Earthquake response analysis of multistorey buildings including foundation interaction

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 3, 65-77 (1974)

EARTHQUAKE RESPONSE ANALYSIS OF MULTISTOREY BUILDINGS INCLUDING FOUNDATION INTERACTION

ANIL K . CHOPRAT AND JORGE A. GUTIERREZ~

Department of Civil Engineering, University of California, Berkeley, California, U.S.A.

SUMMARY An efficient method, based on the Ritz concept, for dynamic analysis of response of multistorey buildings including foundation interaction to earthquake ground motion is presented. The system considered is a shear building on a rigid circular disc footing attached to the surface of a linearly elastic haifspace. In this method, the structural displacements are transformed to normal modes of vibration of the building on a rigid foundation. The analysis procedure is developed and numerical results are presented to demonstrate that excellent results can be obtained by considering only the first few modes of vibration. As the number of unknowns are reduced by transforming to generalized co-ordinates, the method presented is much more efficient than direct methods.

INTRODUCTION

It is well known that the response of structures to earthquake ground motion will be influenced by deformability of the foundation. The significance of foundation interaction in structural response depends on the properties of the structure relative to the foundation. In the present paper, an efficient method of analysis for structure -foundation systems is presented with which the earthquake response of structures including foundation interaction can be determined and the effects of interaction can be studied. Although the basic concepts presented in this investigation apply to all types of linear structure-foundation systems, the system specifically considered is a multistorey building on a rigid disc footing attached to a linear halfspace.

The analysis of structural response to earthquake ground motion is straightforward when the foundation is assumed to be rigid. Well-known efficient methods are available for such analysis. For linear analysis, the mode superposition method is generally most effective. Transformation to modal co-ordinates leads to an uncoupled set of differential equations, one for each normal mode of vibration, identical in form to the equation for a simple one-degree-of-freedom system. Uncoupling of the equations is, of course, a very attractive feature of the modal method. A more significant aspect of the method is that, in general, only a few modal equations need to be solved because response to earthquake ground motion is primarily contained in the lower modes of vibration. Even for a buildmg with many storeys, say 20 or more, the first mode of vibration alone will generally produce satisfactory estimates of response for purposes of design; as few as three or four modes will usually suffice to obtain results accurate within a few per cent?

The governing equations of motion for a structure including foundation interaction as well as the methods of solving these equations are relatively complex. The foundation stiffness and damping terms relating forces and displacements for the rigid disc on a linear elastic halfspace depend on the frequency of excitation. The governing equations for the structure-foundation system must therefore be written in the Fourier-transformed frequency domain or in the Laplace-transformed domain.24 The steady-state response to harmonic ground motion at a particular excitation frequency is determined by solving the frequency domain equations. Response to an arbitrary ground motion is then determined by Fourier transforming the ground motion, determining the steady-state response for a range of frequencies over which the ground motion and structural

t Associate Professor. Graduate Student.

Received 8 November 1973 Revised 20 January 1974

@ 1974 by John Wiley & Sons, Ltd.

3 65

66 ANIL K. CHOPRA AND JORGE A. GUTIERREZ

response has significant components and then performing a Fourier synthesis of the frequency responses to obtain the responses in the time-domain. In practice, these computations are generally performed by Discrete Fourier Transform techniques using the Fast Fourier Transform alg~ri thm.~. This typically involves determining the steady-state responses'for a large number of excitation frequencies. For each frequency, this entails solution of a set of simultaneous algebraic equations, as many as the number of degrees-of-freedom for the structure including those at the structure-foundation interface? Such a procedure, denoted sub- sequently as a direct method, requires a large computational effort for structures with many degrees-of- freedom such as multistorey buildings.

Because much of the complication in the analysis is a result of the frequency dependence, the possibility of approximating the foundation stiffness and damping terms by some representative constant values (independent of frequency) have been inve~tigated.~. Such an approximation leads to differential equations with constant coefficients which permit application of standard methods of structural dynamics. While such approximations may be satisfactory for the analysis of typical multistorey buildings, they may not be adequate for other types of structures, such as concrete gravity dams.1°

The more fundamental and important drawback in the above-mentioned direct method is that it takes no advantage of the important feature that structural response to earthquake ground motion is essentially contained in the first few modes of vibration. Standard modal analysis is not applicable because the building does not possess classical normal modes when foundation interaction is considered, due to the dependence of foundation properties on excitation frequency. Even if the foundation stiffness and damping are approximated by frequency independent values, damping in the structure and foundation will not usually ' be so related as to permit classical normal modes.

