A simple soil±structure interaction model

Applied Mathematical Modelling 24 (2000) 607±635

www.elsevier.nl/locate/apm

A simple soil±structure interaction model Suleyman Kocak a

a,*

, Yalcin Mengi

b

Department of Civil Engineering, Faculty of Engineering and Architecture, Cukurova University, Adana 01330, Turkey b Department of Engineering Sciences, Middle East Technical University, Ankara 06531, Turkey Received 8 October 1998; received in revised form 13 December 1999; accepted 22 December 1999

Abstract A simple three-dimensional soil±structure interaction (SSI) model is proposed. First, a model is developed for a layered soil medium. In that model, the layered soil medium is divided into thin layers and each thin layer is represented by a parametric model. The parameters of this model are determined, in terms of the thickness and elastic properties of the sublayer, by matching, in frequency±wave number space, the actual dynamic sti€ness matrices of the sublayer when the sublayer is thin and subjected to plane strain and out-of-plane deformations with those predicted by the parametric model developed in this study. Then, by adding the structure to soil model a three-dimensional ®nite element model is established for the soil±structure system. For the ¯oors and footings of the structure, rigid diaphragm model is employed. Based on the proposed model, a general computer software is developed. Though the model accommodates both the static and dynamic interaction e€ects, the program is developed presently for static case only and will be extended to dynamic case in a future study. To assess the proposed SSI model, the model is applied to four examples, which are chosen to be static so that they can be analyzed by the developed program. The results are compared with those obtained by other methods. It is found that the proposed model can be used reliably in SSI analysis, and accommodates not only the interaction between soil and structure; but, also the interaction between footings. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Layered soil medium; Parametric model; Dynamic sti€ness matrix; Frequency; Wave number; Rigid diaphragm; Soil±structure interaction; Footing±footing interaction

1. Introduction As is well known, there are two main methods which are being used in the soil±structure interaction (SSI) analysis, namely, substructure and direct methods [1±7]. Also it is known that, due to unbounded nature of soil medium, the computational size of these methods is very large. For this reason it is important to establish some simple SSI models which reduce the computational cost of analysis; but at the same time, accommodate the essential features of interaction. One of such models is cone model [8±10]. In that model, the e€ects of waves in soil are simulated by modelling the soil by elastic conic bars. In this study, a new simple model is proposed for three-dimensional SSI analysis. In this model, it is assumed that the soil has a layered structure and is elastic. Dissipative forces in soil are taken *

Corresponding author. E-mail address: [email protected] (S. Kocak).

0307-904X/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 0 0 ) 0 0 0 0 6 - 8

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into account through the use of hysteretic damping. If desired, nonlinear e€ects in soil may be accounted for by employing equivalent linear method [4]. In the proposed model, the layered soil medium is subdivided into thin layers and each sublayer is represented by a parametric model. Parametric model is composed of some distributed springs and ®ctitious horizontal elements connecting the upper and lower ends of the springs. For vertical motion of the soil, distributed springs are vertical and ®ctitious horizontal element is a membrane. On the other hand, for horizontal motion of soil, springs are horizontal and the ®ctitious horizontal element is a plate under plane state of stress. The parametric model thus developed contains three parameters for vertical case, which are spring constant, membrane force and a mass parameter; and four parameters for horizontal case, which are spring constant, two parameters belonging to ®ctitious plate and a mass parameter. These parameters are related to the thickness and elastic properties of the thin sublayer by matching, in frequency±wave number space, the dynamic sti€ness matrices of the sublayer determined from the model equations and the exact theory. The actual dynamic sti€ness matrices of a thin layer are taken from [11]. Three-dimensional SSI model is obtained by adding the structure to the soil model discussed above and using the ®nite element method. It is assumed that structure is composed of vertical columns, horizontal beams, ¯oors and footings. For the ¯oors and footings of the structure, the rigid diaphragm model is used [12]. The use of rigid diaphragm model for footings facilitates the connection of the structure to soil. Three-dimensional model thus developed accommodates both soil±structure and footing±footing interactions. The structure might have single or continuous or mat foundation(s). The proposed model can be used in static or dynamic or earthquake interaction analysis. In the case of earthquake analysis, it is assumed that seismic environment is generated by vertically propagating waves and the layered soil medium rests on a rigid bedrock, which, in turn, implies that rigid bedrock motion is known. In fact, if the control point is in the site of structure the rigid bedrock motion can be determined through the use of deconvolution. On the other hand, if the control point is at outcrop the rigid bedrock motion would be given by control point motion. The three-dimensional SSI model discussed above has a form suitable for developing a general purpose computer software. In fact such a computer software is developed in this study in FORTRAN 77 for static interaction analysis. The extension of the program to dynamic case will be done in a future study. The proposed three-dimensional interaction model is assessed by applying it to four example problems. These problems are chosen to be static so that they can be analyzed by the developed computer software. In the example problems: 1. a soil layer subjected to a line load, 2. impedance coecients of a strip footing, 3. soil reaction distribution for a rigid strip footing which is under the in¯uence of a line load, and 4. three-dimensional SSI analysis of a one-storey building on a soil layer are considered. Comparisons of the results obtained from the model with those from other methods indicate that the model can be used reliably in static SSI analysis. The study is organized as follows. In Section 2, we de®ne, in general terms, the SSI problem to be modelled. In Section 3, we propose a model for a layered soil medium. In Section 4, we present a procedure for determining the model parameters. Section 5 contains a ®nite element model for three-dimensional SSI analysis, which is based on the proposed soil model. The connection of the structure to soil is also discussed in this section. Example problems are given in Section 6. In the ®nal section, Section 7, we state some conclusions.

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2. General de®nition of the problem We consider the soil±structure system shown in Fig. 1. It is assumed that the structure is composed of vertical columns, horizontal beams, ¯oors and footings. The structure might have single or continuous or mat foundation(s). The soil layer, which is assumed to have a layered structure, rests on a rigid bedrock. To simplify the interaction analysis, the soil material is taken as a linear elastic material and dissipative forces in soil are accounted for through the use of

Fig. 1. Soil±structure system (a) system (b) interaction forces.

