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Compurers & Sfructures Vol 20. No. l-3, pp. 355-363. 1985 0045-7949/85 $3 00 t .OO

Prmted I” the lJ.S.A 0 1985 Pergamon Press Ltd.

MULTIDISCIPLINARY AND INTERACTION PROBLEMS

SEISMIC ANALYSIS OF THREE-DIMENSIONAL SOIL- STRUCTURE INTERACTION SYSTEM ON A

RECTANGULAR BASE

S. SAYLAN and T. G. TORIDIS

The George Washington University, Washington, DC 20052, U.S.A.

and

K. KHOZEIMEH

Smislova, Kehnemui and Associates, Rockville, MD 20852, U.S.A.

Abstract-An analytical procedure is presented for determining the three-dimensional interaction effects of a soil-structure system under seismic excitation. The structure is modeled as an assembly of finite elements representing the three-dimensional superstructure that is attached to a rectangular base on a half-space. Using triple Fourier transform techniques, dynamic displacement equations of the three-dimensional elastic half-space are solved. Boundary conditions on the surface of the half-space are expanded into a Fourier series to generate Dirac delta functions that simplify greatly the inversion of the Fourier transforms. Foundation impedances are then obtained from the dynamic displacement functions of the rectangular bases on the half- space with water table and elastic half-space subjected to harmonic loading. Solutions are presented in closed form. Based on specific values of some of the basic parameters of the problem, numerical results are obtained and presented in the form of a series of curves associated with the dynamic response of a rectangular base. Solutions are also obtained for the soil- structure interaction analysis of three-dimensional structures subjected to earthquake excitation.

INTRODUCTION

The soil-structure interaction associated with earthquake motion has been the subject of several investigations in the past. The study of this phe- nomenon must invariably include the dynamic characteristics of the structure and the foundation medium often treated as an infinite half-space. Among studies of this kind are the works of Reissner [I], Sung [2], Veletsos and Verbic [31, Bycroft [4], Luco and Westman [5], Lysmer [61 and others who simplified the problem by considering a rigid circular slab supported on the surface of an elastic or viscoelastic half-space subject to excitation by harmonic forces. Others, such as Awojobi and Tabiowo [7], Elorduy et al. [8]. Stretenskii [9], Kobori et al. [IO] and Thompson and Kobori [ll], studied the same problem by considering the base slab as a rigid rec- tangle. Other investigators, such as Beredugo and Novak [12] and Urlich and Kuhlemeyer [13], analyzed the problem by including the effect of embedment of the base slab.

Other investigators, including Parmelee et al. [ 141, Jennings and Bielak [ 151, Scavuzzo et al. [ 161, Tsai [17], Lee [18], Liu and Fagel[19], Vaish and Chopra 1201 and Rainer [21], used the foundation impedance approach. They obtained solutions in which the foundation impedance is dynamically coupled with the superstructure. The soil is usually modeled as a two-dimensional half-space and the structure as a lumped mass system or shear-type building. Lys- mer et al. [22], Chopra and Gutierrez [23], Seed et

al. [24] and other investigators used the direct approach. They analyzed the structure and foundation by combining them as an integrated system, and they defined the soil properties based on a finite element model and the superstructure by representing it as a lumped mass system. A few other investigators represented the soil properties with equivalent springs and dampers in the low-frequency range.

As indicated above, most of the previous investigations of soil-structure interaction systems under seismic loads were conducted using a lumped mass model of the superstructure and generally considering no more than two components of earthquake excitation. Furthermore, in those studies based on the impedance approach, the effect of foundation embedment was neglected or estimated in an approximate manner.

In the present study the soil-structure system is generalized to a three-dimensional model with corresponding consistent mass and stiffness properties, and the foundation system is represented by means of a rectangular foundation slab. Moreover, the foundation-soil-foundation interaction effects are introduced in terms of the foundation impedance functions, which are obtained in analytical form.