Recognizing the inherent advantages of modal analysis, their possible application to interacting structure- foundation systems has attracted the attention of a number of researchers. Hadjianll pointed out some of the problems encountered in applying standard modal techniques to the analysis of structure-foundation systems. Tajimi12 formulated the equations for a building on a foundation represented by frequency independent rotational and horizontal springs in terms of the normal modes of vibration of the building on a rigid base. Meek and Ve1etsoslS expanded upon these ideas to develop a method of analysis, using modes of vibration, for a cantilever shear beam on an elastic halfspace foundation. Jennings and Bielak3 showed that the response of a N-storey building-halfspace system, with the base disc having two degrees-of-freedom, can be expressed as the sum of the responses to modified excitation of N + 2 one-degree-of-freedom, viscously damped, linear oscillators resting on rigid ground. However, determination of the parameters of these oscillators for multistorey buildings appears to be rather complicated in general.

Methods, based on extensions of the concepts of modal superposition and the Ritz procedure;O for analysis of earthquake response of buildings including foundation interaction are developed in the present study. It is shown, through numerical results, that the methods developed take full advantage of the fact that structural response to earthquakes is primarily due to the first few modes of vibration.

SYSTEM CONSIDERED

The building-foundation system under investigation is shown in Figure 1. The mass of the N-storey building is considered to be concentrated at the floor levels, the floor systems are assumed to be rigid so that the relative displacements are entirely due to the deformations in the columns, thus leading to a single degree-of-freedom per floor. Viscous damping is assumed to be of such a form that the building on a rigid foundation admits decomposition into classical normal modes.l* The base of the building is assumed to be a rigid circular disc footing of negligible' thickness attached to the surface of a linearly elastic 'halfspace. The assumption of circu!ar base is necessary because most of the available results for dynamic response of a footing on elastic halfspace. are restricted to circular geometry. Limited results that have been obtained for square footings indicate that they are similar to those for circular footings of the same plan area.'6

The building-foundation system will be subjected to the horizontal free-field ground motion : displacement = u,(t), acceleration = cg(t). The displacements of the N floors in addition to the two interaction displacements at the base completely define the displacement response of the system.

EARTHQUAKE RESPONSE ANALYSIS 67

ELASTIC HALF SPACE ( P ,Cs ,V 1

Figure 1 . Idealized building-foundation system

STEADY-STATE EQUATIONS OF MOTION

The equations expressing the dynamic equilibrium of the N floor masses are Miif+CU+Ku = 0

In addition, there are two equations expressing the equilibrium of the building as a whole in translation and rotation:

j=1

In these equations, u is the vector of displacements ui, where uj denotes the displacement at thejth floor due to structural deformations; M is the diagonal matrix of N floor masses mj; K is the stiffness matrix of the building on rigid foundation (the standard tri-diagonal matrix in terms of the storey stiffness kj); C is the viscous damping matrix on rigid foundation; ug is the free-field horizontal ground displacement; u, is the translation of the footing in addition to the free-field motion; 0 is the rotation of the footing; hj is the height of thej th floor above the base; u! = u,+u,+ hj e+u j , is the total horizontal displacement of thej th floor with respect to a fixed vertical axis; rn, is the mass of the footing; Zt = X&,Zj, where Zi is the moment of inertia of the j t h floor mass about a diameter; and V ( t ) and M ( t ) are respectively the base shear and moment, i.e. the interaction forces between the footing and the halfspace.

Equations (I) may be written as Mii + CU + KU + MIGO + MhB' = - Mlii,(t)


where m, = Egomi is the total mass; Zh = J+CjN,omihj2 is the total moment of inertia of all the masses about a diameter of the footing; L:, = xElmihi, each element of vector 1 is unity, h is a vector of heights hi and the superscript T denotes the matrix transposed.

