Response spectrum techniques for asymmetric buildings

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 6. 427-435 (1978)

RESPONSE SPECTRUM TECHNIQUES FOR ASYMMETRIC BUILDINGS AVIGDOR RUTENBERG*

Technion-Israel Instiiute of Technology, Haifa, Israel TZU-I. H S U t AND W. K. TSO$

McMaster University, Hamilion, Oniario, Canada

SUMMARY A scheme is proposed to calculate the effect of torsion on each lateral load resisting element of asymmetrical buildings in the context of the response spectrum technique. The scheme consists of: (i) Obtain the modal shear and torque on the building by the response spectrum technique. (ii) Compute the total modal shear forces on each frame by resolving the modal shear and torque on the building according to principles of structural mechanics. The shears on each frame due to the lateral load effect and torsional effect are combined algebraically. (iii) Obtain the total shear force on each frame by combining the total modal shears on that frame in a root sum square manner. Since the proper phase relationship between the lateral load effect and torsional effect is accounted for on a modal basis, it is believed that the proposed scheme provides a more realistic load estimate on the frames than the conventional approach. An example of a simplified mono-symmetrical frame structure is chosen to illustrate the accuracy of the proposed scheme, using dynamic time-history analysis as a standard for comparison.

INTRODUCTION The response spectrum technique3 is one of the most significant contributions to the seismic analysis of structures. For multi-degree of freedom systems, it has been shown that the square root of the sum of squares (RSS) is the most probable value for any particular earthquake response quantity, provided the natural frequencies are well ~ e p a r a t e dIt . ~follows that in order to estimate the member forces, the structure should be fully analysed for several loading cases, each corresponding to a single mode of vibration. In symmetric structures, where only lateral oscillations are introduced by translational earthquake motions, engineering practice' and building consider the gross response as represented by RSS storey shear forces as the equivalent loading whereby member forces are derived by means of standard structural analysis. Evidently, such a procedure, which requires less computational effort, does not produce the 'most probable' maximum member forces, yet it provides a reasonable correlation between maximum member responses and their responses as obtained from RSS storey shear force^.^ As for the storey shears themselves, the quality of the RSS estimate has been found to be satisfactory for most engineering purposes.s. l3 For asymetric buildings, ground motions induce both lateral and torsional responses. The seismic loading as evaluated by means of the response spectrum technique usually results in providing RSS estimates of the inter-storey shears and inter-storey torque envelope^.^*^* l7 The forces and displacements of a particular frame or assembly are evaluated based on the more unfavourable effect of the combination of RSS shear and RSS torque, or the effects of RSS translation and of RSS rotation.12 Such a technique is intrinsically incorrect since the different phases between rotation and translation in each mode are lost in the process, so that it is likely to overestimate the response of frames which are located away from the reference axis. It is of interest to note that the concept of dynamic eccentri~ity~.'~ used in the static methods of seismic analysis 10.149

* Senior Lecturer.

t

Post-Doctoral Fellow. Professor.

OO98-8847/78/05064)427$0 1.OO @ 1978 by John Wiley & Sons, Ltd.

Received 28 February 1977 Revised 28 July 1977 421

428

A. RUTENBERG, T. I. HSU AND W. K. TSO

has apparently led engineers to treat the RSS lateral and RSS torsional oscillations as two independent quantities. This treatment of torque (or rotation) and shear (or translation) as two independent loading cases can only be justified on the grounds that it requires only two loading cases and that it follows well established static wind load analysis. In the present paper this conventional approach is compared with the more correct RSS scheme in which the modal shear forces on each frame are obtained from the algebraic superposition of the translational and of the rotational effects in each coupled mode, and the estimated response is the RSS of the resulting modal shear forces. This approach can be regarded as an extension of the procedure first proposed by Housner and Outinen." The better agreement of the latter approach with the time-history analysis is demonstrated on a multi-storey example frame building. COMPUTATIONAL PROCEDURES The structural system of many high-rise buildings is typically composed of several planar assemblies such as shear walls, simple or coupled, and frames. This fact permits a rapid evaluation of the building stiffness matrix from the individual stiffness matrices of the planar assemblies, provided the following two assumptions are reasonably satisfied : (a) inplane rigidity of the floor plan and (b) independence of columns for each assembly, i.e. where a column pertains to more than one frame, the errors resulting from allocating its cross-sectional area among the frames are not significant. Since a detailed description of the technique was given by Clough' only the pertinent parts are described here. Consider the multi-storey building whose floor plan is shown in Figure 1. The horizontal equilibrium of the structural system is given by the following relationship :

or

f=kw in which w and f a r e respectively the displacement and force vectors (at floor levels) in the direction of the co-ordinates u, u and 8, and k is the stiffness matrix of the system obtained by adding the contributions of the individual frames by means of elementary transformations.