Applying Fourier analysis and Fourier transform methods, coupled with Dirac delta functions, closed-form solutions are obtained for a half-space that contains the water table and an elastic half- space. Furthermore, the reduction of the mathe-

355

356 S. SAVLAN et al.

matical expressions are based on the use of a Dirac delta function, which constitutes an important sim- plification and permits an easy solution for the general shape of rectangular foundation slabs resting on a three-dimensional half-space.

GOVERNING EQUATIONS

The system under consideration is idealized as an assembly of finite elements representing the superstructure that is attached to a rectangular base resting on an elastic half-space (the soil medium). It possesses a three-dimensional configuration, as shown in Fig. 1, which also indicates the reference axes for the structure and soil medium with common origin at the center of gravity of the rectangular base. The main assumptions made in the analysis are as follows:

1. The base slab is rigid and does not undergo elastic deformations,

2. The input ground motions act at the surface of the soil medium in contact with the infinitesi- mally thin base slab and in the directions of the coordinate axes.

3. The soil-structure interaction system possesses classical normal modes, modal frequencies and modal damping ratios associated with the fixed- base structure.

4. The dynamic force-deformation relationship of the soil medium can be characterized by an independent impedance matrix.

Based on similar assumptions, Tsai et al. [25] derived the equations of motion for the two-dimensional case of plane coupled horizontal translation and rocking interaction system. Kao [261 generalized Tsai et a/.‘~ equations for the three-dimensional case but again with reference to a lumped mass/beamlike structural system. In the present study a more general approach is followed by con-

v tz

Fig. I. Structure-foundation-soil system.

x3

//+-J&d h ‘r 4

X1.

“r

(X2 ,’

x; ;’

Node r

Fig. 2. Distance between center of gravity of base and nodal point r.

sidering a three-dimensional system consisting of an assembly of discrete elements interconnected at nodal points. Each nodal point is allowed all six degrees of freedom.

The differential equations of motion of the nodal point displacements can be expressed in matrix form as

[Ml{;} + [C](i) + [K](x) = {O}. (1)

where [Ml, [Cl and [K] represent the mass, damping and stiffness ,matrices of the fixed-base structure; {E}, {X} and (9 are the total (absolute) and {x}. {x} and {x} the relative displacement, velocity, and acceleration vectors, respectively, of the nodal points with reference to a fixed base slab. The relations between the two sets of vectors are

{X} = {x} + [N]{xo} + {x,} = {g} + ix,},

{X} = {x} + [N]{io} + {&} = {n} + {xX}, (2)

{i} = {i} + [N]{%,} + {x,} = {i} + {x,}.

in which N is the transformation matrix; {x0}. {xc,} and {x0} are the displacement velocity and acceleration vectors, respectively, of the rigidity center relative to the free-field motion; {x,}. {$} and {xX} represent the corresponding vectors of the input free-field ground motion of the base slab.

The equations of motion of the soil-structure system can be written in matrix form in terms of the interaction forces between the structural base and soil medium as follows:

[N][M]{x} + [M,](x) = -{F(t)], (3)

where [RI is the transpose of [Nl, so that with reference to Fig. 2 letting {x,} represent the vector of displacements of node r relative to the structural base

c

100 0 x; -xf 010-x; 0 x: 1

Three-dimensional soil-structure interaction system 3.51

The other quantities in eqn (3) not introduced earlier are defined as follows:

{X} = {io} + {%,}, (5)

where {F(t)} is a time-dependent interaction force vector acting on the center of the rigid base. This can be expressed in terms of the impedance matrix in a simplified manner by considering the off- diagonal elements to be zero, since in certain cases they are not significant. Thus, letting [C,,] and [K,,] represent the diagonal damping and stiffness matrices, respectively, {F(r)} may be represented as

{F(t)} = ]C,,l{x,] + ]K,,l{xo1. (6)

where [M,,] is a diagonal mass matrix containing the mass moments of inertias lumped at the center of the base slab.