For steady-state harmonic vibration at frequency w the interaction forces acting on the footing can be expressed as V(t) = VeiWf, and M(t) = MeiuL and the resulting displacements of the footing as u,(t) = UOeiut and O(t) = 0 eid. For any linear model of the foundation, these forces and displacements are related as

The impedance functions Kvv(w), KMM(w) and K,,,(w) (KMv(w) = KvL$f(w) by reciprocity theorems) may be obtained from solution of two boundary value problems for the foundation, arising from the application of a harmonic horizontal force and a harmonic moment separately to the disc footing. Because the disc is assumed to be rigid, a horizontal uniform translation and a rigid rotation about a horizontal axis perpendicular to the plane of vibration are prescribed under the disc, and the stresses are specified to be zero over the remainder of the surface of the foundation, leading to mixed boundary value problems. Complete solutions for these mixed boundary value problems for a circular disc on an isotropic elastic halfspace have not been obtained to date. Approximate solutions using certain simplifying assumptions regarding the conditions of contact between the disc and the foundation surface, however, have been reported.1s-20

The impedance functions for the elastic halfspace obtained by Veletsos and Weiz0 are used in this investigation. These functions may be expressed as

in which the k's and c's are dimensionless coefficients depending on Poisson's ratio v and the frequency parameter a, = wr/C, where r is the radius of the disc, and C, the shear wave velocity in the halfspace; C, = J(G/p) where G is the shear modulus and p the mass density of the material of the halfspace; i = J(- 1); Kx = 8Gr/(2- v) is the horizontal static force necessary to produce a unit horizontal displacement of the disc with no restriction on the value of the resulting rotation; KO = 8Gr3/3(1 -v) is the static moment necessary to rotate the disc through a unit angle with no restriction on the value of the horizontal displacement. The real parts of the impedance functions represent force components in phase with the displacements, and may therefore be interpreted as dynamic stiffness coefficients for the foundation. The imaginary parts, on the other hand, are force components in phase with the velocities and, when positive, are indicative of energy dissipation by radiation of waves away from the disc into the halfspace; they may therefore be interpreted as foundation damping coefficients. The values of the foundation stiffness and damping coefficients, obtained by Veletsos and Wei, are presented in Figure 2.20

Because the foundation stiffness and damping coefficients are dependent on excitation frequency, it is most convenient to consider the response of the building-foundation system to harmonic ground motion. The base shear V(t) and moment M ( t ) in Equation (2) are related to the interaction displacements by Equation (3). For ground acceleration iig(t) = ei", any response quantity x( t ) can be expressed most conveniently in terms of a complex frequency response function Z(w), i.e. ~ ( t ) = Z(w) ei". Thus for the system under consideration u(t) = ii(w)eid, uo(t) = $ei", O(t) = B(w)eid, V(t) = V(w)ei", M ( t ) = W(w)eid. Substituting these expressions in equations (2) and dropping the eid terms on both sides leads to

(- w2 M + iwC + K) ii(w) - w2 Mlf&(w) - w2 MhB(w) = - M1 - w2 lT Mii(w) - w2 m, zjo(w) - w2L; B(w) + KvV(w) $(w) + KvM(w) B(w) = - m, - w2 hT Mii(w) - w2L; fi0(w) - w2 Ib B(w) + K M ~ ( w ) z$(w) + K M M ( w ) B(w) = -I$

(54 (5b) (54

The complex frequency responses ii(w), fio(w) and 8(w) for a particular value of the excitation frequency w are to be determined from the above simultaneous linear algebraic equations.


A "0 2 4 6

0' ' ' " " ' r

I .o

0.5

0

-0.2 2 4 6 8

Figure 2. Foundation stiffness and damping coefficients (after Veletsos and Weizo)

ANALYSIS FOR ARBITRARY GROUND MOTION

Once the complex frequency response function X(w) for any response quantity x( t ) has been determined, the response to arbitrary ground acceleration ii&) can be obtained by Fourier synthesis of the responses to individual harmonic components as

x( t ) = - 3(w) og(o) ei*dw 2rr s" --m

in which Uff(w) is the Fourier transform of iig(t), i.e.

uff(w) = [iig(t)e-iddt (7)

where d is the duration of ground motion. The Fourier integrals of equations (6) and (7) can be computed very efficiently by the Fast Fourier

Transform (FFT) algorithm? ' wherein they are treated as discrete transforms. The complex frequency responses are required for a large number-a thousand to a few thousand-of excitation frequencies. Because the FFT procedure enables calculation of forward and backward Fourier transforms with relatively little computational effort?' the principal effort in the above analysis often lies in the computation of the complex frequency response functions.