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hysteretic damping. The nonlinear e€ects in soil, if desired, may be included approximately in the analyses by using linear equivalent method [4]. The soil±structure system shown in Fig. 1 is referred to an xi -rectangular …i ˆ 1 ÿ 3† frame in which the (x1 ±x2 ) plane is parallel to the plane of layering. The structure is under the in¯uence of some static and dynamic forces and the rigid bedrock is subjected to an earthquake ground excitation. In the case of earthquake analysis, it is assumed that seismic environment is generated by vertically propagating waves. This assumption facilitates the determination of the earthquake displacement ug at the rigid bedrock (see Fig. 1). In fact, if the control point is at outcrop, the rigid bedrock motion is equal to control point displacement. On the other hand, if the control point is in the site of the structure, the rigid bedrock motion can be determined in terms of control point motion by using deconvolution. For example, for a single elastic, homogeneous soil layer of depth d, the deconvolution formula would be uFg ˆ

uFc cos ad

…1†

which relates the rigid bedrock displacement ug to the control point displacement uc in Fourier transform space. Here, the superscript F designates Fourier transform; a ˆ x=c the wave number in vertical direction, where x is the frequency, and c is given by 8 p < cs ˆ l=q …shear wave velocity† when the vertical waves are S waves; cˆ  : c ˆ p …k ‡ 2l†=q …dilatational wave velocity† when the vertical waves are P waves; p where l is the shear modulus, k …ˆ 2ml=…1 ÿ 2m†† the LameÕs constant, m PoissonÕs ratio, and q is the mass density. It may be noted that when the vertical waves generating the earthquake motion are S waves, then the base displacement ug would be horizontal; on the other hand, when they are P waves, ug would be vertical. The dissipative forces in soil can be taken into account by making the replacement, in Fourier transform space, l ! l…1 ‡ 2izs †;

…2†

where zs is the hysteretic damping and i is the imaginary number. We now start developing the model for the SSI problem stated above. 3. The soil model For developing this model, the soil part of the soil±structure system in Fig. 1 is divided into thin layers as shown in Fig. 2. In the ®gure, the number of sublayers are designated by N; sublayers and interfaces are numbered downwards; the depth, mass density, shear modulus and Poisson's ratio of the ith sublayer are designated respectively by hi ; qi ; li and mi and the layered soil medium is referred to an xi -rectangular …i ˆ 1 ÿ 3† frame in which the (x1 ±x2 ) plane is parallel to the plane of layering and x3 axis is directed downwards. In the proposed model, each thin layer is represented by distributed springs, and two horizontal ®ctitious elements having the same properties which connect the upper and lower ends of the springs. For vertical motion of the soil, distributed springs are vertical and the ®ctitious horizontal element is a membrane (see Fig. 3). Here, one can obtain the Winkler model [13] by ignoring the ®ctitious horizontal element, implying that we disregard the interaction between neighboring springs. The unit cell shown in the ®gure represents the ith thin layer and contains three parameters, which are spring constant Ki (de®ned per unit horizontal area), tension in the

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Fig. 2. Representation of soil medium in terms of thin layers.

Fig. 3. The model for vertical case.

membrane Ci (de®ned per unit length) and mass parameter mi (de®ned per unit horizontal area) and ui3 and F3i designate, for the ith interface, the vertical displacement and force (per unit area), respectively. The model proposed for horizontal motion of the soil is shown in Fig. 4. In this case distributed springs are horizontal (in x1 and x2 directions) and ®ctitious horizontal element is a plate under plane state of stress. To simplify the drawing only the springs in x1 direction are shown in the ®gure. The unit cell representing the ith thin layer in this case contains four parameters, which are

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Fig. 4. The model for horizontal case.

spring constant Ki (de®ned per unit horizontal area), two parameters belonging to ®ctitious plate Ci , Si (de®ned per unit length) and mass parameter mi (de®ned per unit horizontal area); and ui ˆ …ui1 ; ui2 † and F i ˆ …F1i ; F2i † designate, for the ith interface, the horizontal displacement and forces (per unit area), respectively. Here, we note that the parameters (Ki , Ci ) in this case are di€erent from those of vertical case. Here, it should be noted that for the proposed model vertical and horizontal motions are decoupled, which decreases the computational load of the model. However, adding superstructure on the top of the soil model couples the vertical and horizontal motions of soil, due to use of rigid diaphragm model for structure foundation. This will be explained in more detail in Section 5. We now develop the equations of a unit cell of the soil model. 3.1. Equations of unit cell for vertical resistance of soil model A unit cell of the model for vertical soil resistance is shown in Fig. 5, where K, C, m are the parameters of that cell. The upper and lower ends of the cell are numbered, respectively, as 1 and

Fig. 5. A unit cell for vertical case.

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613

2 in the ®gure. The vertical displacement and force of the end i …i ˆ 1; 2† are designated by …u13 ; F31 † and …u23 ; F32 †, respectively. When we consider the free-body diagram of an in®nitesimal membrane element in Fig. 5 and write equilibrium equations (in the sense of D'Alembert principle) we obtain 2 13 2 32 1 3 2 13 2 13 1 ÿ1 u3 u3 u3 F3 4 5 ˆ K4 54 5 ÿ Cr2 4 5 ‡ m4 5 …3† ÿ1 1 F32 u23 u23 u23 which relates the unit cell forces to unit cell displacements. Here dot denotes di€erentiation with respect to time t and r2 is Laplacian operator de®ned by r2 ˆ

o2 o2 ‡ : ox21 ox22

…4†

Unit cell equation, Eq. (3), may be written in Fourier transform space as 2 13 2 32 1 3 2 13 u3 u3 K ÿ mx2 ÿK F3 4 5ˆ4 54 5 ÿ Cr2 4 5; F32 u23 u23 ÿK K ÿ mx2

…5†

where F3i and ui3 …i ˆ 1; 2† stand to Fourier transforms of force and displacements. To simplify the notation, superscript F is not used in Eq. (5) and will not be used hereafter. The hysteretic damping can be introduced into the unit cell equations by making the replacement K ! K…1 ‡ 2izs †

…6†

in Eq. (5). 3.2. Equations of unit cell for horizontal resistance of soil model A unit cell of the model for this case is shown in Fig. 6, where K, C, S and m are the parameters of the cell. In that ®gure (u1 , F1 ) and (u2 , F2 ) denote, respectively, the displacements and forces of the upper and lower ends of the cell. Here, we use the notation ui ˆ …ui1 ; ui2 †;

F i ˆ …F1i ; F2i †

…i ˆ 1; 2†

…7†

for unit cell displacement and forces. It may be noted that, in Eq. (7), superscript designates the end of unit cell and subscript designates the direction of displacement or force.

