Applications of Staggered BEM-FEM Solutions to Soil-StructureInteraction

D.C. Rizos, M. ASCE and Z.Y. WangUniversity of Nebraska Lincoln, Lincoln NE 68588

drizos@unlinfo.unl.edu

Abstract:

The present work introduces a direct time domain BEM-FEM formulation for 3-D Soil-Structure Interaction analysis. The proposed BEM that is based on the B-Spline familyof fundamental solutions computes the dynamic response of the soil domain through asuperposition of the characteristic B-Spline impulse responses. Standard directintegration FEM procedures are used to compute the dynamic response of the structure.A staggered solution process is proposed for the coupling of the two methods. Theproposed methodology is applied on the problem of the dynamic through-the-soilinteraction of massive foundations, and the examples presented in this work demonstratethe accuracy and efficiency of the method.

Introduction

The idea of coupling the FEM and the BEM finds its origins in the work of McDonal andWexley in the beginning of 1970’s on the microwave theory. The first organizedformulation was presented by Zienkiewicz and his coworkers (1977) for analysis ofsolids. Coupled BEM-FEM procedures are of three general types. The first one is aBoundary Element (BE) approach, which considers the Finite Element (FE) subdomainsas equivalent BE subregions by transforming the force-displacement relations of the FEMto “BEM-like” traction-displacement relations. The second approach is the FE, in which,the BE equations are considered as a special case of the FE procedures. StaggeredBEM-FEM solutions have been implemented in fluid-structure interaction analysis. Thecoupling of the FEM with the BEM for wave propagation and soil-structure interactionproblems follows similar procedures. The solutions are obtained in either a direct timedomain, or a frequency (transform) domain approach. Most of the coupled FEM-BEMsolutions reported in the literature are in the frequency domain and adopt the FE or BEapproach. One can mention the work of Bielak et al. (1984), Gaitanaros and Karabalis(1986), Aubry and Clouteau (1992), and Chuhan at al. (1993), among others. Only a fewpublications have dealt with the time domain BEM-FEM techniques for SSI and wavepropagation problems. Karabalis and Beskos (1985) and Spyrakos and Beskos (1986)have reported on 2-D and 3-D flexible foundations following the BE approach. Fukui(1987) and Estorff and Kausel (1989,1990) reported on more generally applicablecoupling formulations for 2-D scattering of anti-plane waves and 2-D plane-strain.This work employs the B-Spline BEM formulation for 3-D wave propagation and SSI inelastic media reported by Rizos (1993) and Rizos and Karabalis (1994,1998). A staggeredsolution algorithm for the coupling with standard FEM processes in the direct timedomain is introduced.

BEM Formulation

The direct time domain BEM employed in this work is developed based on a special caseof the Stokes fundamental solutions for the infinite elastodynamic space that assumes thatthe time variation of the body forces in the domain is defined by the B-Splinepolynomials. The derived B-Spline fundamental solutions accommodate virtually anyorder of parametric representation of the time dependent variables without excessivecomputational effort and implicitly satisfy the continuity conditions of the Stokesfundamental solutions. A detailed formulation and integration of the B-Splinefundamental solutions and of the associated BEM can be found in the work of Rizos(1993), and Rizos and Karabalis (1994, 1998).

Under the assumption of a small displacement field in a linear isotropic andhomogeneous space, the Navier-Cauchy equations of motion can be expressed in theLove integral identity form as,

( ) ( ) ( ) ( )[ ] ( )[ ]{ }∫ −=S

iB

ijiBijiij dStutTtttUtuc ,;,,;,, xîxxîxîî n (1)

where S is the bounding surface of the elastodynamic domain, î, and x represent thereceiver and source points, respectively, and the tensor cij is known as the “jump” termthat depends on the geometric characteristics of the domain in the neighborhood of thereceiver point. The tensors ( )tui ,x , and ( ) ( )tt i ,xn are the displacement and traction fields

of the actual elastodynamic state, and the tensors BijU , and B

ijT are the B-Spline

fundamental solutions of the infinite elastodynamic space (Rizos and Karabalis 1998).Appropriate spatial and temporal discretization schemes along with a transformation oftractions to forces are applied on equation (1) and the system of algebraic equations isderived as

RNfGRNfLGuT +=+= − Nsoil

Nsoil

N ~1** (2)

where T* and U* are coefficient matrices derived on the basis of the B-Splinefundamental solutions and depend only on the first and/or second time steps, L is thetraction-force transformation matrix obtained on a virtual displacement approach andvector RN represents the influence of the past time steps on the current step N and isalways known. Equation (2) can be solved in a time marching scheme to obtain theB-Spline impulse response of the elastodynamic system. The B-Spline impulse responseof all degrees of freedom due to unit force excitations can be collected in a matrix formthat represents the time dependent flexibility matrix, BR(t), of the elastodynamic system.The response u(t) to an arbitrary excitation f(t) can be calculated by an appropriatesuperposition of the B-Spline impulse responses as

( ) ( ) [ ] 1 ,1

and , ,)( 11

11 >

−+++

=∈−= −++

=∑ k

k

ttttttttt kiii

i

j

ijii

LττfBRu (3)

where k is the order of the B-Spline fundamental solutions. Matrices BR are independentof the actual external excitations and need to be computed only once for each soil region.

