Seismic response of multi-storey buildings including foundation interaction and P-Δ effects

Seismic response of multi-storey buildings including foundation interaction and P-A effects K. S. Sivakumaran Department of Civil Engineering and Engineering Mechanics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7

T. Balendra Department of Civil Engineering, National University of Singapore, Republic of Singapore 0511 (Received June 1986; revised May 1987)

This paper presents a method of seismic analysis of multi-storey buildings when both soil-structure interaction and P-A effects have been taken into account. The additional overturning moment at each storey due to P-A effects is replaced by two fictitious horizontal forces acting in opposite directions at the upper and lower ends of each storey. Then, the governing equation for each floor is developed in which these fictitious horizontal forces are incorporated along with seismic inertia forces. Considering the footing as a rigid disc resting on the surface of an elastic half-space, two equations expressing the equilibrium of the building as a whole in translation and rotation complete the set of equations in terms of floor displacements and two rigid body base displacements. Recognizing that the superstructure alone admits classical normal modes, floor governing equations are uncoupled and substituted into the remaining equations, which results in two coupled integro-differential equations in terms of two rigid body displacements. These two equations can be solved using numerical procedures. The method is applied to two unbraced steel buildings and numerical results are presented for different foundation soil conditions. The influence due to P-A effects and soil-structure interactions on the responses of these buildings is discussed.

Keywords: seismic analysis, multi-storey buildings, soil-structure interaction, P-A effects

The analysis of multi-storey buildings to earthquake ground motion is straightforward when the second-order effects, such as P-A effects, P referring to the vertical loads and A to the horizontal displacements, are neglected and the building is assumed to be fixed at the base. Wellknown efficient methods are available for such analysis) Although current building codes 2'3 do not give specific recommendations for evaluating P-A effects in the seismic analysis, when P-A effect is included, the horizontal displacements in taller, multi-storey unbraced buildings may be much larger than the corresponding horizontal displacements calculated ignoring the P-A effects. For this 0141-0296/87/04277-08/$03.00 © 1987 Butterworth & Co. (Publishers) Ltd

reason, the Applied Technology Council (ATC) recommendations 4 for seismic design of buildings require a P-A check as a standard part of earthquake response analysis. Another factor that should be considered in the analysis of building response to earthquake ground motions is the soil conditions at the site. Soft soils have been shown to alter the structural response due to the flexibility and the damping capacity of the foundation. This soil-structure interaction effect will be of slight importance if the foundation rock is firm and the building is relatively flexible. On the other hand, if the structure is stiff and supported on a deep, soft soil layer, considerable energy

Eng. Struct. 1987, Vol. 9, October

277

Seismic response of multi-storey buildings. K. S. Sivakumaran and T. Balendra will be transferred from the structure to the soil and the resulting response of the structure may differ drastically from the estimated response using simplified analysis. Hence, in general, both P-A effects and the interaction effects of a soft surficial soil layer can be important and must be accounted for in the seismic analysis of structures. The first of these, the P-A effect, has been a matter of concern in the determination of the ultimate static load carrying capacity and stability of frames. 5 The P-A terms were found to have considerable influence upon the static effective forces developed and the lateral strength of these structures. Goel 6 evaluated the influence of P-A and axial deformation of columns on the earthquake response of multi-storey structures founded on a fixed base. Based on the results of a limited number of analyses, Goel 6 concluded that the P-A effect influenced the elastic response as much as 10%. More recent analysis 7 for P-A effects in seismic response of buildings confirmed Goel's conclusions. In these analyses, however, the soil-structure interaction has not been considered. Recognizing that soil structure interaction can significantly change the response of a building under earthquake excitation, there has been considerable interest in developing analytical methods to take this effect into account. 8'9't° Difficulties for such an analysis lie in the fact that the stiffness and damping attributed to the soil medium under the footing are frequency-dependent, and that the combined soilstructure system does not possess a set of classical normal modes. I'he present paper oflers an approach in which both PA effects and the soil structure interaction have been taken into account. The influence of the individual effects on the response of the structures can be studied independently, if so desired. Here, the P-A effect is taken into account in the following manner. At a given storey of the building, the additional destabilizing overturning moment due to P-A effect is given by the combination of the sum of all the vertical loads acting above the storey under consideration, and of their interstorey drift. This overturning moment is replaced by a statically equivalent couple consisting of destabilizing horizontal forces applied at the upper and lower ends of the storey. Then, the governing equation for each floor is developed in which these fictitious horizontal destabilizing forces are incorporated along with the lateral inertia, damping and elastic stiffness forces. This process results in a linear system of equations for the superstructure, where the elastic stiffnesses are found to be modified due to P-A effect, In considering the soil-structure interaction, the footing can be considered as a rigid disc resting on the surface of a linear elastic half-space. The appropriate dynamic-force-displacement relationships of a massless rigid disc on a linear elastic half-space can be evaluated by the methods of continuum mechanics.~ ~'~2 Consequently, the interaction should be modelled by a springdashpot device whose stiffness and damping terms are functions of the frequency of excitation. In principle, such a frequency-dependent foundation model could be used directly in a frequency-domain analysis of soil-structure interaction problem. However, Veletsos and Verbic 13 obtained "approxim_ate dynamic-force-displacement relationships in time domain, which are independent of frequency. Neglecting the small coupling between horizontal and rocking motions, such integro-differential forcedisplacement relationships for horizontal translation and rocking motion at time t are given by:

278

Eng. Struct~ 1987, Vol. 9, October

Fo(t) =Kv

[ ~(t) + h,, 1~;" |,~,(t) ]

F

(')

il)

'/

Mo(t)=K4, ( 1 - b , ) 4 ( t i + h l h z ~ ~b(t)+h3(l(~

+/;;

~(t)

r

in which Fo(t ), Mo(t ) and Vo(t), 4(0 are the forces and the displacements corresponding to the horizontal and rocking motions respectively, K,,, K , represent the static stiffnesses, and ~ the shear wave velocity of the half-space and are given by

8Gr Kv = 2

-- ,-'

8Gr 3 Ks = 3(f

-

K =

13)

In equation (3), r is the radius of the rigid disc, G is the shear modulus of elasticity, p the mass density, v the Poisson's ratio of the half-space soil and be, bl, b2 and b 3 are dimensionless constants which depend on he Poisson's ratio v. In this paper, the above integrodifferential force-displacement relationships have been used. Recognizing that the superstructure alone admits classical normal modes, the dynamic equilibrium equations of the superstructure floor masses can be uncoupled and written in terms of normal coordinates. Substitution of the solutions to these uncoupled equations into the remaining two equations, expressing the equilibrium of the building as a whole in translation and rotation, results in two coupled integro-differential equations in terms of the base displacements Vo(t) and qS(t). These two coupled equations are solved using numerical procedures. Back substitutions of the values for Vo(t) and ~b(t) into appropriate relations, the floor displacements, shear and any other relevant parameters are obtained. The proposed method of analysis is used to obtain the response of two unbraced steel buildings of five and twenty storeys subjected to the N-S component of the El Centre 1940 and $69°E component of the Taft 1952 earthquakes and for different soil conditions.

Equations of motions The building-foundation system under investigation is shown in Figure l(a). The mass of the N-storey building is considered to be concentrated at the floor levels, the floor system is assumed to be rigid so the relative displacements are entirely due to the deformation 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 the rigid foundation admits decomposition into classical normal modes. The base of the building is assumed to be a rigid disc resting on the surface of a homogeneous, linearly elastic half-space. The building-foundation system will be subjected to horizontal free-field ground displacement Ug (acceleration Og). In addition to the N horizontal displacements of each floor, the system has two more degrees of freedom, namely, horizontal displacement of the base V0 and the rotation of the base qS. The total horizontal displacement Uj of the jth floor with respect to a fixed vertical reference axis is

U~ = Ug + Vo + h%b + V~

(4)

Seismic response of multi-storey buildings: K. S. Sivakumaran and T. Balendra

m.g [

i- 1 m;g h ; q5

8 ¢..

mlgJ .x_ u_

b [Fo( ~ ) ~

Mo(t)

a Figure 1

(a) Idealized building-foundation system; (b) equivalent lateral force formulation for P-A effects

where Vj denotes the displacement at the jth floor due to structural deformations and J

h * = Z hi

mo(0o + Vo) + {mj)r{uj} + Fo(t) = 0

(8)

I*~J + {mjh*}T{Oj} - {mj}T{hT}g~p- {mj}T{ Vj}g

i=1

where hl is the height of ith storey. The additional overturning moment M~ at each floor, due to P-A effect is given by

(5)

Mj = m*g(uj - u j_,)

where g is the gravitational acceleration and

+ mo(t) = 0

(9)