Returning to eqn (1), premultiplication by [ml and substitution of the modified equation and eqn (6) into eqn (3) leads to

[MO&~ - [~IKIM - [~IUW

+ K,,lh~ + UL.lboI = 0. (7)

The first of eqns (2) and (7) can be combined into a single matrix equation, which in partitioned form is

where

(11)

(12)

As stated previously, it is assumed that the fixed- base structure possesses classical normal modes. Therefore, the modal characteristics of the structure can be represented as

[41TM[41 = [II, (13)

[4l’Kx41 = m3d 7 (14)

[41TM41 = bfl, (15)

in which [4] represents the normalized modal matrix; 0; and pi are the undamped natural frequency and damping ratio, respectively, of the ith mode of the fixed base structure; and [II represents the iden- tity matrix and the right-hand sides of eqns (14) and (15) are diagonal matrices. Letting

Fl = [41TM (16)

and considering the following transformation matrix [T]

(17)

the vectors of transformed variables {u}, {u} and {ii} can be defined as

Using eqns (2), the following relationship between {x} and {g} can be established:

Iu] = [‘Wd, 6) = [‘U{$~ {ii) = [TIM. (18)

Premultiplying eqn (10) by the transpose of [T]

(9) and using eqns (13)-(16) and (18), eqn (10) can be transformed into the following form:

Equation (7) can now be rewritten in terms of {II} [W + rGcil1 + ElI41 = - m, (19)

[M](u) + [C]{u} + [K](u) = [%I][p]{t,}, in which [Cl, [I?], and $1 appearing in eqn (19) are

(10) given below in partitioned form as

[El = - [MO]- “2[41T[ClN

------------____ ’ [Mel- “2(lNlTlC11Nl + [C,,l)[M,I - I” I

(20)

i - Wo- ““141T[KllNl . i -_---------------

- [Mel - “21NlTlKI141 ; Wol- “2(lNlTlK11Nl + lK,,l)lMoI - “’ I (21)

(22)

358 S. SAVLAN et al.

Equations of motion in the frequency domain The differential equations of the soil-structure

interaction system represented by eqn (19) cannot be solved directly because the foundation impedances are frequency dependent. Therefore, the equations are transformed into the frequency domain through a Fourier transformation. Defining the following Fourier transforms, where t and w are, respectively, the related pair of variables in the time and frequency domains

I r _r {q(t)}e-‘“’ dt (23)

{Z(o)} = J- 27F :x {i(t)}e-‘“’ dt I

(24)

Equation (19) now takes the form of

-w’[Il{q} + io[e]{i} + [kl{i} = - [F]{Z}. (25)

The solution for q(o) can be expressed as

in which

{ij(o)l = D-I(w)lMw)l. (26)

and

[H(o)1 = EWl@l (27)

[D(u)] = [--o’[Il + iw[e] + [I?]]-‘. (28)

Response in the time domain The response in the time domain can be obtained

through an inverse Fourier transformation, for example,

{q(t)} = k J:z {?j(w)}eiw’ do. (29)

Using again the transformation matrix [T], {u(o)} becomes

hXw)J = UI[H(~)lM~)I (30)

whose inverse can be expressed in the usual form, so that the absolute displacements are now given

by

{x(t)1 = {u(t,} + {z(t)} (31)

and the corresponding acceleration vector is obtained from

{x(t)} = & I

= [{l} - 02[T][H(w)]]{%(w)}e’w’d do. -r (32)

RECTANGULAR SLAB ON A HALF-SPACE

In many foundations of practical significance, the water table in the soil does not reach the ground

surface. On the other hand, the presence of the water table in the soil mass changes the wave propagation characteristics of the soil medium. Even more significantly, the presence of the water table converts the half-space into a layered system, as depicted in Fig. 3. which also shows the water table surface separating the layers of unsaturated and saturated soil.