DIRECT METHOD

equation (5) represents N + 2 simultaneous algebraic equations in the The coefficients in these equations are complex valued, and the matrix of

For a particular frequency w, unknowns ii(w), G0(w) and &w). coefficients is not banded.

An obvious method is to directly solve the N f 2 equations.* While the computational effort required for one such solution may not be excessive, it becomes large because the complete solution has to be repeated for many values of w. The computational effort would become exorbitant for buildings with many storeys because it increases as the cube of the number of degrees-of-freedom. The need for judiciously reducing the number of degrees of freedom in the analysis of building-foundation systems is therefore apparent.

REDUCTION OF DEGREES-OF-FREEDOM

The number of unknowns and governing equations for the structure-foundation system may be reduced by the well-known Ritz method, which for many years has been extensively applied to the more common dynamic systems which possess classical normal modes. In applying this method to the building-foundation system, two sets of Ritz co-ordinates were considered :

(i) The two degrees-of-freedom-translation and rotation-of the footing along with the first few normal modes of vibration of the building on rigid foundation.

(ii) The first few normal modes of an associated building-foundation system in which Kf(w) is replaced by a frequency independent value, say the static value K,(O).

Methods of analysis using these two sets of Ritz vectors were developed and numerical results obtained.21 Based on that study, it was concluded that the former set of Ritz co-ordinates were preferable; consequently those are the only ones included in this presentation.

Motivation The building displacement is governed by equation (2a) which may be expressed as

Mii + CU + KU = - Mlii,(t) - Mlii,(t) - Mh&t) The left-hand side of this equation is the same as in analysis of buildings on rigid foundation. On the right- hand side, the three terms represent the effective earthquake forces associated with free-field ground motion, additional footing translation due to interaction and footing rotation due to interaction. The distribution

Figure 3.

mi Ug(t)

Distribution of effective

mi Uo(t 1

earthquake F forces Fhi of these forces over the height of the building is shown in Figure 3. The latter two terms vanish for rigid foundations and then equation (8) takes the standard form.

Because the building on a rigid foundation is assumed to possess classical normal modes, the frequencies and mode shapes are solutions of the eigenvalue problem

K+n = 4 M+n (9)


Expressed in terms of the mode shapes, the structural displacements are given by

Introducing the transformation of equation (10) and with the aid of orthogonality properties of normal modes, equation (8) becomes

In equation (1 1)

and L', = +,'m

where tn is the damping ratio for the nth mode of vibration. Lk and L', are the coefficients of generalized dynamic load in the nth mode associated with translation and rotation, respectively, of the base of the building. The distribution of effective forces due to base translation (Figure 3) and variation of modal displacements over the height combine in such a way that the magnitude of Lh, decreases with mode number n. This, of course, is the principal reason that the contributions of higher modes in response to horizontal ground motion are relatively small. Similarly, L', decreases with mode number n; in particular, L', identically vanishes for all modes higher than the first if the shape of the first mode is linear.22 The fundamental mode of vibration of many tall buildings is close to linear so that L; for higher modes will be very small, and LT, will decrease more rapidly than Lk with mode number. It is therefore apparent that the response of multistorey buildings including foundation interaction effects, governed by equation (8), will be primarily due to the first few normal modes of vibration (on rigid foundation).

Method of analysis Returning to the N + 2 simultaneous algebraic equations (equation ( 5 ) ) for the unknown complex

frequency responses ii(w), Co(w) and &o), it is logical then to express the structural displacements in terms of the first Jmodes of vibration, whereJis expected to be much smaller than N,

J

n=l Wt) 2: C Yn(t)+n

which in terms of the complex frequency response functions is J

n =1 Ww) 2 C yn<w>+n

Introducing the transformation of equation (14) into equation (5), pre-multiplying equation (Sa) by +: Mz(-o2+2i5,ww,+w3 ~ n ( w ) - 0 2 L ~ i i o ( w ) - 0 2 L ; O(w) = -L$ n = 1,2, ..., J (15a)

and utilizing the orthogonality properties of modes results in

J

n=l

J

n=l

- w2 2 Lk P,(W) - dm,iio(w) - o 2 L', B(w) + KJ&) z-iO(W) + Kr&f(w) O(,) = - m,

- o2 x L; PJW) - w2L; iio(w) - w2 1, B(w) + K*v(w) iio(o) + KJ.fJ.f(o) B(,) = - L',