In the above equations and in the equations to follow {xj} is a N x 1 vector of quantity x j, where xj denotes that it is associated with the jth floor or jth mode of vibration N

N

I * = ~ Ii

m*-- y m,

j=0

i=j

where m i is the mass of the ith floor. The equivalent lateral forces Fj for each storey applied at the upper and lower ends along opposite directions, are equal to the overturning moment divided by the storey height, thus substituting equation (4) for U~

m']g m~g Fj = ~ V~- ~ Vj_ , "4-m*gc~

(6)

Now, the equation of motion for any floor 'j' can be formulated by expressing the equilibrium of the effective forces associated with each degree of freedom (refer Figure 1 (b)). Thus the equation of motion in terms of the degrees of freedom Vj for each floor mass of the superstructure above the base [ M ] { ~ } + [ C ] { ~ } + I l K ] -- [ K G ] ] { Vj} = -- {m/}({)g + ~'o) --

in translation and rotation

{mjh*}6 + {mj}gc~

(7)

In addition, the equations of motion of the whole building

where Ij is the mass moment of inertia of the jth floor including that of the footing; mo the mass of the footing; [M] the diagonal mass matrix of order N x N; [C] the damping matrix of order N x N; [K] the elastic stiffness matrix of order N x N and [KG] the lateral geometric stiffness matrix. [K] and [Ko] are tri-diagonal matrices whose non-zero terms are

K(j, j) = 12(E/)j/h 3 + 12(EI)j+ tlh3+ x K(j, j - 1) = K(j - 1, j) = - 12(EI)j+ ,/h3+,

(10)

Ka(j, j) = m*a/h j + m*+ ,g/hj+ , KG(j, j - - 1)= KG(j-- l, j)= -m*+ ,g/hj+ x where,j ~< N and (EI)j is the sum of the flexural rigidities of the jth storey columns. When the damping matrix [C] satisfies certain conditions, the simplest of which is when it is linear combination of [M] and [K-I, the system has normal modes of vibration. Assuming this condition is satisfied, the uncoupled equations in terms of normal

Eng. Struct. 1987, Vol. 9, October

279

Seismic response of multi-storey buildings. K. S. Sivakumaran and T. Balendra

coordinates qj are given by

)

~j + 2~/oj~/j + CJJ~qj= - ~i(U, + I?o) - fii~ + '~i.q~b; j = I to N

(1 1)

where

+ Mo(t)= 0 (17) where

and where

{a~} 7'= {{mj}W[73i" ~j} M*

=

Here, mode shape 7~ and the corresponding undamped frequency cot are the eigenvalue solutions arranged in an ascending order and V~ can be related to the normal coordinate q~ by { V~} = [7]{qj}

(13)

Now, the jth mode-displacement and mode-acceleration can be obtained in terms of U0, Vo, q~ and their time derivatives by solving the equations (I 1). Hence:

q~(t) = - J~ L~(z)2~(t - z)dz

(14)

Eli(t) = j'i L1(~)l~i(t - z)dz - Li(t); j =

1 to

{Bj} T = {{m~}T [TUj' fli} = {{mjh*}TCYUj'oQ} ( C j;"~T =

Noting that the interaction force Fo(t ) and moment Mo(t) can be expressed in terms of Vo and ~b (equations (1) and (2)), thus the governing equations of the total system (equations (7), (8) and (9)) with N + 2 degrees of freedom, namely {V~}, Vo, qS, are reduced to two couple integro-differential equations (16) and (I 7) in terms of the base displacements Vo and qS. Here, numerical procedures ~4 are used to obtain a solution. Trapezoidal rule and average acceleration methods have been used to obtain the integrand and the derivatives respectively.

N

where

- 7.- i~clclu-~lng p_~5~effeils ~- t

8.0

Li(t ) = ~j (O'.(t) + Vo(t)) + f l i $ ( t ) - ~ig(~(t) 2~(t) =

{{mjh*}T[y]j'fij}

1 --

-g

e - ~,o,, s i n coo~ t

(15)

E

4.0

(~0Dj 2

~(t) = %

(~LI/)j

e -¢H

cos(~oo/-

g E 8

~)

and where ~oo~ = ~,9~(1 -

8 - 4.0

~)~

~b~= tan-1(1 - 2¢2)/2~ (1 - ~2)~

-8.0

Using equations (4), (13) and their time derivatives, the governing equations (8) and (9) can be written respectively in the form

mo(~Je(t) + (/o(t)) + {A~} r

{;o

2.0

6.0

8.0

t0.0

Time (s)