FRAME A

1"

FRAME

FRAME

Figure 1. Relation between frame and building co-ordinates

Transforming to modal co-ordinates and using the response spectrum technique, the vector of the maximum displacements in mode n, w(") may be given by W ( n )= +cn,pn, S(n) (2) d

RESPONSE SPECTRUM TECHNIQUES FOR ASYMMETRIC BUILDINGS

429

in which +(n) is the nth modal co-ordinate vector, p(n)its modal participation factor and Sp) is the spectral displacement of the nth natural frequency. The modal loads f(n),the modal shear forces SF), SLn) and the modal torque Sy) are then obtained from the following expressions: f ( n ) = kw(n) (3) Sbn) = Tf!n) in which i = u, u, 8, and T is the triangular matrix (numbering floor levels from top to bottom): '1

T=

0 0

(4)

... 0 -

1 1 0 ... 0 1 1 I ... 0 . . . . : : : : 0 1 1 ... ... 1

An estimate of the maximum gross displacements and forces of the building is then given by the RSS (root-sum-square) formula :

wi

=

RSS(w!"))

Si

= RSS(S:n))

This is the stage at which most reports stop, presumably because the evaluation of member forces appears to be quite straightforward. Yet conventional engineering practice and the technical literature12 sometimes follow a different procedure. These two procedures will now be outlined. Starting with the 'correct' approach, the modal displacement of frame p , v z ) , assumed for convenience to lie in the u plane, is given by (n)= v(n) + e ( n )eP (8) vP where ep is the eccentricity of frame p from the reference axis (Figure I), and it follows that

Sg)= Tk

P ~P( n )

(9)

in which k, is the stiffness matrix of frame p . The estimate of the maximum response is then given by:

vprnaX= RSS(v:)) S,

max = RSS(SF))

Obviously, the member forces evaluated from Sp are not the 'most probable' in frames which are not strictly of the shear type, but for shear dependent responses, such as shear and bending moments in columns, the error involved in the procedure has been shown by Burgess5 to be quite small. It is evident that modern code procedures are compatible with this approach. More conventionally, the effects of translation and of rotation are assumed independent, and the maximum response is therefore evaluated as in the case of eccentrically applied wind load :

The difference between the two procedures is illustrated in Figure 2, and it is evident that in the independent treatment of translation and of rotation, the phases between them in each mode are lost, so that the conventional approach will always overestimate the response of assemblies located away from the reference axis.

430

A. RUTENBERG, T. I. HSU AND W. K. TSO

In the following section a simplified mono-symmetric frame structure is chosen to illustrate the two approaches, and their accuracy is discussed by comparing the results with those obtained by means of dynamic time-history analyses. Also, the proposed method is compared with the national building code of Canada15 and the draft U.S. code by the Applied Technology Council, ATC-3.2 The calculations involving the building codes are based on the total base shear of the building being equal to that calculated by the response spectrum technique.

3

j

RSS

t

m

mode i

shear modes i,j

‘mode j

(a) correct RSS superposition

(b) conventional RSS superposition

Figure 2. Correct and conventional RSS procedure for a given frame

COMPARISON O F RESULTS

For the purpose of this study an idealized ten storey shear building structure was considered. The structure comprised several rigid frames located so as to produce an eccentricity between the mass axis and the axis of rigidity. Such an axis can be defined for an asymmetric structural system provided all the structural assemblies are of the same ‘family’ (either frames, or single or coupled shear walls) and they all follow the same law of variation along the height. The basic data on the structure are given in Figures 3 and 4. This structure was subjected to three different earthquake histories : (a) Imperial Valley (El-Centro), 1940, N-S; (b) Taft, 1952, N21E and (c) San Fernando (Wilshire Boulevard), 1971 , N-S. These ground motions were applied parallel to the short plan dimension of the building. In each case the maximum values of the shear forces on the two edge frames were calculated by means of a modal superposition time-history analysis computer program. These results were compared with the estimates computed from the modal maxima as determined from the relevant response spectra using the two RSS procedures: the ‘correct’ and the conventional. These results were further compared with the numerical values obtained by static methods of analysis as given in seismic codes. For this purpose, the National Building Code of Canada15 and the draft U.S. seismic code, ATC-32 were chosen. The basic requirements of the two codes for the design eccentricities are given in Figure 5. In order that the comparison between the results computed by applying the code provisions and by the dynamic analyses may be meaningful, the total base shear values for the two codes were assumed equal to the corresponding RSS values. With the total base shear so adjusted, the linear distribution of the interstorey