FEM Formulation

The FEM system of equations for the structural model is also solved in a time marchingscheme using Newmark’s algorithm. The FEM equations relate incremental forces,

FEiÄf , to displacements FE

iÄu on discrete nodes in the FE model at time interval i, and

are presented symbolically as,

FEi

FEi ÄÄ fuK ˆˆ = (4)

where K̂ is the dynamic stiffness matrix and ^ indicates quantities related to theNewmark’s process.

BEM-FEM Coupling

At the interface of the FE and BE models, the compatibility conditions of thedisplacement vector, u, and force vector, f, need to be satisfied at all time instances tj,

( ) ( ) ( ) ( )jFE

jBE

jFE

jBE tttt intintintint ffuu −== (5)

The coupling of the two models is achieved through a staggered solution approachaccording to which the solution of one method serves as initial conditions to the other atevery time step, as depicted in Figure 1. In view of equation (3) the BEM solverevaluates displacements by a superposition of the B-Spline impulse responses withoutsolving any system of equations. The FEM solver, however, needs to solve for theunknown forces at the BE-FE interface. It is apparent that this coupling scheme increasesthe efficiency of the solution by reducing the computing time. The proposed scheme hasshown superior accuracy and stability for the examples examined so far.

Figure 1 Staggered Solution Scheme

uBE

FEMSolver

Structure

( ) ( )jBE

jFE tt intint uu =uFE

( ) ( )jBE

jFE tt intint ff −=fFE fBE

BEMSolver

Soil

uBE

Numerical Examples

Analysis of Rigid Surface Massive FoundationsThis example examines the through-the-soil interaction of two adjacent massive rigidfoundations. The “structural component” of this soil-structure system pertains only toinertia forces, which are not known a priori, and demonstrates the compatibility of the BEsolver with the proposed solution scheme. The footings are square of side 2b=5 and theirweights correspond to mass ratio M=10. The constants of the surrounding soil mediumare Poisson’s ratio ν = 1/3, mass density ρ=10.368 lb.sec2/ft4 and modulus of elasticityE=2.5898x109 lb/ft2. The surface of the half space is modeled by 8 node quadrilateralelements and each footing covers an area of 4x4 elements. The rigid conditions areimplemented according to the rigid surface element introduced by Rizos (1999). Thefrequency domain solutions are due to harmonic forcing functions of unit magnitudeapplied on one foundation. The B-Spline impulse response matrices of the foundationsystem are obtained only once. Subsequently, for each excitation, the solution is obtainedin the time domain through the procedure outline above and the maximum amplitude ofthe steady state is defined. This approach is very efficient since the BE solver reduces toa mere superposition of pre-computed quantities, as implied by Equation (3). Figure 2shows the maximum amplitude of response of the excited and the unloaded foundationsas a function of the dimensionless frequency. In this example the footings are spaced at adistance d/b=0.25 apart. The results are compared to the ones reported by Huang (1993)and the accuracy is evident. Other modes of vibration as well as distance ratios haveshown the same accuracy.

Figure 2 Through-the-Soil interaction of Massive Rigid Foundations

0.0E+00

2.0E-11

4.0E-11

6.0E-11

8.0E-11

1.0E-10

1.2E-10

0 0.5 1 1.5 2 2.5 3 3.5 4

Dimensionless Frequency ao

Am

plit

ud

e

Excited - Proposed Work Excited - Huang

Unloaded-Proposed Work Unloaded - HuangPosin(ωt)

Half Space

2b

d

Hollow Rigid Surface FoundationThis example examines a square footing of size 2a=5 ft with a centered square hole ofsize 2d for which d/a=0.75. The mass of the foundation varies so that the mass ratio Mtakes the values of M=1,3,5 and 10. The footing rests on the elastic half-space describedin the previous example. A series of harmonic excitations of various frequencies areapplied at its center in order to define the amplitude of the steady state response. Theamplitudes of the vertical mode of vibration are shown in Figure 3 as function of thedimensionless frequency for all mass ratios considered. A comparison with a solutionreported by Huang (1993) for mass ratio M=3 is also shown. All modes of vibration havebeen examined, as well as a number of d/a ratios. The proposed method comparedfavorably and always converged for the considered frequency range.

Figure 3 Vertical Response of Hollow Foundation

Conclusions

A methodology is developed for the efficient coupling of the Finite Element with theBoundary Element Method for 3-D wave propagation and Soil-Structure InteractionAnalysis in the direct time domain. The method uses the newly developed B-SplineBEM along with standard FEM processes. The coupling is obtained through a staggeredscheme, which satisfies the compatibility and equilibrium conditions at the interfaceboundary between the BEM and FEM domains. This article presented the first attempt toimplement the method and the problem of analysis of massive foundations was selected.In such problems, kinematic and inertial interaction effects are present. Although theFEM domain does not contain elastic or damping forces of a real structure, this class ofproblems verifies the suitability of the B-Spline BEM method to such staggered solutionschemes. It has been shown that the proposed methodology is accurate and stable for the

0.0E+00

2.0E-11

4.0E-11

6.0E-11

8.0E-11

1.0E-10

1.2E-10

0 1 2 3 4 5Dimensionless Frequency a0

Am

plit

ud

e (f

t)

M=10 M=5 M=3 M=1 Huang M=3M=10

M=5

M=1

M=3Half Space

Posin(ωt)

2a2d

class of problems considered, and is very efficient. The present formulations will befurther developed to account for elastic and damping forces of a structure.

References

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