Soil-Structure Interaction (SSI) Analysis Using a Hybrid ... · Soil-Structure Interaction (SSI) Analysis Using a Hybrid Spectral Element/Finite Element (SE/FE) Approach flexibility

Summer 2009, Vol. 11, No. 2JSEE

A B S T R A C T

Available online at: http://www.iiees.ac.ir/jsee

This study proposes a new formulation for modeling soil-structure interaction (SSI)problems. In this direct time-domain method, the half-space soil medium is modeledby spectral element method (SEM) which is based upon a conforming mesh oftwo-dimensional quadrilaterals, and the structural frame components are modeledby finite element method (FEM). Formulation and various computational aspects ofthe proposed hybrid approach are thoroughly discussed. To the authors' knowl-edge, this is the first study of a hybrid SE/FE method for SSI analyses. The accuracyand efficiency of the method is discussed by developing a two-dimensional SSI analysisprogram and comparing results obtained from the proposed hybrid SE/FE methodwith those reported in the literature. For this purpose, a number of soil-structureinteraction and wave propagation problems, subjected to various externally appliedtransient loadings or seismic wave excitations, are presented using the proposedapproach. Each problem is successfully modeled using a small number of degrees offreedom in comparison with other numerical methods. The present results agree verywell with the analytical solutions as well the results from other numerical methods.

Soil-Structure Interaction (SSI) AnalysisUsing a Hybrid Spectral Element/Finite

Element (SE/FE) Approach

P. Mirhashemian1, N. Khaji 2*, and H. Shakib3

1. MSc in Earthquake Engineering , Civil Engineering Department, Tarbiat Modares University,Tehran, I.R. Iran

2. Associate Professor, Civil Engineering Dept., Tarbiat Modares University, Tehran, I.R. Iran,* Corresponding Author; email: [email protected]

3. Professor, Civil Engineering Department, Tarbiat Modares University, Tehran, I.R. Iran

Keywords:Soil-structure interaction;Hybrid approach;Spectral elementmethod;Finite element method;Elastodynamics

1. Introduction

The dynamic response of massive structures, suchas high-rise buildings and dams, may be influencedby soil-structure interaction as well as the character-istics of exciting loads and structures. The effect ofsoil-structure interaction is noticeable especially forstiff and massive structures resting on relatively softground. It may alter the dynamic characteristics ofthe structural response significantly. As a result,these interaction effects have to be considered inthe dynamic analysis of structures in a semi-infinitesoil medium [1].

Although a huge number of investigations havebeen conducted for soil-structure interaction analy-sis, depending on the modeling method for the soilregion, all methods may be classified into two maincategories. One method, the so-called direct method,evaluates the dynamic response of structure andits surrounding soil in a single analysis step, by

subjecting the combined soil-structure system to adynamic excitation. Another one which is termed assubstructure method is based on the principle ofsuperposition and the analysis is performed inseveral steps [2].

Various types of numerical methods such asfinite element method (FEM), boundary elementmethod (BEM), and hybrid techniques are commonlyused to model soil-structure interaction effects. Theuse of FEM is advantageous as the proceduresare versatile in nature and well-established (see forinstance Refs. [3-6] among others). This methodoffers efficient advantages in the treatment ofvarious aspects of soil domain such as arbitrarygeometries, soil layering, material non-linearities,anisotropies and inhomogeneities. Nevertheless, dueto its inherent difficulty in analyzing media of infiniteextent, the radiation boundary conditions at infinity

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P. Mirhashemian, N. Khaji, and H. Shakib

have also been employed to overcome the problemof wave reflection and radiation on the far-fieldboundaries of the soil domain [7-10]. However,FEM remains a “computationally expensive”approach for elastody-namic and SSI problems [5,11-13]. Apparently, one reason is that low-orderFEM exhibit poor dispersion properties [14], whilehigher-order finite elements cause some troublesomeproblems like the occurrence of spurious waves.

Alternatively BEM which satisfies automaticallythe radiation boundary condition at infinity, requiressubstantially reduced surface discretizations even for3D problems, and is appealing alternative to FEMfor soil-structure systems [15-17]. However, BEMobviously suffers from two disadvantages. The firstone is that although the coefficient matrix of thesystem is much smaller than those of FEM, thismatrix is non-symmetric, non-positive definite andfully populated. The second disadvantage is thatBEM is not appropriate in the treatment of materialnonlinearities, anisotropies, and inhomogeneitiesof soil domain, and arbitrary complex half-spacegeometries.

