SEISMIC ANALYSIS OF SPHERICAL TANKS INCLUDING FLUID-STRUCTURE-SOIL INTERACTION

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1 3 th World Conferene on Earthquake Engineering Vanouver, B.C., Canada August -6, 2004 aper o. 84 SEISMIC AALYSIS OF SHERICAL TAKS ICLUDIG FLUID-STRUCTURE-SOIL ITERACTIO T.L. Karavasilis, D.C. Rizos 2, and D.L. Karabalis SUMMARY The aurate and effiient, from a pratial point of view, analysis of spherial tanks is the primary motivation for this work. The proposed methodology implements: (a) the Boundary Element Method (BEM) for the simulation of the soil-struture interation, (b) a disrete 2-DOF mehanial system for the approximate omputation of the impedanes of a massive, rigid, ring foundation, a usual feature of a spherial tank, and () a FEM simulation of the struture and fluid regions. The above modules are oupled via the pre- and post-proessing apabilities of the widely used general purpose program ASYS. The entire analysis is performed diretly in time domain assuming linear elasti struture and soil region. ITRODUCTIO Mehanial systems onsisting of strutures, fluid, and soil regions are representatives of multi-phase systems. The omprehensive and aurate analysis of multi-phase strutural systems in dynami environments has substantial impat in the strutural design proedures, espeially whenever relative motion among the phases is expeted. Suh ases an be lassified into problems with large relative motion governed by flow harateristis, problems of short duration, and problems of long duration with limited fluid displaement. The latter inludes suh typial Fluid-Struture-Soil-Interation (FSSI) phenomena as the dynami response of offshore strutures, nulear reator omponents, liquid filled tanks, and dams subjeted to waves or earthquakes. Information pertaining to the developments in the fields of dynami interation problems an also be found in the review artiles, and referenes therein, of Rizos and Karabalis [3], Beskos [2,3], and Makerle [8], among others. The analysis of spherial tanks is the primary motivation for this work. To this end, the main emphasis lies on a pratial, yet aurate, proedure to simulate the soil-foundation interation while taking into onsideration, through standard FEM proedures, the fluid-struture interation. The proposed methodology implements: (a) the Boundary Element Method (BEM) for the simulation of the soil- Department of Civil Engineering, University of atras, atras, GREECE 2 Department of Civil Engineering, University of South Carolina, Columbia, SC 29208, USA