Fig. 6. A unit cell for horizontal case.

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The lower and upper ends of the distributed horizontal springs are connected by ®ctitious plates which are under plane state of stress. When we consider the free-body diagram of in®nitesimal plate element in Fig. 6 and write equilibrium equations (in the sense of D'Alembert principle) we obtain 2 13 2 32 1 3 2 13 2 13 2 13 1 ÿ1 ui ui ui Fi D C 4 5 ˆ K4 54 5 ÿ r2 4 5 ÿ Soi 4 5 ‡ m4 5 …i ˆ 1; 2† …8† 2 Fi2 ÿ1 1 u2i u2i u2i D2 which relates horizontal forces and displacements of unit cell. Here, oi ˆ o=oXi (partial derivative with respect to xi ) and Dk is dilatation at kth end of unit cell de®ned by Dk ˆ o1 uk1 ‡ o2 uk2

…k ˆ 1; 2†

…9†

An important point is now in order. For the horizontal resistance of the soil model, we assume that the ®ctitious horizontal element connecting the springs is a plate which is under the in¯uence of plane state of stress. Thus, the deformation of this plate would involve two independent elastic constants, for example elasticity modulus and PoissonÕs ratio. This explains why Eq. (8) contains two independent parameters C and S, of course, in addition to the spring constant K and mass parameter m. Unit cell equation, Eq. (8), may be written in Fourier transform space as 2 13 2 32 1 3 2 13 2 13 ui ui ÿK Fi K ÿ mx2 D C 4 5ˆ4 54 5 ÿ r2 4 5 ÿ Soi 4 5 …i ˆ 1; 2†: …10† 2 2 Fi2 u2i u2i ÿK K ÿ mx2 D The hysteretic damping ratio can be introduced into the unit cell equations by using the correspondence principle in Eq. (6). 4. Determination of model parameters Unit cell Eqs. (5) and (10) contain, respectively, three and four parameters, which are for vertical case : Kv ; Cv ; mv ; for horizontal case : Kh ; Ch ; S; mh ;

…11†

where the subscripts v and h are used, respectively, for vertical and horizontal cases. In this section, the parameters of the proposed model will be determined, in terms of the thickness h, mass density q, and elastic properties l and m of the sublayer (see Fig. 7) by matching, in frequency±wave number space, the dynamic sti€ness matrices of a thin sublayer (subjected to plane strain and out-of-plane deformations), determined from the model equations and exact theory. Here it may be noted that plane strain and out-of-plane deformations in soil are produced, respectively, by P, SV and SH waves. The actual dynamic sti€ness equations can be found in [11] for a thin sublayer undergoing plane strain and out-of-plane deformations. These equations for plane strain and out-of-plane cases can be written in the common form F ˆ D U;

…12†

where F ; U and D are, respectively, force, displacement vectors and dynamic sti€ness matrix which are de®ned by, for plane strain and out-of-plane cases,

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Fig. 7. A typical thin layer.

for plane strain case:  F ˆ F11 iF31 F12

iF32

T

 u ˆ u11

;

iu13

u21

iu23

T

…13†

and D ˆ G ‡ Bk ‡ Ak 2 ÿ x2 M;

…14†

where k is being wave number in x1 direction (see Fig. 7) and 3 2 l 0 ÿl 0 A 0 ÿA 7 16 7 6 0 Gˆ 6 7; h 4 ÿl 0 l 0 5

2

0

ÿA

0

E

0

0

P

P

0

0

ÿE

16 6 E Bˆ 6 24 0 ÿP

0

ÿP

0

2A 0 A 6 h 6 0 2l 0 Aˆ 6 64 A 0 2A 0 l 0

M ˆ

2

qh 6 60 6 6 41 0

3

0 7 7 7; ÿE 5

2

2

A

0

3

l 7 7 7; 0 5 2l 0

3

0

1

2

0

0

2

17 7 7; 05

1

0

2

…15†

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in which Aˆ

2…1 ÿ m† l; 1 ÿ 2m

Pˆ

1 l; 1 ÿ 2m

for out-of-plane case:  T  F ˆ F21 F22 u ˆ u12

u22

Eˆ

4m ÿ 1 l; 1 ÿ 2m

…16†

T

…17†

and D ˆ G ‡ Ak 2 ÿ x2 M; in which

 l 1 Gˆ h ÿ1

…18†

 ÿ1 ; 1

 lh 2 Aˆ 6 1

 1 ; 2

 qh 2 Mˆ 6 1

 1 : 2

…19†

In Eq. (14), B de®nes the coupling between horizontal and vertical motion of the sublayer for plane strain case. In this study, it is assumed that the free-®eld system is subjected to vertical waves and thus, the coupling described above can be assumed to be small. In view of this assumption, Eq. (14) can be written as D ˆ G ‡ Ak 2 ÿ x2 M

…20†

when B is ignored. To obtain the sti€ness equations of the proposed model for plane strain and out-of-plane cases, we ®rst note that model displacements are dependent only on x1 and t, i.e., uik ˆ uik …x1 ; t† …i ˆ 1; 2; k ˆ 1 ÿ 3†:

…21†

When the unit cell equations, Eqs. (3) and (8), are written in Fourier transform space (with respect to time t and x1 ) in view of Eq. (21), one can obtain the dynamic sti€ness matrix of the model again in the form of Eq. (20), but, in this case G, A, M matrices are de®ned by for plane strain case: 2

Kh 6 0 Gˆ6 4 ÿKh 0

0 Kv 0 ÿ Kv

2

…Ch =2† ‡ S 6 0 Aˆ6 4 0 0 2

mh 6 0 M ˆ6 4 0 0

0 mh 0 0

ÿ Kh 0 Kh 0 0 Cv 0 0

0 0 mh 0

3 0 ÿ Kv 7 7; 0 5 Kv

0 0 …Ch =2† ‡ S 0 3 0 0 7 7; 0 5 mh

3 0 0 7 7; 0 5 Cv

…22†

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for out-of-plane case:   ÿ Kh Kh ; Gˆ ÿKh Kh