Coupling finite element and boundary elementmethods retains the advantages of each method andeliminates their disadvantages. Various coupled FE/BE approaches have been suggested in the literaturefor problems in dynamic and seismic analysis ofcoupled soil-structure systems, see for exampleRefs. [18-24].

Spectral element method (SEM) provides ahigh-order technique, which allows obtaining thesame accuracy as low-order methods (such asFEM) by using a reduced number of grid points, thusgiving rise to a significant efficiency in computationalresources. SEM, which enjoys all advantages ofclassical FEM (e.g., well-suited to handle complexgeometries and nonlinear conditions), has beenbenefited by special sort of interpolation functions.These interpolation functions enable SEM to trans-fer a wide range of wavelengths through elementswith higher accuracy and lower computationalefforts compared to classical FEM. SEM whichwas originally introduced in computational fluidmechanics [25] is currently being implemented inelastodynamics problems, see for instance Refs.[26-28] among others. Nevertheless, little effortshave been achieved for the analysis of soil-structuresystems using SEM [29].

In the past decade, SEM, which is especially

appropriate for seismic wave propagation problems,has experienced significant developments. As aresult, it seems that the soil-structure interactionwhile including the wave propagation considerationswould be efficiently investigated by SEM. On theother hand, structural system effects may be accu-rately modeled using FEM. Consequently, a coupledSEM-FEM approach seems to exhibit an attractiveand efficient tool for the soil-structure interactionanalyses.

In this study, a combination of SEM and FEM isproposed for general two-dimensional analysis ofsoil-structure interaction problems under dynamicloading and elastodynamic problems. In order toobtain an approach applicable to nonlinear problems,a formulation that works directly in the time domainis developed. Employing the well-known directmethod, the system is divided into a homogeneous,linear elastic half-space that is modeled using SEMand structural system, which is modeled by FEM.Before combining these two methods, SEM isverified by analyzing some example problems. Theresults obtained are compared with those from theliterature. Afterwards, a soil-structure interactionmodel is developed by coupling the two methods.The coupled model is then applied to severalnumerical examples to illustrate how the SE/FEcoupling may be achieved and how accurate theresults are. It is found that the results agree wellwith the literature.

2. Problem Formulation

A numerical procedure based on the hybridformulation, which combines SEM for the half-spacesoil domain and FEM for the superstructure isdeveloped in the time domain, see Figure (1).

2.1. Spectral Element Method Formulation

In the early 1980s, spectral element method(SEM) was initially developed by Patera [30] in thefield of fluid dynamics. This method was furtherdeveloped and found many applications for modelingseismic wave propagation, e.g. [26-28]. In thissection, a brief review of some important aspects ofSEM, in particular those that would be subject tomodification for their application to the coupledsoil-structure system are briefly presented.

SEM is based upon a weak formulation ofthe equations of motion. This method combines the

JSEE / Summer 2009, Vol. 11, No. 2 85

Soil-Structure Interaction (SSI) Analysis Using a Hybrid Spectral Element/Finite Element (SE/FE) Approach

flexibility of a finite element method with the accu-racy of a pseudo-spectral method. Consequently,the SEM formulation process of stiffness and massmatrices is analogous to the classical FEM formula-tion. The main differences between SEM and FEMmay be summarized in three main aspects: thepolynomial degree of the basis functions, the choiceof integration rule, and the nature of the time-march-ing scheme.

A crucial idea of SEM is adoption of specificshape/basis functions which is illustrated here as thefirst main difference between SEM and FEM. Forany given quadrilateral element, the relation betweena point displacement field ),( ηξu

rwithin the element

and a nodal point in the master square may beapproximated as the following form:

∑+

=

ηξ=ηξ1

1

),(),(l rr n

aaa uNu (1)

where the local shape functions ),( ηξaN are prod-ucts of Lagrange polynomials, and au

r denotes nodal

degrees of freedom. Firstly, the degree ln of theLegendre polynomials has to be chosen. The )1( +lnLagrange polynomials of degree ln are defined interms of )1( +ln internal local nodes 11 ≤ξ≤− p ,

1,...,2,1 += ln p . These local nodes are placed atspecial positions called Legendre-Gauss-Lobatto(LGL) points. These correspond, in a normalized 1Dsituation [-1, 1], to the zeroes of ),(ξ′

lnP the firstderivative of the Legendre polynomial of degree ,lnand the extremes of the interval