2 struture interation, (b) a disrete 2-DOF mehanial system for the omputation of the impedanes of a ring foundation, a usual feature of a spherial tank, and () a FEM simulation of the struture and fluid regions. The above modules are oupled via the pre- and post-proessing apabilities of the widely used general purpose program ASYS. The entire analysis is performed diretly in time domain assuming linear elasti struture and soil region. However, taking into aount nonlinear behavior, both material and geometri, of the struture is a straightforward matter within the framework of the proposed proedures. The FEM methods are well-established proedures (e.g. [,5]). These methods routinely aommodate omplex geometries and non-linear phenomena. The matries assoiated to the algebrai system of equations are symmetri, banded and positive definite, allowing the implementation of effiient solution algorithms. The nature of the FEM implies that the solution domain must be finite. However, ases involving infinite media, suh as the soil region, annot be represented aording to lassial FEM proedures. The treatment of the radiation ondition, imposed by the infinite extent of the soil medium, has lead to the development of numerous speial types of radiating, transmitting or absorbing elements for use predominantly in the frequeny domain FE method, e.g. Rizos and Karabalis [3]. Boundary Element Methods for SSI analysis are based on integral formulations and fundamental solutions of the wave propagation problem. Suh formulations satisfy impliitly the radiation boundary ondition of infinite domains, and are appealing alternatives to the FEM. They require disretization of the boundary of the solution domain only. Consequently, BEM methods redue the dimensionality of the problem by one, yielding smaller systems of algebrai equations. However, the matries assoiated with the BEM algebrai system of equations are, in general, not symmetri and fully populated. BEM an aommodate nonlinear phenomena, but in this ase additional volume disretizations are required. Comprehensive reviews on BEM are reported by Beskos [2,3]. Time domain BEM solutions for 3-D problems assoiated with the proposed formulation have been presented by Karabalis and Beskos [4, 5, 6, 7], Rizos and Karabalis [,2], Rizos [0] and Rizos and Loya [4], among others. SOIL-FOUDATIO ITERACTIO BY BEM-FEM A oupled BEM-FEM methodology suitable for 3-D dynami SSI analysis and wave propagation is proposed for the simulation of the soil-foundation system. The proposed FEM-BEM is derived in time domain based on the BEM formulations presented by Rizos [9] and Rizos and Karabalis [,2] and standard FEM analysis proedures for solids and strutures. The foundation is modeled by finite elements, while the linear soil region, of semi-infinite extent, is modeled by boundary elements. The BEM method is used to ompute the B-spline impulse response of the surfae of the BE domain, in general, and of the BE-FE interfae, in partiular. In ontrast to other formulations reported in the literature, the oupling of the two methods is performed in a staggered solution approah in diret time domain. Aording to this, the solution of one method serves as initial onditions to the other at every time step, as demonstrated in Figure. Although the BEM-FEM methodology has been developed for the analysis of flexible strutures resting on an elasti half-spae, in this work its usefulness is limited to the omputation of the response of a rigid ring foundation, whih is the usual ase in onjuntion with spherial tanks. B-spline BEM Formulation This setion presents a brief overview of the diret time domain Boundary Element Method (BEM) formulation in onjuntion with the B-spline fundamental solutions that are derived from the Stokes fundamental solutions for the infinite elastodynami spae assuming linear and homogeneous material. The B-spline fundamental solutions aommodate virtually any order of parametri representation of the

3 time dependent variables without exessive omputational effort and impliitly satisfy the ontinuity onditions of the Stokes fundamental solutions. A detailed formulation and integration of the B-spline fundamental solutions and of the assoiated BEM an be found in the work of Rizos [9], Rizos and Karabalis [,2], and Rizos and Loya [4]. In the present outline symbols in boldfae indiate vetors or tensors, dots indiate differentiation with respet to time and indiial summation notation is adopted where appropriate. External exitation/ Ground motion FEM Fore equilibrium Equation (3) BEM FEM analyzer Equation () BEM analyzer Equation (8) u FEM External exitation Compatibility of displaements Equation (2) FEM BEM u BEM time i i+ Figure. Shemati representation of the proposed BEM-FEM oupling sheme. Following well-established proedures, the Love integral identity form of the avier-cauhy governing equations of motion for a linear isotropi and homogeneous elastodynami medium an be expressed in a disrete form, in spae and time, at time instant t, as M ( ), M ( t ) U( t ) t ( ) ( τ ) T( t ) u( τ ) u = n n n n n () n= t n+ + t n+ 2 + L+ t n+ k where τ n = k The disrete form of Eq. () implies that: (i) B-spline fundamental solutions of the k th order for the infinite elastodynami spae are adopted. The support of the assoiated B-spline polynomials is t. The summation indiated in Eq. () refers to disrete time steps, and M depends on the order of the B-spline fundamental solutions. For example for the ubi B-spline fundamental solutions M=+2. (ii) The temporal disretization of the field variables is based on B-spline funtion approximation shemes of the same order as the B-spline fundamental solutions. Eq () is evaluated at time knot t and the assoiated time knot sequene is