Ch =2 Aˆ 0

 0 ; Ch =2



mh Mˆ 0

 0 : mh

617

…23†

The model parameters may be obtained by matching the G, A and M matrices given in Eqs. (15) and (19) and Eqs. (22) and (23) which belong, respectively, to the exact theory and model. If o€-diagonal terms in A and M matrices in Eqs. (15) and (19) are lumped on the diagonal terms, the model parameters may be found as l lh qh ; mh ˆ ; Kh ˆ ; Ch ˆ lh; S ˆ h 2…1 ÿ 2m† 2 2…1 ÿ m†l lh qh Kv ˆ ; Cv ˆ ; mv ˆ ; …1 ÿ 2m†h 2 2

…24†

where mh and mv represent the mass values of the thin layer lumped at its upper and lower ends. 5. A ®nite element model for three-dimensional SSI analysis In this section, the structure will be added to soil foundation through the use of ®nite element method and the SSI model thus developed will be described brie¯y. In the development of this SSI model we do the following assumptions: 1. For the soil we use the model proposed in Section 3. 2. The structure is composed of vertical columns, horizontal beams, ¯oors and footings. The structure might have single or continuous or mat foundation(s). 3. We use rigid diaphragm model for ¯oors and footings [12]. The use of rigid diaphragm model for footings facilitates the connection of structure to soil foundation. 4. The foundation of the structure is perfectly bonded to soil foundation. The ®nite element model is based on writing sti€ness and mass matrices for soil, beam, column and plate elements and combining them by coding technique, in view of rigid diaphragm assumption, so that the equilibrium equations (in the sense of DÕAlembert principle) and compatibility conditions at the nodes are satis®ed. A special care will be given to the connection of structure to soil foundation. As the equations of beam, column and plate elements are well established and the use of rigid diaphragm model in the formulation of three-dimensional response of structures is well known in literature, in what follows we develop only, based on the proposed model in Section 3, the sti€ness and mass matrices for soil elements and emphasize the procedure by which the soil foundation and structure can be connected. Since the equations of the proposed model are uncoupled for vertical and horizontal cases, we present the sti€ness and mass matrices of soil elements for the aforementioned cases separately. However, it should be noted that when the soil elements are combined with the structure the vertical and horizontal degrees of freedoms (DOFs) of soil elements would become coupled due to the presence of the structure foundation which is modelled by using rigid diaphragm assumption. 5.1. The soil element for vertical case A three-dimensional element associated with a unit cell of the soil model undergoing vertical displacements is shown in Fig. 8. As the soil model eliminates the dimension in vertical direction, the element in the ®gure should be taken as dimensionless in that direction. The ®ctitious vertical

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Fig. 8. Three-dimensional soil element (vertical case).

dimension is used in the ®gure to di€erentiate the upper and lower ends of the cell. The horizontal faces of the elements are taken as quadrilateral and the corners of the element are numbered as shown in the ®gure. In the formulation we use bilinear interpolation functions for both geometry and vertical displacement distributions …ui3 …x1 ; x2 † …i ˆ 1; 2†† over the upper and lower faces of the element. When we use virtual work principle for the unit cell equations given in Eq. (5) and follow the procedure used in conventional ®nite element formulation, we obtain the dynamic sti€ness matrix as, in Fourier transform space,   ÿ K~ K~ ‡ C~ ÿ x2 M …25† Dˆ K~ ‡ C~ ÿ x2 M ÿK~ for the three-dimensional soil element. In Eq. (25) the ®rst and second rows represent, respectively, the force DOFs of the upper and lower faces, and the ®rst and second columns represent the displacement DOFs of the upper and lower faces. Consequently, the o€-diagonal matrices describe the coupling between the upper and lower faces of the element. The matrices ~ C~ and M in Eq. (25) are de®ned as K; Z ~ K ˆ Kv N N T dA; A " # Z T o N 1 …26† dA; C~ ˆ Cv ‰ o1 N o2 N Š o2 N T A Z M ˆ mv N N T dA; A

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619

where A is the area of the horizontal face of the element and N is a vector of the dimension …4  1† containing bilinear interpolation functions. The elastic and dynamic e€ects are both included in the dynamic sti€ness matrix in Eq. (25). The hysteretic damping can be introduced into the model through the use of Eq. (6). 5.2. The soil element for horizontal case A three-dimensional element associated with a unit cell of the model undergoing horizontal displacements is shown in Fig. 9. As stated for vertical case, ®ctitious vertical dimension is used to di€erentiate upper and lower ends of the cell. The horizontal faces of the element are taken as quadrilateral and the corners of the element are numbered as shown in the ®gure. As in vertical case, here again we use bilinear interpolation functions for both geometry and horizontal displacement distributions …ui1 …x1 ; x2 †; ui2 …x1 ; x2 †; …i ˆ 1; 2†) over the upper and lower faces of the element. When we use virtual work principle for the unit cell equations given in Eq. (10) and follow the procedure used in conventional ®nite element formulation, we obtain the dynamic sti€ness matrix as, in Fourier transform space, 2 3 A1 ‡ A2 ÿB 5 Dˆ4 …27† T A1 ‡ A2 ÿB for the three-dimensional soil element. In Eq. (27) the ®rst and second rows represent, respectively, the force DOFs of the upper and lower faces, and the ®rst and second columns represent the displacement DOFs of the upper and lower faces. The matrices A1 ; A2 and D in Eq. (27) are de®ned as

Fig. 9. Three-dimensional soil element (horizontal case).