1,1)( P of zeros n −ξ′ Ul (2)

Figure 1. Soil-structure media.

which means that one has )1( +ln LGL points for apolynomial of degree ln . By construction of a weakform (similarly as in FEM), element mass andstiffness matrices may be defined [31]. In otherwords, instead of using the strong form of theequations of motion, one can use an integrated form(i.e., weak form such as weighted residual approach).This is accomplished by weighting the equations ofmotion with an arbitrary test vector (the variation ofdisplacement function is chosen here), integratingby parts over the domain volume, and imposingappropriate boundary conditions. To solve the weakform of the governing equations, integrations overthe domain volume and boundary are subdivided interms of smaller integrals over the volume andsurface elements, respectively. Transformation to theglobal coordinate system and the assembly processare the same as in FEM. Finally, a wave propagationproblem is reduced to the well-known ordinarydifferential equations, which can be written in amatrix form:

FaKaCaMrr&r&&r =++ ][][][ (3)

where [M], [C] and [K] are, respectively, the globalmass, damping and stiffness matrices, and F

r is a

vector of the time dependent excitation signal. Theglobal matrices are equivalent to the sum of integralsover the set of the elements of the entire domain.They are built through the assembly process of theelement matrices (direct stiffness method).

The second difference between SEM and FEMconcerns to the numerical integration of elementmatrices. Let us consider the standard FE formulaeof element characteristic matrices as:

Ω

Ω

Ω

Ω

dBDBK

dNNM

e

e

Te

Te

]][[][][

][][][

∫∫

=

ρ=

(4)

in which [B] is strain-displacement transformationmatrix, [D] denotes material stiffness matrix, [N]indicates shape function matrix, and ρ is the massdensity. Each of integrations over the elements eΩare usually calculated by Gauss quadrature in FEM.In SEM, integrations may be approximated usingthe LGL quadrature rule instead

( )

∑ ∑

∫ ∫∫+

=

+

=

− −

ωω

≈ηξηξηξ=

1

1

1

1)(

1

1

1

1),(),()(

l l

rr

n

p

n

qpqepqqp

e

Jf

ddJxfdxdyxf

eΩ

(5)



in which )(xfr

indicates the integrand componentsof Eq. (4), pω and qω are the weights associatedwith the LGL points of integration, and =)( pqe J

),( qpe J ηξ is the Jacobian of mapping from theelement eΩ to the reference domain. To integratethe functions and their partial derivatives over theelements, the values of the inverse Jacobian matrixneed to be determined at the 2)1( +ln LGL integra-tion points for each element. The quadrature weights,which are independent of the element, are determinedfrom the following equation:

1,...,2,1,)()1(

22 +=

ξ+=ω l

ll l

n a Pnn an

a

(6)

As a result of the Lagrange interpolants selectionat the GLL points, and its relation to the GLL integra-tion rule, SEM shows an important property thatleads to the diagonal form of the mass matrix, whichallows a crucial reduction of the complexity andthe cost of the algorithm [28]. In this manner, thediagonal mass matrix is obtained naturally incomparison with mass matrix lumping techniques,which incorporate considerable errors.

In order to take full advantage of the propertydescribed above, time-discretization of Eq. (3) isachieved based upon a Newmark scheme (centraldifference method) which, may be considered as thethird difference between SEM and FEM. For SEM,this leads to simple explicit time schemes, as opposedto the numerically more expensive implicit timeschemes used in FEM [31].

2.2. Coupled SEM-FEM Formulation

Formulation of equations of motion for a soil-structure interaction analysis has been frequentlyexplained in the literature, see e.g. [1], and thereforeis only briefly summarized in this section. Considerthe dynamic response of a structure on a semi-infinite soil medium as shown in Figure (1) in which,the structure and soil media are to be modeled byFEM and SEM, respectively. Following a commonnotation used in the SSI literature, the nodes associ-ated with the structure are identified with “s”, thosethat lie along the soil-structure contact zone are “c”nodes, and the other nodes within the soil mediumare identified with “m”, see Figure (1). From thedirect stiffness method in finite element analysis, thedynamic equilibrium of the system may be writtenin the following sub-matrix form:

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]