4 t n k even k t n = n = 0,, 2, L,, L, M (2) t ( n 0.5) k odd k where t is the support of the adopted B-spline polynomials. (iii) The surfae S of the elastodynami domain is disretized in a total of E surfae elements for a total of boundary nodes. The present work implements higher order 3-D surfae elements in an isoparametri formulation. The tensor ij in Eq. () is known as the jump term and depends on the loation of a disrete node with respet to the solution domain and its boundary, and the geometri harateristis of the domain in the neighborhood of the node. The oeffiient matries U(t) and T(t) are of size 3 x 3 and represent the influene of a soure boundary node on a reeiver boundary node. These matries are evaluated based on integrals of the B-spline fundamental solutions over the surfae of eah element for all boundary nodes. Vetors t (n) and u are of size 3 x and represent the omponents of the tration and displaement fields, respetively, at every boundary node due to a single B-spline impulse exitation applied at time τ. As reported in Refs [0] and [4], nodal fores, p, an be related to trations, t (n), through a transformation matrix L that is developed based on a virtual displaement approah, allowing Eq. () to be expressed in terms of nodal fores, p, as ( t ) = ( G( tn ) p( τ n ) T( tn ) u( τ n )) u (3) n= A T where G = UL and L = s sdsa. The integration is over the area, S a, of eah element, the Sa a= summation indiates superposition over the number of elements, A, that ontain a soil node, and s are the shape funtions of the spatial disretization. The impulse response of the elastodynami system an be obtain from Eqs () or (3). To this end, a B-spline impulse fore, or tration, exitation of duration t is applied in the diretion of eah degree of freedom, j. Eqs () or (3) are solved in a time marhing sheme for the displaements, u, whih represent the B-spline impulse response vetor, b. These impulse responses an be olleted in matrix form as B ( ) B = [ b, b, K, b Kb ] t 2 j, 3 j = (4) The dimension of matrix B(t ) is 3 x 3, in general, and eah olumn, b j, represents the impulse response of all degrees of freedom at time step t due to a unit tration, or fore, applied at degree of freedom j. This matrix is a harateristi of the system and needs to be omputed only one for the speifi geometry of the free surfae of the elastodynami domain. In the general ase, the B-spline impulse response is omputed for all degrees of freedom of the problem and the impulse response matrix, B, is square. In most pratial ases, however, the impulse response of the system needs to be omputed only for those degrees of freedom that are expeted to arry fores (ontat loads or inident fields) of the arbitrary exitation, yielding a retangular matrix or a square matrix of signifiantly redued size, as shown in Ref. [4]. Response to an arbitrary transient exitation

5 One this harateristi response of the system is known, the system an be analyzed for any arbitrary transient exitation, f(t). Let f(t) represent the vetor of arbitrary transient external exitations applied at n the boundary nodes of the elastodynami system and f = f ( t = tn ) the value of the exitations at time step t n. Furthermore, let R represent the response of the elastodynami system to the exitation f(t) at time step t. If the B-spline impulse response, B, matrix of the system is known, the alulation of R redues to a mere superposition of the B-spline impulse responses as + n n+ 2 R = B f (5) n= This approah is very effiient espeially when multiple load ases are onsidered. Eq. (5) an be speialized for the proposed oupling sheme. To this end, the vetor of arbitrary exitations f = { } T, 0 onsists of nodal fores, applied at the ontat interfae between the BE and FE domain only, while the free field is not loaded. The response R of the BEM region pertains to nodal displaements u = { u, u } T f due to the ontat fores at time step. In view of the separation into ontat and free field degrees of freedom, Eq. (5) an be expressed as u + = u f n= B B f B B f ff n 0 + n= + n= n+ 2 n u = B (6a) n+ 2 n u = B (6b) f f n+ 2 It is evident that only that part of the B-spline impulse response matrix that orresponds to the ontat degrees of freedom is required. In addition, the response of the free field an be omputed independent of that of the ontat area. For this reason only Eq. (6a) shall be onsidered in the proposed oupling sheme. The interfae fores,, however, depend on the response of the FE region and are not known in advane. Consequently, Eqs (6) are in an impliit form. The orresponding expliit sheme is obtained by interpolating the interfae fores at time t using B-spline polynomials. For the ubi B-spline polynomials (k=4) implemented in this work, the interfae fores are interpolated as + = B 4 o = B 4 o 2 + B 4 o 4 + = (7) Introduing Eq. (7) into (6a) and separating known from unknown quantities with respet to time step, the latter an be expressed as