43 1

RESPONSE SPECTRUM TECHNIQUES FOR ASYMMETRIC BUILDINGS

A

Y

x

OT LL

B

t

DIRECTION OF EXCITATION

NO. OF STORY=lO STORY HT. =11 FT TOTAL WT. = 12,000 KIP INTERSTORY LATERAL STIFFNESS44.440 KIP/FT INTERSTORY TORSIONAL STIFFNESS ~ 7 5 x1$ 6 KIP-FT/RAD ECCENTRIC~TY = I S FT DAMPING =2% FOR ALL MODES EXAMPLE 10 STORY UNIFORM SHEAR BUILDING 1

e

Figure 3. Typical floor plan : 10-storey asymmetric uniform building

!!l V

e

MODE 1

T=1.04 sec MPFS-1.52

e

V

MODEZ T=0.70 sec MPF4.58

MODE 3

MODE 4

T=036 sec

T 4 2 4 s8c

MPF=0.48

MPF=O.lS

Figure 4. Dynamic properties of building in Figure 3

+

NBCC (1975) : e, = 1*5e 0.05Dn or e. = 0-5e-0.05Dn ATC-3 (1976):

e, = ek0.05Dn

Figure 5. Code requirements for design eccentricities

bG!!x!i T 4 2 3 sec MPF=-O.25

2-

4-

6-

8-

0

0

I

I

2

I

3

2

I)

3

( FRAME

I)

4

-PROPOSED METHOD

I

4

CONVENTIONAL APPROACH

--NBCC ( 1975 ---- ATC-3 ( 1976

TOTAL SHEAR-KIPSx 103

i

( FRAME

TOTAL SHEAR-KIPS x Id

1

----

DYNAMIC ANALYSIS

-PROPOSED METHOD

Figure 6. El Centro record; Frame I, comparison of results: (a) proposed us conventional, (b) proposed us codes

v)

e

z

0

10

EL CENTRO 1940 NS

2-

4-

6-

2

I

3

I I

I L 1

I

1

I

( FRAME

TI

1

4

I

---- ATC-3

I

4

NBCC ( 1975 ) ( 1976 )

-PROPOSED METHOD

I1 1

1 2 3 TOTAL SHEAR-KIPS x 103

I

( FRAME

TOTAL SHEAR-KIPS x 103

1

L---

I

I

Figure 7. El Centro record; Frame 11, comparison of results: (a) proposed us conventional, (b) proposed us codes

k

0

z > LT

0

2-

4-

1

DYNAMIC ANALYSIS

-PROPOSED METHOD ---- CONVENTIONAL APPROACH

EL CENTRO 1940 NS

P

?

h)

W

433

RESPONSE SPECTRUM TECHNIQUES FOR ASYMMETRIC BUILDINGS

..

r-I

e

w

w

b

N

'ON AHOlS

'ON AUOIS

rr)

V

r"

cy

07

0,

w0

a

z

h

Ld

v

c

F

'ON AUOlS

'ON AUOIS

lo3

1.5

- NBCC ( 1975 1 ----. ATC-3 ( 1976 1

Figure 10. San Fernando record; Frame I, comparison of results: (a) proposed us conventional, (b) proposed us codes

TOTAL SHEAR-KIPS x 103 ( FRAME I )

0

tcn

cn

4

Zi

Z

2

--

i

2.0

-PROPOSED METHOD

TOTAL SHEAR-KIPS x ( FRAME I )

1.o

I

P

0.5

I

I I

L- 1

I

0

*

1-!

-_DYNAMIC ANALYSIS -PROPOSED METHOD ---- CONVENTIONAL APPROACH

0

10

2-

8

'"1

SAN FERNANDO 1971 WILSHIRE BVLD. NS

I

0.5

I

1.o

1.5

I

--_- ATC-3 ( 1976

-_NBCC I1975 1

I

2.0

-PROPOSED METHOD

I

I

I L-1

TOTAL SHEAR-KIPS x Id ( FRAME I1 Figure 11. San Fernando record; Frame 11, comparison of results: (a) proposed us conventional, (b) proposed us codes

2-

L..

SAN FERNANDO 1971 WILSHIRE BVLD. NS

?