=

+

+

m

c

s

m

c

s

mmmc

cmcccs

scss

m

c

s

mmmc

cmcccs

scss

m

c

s

mm

cc

ss

F

F

F

a

a

a

KK

KKK

KK

a

a

a

CC

CCC

CC

a

a

a

M

M

M

r

r

r

r

r

r

&r&r&r

&&r&&r

&&r

0

0

0

0

00

00

00

(7)

where the characteristic matrices at the contactnodes are the sum of contribution from the structure(s) and the soil medium (m)

][][][

][][][

][][][

)()(

)()(

)()(

mcc

scccc

mcc

scccc

mcc

scccc

KKK

CCC

MMM

+=

+=

+=

(8)

Also in Eq. (7), ar indicates the vector of absolute

displacements of the structure. If both earthquakeexcitation and externally applied transient loadingare considered, the right hand side of Eq. (7) can beexpressed by corresponding nonzero terms. Thisforce vector is included for completeness but is notcompletely considered in the numerical examplespresented later in this paper. In this study, twodifferent load conditions are taken into account: soilmedium subjected to seismic waves only, conse-quently the external forces on the structure are setequal to zero in Eq. (7); and structures subjected toexternal loading only to investigate the effects ofwave propagation in an elastic half-space.

3. Illustrative Numerical Analyses

The aforementioned methodology has beenimplemented in a two-dimensional time-domain SE/FE code in which, a library of spectral quadrilateralelements with various orders is coupled with frame(or, beam-column) finite elements. In order to vali-date the nature and general behavior of the method,some numerical examples have been considered. Aplane strain condition of soil domain is assumed forall examples. No physical damping (i.e., pure elasticmaterial behavior) is considered in all hybrid



analyses. Since a finite size of soil domain is regar-ded in the application of a domain type method suchas SEM, a customary approach for dealing withinfinite media consists of introducing truncatedboundaries and setting fictitious absorbing conditionson them. To ensure that all energy arriving at thedomain boundaries is absorbed, a special frequencyindependent non-reflecting viscous boundary condi-tion has been widely used for various elastodynamicand soil-structure interaction problems [32-33]. Toimplement this boundary condition, two normal andtangential dashpots are defined at each node alongthe lateral and base boundaries of the soil domain.The normal dashpots are assigned to absorb thereflected compressive waves while the tangentialones are set to absorb the reflected shear waves.These dashpots must be placed far away fromthe initially disturbed region, as they are usuallycompetent to transmit plane or cylindrical waves.Consequently, a large-scale domain should bediscretized by SEM. Nevertheless, as shown in thefollowing examples, SEM is capable to model alarge-scale domain with considerably few degreesof freedom compared to other numerical methods.This feature of SEM compensates the necessity ofmodeling a large-scale domain, and let the approachto be still efficient.

3.1. Analysis of Elastic Half-Plane

As the first example, a linear elastic half-plane isexamined to explain the applicability and accuracyof the present SE method, in calculating the free-fieldmotion caused by vertical propagating incident SVwave of the Ricker type:

)5.0(3

)()21.()( 22

−π=τ

τ−τ−=

t

pexftf

v

axm

(9)

where axm f is the maximum amplitude of the time

history, which is selected as 0.0005m in this example,see Figure (2). Figure (3) represents the geometryused for this problem in which 25 spectral elementsare considered. The material properties of themedium are: the shear wave velocity of the mediumis 267m/s, the mass density is 2.22t/m3, and thePoisson's ratio is 0.33.

To examine the convergence of SE modeling,three different discretized meshes are chosen.Each mesh consists of square spectral elements,see Figure (3), with a specific degree of Lagrange

Figure 2. Load function time history of the incident wave (Rickerwavelet).

Figure 3. Geometry and discretization of the half-plane prob-lem for which a typical third order ( ln =3 ) spectralelement is shown in more detail.

Figure 4. The SEM results of horizontal displacement timehistories of the half-plane in the case of an incidentSV wave for three types of spectral elements withspecified order of Lagrange polynomials.

polynomials, .ln The time histories of horizontaldisplacement at the ground surface are investigated.The numerical results by SEM are compared withthat of analytical solution, in the case of an incidentSV wave. Figure (4) presents the results for threetypes of spectral elements inspected in this example.



From Figure (4), one may easily observe that thehigher-order elements give better results so that theresults from the third order spectral elements andanalytical approach are almost identical. As expectedin the case of vertically propagating shear waves inan elastic half-plane, the total horizontal displacementof the free surface is twice the incident waveamplitude.