0.4

}

((Jo(z) + ¢o(t))#3(t - z)dz

4.0

---

A

Including P - A effects

0.2 yf~

0

d .9

0.0

(16)

l*~(t)+ {Bg}W{fi (Uo(z)+ 9o(Z))#j(t- z)dz}

-0.2

-0.4 2 i0

'

' 4.0

'

6 i0

-L-

8 i0

10.0

Time (s)

Figure 2

Rigid b o d y motions of a t w e n t y - s t o r e y building on soft clay ( V s = 6 O m s 1, El Centro 1 9 4 0 )

280

Eng. Struct. 1987, Vol. 9, October

Seismic response of multi-storey buildings." K. S. Sivakumaran and T. Balendra i

---

i

Including P - & effects

---

Foundation: sand and gravel

// 4

V~ =

60 ms-1~

Including p_A effects

Foundation: sand and gravel

5

"

V,

= 400

ms -~

v,~,o~= =

_4

~3

~3

o

,Oms O

O3

2

,\

2~ 1 i

i

i

100

i

i

200 Displacement (ram)

i

i

i

- - - - - Including P - ~ effects

i

i

i

300

500

1000

,

i

i

,

Foundation: soft clay

i

1500 Shear (kN)

i

i

2000

2500

i

- - - - - Including P - & effects

Foundation: soft clay

/Z V~ = 60 ms -1 - - ~ - - - ~ V ~

4

z

= 400 ms -1

= 400 ms -1

f

9

9 Vs = 60 ms-1

2

~,

"XX~x'~

, I

I

IO0

i

i

---

I

i

Including p _ A effects

I

300 i

I

a

0

500

l

i

1000 1500 Shear (kN)

i

i

2000

2500

]

Foundation: sand and gravel

v, ~2ooms-' /

4

l

200 Displacement (mm)

'\\

\

---

Including P - & effects

Foundation: sand and gravel

.Jj"

~3 £

'VV,='OOm,-'

2

\\ 25

50 75 Displacement (mm)

i

----

i

i

Including P - ~ effects

4

v,=oOms-'7 "

~ V s

0

i

Foundation: soft clay

/ m

100

----

I

I

I

250

500

750 Shear (kN)

I

1

I

1000

1250 '

15'o 0

i

i

I

Including p _ A effects

Foundation: soft clay

,Z-/" ~ 200

/

ms

1 / ~ . , ~

..f.~

. " \ X',,

0

v, = 60 ms-~

2

'\

\\\\ \~ ~-~--v~

= 400 ms -1

1

I

i 7=5 , , , i 1250 1500 1000 50 100 L 0 250 500 750 Displacement (ram) D Shear (kN) Figure 3 Effect of P-& on the maximum floor displacement relative to the base and the maximum storey shear of a five-storey building. (a) El Centro 1940 excitation; (b) Taft 1952 excitation

0

,

25

Eng. Struct. 1987, Vol. 9, October

281

Seismic response of multi-storey buildings. K S. Sivakumaran and T. Balendra . . . . . . .

---

q ~

. . . . . .

~

. . . . . . . .

Including P - A effects

~

. . . . . . . . . .

F . . . . . . . .

. . . . . . . .

Foundation: sand and gravel

.......

l

i

l

. . . . . .

- - - Including P-z& effects

t

.

.

.

.

Foundation: sand and gravel

20' V, = 60 ms 1 Vs = 200 m s - t / . /

16

z'/

~--~~----~L~V~

~

= 200 ms 1

16

////~'

~12 U~

8

I

0

I

100

I~_

I

, p l A effects -r Including

1 /

X/

16~S

400

/V

-K/

//

I

---

6000 Shear (kN)

I

8000

I

10 000

I

Including p--A effects

I

Foundation: soft clay

20

200 ms'

. 16

Z /

=>

Vi\v,:

}12 //"

4000

-~J'

.,///~ = 60 ms-1 ,,~,~././..J~ "/>"

2000

'Foundation: soft clay 1 Vs = 200 ms/~/" I ~ z /'{'~/"

20 I -

12 b Iv,

l

200 300 Displacement (rnm)

,o0 ms'

v, = 400 ms ' 8

/

I

0

100

----

I

I.__

200 300 Displacement (mm)