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2

K~ ‡ …Ch =2†C~11 6 ‡…Ch =2†C~22 ‡ S C~11 6 A1 ˆ 6 4 S C~12 

ÿx2 M A2 ˆ 0 

K~ BˆB ˆ 0 T

S C~21 K~ ‡ …Ch =2†C~11 ‡…Ch =2†C~22 ‡ S C~11

3 7 7 7; 5

…28†

 0 ; ÿ x2 M  0 ; K~

in which Z ~ K ˆ Kh NN T dA; Z A C~ij ˆ oi Noj N T dA A Z M ˆ mh N N T dA:

…i; j ˆ 1; 2†;

…29†

A

In Eq. (28), the ®rst row and column of the matrices represent the DOFs in x1 and the second row and column represent the DOFs in x2 direction. The hysteretic damping again can be introduced into the model through the use of Eq. (6). 5.3. The connection of the structure to soil In this section, the connection of the structure to soil foundation is discussed very brie¯y. In the SSI system considered in this study, the foundation of the structure is assumed to be perfectly bonded to soil foundation which implies that the structure and soil, at the points of contact area, would move together. To satisfy this continuity condition between the structure and soil foundation, the equations of soil elements should be modi®ed when they have nodes over the contact area. For vertical case, the modi®ed equations can be obtained by introducing rotational DOFs about x1 and x2 axes, in addition to vertical DOF, for the nodes lying over contact area. On the other hand, for horizontal case, the modi®ed equations can be derived by imposing rigid diaphragm constraints, for horizontal DOFs, to the nodes lying over the contact area. To explain the connection more explicitly, we refer to Fig. 10 showing the ®nite element mesh of a soil foundation referred to an x1 x2 x3 global coordinate system. In the ®gure, the shaded regions represent the contact areas of the footings of the superstructure; M represents the master point of rigid diaphragm model used for each contact area; D1 ; D2 and Dh represent, respectively, the displacements in x1 , x2 (horizontal) directions and rotation about x3 (vertical) axis. The horizontal displacements (d1 ; d2 ) of an arbitrary point in contact area are related to master point displacements (D1 ; D2 ; Dh ) by, in view of rigid diaphragm assumption, d1 ˆ D1 ÿ …x2 ÿ xM 2 †Dh ; d2 ˆ D2 ‡ …x1 ÿ xM 1 †Dh ;

…30†

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Fig. 10. Modelling the footings by rigid diaphragms.

M where (xM 1 ; x2 ) and (x1 ; x2 ) denote, respectively, the horizontal coordinates of the master point M and arbitrary point. We now discuss the connection procedure for horizontal and vertical cases separately. (a) Horizontal case: Horizontal displacements of the nodes lying over the contact area are related to the horizontal displacements and rotation (about vertical axis) of the master point by Eqs. (30). By using these constraints in Eq. (27), the modi®ed equations can be obtained, for horizontal case, for the soil elements having nodes over the contact area. It should be noted that these modi®cations should be done not only for the element A which lies completely in the contact area; but, also for the neighboring elements B and C which have nodes in that area (see Fig. 10). (b) Vertical case: As stated previously, the connection for this case could be done by introducing rotational DOFs about x1 and x2 axes for the nodes lying over the contact area (see Fig. 10), which dictates that we modify Eq. (25) for the elements having nodes on that area. To simplify the formulation and the computer implementation, this modi®cation is done in the present study not only for the aforementioned elements; but, also for the elements outside the contact area. The modi®ed form of the equations for horizontal and vertical cases are not given here not to lengthen the paper.

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5.4. Information about the computer program A computer program is developed, based on the proposed model, for three-dimensional SSI analysis. The program is written in FORTRAN 77. Though the proposed model accommodates both static and dynamic e€ects, presently the computer program is developed for static interaction analysis and the extension of the computer program to dynamic case will be done in a future study. The system sti€ness matrix is formed as a skyline matrix which is stored in out-of-core by blocks and the solution is obtained by LDU decomposition algorithm [14]. The numbering of the DOFs of the nodes is done automatically by the program so that the computational load of the solution of the system equations is minimum. The program has some modules working sequentially. The input ®le has some blocks containing information about system, coordinates, layers, etc., similar to those of SAP90 [15]. The output ®les have a structure easy to interpret. In the program, only the upper surface of soil is discretized. The ®nite element meshes of the layers underneath are generated by projecting the nodes of the top surface on horizontal layers and numbered by the program automatically. With this generation facility the ®nite element mesh for the whole domain is assembled by the program, copying the ®nite element mesh of the top soil layer like slices of meshes for as many of sublayers as de®ned. Therefore, user do not have to mesh the layers underneath the top of soil resulting minimum data preparation e€ort. Once the mesh on the top of soil is de®ned the other nodes of the sublayers are generated by the program provided that the number of layers is de®ned by the user. However, for the de®nition of superstructure the complete ®nite element mesh has to be entered by the user. The computer software can be obtained, together with its manual, from the authors upon request.

6. Example problems In this section, with the object of assessing the model we analyze some example problems, and compare the results obtained from the model with those from other methods. The example problems considered in this section are: 1. a soil layer subjected to a line load, 2. impedance coecients of a strip footing, 3. soil reaction distribution for a rigid strip footing which is under the in¯uence of a line load, 4. three-dimensional SSI analysis of a one-storey building on a soil layer. Problems 1, 2 and 3 are two-dimensional and involve only the soil foundation without superstructure on it. As the mesh of the top surface in these problems is two-dimensional and has a simple form, to save space, it is not given in the paper. However, for Problem 4, which is threedimensional and contains a superstructure, all the details regarding the mesh used in the analysis are given. As mentioned before, program discretizes the horizontal layers of soil model automatically. 6.1. Problem 1: a soil layer subjected to a line load We consider an elastic layer resting on a rigid bedrock and subjected to a vertical or horizontal line load. The vertical loading case is shown in Fig. 11. The layer has a depth of H and is referred to an x1 x2 x3 global coordinate system. The vertical line load P extends to in®nity in

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Fig. 11. An elastic layer subjected to a vertical line load.

out-of-plane direction (along x2 axis). In horizontal loading case, the line load P acts in x1 direction. The shear modulus and Poisson's ratio of the layer are designated by l and m in the ®gure. The analyses of the problems stated above are performed by the computer program developed in this study and the results are compared with the ones obtained from boundary element method (BEM) [16±20], in terms of nondimensional (ND) variables, in Figs. 12 and 13 (for the Poisson ratio m ˆ 0.2). It may be noted that BEM is a semi-analytical method whose accuracy is much better than FEM, in particular, for dynamic problems and for problems with the solution domain extending to in®nity in some directions. The accuracy of the boundary element results in this problem is insured by increasing the number of boundary elements employed in the analysis until the convergence of the results is achieved. In the model analyses, the model parameters are determined through the use of Eq. (24). In Figs. 12 and 13, ND vertical and horizontal displacements (u3 ) and …u1 † are de®ned by …u3 ; u1 † ˆ

l …u3 ; u1 † P

…31†

and x1 ˆ

x1 a

in which a represents a characteristic length. The results are obtained when H ˆa and are shown in the ®gures only for x1 6 0 due to the symmetry of the problem.