3.2. Analysis of Two-Layered Elastic Half-Plane

In this example, the application of the presentedSE method in performing site response analysis of atwo-layered half-plane is illustrated. In Figure (5),a soft layer with a height of 10m rests on a stifferhalf-plane whose 50m in depth direction is modeledin this example. For this problem, the comparison ofthe results is made with the results based upon ahybrid BE/FE method [23]. Therefore, we selectedthe same geometry, see Figure (5), with the sameshear wave velocities of 70.5m/s and 141m/s forthe upper soft layer and the half-plane medium,respectively. Furthermore, the material propertiesof both layers are as follows: the Poisson's ratiois 0.33, and the mass density is 2.0ton/m3. Thehalf-plane is excited by the same vertically incidentSV wave as Figure (2). Similar to the first example,various meshes were inspected among which, theresults corresponding to the converged mesh arepresented here. These converged meshes are thoseof the minimum order that present the most accu-racy in comparison with the hybrid BE/FE method[23]. The mesh includes 36 square spectral elementswhose sides are shown in Figure (5).

Figure 5. Geometry and discretization of the two-layered half-plane problem.

The time variation of horizontal displacement atpoints O and I are shown in Figures (6) and (7),respectively. Figures (8) and (9) present the hori-zontal acceleration time histories at the same pointsof Figures (6) and (7), respectively. As it is obvious

Figure 6. The SEM results of horizontal displacement time his-tory at point O of the upper layer for the spectralelements of order 4.

Figure 7. The SEM results of horizontal displacement timehistory at point I of the upper layer for the spectralelements of order 4.

Figure 8. The SEM results of horizontal acceleration time his-tory at point O of the upper layer for the spectralelements of order 4.



Figure 9. The SEM results of horizontal acceleration timehistory at point I of the upper layer for the spectralelements of order 4.

Figure 10. The considered geometry of a half-plane domainsubjected to surface traction.

Figure 11. Horizontal and vertical components of displacementat observation point A due to a surface traction.

from these figures, excellent agreement can beobserved between the results of the present workand the hybrid BE/FE method [23]. It is worthwhileremarking that the results shown in these figuresare pertaining to very coarse meshes in which fewdegrees of freedom are involved.

3.3. Half-Plane Domain Subjected to SurfaceTraction

The response of a half-plane domain excited bya surface traction is examined in this example.Solutions based upon a BE/FE approach [34] areavailable for comparison. Figure (10) depicts themodeled elastic region, which is loaded by a uni-formly-distributed ramp function, ),(tP as:

≥

≤≤

≤

=

ttP

ttttP

t

tP

∆

∆∆

;

0;)/(

0;0

)(

0

0 (10)

in which, 0P is 1000N/m, and ∆ t is 0.0045 second.The material constants are as follows: the Young’smodulus E = 2.66x105kN/m2, Poisson’s ratio is 0.33,and the mass density is 2.0ton/m3. In Ref. [34], theconsidered domain of Figure (10) was discretizedinto a FE subdomain (Region 1) and a BE subdomain(Region 2). In the present work, using a meshconsisting of 12 square spectral elements, SEMapproach yields results, see Figure (11) which arealmost identical to the results obtained by the BE/FE approach. It is worthwhile remarking that theresults of SEM approach are pertaining to a simplerformulation in comparison with the formulation ofthe BE/FE approach.

3.4. Single Degree of Freedom (DOF) Structure

The application of the present approach to theinteraction between a flexible structure and itssupporting elastic half-plane soil is examined in thisexample. The result of the present approach iscompared with the lumped parameter approach[35] in which the supporting half-plane is replacedby equivalent dashpots and springs. Furthermore,the comparison of the results is made with theresults based upon a hybrid BE/FE method [21].Figure (12) represents the single degree of freedomstick model of the structure and its supportinghalf-plane. The same material properties of Ref.[21] are selected as follows: the Lame’ constantsλ= 24081 and µ = 16054, and the mass density ofthe elastic half-plane is 0.002006. Furthermore, thelumped mass is equal to 10 and is located at thestory level with the height of L = 3. The lateral



rigidity of the structure is EI = 1.8x106; all quantitiesin consistent units. The right hand side of Eq. (7)is modified as the following form to represent theapplied harmonic horizontal excitation at the masslevel

)14.14()(;

0

0

)(

tnsitP

tP

F

F

F

m

c

s

=

=

r

r

r

r

r

(11)

The response of the structure is obtained using amesh including four square spectral elements tomodel the elastic half-plane, and one frame finiteelement for the single DOF structure. The resultscorresponding to fourth and fifth degree of Lagrangepolynomials ( =ln 4 and 5) are represented inFigure (13). As may be observed from these figures,good agreement between the results of the proposedSE/FE approach and the results obtained by theBE/FE [21] and simplified [35] approaches indicatesthe validity of the present algorithm.