6 u u = 2 ( 2B + B ) = F + H + + n= 3 B n n+ 2 B (8) where 2 2B B F = + represents the flexibility matrix of the ontat interfae of the BEM region and + n n+ 2 H = B B is a vetor representing the influene of the response history on the n= 3 urrent step. Eq. (8) is used in the proposed oupling sheme and represents the BEM solver. It is evident that if the ontat fores are provided by the FE solver, at every step, the BE solver evaluates the orresponding displaements of the interfae without solving a system of algebrai equations. Furthermore, if the response of the free field is of interest, it an be omputed by a simple evaluation of Eq. (6b). Time Domain FEM Formulation This setion presents an outline of the FE proedures as implemented in the present work in a manner suitable for oupling with the BEM. The FEM is used to model the strutural omponent in a soil-struture system subjeted to transient exitations. The governing equations are integrated diretly in time using two different time stepping algorithms, i.e. the bakward differene sheme and the ewmark-β method. Governing Equation The governing equations of motion of a multi-degree of freedom system, as derived in a semi-disrete FE approah, is expressed in matrix form as, () t + Cu& () t + Ku() t f () t M u& = (9) These are seond order differential equations with M, C, and K being the mass, damping and stiffness matries of the struture, respetively. Vetor u(t) represents the displaement field at the nodal points of the system, f(t) represents the external exitation vetor and dots indiate derivatives with respet to time. In a general soil-struture system the struture is supported by the elasti soil medium at the foundation level. Exitations of the supports of a struture pertain to time histories of foundation displaements presribed at the foundation level. The external loads onsist of time histories of nodal fores and moments, applied at the strutural degrees of freedom. Time stepping algorithms The diret numerial integration in time of Eqs (9), may be aomplished by standard finite differene shemes. This work implements the ewmark-β and a bakward differene sheme sine both are expressed in expliit forms. The former is a preferred approah beause of its ontrolled stability harateristis and its ability to aommodate iterative shemes for the solution of nonlinear dynami systems. The bakward differene sheme, however, shows simpler implementation. Appliation of any of these these shemes to Eqs (9) yields expliit, fully disrete forms of the FE equations expressed symbolially as Du = f (0)

7 where the dynami matrix, D, and the equivalent fore vetor, f, depend on the diret time integration sheme and mathematial expressions are widely available in the literature, e.g. Bathe []. It is noted that the dynami matrix, D, is independent of time and for linear appliations needs to be evaluated only one. In terms of the degrees of freedom in ontat with the BE domain, Eq (0) an be expressed as D D f D D f ff u u f f = f f () subsript indiates the degrees of freedom at the ontat interfae, and f pertains to the remaining, unsupported degrees of freedom. Furthermore, the fore vetor f onsists of the known external exitations, f (ext), applied diretly on the interfae degrees of freedom, as well as the unknown ontat fores,, exerted by the supporting medium, i.e., f = f( ext) +. rovided that the displaements u are furnished by the BEM solver as initial onditions at step, Eqs () an be solved for the ontat fores. Coupled BEM-FEM Formulation The BE and FE domains are oupled at the interfae level as depited in Figure. Consequently, the time marhing Eqs (9) and () annot be solved independent of eah other. The proposed solution sheme follows a staggered approah that omputes the response of the BE-FE system by enforing the equilibrium and ompatibility equations expliitly at the interfae degrees of freedom and at every time (BEM ) (FEM ) step of the solution. Let u and u represent the displaement vetors at the ommon nodes in (BEM ) (FEM ) the BEM FEM regions, and and the assoiated fore vetors. Assuming the two regions remain always in ontat, the ontinuity of nodal displaements and equilibrium of fores are expressed as u = u (2) ( BEM ) ( FEM ) ( BEM ) ( FEM ) + = 0 (3) In view of the BEM Eqs (8), and the FEM Eqs (), whih are repeated here for ompleteness, ( BEM ) ( BEM ) u = F + ( FEM ) ( FEM ) u f = D ff f f H ( FEM ) ( D u ) ( FEM ) ( FEM ) ( FEM ) f f ( ext) f (4) (5) = D u + D u f (6) the staggered solution approah an be formally defined. At every time step,, of the solution the FE ( FEM ) ( FEM ) region is loaded by the external fores f f, and f( ext ) while the displaements of the interfae ( FEM ) nodes, u, are presribed as initial onditions at time t=0 or as provided by the BE solutions along with the ompatibility Eq. (2). Subsequently, the ontat degrees of freedom are fixed at the new loation and the FE solver, Eqs (5), omputes the response of the struture as well as the fores at the interfae nodes, Eqs (6). Subsequently, Eq. (3) omputes the orresponding fores at the interfae nodes in the BE region. Based on these fores, the BE solver, Eqs (4), evaluates the displaements at the