…32†

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Fig. 12. Vertical displacement distribution on soil surface (vertical line load).

Fig. 13. Horizontal displacement distribution on soil surface (horizontal line load).

The model prediction for the vertical displacement distribution of soil surface for vertical line load problem is obtained by taking the number of sublayers as NS ˆ 1; 2; 4; 10 and compared with the BEM results in Fig. 12. From the ®gure it may be observed that when the number of sublayers (NS) increases the model prediction approaches that of BEM. Fig. 13 gives the same comparison for horizontal line load problem, which shows that the observation made for vertical case holds also for this case.

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6.2. Problem 2: impedance coecients of a strip footing We consider a rigid strip footing of width B perfectly bonded to an elastic layer of depth H (see Fig. 14). The layer rests on a rigid bedrock. The rigid strip footing (which extends to in®nity in out-of-plane direction) is subjected to static horizontal and vertical forces F1 ; F3 and moment M at its center as shown in the ®gure. The strip footing system is referred to an x1 ±x3 coordinate system in which x1 axis is horizontal and x3 is directed downwards. The thickness of the footing will be ignored in the analysis. The static impedance relation for the rigid strip footing would be in the form 2 3 2 32 3 0 SHM SHH D1 F1 4 F3 5 ˆ 4 0 SVV 0 54 D3 5; …33† SMH M 0 SMM a where SHH , SVV and SMM represent, respectively, the static horizontal, vertical and rocking impedance coecients and SHM ˆ SMH designates the coupling impedance coecient between the horizontal and rocking motions of the footing; a, D1 and D3 are, respectively, the rotation, and the horizontal and vertical translations of the rigid footing. The coupling impedance coecient SHM will be disregarded in the analysis as its value is small compared to those of the other impedance coecients. The vertical and horizontal impedance coecients in Eq. (33) are computed by the proposed model by subdividing the soil layer into thin layers and the variations of these coecients with the layer depth are compared with those obtained from ``BEM'' in Figs. 15 and 16. In the model analyses (carried out by the computer program developed in this study) the rigidity of footing is accounted for by choosing large values for its bending and axial rigidities. In the ®gures, the ND depth is given by H …34† Hˆ B and ND impedance coecients are de®ned by   SHH SVV ; ; …35† …S HH ; S VV † ˆ l l where l is shear modulus of soil layer.

Fig. 14. Strip footing problem.

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Fig. 15. Variation of vertical impedance with depth.

Fig. 16. Variation of horizontal impedance with depth.

The model results are obtained when each layer of depth B is subdivided into NS ˆ 20 sublayers and parameters are determined through the use of Eq. (24) and PoissonÕs ratio is taken to be m ˆ 0:2. Figs. 15 and 16 show, respectively, the variation of vertical and horizontal impedance coecients with the depth of soil layer. From the ®gures one can see that model results are fairly close to those of BEM.

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6.3. Problem 3: soil response under a rigid strip footing As in Problem 2, here we consider a rigid strip footing of width B which lies on the top surface of an elastic layer of depth H (see Fig. 17). The base of the layer is rigid. The system is referred to a rectangular xi -frame in which the (x1 ±x2 ) plane coincides with the top surface of the layer and x3 axis is directed downwards. The strip footing is subjected, along its center line, a vertical uniform line load P as shown in the ®gure. In the analysis, we ignore the thickness of the footing. The object in this problem is to determine the soil reaction underneath the footing by using the proposed model and compare it with that obtained by the BEM. We do the computations in terms of ND variables by choosing H ˆ4 B and PoissonÕs ratio m ˆ 0:2. The model parameters are determined through the use of Eqs. (24). In the model analyses (carried out by the computer program developed in this study) the rigidity of footing is accounted for by assigning a large value for its bending rigidity. The soil reaction distribution is obtained by the model through the subdivision of the soil layer into Hˆ

NS ˆ 1; 2; 8; 32; 64

…36†

sublayers and is compared with BEM results in Fig. 18. In the ®gure, ND horizontal coordinate x1 is given by x1 x1 ˆ …37† B

Fig. 17. Rigid strip footing.

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Fig. 18. Soil response distribution under strip footing.

and ND soil reaction p is de®ned by pB …38† P in which p is the soil pressure acting on the footing. The ®gure shows that as the number of subdivisions NS, used in approximating the layer response, increases the model prediction for the soil reaction distribution approaches that obtained by BEM, and that the model is capable to describe the singularities in the soil pressure along the edges of the rigid strip footing. pˆ

6.4. Problem 4: one-storey building supported by soil layer We consider a one-storey building having the dimensions, and material and sectional properties shown in Fig. 19. The one-storey building has an elastic ¯oor surrounded with beams, which stands, at its corners on columns supported by rigid square footings. The soil layer overlies a rigid bedrock. The soil±structure system is referred to a global X1 X2 X3 coordinate system as shown in the ®gure, where X1 X2 coincides with the top surface of the layer and X3 axis is directed upwards. The object in this problem is to analyze the soil±structure system, described above, under the in¯uence of some loadings by the developed computer program (which is based on the proposed model) and compare the results with those of SAP90 [15]. SAP90 is a widely used structural analysis program and it can accommodate discrete springs on any point of structure. Using this program one can model soil with discrete springs as in Winkler Model. For the purpose of comparing our program (based on the proposed model) with a software which is practically used by engineers, this problem is chosen. In the analysis by the proposed model, the soil layer is divided into NS ˆ 4 sublayers which implies that each thin layer has a depth of hˆ

H ˆ 0:5 m: NS

…39†

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Fig. 19. Soil±structure system considered in Problem 4.