3.5. Two Degrees of Freedom Structure

In this example, the application of the hybrid SE/FE approach to the interaction between a flexibletwo degrees of freedom structure and its supportinghalf-plane medium is investigated. The result of thepresent approach is compared with the results of ahybrid BE/FE method [21].

Figure (14) represents the two DOF stick modelof the structure and its supporting half-plane. The

Figure 12. Geometry and discretization of the 1DOF lumpedmass structure resting on an elastic half-plane.

Figure 14. Geometry, discretization, and applied loading of the2DOF lumped mass structure resting on an elastichalf-plane.

Figure 13. Time variations of horizontal displacement at thelevel of lumped mass: (a) fourth order and (b) fifthorder spectral elements are used for comparisonof three methods.

same material properties of Ref. [21] are chosen asfollows: the Lame constants λ = 6629990 and µ =3315000, and the mass density of the elastic half-plane is 0.000282. Moreover, the lumped masses ofeach degree of freedom are equal to 10 and arelocated at the story levels with the height of L = 12.The lateral rigidity of the structure is EI = 1.55x1012;all quantities in consistent units. No structuraldamping is considered in this example. For thisexample, the right hand side of Eq. (7) is modified



as the following form to represent the trapezoidalexcitation, see the inset of Figure (14), applied atthe top mass level

=

=

0

)()(;

0

0

)(tP

tF

tF

F

F

F

s

s

m

c

sr

r

r

r

r

r

r

(12)

The response of this structure is computed usinga mesh of eight square spectral elements to modelthe elastic half-plane, and two frame finite elementsfor the 2DOF structure. The response of the soil-structure system is firstly obtained using a fixedbase analysis to validate the FEM formulation ofthe present approach, see Figure (15). As expected,this analysis represents the condition in which thestructure experiences undamped oscillations in theinitial period of 1.08ms (forced vibration) followedby the rest period (free vibration) of the response.For the whole soil-structure system, the horizontaldisplacements of the first and second lumped

Figure 15. Horizontal displacement time histories of the fixedbase 2DOF structure at (a) the first and (b) the sec-ond lumped mass levels.

Figure 16. The coupled SEM/FEM results of horizontal displace-ment time histories at (a) the first and (b) the secondlumped mass levels.

masses corresponding to ln = 2 are represented inFigure (16). As can be seen from this figure, thegeneral trend of two approaches presents a reason-able agreement from engineering point of view.The discrepancy between two approaches may beattributed to the following features:a) The proposed SE/FE approach is two-dimen-

sional, whereas the BE/FE approach [21] isthree-dimensional, and

b) The principle difference between the SE and BEformulation of the present approach and the BE/FE one, respectively.

3.6. Five Story Frame Structure

In the last example, the application of the pro-posed approach to a more practical problem ofinteraction between a typical two-span five-storyframe consisting of twenty-five frame finite elementsand its supporting half-plane soil is studied, seeFigure (17). The following properties are selectedfor all structural elements: lateral rigidity EI = 2.52 x

1010, mass density ρ = 6000, and cross-section area



Figure 19. The horizontal displacement time histories at thefifth story level (point A) for three types of spectralelements with specified order of Lagrange polyno-mials.

Figure 20. The horizontal displacement time histories of pointA for two cases of far-field boundary conditions.

A = 1.00, all quantities in consistent units. No struc-tural damping is considered in this example. Inaddition, the following parameters are used forhalf-plane medium: EI = 2.6 x 109, ρ = 2000, and v =0.33. Figure (18) depicts the structural system,excited by a concentrated horizontal loading, see theinset of Figure (18), resting on a half-plane whichconsists of fourteen spectral elements (20 x 70m2)with a certain degree of Lagrange.

