deflections in the p, q directions, respectively and 0 denoting the rotation, the mode

Equations of motion presented above [in eqn (23)] can be conveniently programmed on a microcomputer to obtain: (a) response of the building to static loads, in which case I?~= 0 (b) free vibration analysis of the building when Q,=O and (c) dynamic response of the building for applied dynamic loads Q,. For case (c), the natural frequencies determined in case (b) will be used together with the mode superposition method, where the first few modes alone will be sufficient [1 11. A special case of (c) will be the response to ground excitation such as earthquakes. In this case Ql will be given by [9] Q, = -I%~

(32)

Table 1. Mode shapes 71

Mode No. 1 2 3 4 5 6 I 8 9

V 0.52 0 0.35 0 1.0 -0.49 0 0.20 0.02

V 0 0.48 0 -0.56 0 0 0.03 0 0

72

8 0.01 0 0.05 0 0.02 -0.03 0 0.01 0

V

V 0.90 0 0.82 0 0.38 0.42 0 -0.78 -0.12

0 0.89 0 0.53 0 -0”.17 0 0

73

e 0.02 0 0.10 0 0.15 0.05 0 -0.06 0

V

V 1.0 0 1.0 0 0.29 1.0 0 1.00 1.00

0 1.0 0 1.0 0 0 1.0 0 0

e 0.02 0 0.11 0 0.19 0.07 0 -0.14 0.02

Microcomputer

analysis of torsionally coupled multistorey buildings

1181

Table 2. Static deflections oft -0.057

-0.088

t;

-0.0063 0.269 x 10-s

-0.01154 0.469 x 10-J

2

2

z,-

I I

-0.088

I(

-

MOOE

-.-

MODE 3

---

MdDE

0.001154 0.563 x 103

I

2

-

1

5

HOOE

2

-.-

MOOE

L

---

MODE

1

t :1 A’

/’

-

MODE

-‘-

MODE 8

---

MODE 9

I

( t, I

:

\/

‘I-

\’

t oP.U lal

DEFLECTION

u

lbl

OEFLECTION

v

Fig. 5. Mode shapes of frame (translation in z-p, z-q planes): (a) deflection u, (b) deflection u, (c) deflection u.

0.5

(01

DEFLECTION

1

Y. ” v

-0.1 (bl

0 OEFLECTION

x, u u

Fig. 6. Lateral deflections of frame at 1 and A: (a) deflection u, (b) deflection u.

6

1182

DAW, P. TXAMBIIUTNAM and H. MAX BNINE

where 6jg= (i&,i;,, 0) is the base acceleration vector. Equation (32) follows from eqn (14) when the components of the base velocity GJ#are added to the right hand sides to give the absolute velocities at any point Gi. Interaction between the lateral and torsional components of the response are clearly evident from the stiffness and mass matrices. The stiffness sub-matrix in equation (8) and the mass sub-matrix in equation (21) display the coupling effects so anticipated. Number of degrees of freedom in the anafysis is 3n where n is the number of storeys. For buildings with SO-60 storeys the number of degrees of freedom are such that they permit microcomputer analysis. Damping is not included in the above analysis, where the authors are primarily concerned in including the effects of torsional coupling. A decision has to be made whether or not to include shear effects given by the factor 4. When the buildings have vertical members which are predominantly columns, 4 is very small for such members and hence can be neglected. If Poisson’s ratio v is taken as ~~0.2 (a value suitable for concrete), and the shear area a, is taken as equal to 1.2 times the actual area, 4 defined in equation (4) becomes

where d is the depth of the section having a height L. For columns (of usual dimensions), d/L will be %O.1 + 0.3 resulting in a small value for r#~.But for shear walls and service cores, contribution from shearing effects will be sibilant as the ratio d/L can exceed unity. For such members it will be incorrect not to include shear effects. Finally when shear walls and service cores dominate the behaviour of the building even the inclusion of Q, in the analysis will give only an approximate

solution. In this case, the floors will be unable to restrain the rotation of the vertical members at the floor levels and this effect has to be given due co~ideration, Work is underway simplifying the analysis of buildings whose behaviour is dominated by shear walls and/or service cores using the procedures developed herein as the framework. REFEREE

1.

R. S. Ayre, Interconnection of translational and torsional vibrations in building. Bull. Seism. Sot. Am. 28,

89-130 (1938). 2. A. M. Chandler and G. L. Hutchinson, Evaluation of code torsional provisions by a time history approach. Ebrthq. Eng. and Struct. Dys. 15,491-516 (1987). 3. G. L. Hutchinson and A. M. Chandler, Code design provisions for torsionally coupled buildings on elastic foundations. Eurthq. Eng. and Struct. Dyn. 15, 517-536 (1987). 4. T. Tsicnias and G. L. Hutchinson, Soil structure interaction effects on the steady state response of torsionally coupled buildings. Earthq. Eng. and Struct. Dyn, 12, 237-262 (1984). 5. A. C. Heidebrecht and B. Stafford Smith, Approximate analysis of tall wall-frame structures. J. Struct. Div. ASCE 99, 199-221 (1973). 6. C. L. Kan and A. K. Chopra, Elastic earthquake analysis of torsionally coupled multistorey buildings. Eurthq. Eng. and Struct. Dyn. 5, 395-412 (1977). 7. T. Balendra, S. Swaddi~d~~n& S. T. Quek and S. L. Lee, Approximate analysis -of -noetic buiI~ng. J. Struct. Ens. ASCE 110.2056-2072 (19841. 8. V. W. T. Ch&ng and W. i. Tso, Lateral load analysis for buildings with setback. J. Struct. Eng. ASCE 113, 209-227 (1987). 9. K. M. Dempsey and H. M. Irvine, Envelopes of maximum seismic response for a partially symmetric single storey model. Earthq. Eng. and Struct. Dyn. 7, 161-180 (1979). 10. A. Ghali and A. M. Neville, Structural Analysis-a Unified CIassicai and Matrix Approach. Intext Educational Publishers, Toronto (1972). 11. R. W. Clough and J. Penzien, Dynamics of Structures. McGraw-Hill, New York (1975).