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The model parameters can be obtained through the use of Eqs. (24), in view of soil properties given in Fig. 19 and Eq. (39), as Kv ˆ 53333:3 kN=m3 ; 3

Kh ˆ 20000 kN=m ;

Cv ˆ 2500 kN=m; Ch ˆ 5000 kN=m;

S ˆ 4166:67 kN=m:

…40†

In the analyses performed by the model, the ¯oor and footings of the system are modelled as rigid diaphragms. The network used in the analysis by the proposed model is shown in Fig. 20. In the analysis by SAP90, we put springs at the lower ends A, B, C and D of the columns (see Fig. 19) to describe the in¯uence of the soil on the response of the structure. Each of the column ends A, B, C and D have six springs associated with its three translational and three rotational DOFs. In the present analysis these spring constants are determined through the use of BEM by considering a square rigid footing of side 1 m overlying on an elastic layer of depth 2 m. Their computed values are: Kx ˆ Ky ˆ 37800 kN=m …translational springs in X1 and X2 directions †; Kz ˆ 28400 kN=m …translational spring in X3 direction†; ax ˆ ay ˆ 6200 kN m=rad …rotational springs about X1 and X2 axes†; az ˆ 4780 kN m=rad

…41†

…rotational spring about X3 axis†:

It may be noted that, in computing these spring constants, the interactions between the footings are ignored. In this problem, we analyze the soil±structure system under consideration for three di€erent loadings. They are: Loading 1: A 10 kN horizontal concentrated force in X1 direction is applied to the master point of the ¯oor which is chosen to be at the centroid of the ¯oor (joint number 527 in Fig. 20). Loading 2: A 100 kN vertical concentrated force is applied to the joint 442 in Fig. 20 (which acts on the column A in Fig. 19). Loading 3: A 100 kN vertical concentrated force is applied to the joint 462, and at the same time, a 10 kN horizontal concentrated force in X1 direction and a 7.5 kN m moment about X3 axis are applied to the master point of the ¯oor (joint number 527 in Fig. 20). 6.4.1. Loading 1 As stated previously, this loading involves a 10 kN horizontal concentrated force in X1 direction applied to the master point of the ¯oor (joint 527 in Fig. 20 which is at the centroid of the ¯oor). In this loading, we study only the rotations (about vertical axis, X3 ) of the footings produced by the aforementioned applied horizontal force. SAP90 results yield no rotations for the footings, which may be anticipated in view of the symmetry of the structure and of the fact that we ignore the interaction between the footings in SAP90 analysis. On the other hand, as the proposed model permits the footings to interact with each other, it predicts the rotations of the footings as sketched in Fig. 21 showing that the front (A, B) and rear (C, D) footings rotate in opposite directions of the amount h3 ˆ 0:2872  10ÿ5 rad:

…42†

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Fig. 20. Finite element network used in Problem 4.

631

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Fig. 21. Rotational distortion of footings.

To check the order of the amount of rotation in Eq. (42), we approximate the layer by a halfspace and consider the top surface on which four horizontal loads (in X1 direction) of 2.5 kN are acting at the points A, B, C, D (see Fig. 19). The rotation at one of these points can be obtained by computing, through the use of CerrutiÕs solution [21], the rotation at that point produced by the horizontal forces at the other points and superposing the results. The rotational distortion of the footings obtained using this procedure is h3 ˆ 0:2990  10ÿ5 rad

…43†

which is slightly higher than that in Eq. (42) (which was found by the model). This small di€erence may be expected in view of the fact that the value in Eq. (43) is obtained for a half-space while the supporting soil medium in our soil±structure system is a layer of thickness 2 m. 6.4.2. Loading 2 A 100 kN concentrated vertical force is applied at the joint 442 in Fig. 20 (on the column A). We analyze this system by the computer program developed in this study and SAP90 and compare the results in Tables 1 and 2, where E designates the relative error between SAP90 and model results de®ned by Eˆ

jVmodel ÿ VSAP j ; MaxjVSAP j

…44†

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Table 1

Column element forces (Loading 2) Element #

Node #

1

133 442 141 450 301 514 309 522

2 3 4

Moment (kN m)

Shear (kN)

Normal (kN)

SAP

Model

E (%)

SAP

Model

E (%)

SAP

Model

E (%)

)0.14 2.26 )1.07 )1.04 0.14 2.24 )1.07 )1.02

)0.36 1.58 )0.89 )0.68 0.20 1.70 )0.87 )0.67

10 30 8 16 2 24 9 16

)0.71

)0.41

42

)96.4

)97.4

1

0.71

0.52

27

)2.77

)1.93

1

)0.70

)0.63

10

)2.77

)1.93

1

0.70

0.51

27

1.96

1.39

1

Table 2

Beam element forces (Loading 2) Element #

Node #

SAP

Model

E (%)

1 8 9 16

442 450 514 522

2.25 2.26 )1.14 )1.14

1.57 1.69 )0.74 )0.75

30 25 18 17

where Vmodel and VSAP are element forces computed, respectively, by model and SAP90, and MaxjVSAP j is the absolutely maximum value of VSAP obtained by considering the whole system. A study of Tables 1 and 2 indicates that the results obtained by SAP90 and model do not di€er much, implying that the interaction between the footings may be ignored with respect to the element forces for the problem under consideration. 6.4.3. Loading 3 In this loading case, a 100 kN vertical concentrated force is applied to the joint 462, and at the same time, a 10 kN horizontal concentrated force in X1 direction and a 7.5 kN m moment about X3 axis are applied to the master point of the ¯oor (joint number 527 in Fig. 20). The comparison of some column and beam element forces obtained by the model and SAP90 are given in Tables 3 and 4. Studying the relative error E, we see that, for the reason stated for Loading 2, the two results do not di€er much for this loading too. Table 3

Column element forces (Loading 3) Element #

1 2 3 4

Node #

133 442 141 450 301 514 309 522

Moment (kN m)

Shear (kN)

SAP

Model

E (%)

)1.25 7.07 )7.59 )15.2 )2.15 )0.04 )4.05 )6.75

)0.08 8.38 )9.81 )14.8 )2.26 )0.35 )5.20 )6.60

8 9 15 3 1 2 8 1

SAP

Normal (kN)

Torsional moment (kN m)

Model

E (%)

SAP

Model

E (%)

SAP

Model

E (%)

1.94

2.77

11

)54.6

)55.4

1

0.12

0.10

16

)7.61

)8.20

8

)20.6

)20.0

1

0.12

0.10

16

)0.73

)0.64

1

)15.3

)15.4

1

0.12

0.10

16

)3.60

)3.93

4

1

0.12

0.10

16

)9.93

)9.26

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Table 4

Beam element forces (Loading 3) Element #

Node #

1 4 8 9 11 16 20 27

442 445 450 514 516 522 469 477

Moment (kN m)