In order to show the convergence of resultstowards an exact solution, various meshes associ-ated to ln = 4, 6 and 8 are inspected whose resultsare presented in Figure (19). This figure indicatesthat the result corresponding to ln = 8 representsthe converged values of horizontal displacement.Furthermore, as this method considers soil-structureinteraction effects, damped motion of the structureresponse due to the radiation damping in the half-plane is clearly observed. In addition, the efficiencyof the viscous boundary condition employed inthe present research is examined in Figure (20).Obviously, the results of fixed (or, without viscousdampers) boundary conditions show significanterrors due to the problem of wave reflection on thefar-field boundaries of the half-plane domain.

To compare the responses of the soil-structuresystem according to variations in the stiffness ofhalf-plane domain, a time history analysis fordifferent Young's modulus is carried out using theproposed hybrid approach.

Figure 17. Geometry and applied loading (at point A) of thetwo-span five-story frame structure.

Figure 18. Geometry, discretization, and applied loading ofthe two-span five-story frame structure and itsbeneath elastic half-plane.



Figure 21. The horizontal displacement time histories at thefifth story level (point A) for various Young's modu-lus of half-plane medium.

be successfully modeled with few number of DOFs,preserving high accuracy and efficiency.

Further development of the proposed method isnecessary, as this method will be much moreefficient for the analysis of large three-dimensionaldomains. Moreover, this method has the advantageover other numerical methods in that it provides adirect time domain approach of obtaining the timehistory of the response. Using this advantage, thenew SE/FE method is appropriate to handle non-linearities and inhomogeneities of soil domain. Theaforementioned extensions of the present methodare under investigation whose results will be offeredin a future publication.

References

1. Wolf, J.P. (1995). “Dynamic Soil-StructureInteraction”, Prentice-Hall, NJ.

2. Wolf, J.P. and Song, C. (2002). “Some Corner-stones of Dynamic Soil-Structure Interaction”,Engineering Structures, 24, 13-28.

3. Chopra, A.K. and Chakrabarti, P. (1981). “Earth-quake Analysis of Concrete Gravity DamsIncluding Dam-Water-Foundation Rock Interac-tion”, Earthquake Engineering and StructuralDynamics, 9(4), 363-383.

4. Mroueh, H. and Shahrour, I. (2003). “A Full 3-DFinite Element Analysis of Tunneling-AdjacentStructures Interaction”, Computers and Geo-technics, 30, 245-253.

5. Ju, S.H. (2003). “Evaluating Foundation Mass,Damping and Stiffness by the Least-SquaresMethod”, Earthquake Engineering and Struc-tural Dynamics, 32(9), 1431-1442.

6. Maki, T., Maekawa, K., and Mutsuyoshi, H.(2006). “RC Pile-Soil Interaction AnalysisUsing a 3D-Finite Element Method with FibreTheory-Based Beam Elements”, EarthquakeEngineering and Structural Dynamics, 35(13),1587-1607.

7. Guddati, M.N. and Tassoulas, J.L. (1999).“Space-Time Finite Elements for the Analysisof Transient Wave Propagation in UnboundedLayered Media”, International Journal ofSolids and Structures, 36, 4699-4723.

Figure (21) shows the plots of time histories ofhorizontal displacement at the fifth story level forvarious Young’s modulus of soil medium. It isobserved that, as the Young’s modulus of half-planedecreases, the effect of soil-structure interaction onthe response of whole system is more considerable.Moreover, this effect considerably increases thenatural frequency of whole system, as expected.

4. Conclusions

The present research proposes a hybrid high-order (SEM)-low-order (FEM) formulation for thetransient analysis of interaction of soil-structurecoupled media. As realized from the detailed SEMformulation, the only discrepancies between SEMand classical FEM could be summarized in threefeatures: adoption of specific high-order shapefunctions, specific integration rule, and the nature ofthe time-marching scheme. These special featuresenable SEM to overcome some inherent difficultiesof other numerical methods (e.g., FEM and BEM)for which, illustrative discussions have been providedin this paper. In the proposed approach, the finitedomain, which would be considered as the structure,is represented by FEM and the half space mediumof soil is modeled by SEM. Dynamic mass, damping,and stiffness matrices corresponding to relateddomains of the SSI system are derived and combinedby using direct method. Several numerical studiesare performed to verify and examine the applicabilityof the hybrid model and the developed computerprogram. Comparison is made among differentanalytical and numerical (e.g., FEM and BEM)approaches to show that the selected problems may



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Soil-Structure Interaction (SSI) Analysis Using a Hybrid ... · Soil-Structure Interaction (SSI) Analysis Using a Hybrid Spectral Element/Finite Element (SE/FE) Approach flexibility