(1 5b)

(154

For a particular excitation frequency o, equations (15) represent J + 2 simultaneous algebraic equations in the unknowns Pn(w), n = 1,2, ..., J ; iio(o) and &w). It was argued earlier that sufficiently accurate solutions


for structural displacements should result even when J, the number of modes included, is much smaller than N, the number of degrees-of-freedom of the building excluding those at the base. If this is indeed the case, the computational effort required for solution of equations (1 5) will be considerably smaller than for the direct method wherein N S 2 equations had to be solved. The efficiency in the solution process is of great significance because the process has to be repeated for many values of w. If all the modes are included, equations (15) would, of course, lead to the exact results.

The form of equations (15) permits an especially efficient solution. Each generalized co-ordinate P,(w) can be expressed in terms of iio(w) and g(w) from equations (15a). Substitution of these expressions into equations (15b) and (15c) leads to two equations in the unknowns C,,(w) and &w) which can be solved either explicitly3 or numerically.

Numerical results To evaluate the effectiveness of this analysis procedure, in which the equations are transformed to the

normal modes of the building on rigid foundation, results of analysis of a building-foundation system are presented in this section.

In order to test the capabilities of this method under rather severe conditions, the selected system must satisfy two criteria: the building must have a large number of storeys, and the properties of the building relative to the foundation must be such that foundation interaction has important effects on structural response. The two requirements are mutually opposing in the sense that, in general, the longer the periods of vibration of the building, the less significant are the effects of foundation interaction on building response. For the purposes of this investigation, it was important to satisfy both the criteria and this was achieved by selecting a building with many storeys but having unrealistically short periods of vibration; alternatively, values of periods of vibration typical of tall buildings could have been chosen together with unrealistically small shear wave velocities for the foundation material.

m I:=

(v

W / W I

Figure 4. Effects of foundation interaction on building response


- -

- 3 Number of Modes Included

- L- - ;

I I

5

A 25-storey building having the same lumped mass at all floors and the same stiffness for all storeys with a 0.5 sec fundamental period of vibration was considered. The damping matrix C was taken as a linear combination of the mass and stiffness matrices, M and K ; the coefficients for M and K were selected to produce damping ratios of 5 per cent in the first and fifth modes of vibration of the building on rigid foundation. The height of all storeys was 12 ft. The diameter of the building was 100 ft. The footing mass was taken as half of the floor mass. For the material of the halfspace, the shear wave velocity was selected as 1,000 ft/sec and the Poisson’s ratio as 4.

That the properties of the selected system are such that effects of foundation interaction are significant in the building response is evident from Figure 4, wherein the complex frequency response Zz5 at the top floor is presented from two analyses, one assuming the foundation as rigid and the other including foundation interaction by the direct method (equations (5)). In this case, foundation interaction lowers the fundamental resonant frequency of the building to about one half the value on rigid foundation.

$ 8 I:=!

Q 0 a 6 e .- - 5 4

2

0

n N I: =I

.YY . I 2 3 4 5 6 7

w/wI

Figure 5. Evaluation of analysis technique using normal modes of the building on rigid foundation-response 8,, due to i ig( t ) = l e i ~ t

The ‘exact’ response of the building-foundation system, as obtained from the direct method (equation ( 5 ) ) , is compared with approximate results obtained by considering relatively few modes of vibration in equations (15). The complex frequency response functions, i.e. the responses to ii,(t) = 1 eid, Gz5 for the top floor, CIS for the thirteenth floor, Zl for the first floor, Eo for the base translation and 6 for the base rotation are presented in Figures 5-9. The ‘exact’ response was obtained by the direct method which involved solving 27 simultaneous equations for each value of the excitation frequency. Including 1,3 and 5 modes of vibration, approximate responses were computed from equations (15) by respectively solving 3, 5 and 7 equations.