Saturated soil is a mixture of soil skeleton and water filled with voids and is referred to as a porous soil medium with pore water flow. Brandt [27] and Biot [28, 291 studied the elastic waves propagated in saturated porous materials. Ishihara 1301 devel- oped the wave equations for the low-frequency range. Kao [261 obtained the displacement and stiffness coefficients in closed integral form for a porous elastic half-space.

In the present study the influence of the boundary between the unsaturated and saturated soil is as- sessed. Analytical expressions are derived for a system composed of a dry homogeneous elastic soil layer on a porous elastic half-space that is com- pletely saturated by water. In formulating the be- havior of the soil, the usual assumptions are made, including the assumptions that the material is homogeneous and isotropic, the strains are small, the soil skeleton in the layer below the water table is com- pletely saturated and water is incompressible, the soil and water move with the same ground acceleration, body forces are negligible and the medium obeys Darcy’s law.

Based on the above assumptions, the strain-displacement and stress-strain relationships are

et, = t (Ui.1 + It,,,) (33)

UlJ = 21*e,, + (he - p)6,,, (34)

in which e,, and u,, (i, j = 1, 2. 3) are the components of the strain and stress tensors, respectively, ui is the displacement component along the ith di-

k-b&--b+

‘Y

1 Qlt) Exciting Force

I x PVfl t

\W

tz Saturated Half-Space

Fig. 3. Geometry of the foundation-ground system.

Three-dimensional soil-structure interaction system 359

rection, A and p. are Lame’s constants, p represents the pore water pressure and 6, is the Kronecker delta. The equation of motion in terms of the dila- tation e can be expressed as [26]

(A + 2p.)VZe - V’p = pl”, (35)

where V’ denotes the Laplacian operator and p the mass density of saturated soil. Differential equations can also be formulated in terms of the displacement components, so that letting ii represent the displacement in the x direction (u = II, for i = 1)

p-V2u + (A + @le., - p,.r = pti. (36)

Based on Darcy’s law and considering the water to be incompressible, the following relationship can be obtained:

vlp = f &, (37)

in which K is the permeability coefficient per unit weight of water. Combining eqns (35) and (371, the result is [26]

V2 _

in which C1 is the diIatationa1 wave velocity represented by

(39)

Solution of the wave equations Taking the triple Fourier transform with respect

to X, y and t. eqn (38) is transformed to 1261

V’p- i ;--i.,=O 1Y(A + 2~) Cf ’ (40)

where

v2t; = _& _ f-32 + d?

, c dz” >

$ (41)

F = iwP (42) ; = -W”* (43)

and

x e(x, y, z, r)ei(W+p.v+wt’ dx dy dt. (44)

Substituting eqns (41) through (44) into eqn (40), the solution for Z is obtained as

e zz A,e-r]z + B1 erJz, (45)

in which

02 r: = 012 + @ - - +

iw

c: K(h + 21.L)’

Re (rl) > 0. (48)

As the depth of the soil medium z -3 a, P must be finite, so that B, = 0. Therefore

ii; = Ale-‘Iz fi (47)

where Al is an arbitrary function to be determined from boundary conditions. The inverse triple Fou- rier transform of 1; is

A,e-‘IZ+“““+YP+tw) da dfi dw. (48)

The solution for the pore water pressure is obtained in a similar manner, so that the transformed pore pressure is expressed as

p= iA,w r1 --Rz K(r: - r:,

e-nz _ -e r2 1 (49)

in which

r: = cc2 + p2, r2 > 0. (50)

The solution for the displacements can be obtained in a similar manner. Thus, for example, in the case of displacement u referring to eqn (36), the solution for the transformed variable becomes

x i(A + p) + [

w K(r: - r$) I

e-W

Q - i*(d - r:,

B1

X i(A + p) + QI 01

K(r: - rf) 1 ertZ +

F(rZ - $)

x [ASee *’ + Asem’], (51)

where AZ, A3, AS and A6 represent additions coefficients to be determined, and r$ = a2 + p* - (p/ ~)w*, Re(r3) > 0. The transformed displacements along the other two coordinate directions can be obtained in a similar manner.