8 interfae nodes, whih, in view of Eq. (2), beome the new initial onditions for the FE solver and the solution moves to the next time step. Cases involving diret oupling of finite elements that have both translational and rotational degrees of freedom with boundary elements that have only translational degrees of freedom ould be handled routinely by the proposed methodology through ondensation of the rotational degrees of freedom. Further details on the numerial handling of suh ases an be found in Rizos [0]. Rigid strutures A speial ase The analysis of rigid strutures resting on elasti half-spae is distinguished by the absene of elasti and damping fores. The dynami equilibrium of the struture is expressed as f = Mu& = + f (7) i ext where f i are the inertia fores based on the total displaements u, are the ontat interation fores applied at the interfae between the soil and the struture, and f ext pertains to the externally applied loads. In Eq. (7) it has been assumed for simpliity that all strutural degrees of freedom are in ontat with the BE surfae. The general approah introdued earlier implies that the BE solver omputes the displaement vetor u while the FE solver solves for. However, the FE solver of the proposed method beomes unstable for this lass of problems. In partiular, it has been observed that either time integration sheme of the FE solver yields divergent solutions for the veloity and aeleration of the FE sub-domain. Stabilization of the proposed sheme requires that the ontat fores need to be impliitly aounted for in the dynami equilibrium equation. To this end, it is reognized that the vetor of total displaements onsists of two distint omponents, i.e. ( BEM ) u = δ + d (8) The first omponent, ä, pertains to the ontribution of the kinemati interation effets at the urrent time only, and the seond omponent, d, is the ontribution to the total displaement from the history of the response. In view of Eqs (2), (3) and (4) the displaement omponent, ä, due to the kinemati interation an be omputed as δ = F ( FEM ) ( BEM ) = F δ = K int δ (9) where K int is onsidered as the equivalent stiffness matrix of the BE-FE interfae. Substitution of Eqs (8) and (9) into the dynami equilibrium Eq. (7) yields M & δ K δ = Md& int (20) + ext and, thus, the FE equations are augmented by the equivalent stiffness matrix of the soil. Appliation of the diret integration shemes on Eqs (20) yields the stabilized form of the FE solver and the general approah disussed earlier. One the displaement ä is omputed, the ontat fores are omputed as ( BEM ) = K δ. This sheme is applied to the problem of rigid massive foundations. int UMERICAL RESULTS FOR AULAR FOUDATIOS

9 The above formulation has been applied to the determination of the impedanes of an annular, rigid, massive foundation. The results for all six degrees of freedom, in frequeny and time domains, are shown in Figure 2. These results orrespond to soil elasti onstants: E=78000 ka, ρ=834.8 kg/m 3, and ν=/3. The foundation has a outside radius of R o =.06m, thikness R=2.5m, height h=2m, and mass density ρ=2550kg/m 3. Figure 2. Impulse responses versus time and impedanes versus frequeny of a Massive, rigid, ring foundation.