Torsional moment (kN m)

SAP

Model

E (%)

SAP

Model

E (%)

)1.14 17.95 12.79 2.47 4.60 4.55 16.68 )4.14

)2.48 17.53 12.34 2.15 4.39 4.46 16.40 )4.04

7 2 2 2 1 1 2 1

5.85 )2.12 2.70 )1.69 )0.41 )2.0 2.31 )0.56

5.89 )2.16 2.71 )1.76 )0.45 )1.96 2.33 )0.58

1 1 0 1 1 1 1 1

7. Conclusions In the study, ®rst a model is proposed for layered soil foundations; then, based on this soil foundation model, a ®nite element model is proposed for the three-dimensional SSI analysis. Now, in what follows, we repeat some important points which are already stated in the previous sections and draw some conclusions from the ®ndings of the study: (a) In the study, the model parameters are obtained through the match of dynamic sti€ness matrices of a sublayer as determined from the exact theory and the model. Alternatively, it is also possible to determine the soil model parameters via optimization, which involves selection of a base problem and ®tting, through optimization, the model and exact responses for that base problem. This alternative approach may be the subject of a future study. (b) As noted in Problem 4 of Section 6, the model proposed in the study for SSI analysis accommodates not only the interaction between the structure and soil; but, also the interaction between the footings. This makes the proposed model general and suitable for the analysis of structures having multiple footings or a mat foundation, etc. (c) In the proposed model, we use the rigid diaphragm assumption for both the ¯oors and footings, each footing having a separate master point. The use of the rigid diaphragm assumption for footings facilitates the connection of the structure to the soil foundation. (d) The model accommodates the layered structure of the soil foundation through the representation of sublayers by unit cells each having di€erent model parameters. (e) The results of Loading 2 in Problem 4 of Section 6 indicate that the interaction between the footings may be ignored, with respect to the element forces, for the one-storey structure analyzed in that problem. We think that this result is due to the symmetry of the one-storey structure. We expect that the footing±footing interaction e€ects may become important for a general nonsymmetric structure, in particular, in dynamic analysis. We plan to study this aspect of interaction analysis in future. (f) As stated previously, SAP90 can be used in SSI analysis only when the structure has rigid footings, not continuous beam or mat foundations. Under these conditions, in order to use SAP90 in the interaction analysis, we ®rst determine the spring constants for each of the footings using ``BEM''; then, attaching these springs to the base levels of the columns, we analyze the structure. We note that this kind of analysis ignores the footing±footing interaction e€ects. In view of the arguments given above, we can summarize the shortcomings of using SAP90 in SSI analysis as: 1. it can be used only if the structure has rigid footings, 2. it requires a separate analysis by ``BEM'', and 3. it ignores the footing±footing interaction e€ects. The model proposed in this study eliminates all these shortcomings.

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(g) The model is proposed for the three-dimensional dynamic SSI analysis; but, the computer software, as stated above, is developed for static case. The program will be extended to dynamic case in a future study. References [1] [2] [3] [4] [5] [6] [4] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

J.P. Wolf, Dynamic Soil±Structure Interaction, Prentice-Hall, Englewood Cli€s, NJ, 1985. J.P. Wolf, Soil±Structure Interaction Analysis in Time Domain, Prentice-Hall, Englewood Cli€s, NJ, 1988. R.W. Clough, J. Penzien, Dynamics of Structures, second ed., McGraw-Hill, Tokyo, 1993. J. Lysmer, T. Udaka, C. Tsai, H.B. Seed, FLUSH: a computer program for approximate 3D dynamic analysis of soil±structure problems, Report of Earthquake Engineering Research Center, University of California, Berkeley, Report No. EERC75-30, 1975. S. Gupta, T.W. Lin, J. Penzien, C.S. Yeh, Hybrid modelling of soil±structure interaction, Report of Earthquake Engineering Research Center, University of California, Berkeley, Report No UCB/EERC-80/09, 1980. A. Gomez-Masso, J. Lysmer, J.-C. Chen, H.B. Seed, Soil structure interaction in di€erent seismic environments, Report of Earthquake Engineering Research Center, University of California, Berkeley, Report No UCB/EERC79/18, 1979. J.A. Gutierrez, Substructure method for earthquake analysis of structure-soil interaction, Report of Earthquake Engineering Research Center, University of California, Berkeley, Report No EERC 76-9, 1976. J.W. Meek, J.P. Wolf, Cone models for homogenous soil, part I, J. Geotech. Eng. 118 (1992) 667. J.W. Meek, J.P. Wolf, Cone models for homogenous soil, part II, J. Geotech. Eng. 118 (1992) 686. J.P. Wolf, W. Meek, Cone models for a soil layer on a ¯exible rock half-space, Earthquake Eng. Struct. Dyn. 22 (1993) 185. E. Kausel, J.M. Roesset, Sti€ness matrices for layered soils, Bull. Seismol. Soc. Am. 71 (1981) 1743. A. Ghali, A.M. Neville, Structural Analysis, Chapman and Hall, London, 1978. E. Winkler, Die Lehre von Der Elastizitaet und Festigkeit, Dominiqus, Prag, 1867. G. Dhatt, G. Touzot, G. Cantin, The Finite Element Method Displayed, Wiley, Canada, 1985. E.L. Wilson, A. Habibullah, SAP90 Structural Analysis Programs, Computers and Structures, Berkeley, CA, 1984. Y. Mengi, A.H. Tanrikulu, A.K. Tanrikulu, Boundary Element Method for Elastic Media: An Introduction, METU Publications, Ankara, 1994. P.K. Banerjee, The Boundary Element Method in Engineering, McGraw-Hill, London, 1994. G.D. Manolis, D.E. Beskos, Boundary Element Method in Elastodynamics, Unwin Hyman, London, 1987. A.A. Becker, The Boundary Element Method in Engineering: A Complete Course, McGraw-Hill, London, 1992. J. Trevelyan, Boundary Elements for Engineers: Theory and Applications, Computational Mechanics Publications, Southampton-Boston, 1994. Y.C. Fung, Foundations of Solid Mechanics, Prentice-Hall, Englewood Cli€s, NJ, 1965.