Consider now the dry layer of soil material between the water table and the ground surface: in this case the water pressure p = 0 and permeability coefficient K tends to infinity. Therefore, U sim- plifies to

h(A + JL) - Et(rj _ rz) (Ate-‘fz + &PI. (52)

360 S. SAYLAN~ nl.

The other displacement components can be expressed in a similar manner. In the case of the saturated soil medium, as the depth of the soil medium tends to infinity the displacements and pore water pressure must be finite. Therefore, A3 = A6 = B, = 0, so that in the case of si

U = A2e-‘= _ 4, I*(4 - r:, i(h + k)

1

iP + ~p;~_ ,.:) e-“” + p(r; _ r:)A5e-ni. (53)

It should be noted that some of the coefftcients appearing in the above equations can be interrelated through the use of boundary conditions. For example, one such condition corresponds to the rate of change of the pore pressure with depth, which assumes the value of zero at the level of the water table (see Fig. 3) that separates the unsaturated and saturated layers of soil.

The transformed displacement components in the other coordinate directions can be obtained in a similar manner [31]. Then, using the stress-strain relations that incorporate the pore water pressure and that are expressed in terms of derivatives of the displacement functions, the components of the stress tensors can be obtained [26, 311. For example, the transformed stress components EXz and a:, corresponding to the unsaturated and saturated soil media, respectively, are represented as

- 2Alctrl w

K(r: - r:)(d - r:) e-l2Z

0 - K(r: - &(r: - t-g)

e--T*.?

0 + 14 c-,,L

(r: - 4) I> (55)

where V represents the compression wave velocity of the soil medium. The remaining stress components can be expressed in a similar manner [31].

The two sets of displacement and stress components obtained in this manner are related to each other on the common boundary between the unsaturated elastic layer and the saturated half-space

that is the surface of the water table. Such relations lead to equivalent constraint conditions, so that all coefficients appearing in the various equations can be expressed in terms of three arbitrary coefficients (functions). In turn these functions can be determined as follows. First, the transformed stress components are evaluated at the ground surface of the soil medium (z = 0). Then the resulting expressions for stresses are made to correspond to the specified contact stresses at the interface of the loading area. This leads to an eigenvalue problem that yields the frequency equation for the Rayleigh surface wave

WI. The displacement components of the ground sur-

face can now be found. Then taking the inverse triple Fourier transform, the solutions for the displacement components at the ground surface subject to various types of motion are obtained in closed form, i.e. in the form of integrals. Thus, in the case of vertical or rocking motion, for example, the displacement component u(0) at the ground surface is expressed as [26]

u(0) =

ei(ux+py+w’)dod13dw, (56)

in which E,,(O) represents the transformed normal stress component in the z direction, F(cw, p, w) is the function appearing in the frequency equation for the Rayleigh surface wave and

C2 = shear wave velocity = J

cl, (59) P

K’ = K($“- r;)

K2 = A + p,, (61)

and N is the ratio of shear wave velocity to compression wave velocity of the soil medium. The other displacement components are found using the same procedure. Finally, similar expressions are derived for translational motions in the x and y directions and torsional motion about the z axis.

Contact stresses

When a foundation base rests on the ground, the boundary conditions are represented as prescribed

Three-dimensional soil-structure interaction system 361

stresses that are distributed only under the loading area of the foundation base and that are zero outside the loading area. Reissner [I] assumed the stress dist~bution under a circular slab subjected to a vertical harmonic load to be uniform. Similarly, Sung 123 studied the problem for three types of contact stress distributions under a circular base. In the present study it is assumed that the stress distribution under the foundation base is either uniform or linearly proportional to the distance from the ro- tation axis, depending on the type of excitation. To simplify the analysis, Dirac delta functions are used by expanding the contact stresses into Fourier series coupled with the use of Fourier transform methods.