10 DISCRETE MODEL FOR SOIL-FOUDATIO IMEDACES In order to minimize the omputational effort, a set of disrete foundation springs and dashpots are introdued at the base of the foundation. They represent the disrete impedanes of the soil-foundation system for all possible degrees of freedom of a rigid ring foundation of the type under onsideration. However, the proposed proedure is general and an be applied for any type of foundation. Aording to this, the numerial results obtained in the previous setion by the BEM-FEM methodology for the time and frequeny domain responses of an ring foundation provide the basis for the determination of a 2-mass disrete model as the one shown in Figure 3. The omputation of the proper ombinations of mass, stiffness and damping onstants is obtained via an automati trial-and-error proess programmed diretly in the ADL programming language within ASYS. Figure 3. Disrete model for the omputation of soil-foundation impedanes. The results of this proedure are shown in Figure 2 for all possible modes of vibration of the rigid ring foundation under onsideration. The orresponding disrete parameters (masses, springs and dampers) are listed in Table. Table. Disrete parameters for approximate impedanes in all six degrees of freedom (units in k, m, and se) K C K2 C2 K3 C3 M M2 Vertial Horizonta l Torsional Roking AALYSIS OF SHERICAL TAK Some of the numerial results obtained from the analysis of a typial spherial tank using the proposed hybrid methodology are shown in this setion. The outside radius of the tank is 20m, its average thikness is 4m, and its equator stands at an elevation of 2.5m. It is supported by olumns of tube setion with an outside diameter of 94.4mm and thikness 2.7mm. Lateral stiffness is provided by pairs of braings eah one of whih has a length of 8m and a ross setion of 30x350mm. The tank is assumed to ontain propylene with bulk modulus 2Ga, and mass density 522kg/m 3. The FE model for the struture, shown in Figure 5, onsists of 524 elements, inluding fluid elements, and 5328 nodal points.

11 Figure 5. FEM model of spherial tank. The results from the modal analyses of the spherial tank assuming fixed base onditions or SSI are presented in Tables and 2 for different levels of liquid fullness. Table 2. Eigenvalues for fixed base onditions erentage of fluid fullness Φ period, se (% modal mass partiipation) 50% 5 (4.7%) 75% 4.5 (26.6%) 98% 0.44 (98%) Φ2 period, se (% modal mass partiipation) 0.25 (50%) 0.32 (69%) 0.8 (2%) Φ3 period, se (% modal mass partiipation) 0.8 (8.3%) 0.8 (4.4%) 0. (0.00%) Table 3. Eigenvalues inluding SSI erentage of fluid fullness Φ period, se (% modal mass partiipation) 50% 5 (4.7%) 75% 4.5 (%) 98% 0.52 (59%) Φ2 period, se (% modal mass partiipation) 0.3 (45%) 0.32 (45%) 0.8 (40%) Φ3 period, se (% modal mass partiipation) 0.8 (39%) 0.8 (42%) 0.2 (0.00%) Subsequently, the transient response of the given struture to an artifiial seismi exitation, shown in Figure 6, is omputed. The seismi exitation is ompliant to the spetral requirements of the Greek Seismi Code, for soil ategory C. The response of the spherial tank under onsideration to the above seismi exitation is depited in Figures 7 and 8 in terms of base shear and displaements at the top of the tank vs time. The base shear variation of Figure 7 onfirms the well aepted notion that for the same struture SSI yields smaller