RESPONSE SPECTRUM TECHNIQUES FOR ASYMMETRIC BUILDINGS

435

shears was computed by the formula:

in which W is the weight of the floor, h the storey height and So the total base shear. The total torque was computed using the formulas in Figure 5, and then statically distributed among the participating frames. Results for the two edge frames (I and I1 in Figure 3) are given in Figures 6-1 1. A comparison of the results in part (a) of the figures demonstrates the accuracies of the proposed approach over the conventional one for excitations by each of the three earthquake records considered. The comparisons with code values as shown in part (b) of these figures indicate that in most of the cases considered, the NBCC15 gives a better estimate of the interstorey shears than ATC-3.2 However, the main limitation of the two static methods, as compared with the ‘correct’ RSS procedure, lies in their inability to predict the response in a consistent manner. CONCLUSIONS The graphical results presented in the foregoing section have indicated that the root-sum-square (RSS) formula, when correctly applied, gives a good estimate for the maximum response of asymmetric buildings. The conventional method, which obtains this response from the worst combinations of RSS lateral and RSS torsional loadings, generally tends to overestimate the response. This effect becomes more pronounced with increasing distance from the reference axis. The results given herein have also demonstrated the limitations inherent in the static methods of analysis. Since the RSS procedure as applied in this paper to asymmetric buildings is basically similar to the one recommended by modern seismic codes, the results presented here should encourage practising engineers to extend the use of the response spectrum technique to more complicated structures. ACKNOWLEDGEMENTS

The authors wish to acknowledge the support of the National Research Council of Canada and the VicePresident for Research Fund of the Technion, Haifa, Israel, for the work presented herein. REFERENCES 1. Applied Technology Council, ‘An evaluation of a response spectrum approach to seismic design of buildings’, ATC-2,

San Francisco (1974). 2. Applied Technology Council, ‘Working draft of recommended comprehensive seismic design provisions for buildings’, ATC-3-04, San Francisco (1976). 3. M. A. Biot, ‘A mechanical analyzer for the prediction of earthquake stresses’, Bull. Seismo. SOC. Am. 41, 151-161 (1941). 4. J. I. Bursamente and E. Rosenblueth, ‘Building code provision on torsional oscillations’, Proc. 2nd World Con/. Earthq. Engng, Tokyo (1960). 5 . D. N. Burgess, ‘Modal analysis of multistory framed structures subjected to earthquake effects’, Ph. D. Thesis, University of Illinois, Urbana, 1966. 6. R. W. Clough, ‘Earthquake analysis by response spectrum superposition’, Bull. Seismo. SOC.Am. 52, 647-660 (1962). 7. R. W. Clough, ‘Earthquake resistant design of tall buildings’, in Tall Buildings, Planning, Design and Construction; Proc. Symp., Vanderbilt University, Nashville, Tenn. (1974). 8 . R. E. Gibson, M. L. Moody and R. S. Ayre, ‘Response spectrum solution for earthquake analysis of unsymmetrical multistoried buildings’, Bull. Seismo. SOC.Am. 62, 215-229 (1972). 9. L. E. Goodman, E. Rosenblueth and N. M. Newmark, ‘Aseismic design of firmly founded elastic structures’, Trans. ASCE, 120, 782-802 (1955). 10. J. B. Hoerner, ‘Modal coupling and earthquake response of tall buildings’, Report EERL 71-07, Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena, 1971. 11. G. W. Housner and H. Outinen, ‘The effect of torsional oscillations on earthquake stresses’, Bull. Seismo. SOC.Am. 48, 221-229 (1958). 12. A. W. Irwin, ‘Analysis of tall shear wall buildings including in-plane floor deformations’, Build Int. 8, 43-56 (1975). 13. R. L. Jennings and N. M. Newmark, ‘Elastic response of multistory shear beam type structures’, Pror. 2nd World Con/. Earthq. Engng, Tokyo (1960). 14. C. L. Kan and A. K. Chopra, ‘Coupled lateral-torsional response of buildings to ground shaking’, Report No. EERC-76-13, College of Engineering, University of California, Berkeley, Calif., 1976. 15. National Research Council of Canada, Nurional Building Code of Canada, Ottawa, pp. 158-164 (1975), and supplement no. 4 Commentary K. 16. N. M. Newmark and E. Rosenblueth, Fundamentals of Earthquake Engineering, Prentice-Hall, Englewood Cliffs, N.J.. 1971. 17. J. Penzien, ‘Earthquake response of irregularly shaped buildings’, Proc. 4th World Con/. Eurrhq. Engng, Santiago, Chile (1969).