Considering a uniform stress distribution for the translational motion in the vertical z direction, the normal stress uZZ and the transformed stress EZZ are expressed as

o;, = qoe-‘w*, IxI<b, IYI <c (62)

Tf,, = (273)30 q&w - Co*) I S(a - O)S(f3 - 0)

sin (T) sin’;?)

i: t?l=t

ii n==l mn

x [s(+) +s(a+y)]

x [s(P-F) +S(P+y)])’ (6%

in which q. is the amplitude of the contact stresses with frequency WO; 6 designates a delta function; CX, l3 and W, as before, correspond to the Fourier transform parameters with respect to X, y and r, respectively; andj is an integer associated with the spacing of base slabs. Similar expressions can be written for translational motions in the x and y directions. The entire process can be repeated for the assumed linear stress distributions [26, 311.

Dynamical ground compliances A dynamical ground compliance is the ratio of

the foundation displacement in the direction of excitation to a harmonic disturbing force or moment acting on the foundation. The inverse of the compliance leads to the dynamical impedance. The forces (moments) acting on the rigid foundation are represented as follows:

Q,(t) = Qroe’OO1,

M,(t) = MrOetw’,

r = x, y, z

r = x, y, 2

64

(65)

where Qro and M,.o represent the amplitude of the disturbing force and the moment, respectively.

Considering now the horizontal motion in the x

direction

e* = 4bcq,

and the compliance F,, is expressed as

F = u(O, 0, 0, f) xx 4bcqoeiuo’ ’

f66)

(67)

Based on the solution for displacement component u discussed earlier, the dynamical compliance in closed form is written as follows:

1 mm F, = -

IS Ti;AO)

8apbcqo -0~ -OE r3e, BP wo)

x

{

2ih*ff%:(r: - r:,

IJ;

[

Klrl K, + iK2

(r: - &2 r: - r$ 1 i- 4132r,r3(jZ - h2) - (rf f p2)

X [j’r: + (j” - 2h*)r:]

- ?!y! [(r$ + p*)

x [(r: - r&r2 - j*r:l

- 2P*r,r& - r: - j*)] 1

da dl3. (68)

The remaining compliance functions can be obtained in a similar manner.

The dynamical solutions presented above can be specialized by setting the dimensionless frequency a0 - 0, where

(10 = 6h&/C. (69)

Applying L’hospital’s rule, numerical solutions for the static displacements and stiffness can be obtained 1311. These quantities are dependent on several parameters, including Poisson’s ratio, the aspect ratio and the footing spacing ratio. If the periods T, = 2jb and T, = 2jc of the double Fou- rier series are small, the structure-soil-structure interaction effects become more prominent because of influences from adjacent structures.

Figure 4 shows a plot of the vertical stiffness coefficient K, rendered dimensionless as a function of the parameter j, which is a measure of spacing of base slabs. These results correspond to a rigid rectangular base on an elastic half-space and are compared with results given by Kao [26] and, for largej, with those of Barkan [32]. It can be observed that good correlation exists between the various results. The difference between the curve obtained in this study and that of Kao is that Kao neglected the

362

, K,/GVE

S. SAYLAN et trl.

t2 . Present Study

a Present Study

0 KaoI261

. Barkan [321

d:Aspect ratlo YzO.25

I I I I I J 10 20 30 40 50 60

Fig. 4. Variation of K, as a function of spacing of base slab.

cross-product terms in the Fourier series expan-

sion, but they were included in the present study.

In the case of the dynamic response of a rectangular base, numerical results are obtained for some specific values of the basic parameters and are presented in the form of a series of curves. For example, Fig. 5 depicts the plot for a square base of the rocking compliance functions f, and f? as a function of the dimensionless frequency ao. The results of the present three-dimensionless analysis are compared with the work of Kobori et al. [lo] and Elorduy et al. [8]. In general, the correlation between the three sets of results is rather favorable.