12 values than those produed by fixed base boundary onditions. The opposite should be true for the displaement vs time plot. Figure 8 verifies this assumption, too. ARTIFICIAL ACCELEROGRAM Soil C Ground Aeleration 0,3 0,25 0,2 0,5 0, 0,05 0-0,05-0, -0,5-0,2-0, Time (se) Figure 6. Artifiial aelerogram used for the analysis of spherial tank COCLUSIOS An effiient hybrid BEM-FEM methodology in diret time domain is developed for the omputation of the dynami harateristis of massive foundations of arbitrary shape. The rigorous results obtained by the above methodology have been used for the determination of the disrete impedanes of a rigid, ring foundation, on the basis of a 2 DOF model. Finally the omplete seismi analysis of a typial spherial tank is attempted taking into onsideration fluid-struture-soil interation phenomena in diret time domain. For the struture fluid portion of the entire model, standard FEM modeling is used oupled to the ASYS pre- and post-proessing modules. The proposed methodology an be easily extended to nonlinear dynami analyses. ACKOWLEDGEMETS The work desribed in this paper has been developed in the framework of IDETH rojet (EVG-CT ) whih is supported by the Environment rogramme, Global Change and atural Disasters of the European Commission, Researh Diretorate General.

13 BASE SHEAR TIME HISTORY ( 50% perentage of fullness ) Base Shear (K Time (se) Fixe d Ba se S.S.I. BASE SHEAR TIME HISTORY ( 75% perentage of fullness ) Base Shear (K Time (se) Fixe d Ba se S.S.I. BASE SHEAR TIME HISTORY ( 98% perentage of fullness ) Base Shear (K Time (se) Fixe d Ba se S.S.I. Figure 7. Base shear (k) versus time for various levels of fluid fullness.

14 Displaement of Sphere Time History (50% perentage of fullness) Displaement ( Fixed 40Base Time (se) S.S.I. Displaement of Sphere Time History ( 75 % perentage of fullness) Displaement ( Time (se) Fixed Base S.S.I. Displaement of Sphere Time History ( 98 % perentage of fullness) Displaement (m Time (se) Fixed Base S.S.I. Figure 8. Displaement (m) at the top of the spherial tank versus time for various levels of fluid fullness

15 REFERECES Bathe K-J. Finite element proedures in engineering analysis. Englewood Cliffs, J: rentie- Hall, Beskos DE. Boundary element methods in dynami analysis. Applied Mehanis Review 987; 40(). 3 Beskos DE. Boundary element methods in dynami analysis: art II ASME Applied Mehanis Reviews 997; 50(3): Karabalis DL, Beskos DE. Dynami response of 3-D rigid surfae foundations by time domain boundary element method. Earthquake Engineering and Strutural Dynamis 984; 2: Karabalis DL, Beskos DE. Dynami response of 3-D flexible foundations by time domain BEM and FEM. Soil Dynamis and Earthquake Engineering 985; 4: Karabalis DL, Beskos DE. Dynami response of 3-D embedded foundations by the boundary element method. Computer Methods in Applied Mehanis and Engineering 986; 56: Karabalis DL, Beskos DE. Dynami soil struture interation. Beskos DE. Editor. Boundary Element Methods in Mehanis. Amsterdam: orth Holland, Markerle J. FEM and BEM in geomehanis: foundations and soil-struture interation a bibliography ( ). Finite Elements In Analysis and Design 996; 22(3): Rizos DC. 3-D diret time domain BEM in elastodynami. h.d Thesis, Columbia: Univ. of South Carolina Rizos DC. A rigid surfae boundary element for 3D soil-struture interation analysis in the diret time domain. Computational Mehanis Rizos DC, Karabalis DL. An advaned diret time domain BEM formulation for general 3-D elastodynami problems. Computational Mehanis 994; 5: Rizos DC, Karabalis DL. A time domain BEM for 3-D elastodynami analysis using the B-Spline fundamental solutions. Computational mehanis 998; 5: Rizos DC, Karabalis DL. Soil-fluid-struture interation. Kausel E, Manolis GD. Editors. Wave Motion roblems in Earthquake Engineering. Southampton: WIT ress, Rizos DC, Loya KG. Dynami and seismi analysis of foundations based on free field B-Spline harateristi response histories. Journal of Engineering Mehanis ASCE Zienkiewiz OC, Taylor RL. The Finite Element Method vol., Basi Formulation and Linear roblems. London: MGraw-Hill, 989.