Including P-.A effects

I

400

0

a

Foundation: sand and gravel

I

I

2000

4000

r

i

1

6000 Shear (kN)

l

I

8000

10 000

r

r

i

- - - - - Including p--A effects

Foundation: sand and gravel

20 20

V, = 60 ms-'

16

/ / ~

/

/~"

Vs = 400 ms

16

v, = 4o0 ms-~

~12

,/ /

8 4

/

~

"

I

0

~

I

2o~ 161~128 4

8

I

50

I"

"V~ = 200 ms 1

I

100 150 Displacement (mm) I

Including p_A effects

/

/ j r

~-~-

--

I

I

I

I

2 00

1000

2000

I

I

i

Vs = 60 ms -I

----

f~>Z

v, = 4ooms-;

I

~

I

Including p--A effects

I

4000

5000

F

I

Foundation: soft clay

2(

{

1E

~

V

s = 400 ms --1

v, = 200 ms,-/

T~ "~..

~12 8

Foundation: soft clay

4 1

0

I

3000 Shear (kN)

"\

I

I

*

I

I \

I

I

100 150 200 b 0 1000 2000 3000 4000 5000 Displacement (mm) Shear (kN) Figure 4 Effect of P-& on the maximum floor displacement relative to the base and the maximum storey shear of a twenty-storey building. (a) El Centro 1 940 excitation; (b) Taft 1952 excitation

282

50

I

Eng. Struct. 1 9 8 7 , Vol. 9, O c t o b e r

- -

Seismic response of multi-storey buildings: K. S. Sivakumaran and T. Balendra Solving the equations for Vo and q~ using time steps, time history for base displacements, deflections and shear can be obtained. For brevity, details of these procedures are not presented herein. Note that in equations (16) and (17) the terms containing gravitational acceleration g represent the P-A effects. Therefore, if the value of 9 is assigned to zero, the above analysis neglects the P-A effect. The response of the building without the soil-structure interaction can be obtained either by setting V0, tk and their time derivatives equals to zero in equation (11) or by assigning a high value for shear wave velocity Vs. Therefore, in this method of analysis the effects due P-A and soil structure can be obtained independently.

Numerical examples Two steel buildings of five and twenty storeys are analysed for the P-A effects by the proposed method. These buildings are fully moment resistant, unbraced frames, square in plan and have bay widths of 12 m and 2 4 m respectively. Each floor has a uniformly distributed gravity load of 4.8 kPa and each storey height is 3.6 m. The moment of inertia of the columns and other pertinent structural data were obtained from reference 9. The fundamental frequency of these buildings, assumed to be on a fixed base and neglecting P-A effects, is 1.15 Hz and 0.5 Hz respectively. The damping ratio in the fundamental mode of the superstructure on rigid foundation is taken to be 2 %. In order to study the building response, the shear wave velocity of the half-space material was selected as 60 m s- t, 200 m s - 1,400 m s- 1 and 800 m s- 1. Also two different half-space materials namely sand and gravel (p = 2000 kg m -3, v = 0.33, b 0 =0.65, b 1 =0.5, b 2 =0.8, b 3 = 0.0)and soft clay (p = 1750 kg m-3, v = 0.45, bo = 0.6, b~ =0.45, b2=0.8, b3=0.023) were considered. The responses of these buildings, with and without the P-A effects, to the N-S component of the El Centro 1940 and $69°E Taft 1952 earthquakes are presented in Figures 2-4 for various values of shear wave velocity and foundation soil medium. Figure 2 shows the effect of P-A on the base displacement (rigid body motions) of the twenty-storey building on soft clay of shear wave velocity 60 m s- 1 subjected to E1 Centro 1940 earthquake. In this case, inclusion of P-A effects tend to oscillate the time history of rigid body displacements about the time history without P-A effects and an accumulation of the fluctuation with time is noted. The maximum base displacement is only about 4 % of the maximum displacement of the 20th floor. For higher values of soil shear wave velocity, therefore approaching rigid foundation conditions, the base displacements and base rotations are very much reduced. Figure 3 and Figure 4 show the seismic responses of five- and twenty-storey buildings respectively. Although, the responses of these buildings were obtained for four different shear wave velocities, the responses for V~= 400 m s- t and V~ = 800 m s- 1 were found to be almost the same, indicating that the soil-structure interaction is negligible for foundation medium with shear wave velocity greater than 400 m s- 1. Hence, here the responses for V~--- 800 m s- t are not included in the figures. Karasudhi et al. 9 obtained the responses of the above structures subjected to E1 Centro 1940 excitement, excluding the P-A effects but including the soil-structure interactions. Wherever comparisons possible, the present results agree well with the