Considering the dynamic response of a complete soil-structure interaction system, in order to allow for a comparison of the present three-dimensional analysis with the results of previous analyses, the present three-dimensional structural model [30] is specialized to a two-dimensional spring-mass system, shown in Fig. 6. The structure that was previously studied by Tseng [33] is subjected to a time- dependent acceleration input at the base. Figure 7 shows a plot of the acceleration response history at mass point 2 of the structure in the horizontal direction. Plotted in the same figure is the response history curve obtained by Tseng [33]. It can be observed that there is fairly close correlation between the two response curves in the peak response re- gion. which is in general of greater significance.

Finally, a three-dimensional soil-structure inter-

6

Fig. 6. Two-dimensional soil-structure interaction model.

- Present Study -- Tseng 1331

I I I I I I I I I

0.0 1.0 20 30 40 50 60 70 8,O Time (Second I

Fig. 7. Absolute acceleration response time histories at

the mass point No. ?. of two-dimensional structure.

t_18ft --_I

Fig. 8. Three-dimensional space frame structure.

o Kobw eta1 1101 A Elordy et al.[81 . Present Study

0 SOII-Structure lntemction

A Flxed-Base Structure [ 341

Time (Second1

Fig. 5. Rocking compliance function .fl and f2 for square Fig. 9. Relative displacement time histories of the second base. story in the x direction.

Three-dimensional soil-structure Interaction system 363

action analysis is performed for a simple space frame, depicted in Fig. 8. This two-story structure has base dimensions of 18 x 18 ft and story heights of 12 ft. The properties of the structure are given in Ref. [30]. The input ground motion is the same as the one used by Toridis et al. [34], who consid- ered a fixed-base idealization. The displacement time history of the second story in the x direction is presented in Fig. 9 for both studies. An exami- nation of the figure indicates the significance of including interaction effects, particularly for the peak dynamic response.

15.

16.

17.

18.

19.

Acknowledgement-The research reported in this study was partially supported by the National Science Foun- dation Research grant No. CEE-8219357.

20.

REFERENCES

21.

22. 1. E. Reissner, Stationare Axialsymmetrische Durch

eine Schuttelnde Masse Erregte Schwingungen eines Homogenen Elastischen Halbraums. Ingeniur Archiv. 7 (1936). 23.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

T. Y. Sung, Vibrations and semi-infinite soils due to periodic surface loading. Special Tech. Pub]., ASTM No. 156 (1953). A. S. Veletsos and B. Verbic, Vibration of viscoelastic foundations. Int. J. Earthquake Engng Struct. Dyn. 2 (July-Sept. 1973). G. N. Bycroft, Forced vibrations of a rigid circular plate on a semi-infinite elastic space and on elastic medium. Phil. Trans. R. Sot. London Ser. A 248 (1956). J. E. Luco and R. A. Westman, Dynamic response of a rigid footing bonded to an elastic half-space. Trans. ASME J. Appl. Mech. 39 (1972). L. Lysmer, Vertical motion of the rigid footings. Ph.D. dissertation, University of Michigan (1965). A. 0. Awojobi and P. H. Tabiowo, Vertical vibration of rigid bodies with rectangular bases on elastic media. Znt. J. Earthq. Engng Struct. Dyn. 4 (1976). J. Elorduy, J. A. Nieto and E. M. Szekely, Dynamic response of bases of arbitrary shape subjected to periodic vertical loading. Proc. Znt. Symp. Wave Prop- agation and Dynamic Properties of Earth Materials. University of New Mexico Press, New Mexico (1967). L. N. Stretenskii, Elastic waves arising from normal stresses applied to the surface of a half-space. SIAM, Muskhelisvilli Anniv. PA (1961). T. Kobori, R. Minai, T. Suzuki and K. Kusakabe, Dynamical ground compliance of rectangular foundation. Proc. Sixteenth Japan National Congress for Applied Mechanics (1967). W. T. Thompson and T. Kobori, Dynamical compliance of rectangular foundations on an elastic half- space. J. Appl. Mech. Ser. E 30 (1963). Y. 0. Beredugo and M. Novak, Coupled horizontal and rocking vibration of embedded footings. Can. Geotech. J. (Nov. 1972). C. M. Urlich and R. L. Kuhlemeyer, Coupled rocking and lateral vibrations of embedded footings. Can. Geotech. J. 10 (1973). R. A. Parmelee. D. S. Perelman, L. S. Lee and L. M.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