The results presented in Figures 5-9 demonstrate that very few modes suffice to produce an excellent approximation to the ‘exact’ response. The quality of the results varies somewhat with the degree-of-freedom being considered, depending on the relative significance of various modes for response at that particular location. If the range of interest of excitation frequency is restricted to slightly beyond the fundamental

4

74

a.

ANlL K. CHOPRA AND JORGE A. GUTIERREZ

9

4

3

2 I 0 I 0

Number of Modes Included Exact

I 2 3 4 5 6 7 W/Wl

0 I 2 3 4 5 6 7 W/W,

Figure 6. Evaluation of analysis technique using normal modes of the building on rigid foundation-response zl3 due to ii,(r) = 1 eiwt

Number of Modes Included, "F .6 I .5

.4

.3

.2

.I

n - 0 1 2 3 4 5 6 7

cri/W,

EARTHQUAKE RESPONSE ANALYSIS 75 ! I

0

O h

-l n;"""' Number of Modes Inclued

I 2 3 4 5 6 7 w / w ,

1 2 3 4 5 6 7 w/w I

Figure 8. Evaluation of analysis technique using normal modes of the building on rigid foundation-response Zo due to ii ,(t) = l e i ~ t

1 Number of Modes Included

5 Exact

L . , < , - - - .- - /. - . 4&

I 2 3 4 5 6 7 w / w ,

Figure 9. Evaluation of analysis technique using normal modes of the building on rigid foundation-response 8 due ii,(t) = 1 e'w*

to


resonant frequency, the first mode alone is sufficient to accurately express the building response. In general, no more than the lower modes having frequencies below a certain maximum excitation frequency of interest need to be included. It is not necessary even to consider all of these lower modes because response to arbitrary earthquake ground motion will be primarily due to those harmonic components which have frequencies close to the first two or three resonant frequencies, where large amplifications in the building response occur (Figures 5-9). Errors at higher frequencies in the approximate results are especially of little consequence in displacement responses (given by dividing the complex frequency response for accelerations by the square of the excitation frequency) because the displacement amplitudes decrease rapidly with excitation frequency, and hence their contributions to the earthquake response are extremely small.

CONCLUSIONS

A very effective method for analysis of dynamic response of multistorey buildings including foundation interaction to earthquake ground motion has been presented. The key idea, which is based on the Ritz concept, is the transformation of structural displacements to a small set of generalized co-ordinates. For the system considered-a multistorey building on a rigid footing attached to the surface of a linearly elastic half space-it has been demonstrated that the family of Ritz functions which includes the first few normal modes of the building on fixed base and translation and rotation of the base is very effective in reducing the number of unknowns. Although standard modal analysis does not apply to building-foundation systems, the method developed takes full advantage of the important feature that structural response to earthquakes is primarily due to the first few modes of vibration.

ACKNOWLEDGEMENTS

This study was supported in part by National Science Foundation Grants GI-31883 and GI-36387 and by the Industrial Affiliates of the Earthquake Engineering Research Center, University of California, Berkeley.

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Wld Conf. Eartha. Enanp. 1327-1343. Tokvo (1960). , - . , 5. L. A. Rosenberi ‘0; the interaction between ground and structure during earthquake’, Bull. Int. Inst. Seism. Earthq.

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11. A. H. Hadjian, ‘Earthquake forces on equipment in nuclear power plants’, J. Power Div., A X E , 97, 649-665 (1971). 12. H. Tajimi, ‘Discussion: building-Foundation interaction effects’, J. Engng Mech. Divi., ASCE, 93, EM6, 294-298 (1967). 13. J. W. Meek and A. S. Veletsos, ‘Dynamic analysis and behavior of structurefoundation systems’, Report No. 13, Dept.

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Computation, 19, 297-301 (1965).

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18. I. E. Luco and R. A. Westmann, ‘Dynamic response of circular footings’, J. Engng Mech. Div., ASCE, 97, 1381-1395 (1971).

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20. A. S. Veletsos and Y. T. Wei, ‘Lateral and rocking vibration of footings’, J. Soil Mech. Found. Div., ASCE, 97,1227-1248 (1971).

21. A. K. Chopra and J. A. Gutierrez, ‘Earthquake Analysis of multistory buildings including foundation interaction’, Report No. EERC 73-13, Earthq. Engng Research Center, Univ. of California, Berkeley, California (1973).

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Earthquake response analysis of multistorey buildings including foundation interaction

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