results presented by Karasudhi et al. 9 From Figures 3 and 4 the following observations can be made. Contrary to intuition, the maximum floor displacements in most of the cases considered herein are reduced due to P-A effects, although the influence in displacements in the lower storeys is small. Due to P-A effect, a maximum reduction of about 8 % is noted on the twenty-storey building on sand and gravel V~= 60 m s-1, subjected to $69°E Taft 1952 accelerogram. Among the few cases where increases in displacements due to the inclusion of P-A effect were noted the maximum increment of 4 % is observed on the 20th floor of the twenty-storey building on soft clay, V~ = 400 m s 1, El Centro 1940 excitation. The maximum displacement, in general, increases with the increasing shear wave velocity. Also, the deformations of the structure on soft clay, therefore higher Poisson's ratio, is in general more than the one founded on sand. However, this difference is insignificant for higher values of shear wave velocity. Although, it appears that it is difficult to predict the effect of P-A, it is seen that for specific soil conditions and excitement, the displacements either increased or decreased in all the floors. However, this is not the case with the variations in shear. Especially from Figure 4, shear variations due to P-A effect show a mixed trend of being less in some storeys and more in others. Maximum increase in shear of about 5 % is noted on the 14th storey of the twenty-storey building on soft clay, V~= 200 m s- 1, subjected to El Centro 1940. Also, a maximum reduction in shear of about 4.5 % is noted at some storeys. On the basis of these results, it appears that in typical unbraced, multi-storey steel frames the P-A effects influence the elastic responses as much as 8 %. Since it is difficult to predict whether the P-A effect increases or decreases the deflections from a simple analysis, in the event that it is necessary to conduct a P-A check, it is advisable to determine the P-A effects by rational analysis.

Acknowledgements The authors wish to acknowledge the assistance provided by Mr Philip N. Kelly who participated in this project as a recipient of Natural Sciences and Engineering Research Council Undergraduate Summer Research Award. This research has been supported by Natural Sciences and Engineering Research Council of Canada grant No. U0384.

References 1 Clough,R. W. and Penzien,J. 'Dynamics of structures', McGrawHill Co., New York, USA, 1975 2 International Conferenceof Building Officials, Uniform Building Code, 1979 3 NationalBuildingCode of Canada, National Research Council of Canada, Ottawa, Canada, 1985 4 Tentative Provisionsfor the Developmentof Seismic Regulations for Buildings, ATC 3-06, Applied TechnologyCouncil, Palo Alto, CA, USA, 1984 5 Vandepitte,D. 'Non-iterativeanalysisof frames includingthe P-Aeffect',J. Constructional Steel Research, 1982,2(2),3-10 6 Goel,S. C. ~P-Aand axial column deformationin aseismicframes', J. Struct. Div., ASCE, i969, 95(ST8), 1693-1717 7 Neuss,C. F. and Maison, B. F. 'Analysis for P-A effectsin seismic response of buildings', Computers and Struct., 1984, 19(3),369-380 8 Chopra,A. K. and Gutierrez,J. A. 'Earthquake responseanalysisof multistoreybuildingincluding foundationinteraction', Earthquake Eng. Struct. Dyn., 1974,3, 65-77

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Seismic response of multi-storey buildings. K. S. Sivakumaran and T. Balendra 9 10 11

284

Karasudhi, P., Balendra, T. and Lee, S. L. 'An efficient method of seismic analysis of structure-foundation systems', Geotechnical Eng., 1975, 6(2), 133-154 Lin, Y. K. and Wu, W. F. 'A closed form earthquake response analysis of multistorey building on compliant soil', J. Struct. Mech., 1984, 12(1), 87-110 Savuzzo, R. J., Bailey, J. L. and Raftpoulos, D. D. "Lateral structure

Eng. Struct. 1987, Vol. 9, October

12 13 14

interation with seismic waves', ./. Applied Mech., 19"71,38, 125 134 Wolf, J. P. "Dynamic soil-structure interaction', Prentice-Hall Inc., New Jersey, USA, 1985 Veletsos, A. S. and Verbic, B. 'Basic response function for clasdc foundations', J. Eng. Mech. Div., ASCE, 1974, 100, 189 202 Craig, R. R. 'Structural dynamics, an introduction to computer methods', John Wiley and Sons, New York, 1981