Keer, Seismic response of structure-foundation system. J. Engng Mech. Div. ASCE 94 (Dec. 1968). P. C. Jennings and J. Bielak, Report on dynamics of building-soil interaction. EERL 72-01, Pasadena, Cal- ifornia, Caltech (1972). R. J. Scavuzzo, D. D. Raftopoulos and J. L. Bailey, Lateral structure-foundation interaction of structures with base masses. Bull. Seism. Sot. Am. 6 (Apr. 1972). N. C. Tsai. Modal damping for soil-structure interaction. J. Engng Mech. Div. ASCE 100 (Apr. 1974). T. M. Lee, Surface vibration of a semi-infinite viscoelastic medium. Proc. Znt. Symp. on Wave Prop- agation and Dynamic Properties of Earth Maferials. University of New Mexico Press, New Mexico (1967). S. C. Liu and L. W. Fagel. Earthquake interaction by fast Fourier transform. J. Engng Mech. Div. ASCE 97 (EM4) (Aug. 1971). A. K. Vaish and A. K. Chopra, Earthquake analysis of structure-foundation systems, EERC 73-9. Berke- ley, University of California (1973). J. H. Rainer, Structure-ground interaction in earth- quakes. J. Engng Mech. Div. ASCE 97 (Oct. 1971). J. Lysmer, T. Uduka, F. C. Tsai, B. H. Seed and B. H. Flush, A computer program for approximate 3-D analysis of soil-structure interaction problems. EERC 75-30, Berkeley, University of California (1975). K. A. Chopra and J. A. Gutierrez, Earthquake response analysis of multistory buildings including foundation interaction. Znt. J. Earthq. Engng Struct. Dyn. 3 (1974). B. H. Seed, J. Lysmer and R. Hwang, Soil-structure interaction analyses for seismic response. J. Geotech. Enww Div. ASCE 101 (Mav 1975). N.C.-Tsai, M. S. Niehoff and A. H. Hadjian, The use of frequency-independent soil-structure interaction parameters. Nucl. Engng Des. 31 (1974). S. R. Kao, Seismic analysis of structures with rectangular bases on porous elastic soil medium. D.Sc. dissertation, The George Washington University (1978). H. Brandt, A study of the speed of sound in porous granular media. J. Appl. Mech. Trans. ASME 22, 479-486 (1955). M. A. Biot, General theory of three-dimensional con- solidation. J. Appl. Phys. 12 (1941). M. A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. J. Acoust. Sot. Am. 28 (March 1956). K. Ishihara, Propagation of compressional waves in a saturated soil. Proc. Znt. Symp. an Wave Propa- gation and Dynamic Properties of Earth Materials, University of New Mexico Press, New Mexico (1968). S. Saylan, Seismic analysis of three-dimensional soil- structure interaction system with a rectangular base. D.Sc. dissertation, The George Washington Univer- sity (1982). D. D. Barkan, Dynamics of Bases and Foundations. McGraw-Hill, New York (1962). W. S. Tseng, Soil-structure interaction analysis using fast Fourier transform method. The Theoretical Man- ual for Computer Program. CE 933, Bechtel Power Corp., San Francisco, California (1974). T. G. Toridis. K. Khozeimeh and 0. A. Fettahlioglu, Dynamic response of structures subjected to earthquake excitation. Proc. Second Iranian Congress of Civil Engineering. Pahlavi University, Shiraz. Iran (May 1976).