BEM FEM Soil Structure

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Applications of Staggered BEM-FEM Solutions to Soil-Structure

Interaction

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

[email protected]

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 a

superposition 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. The proposed methodology is applied on the problem of the dynamic through-the-soil

interaction 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 and

Wexley in the beginning of 1970’s on the microwave theory. The first organizedformulation was presented by Zienkiewicz and his coworkers (1977) for analysis of

solids. Coupled BEM-FEM procedures are of three general types. The first one is aBoundary Element (BE) approach, which considers the Finite Element (FE) subdomains

as equivalent BE subregions by transforming the force-displacement relations of the FEM

to “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. The

coupling of the FEM with the BEM for wave propagation and soil-structure interaction problems follows similar procedures. The solutions are obtained in either a direct time

domain, 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 BE

approach. 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 few

publications have dealt with the time domain BEM-FEM techniques for SSI and wave propagation 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 applicable

coupling 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 in

elastic media reported by Rizos (1993) and Rizos and Karabalis (1994,1998). A staggeredsolution algorithm for the coupling with standard FEM processes in the direct time

domain 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 that

the time variation of the body forces in the domain is defined by the B-Spline polynomials. The derived B-Spline fundamental solutions accommodate virtually any

order of parametric representation of the time dependent variables without excessivecomputational effort and implicitly satisfy the continuity conditions of the Stokes

fundamental 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 and

homogeneous space, the Navier-Cauchy equations of motion can be expressed in theLove integral identity form as,

( ) ( ) ( ) ( ) ( )[ ]{ }∫ −=S

i

B

iji

B

ijiij dS t ut T t t t U t uc ,;,,;,, x î xx î x î î 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” term

that depends on the geometric characteristics of the domain in the neighborhood of the

receiver point. The tensors ( )t ui ,x , and ( ) ( )t t i ,xn are the displacement and traction fields

of the actual elastodynamic state, and the tensors B

ijU , 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 of

tractions to forces are applied on equation (1) and the system of algebraic equations isderived as

RNf GRNf LGuT +=+= − N

soil

N

soil

N ~1** (2)

where T* and U

* are coefficient matrices derived on the basis of the B-Spline

fundamental solutions and depend only on the first and/or second time steps, L is the

traction-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 is

always known. Equation (2) can be solved in a time marching scheme to obtain the B-Spline impulse response of the elastodynamic system. The B-Spline impulse response

of 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

1

1 >−

+++=∈−= −++

=∑ k

k

t t t t t t t t t k iii

i

j

i

jii

L

ττf BR u (3)



where k is the order of the B-Spline fundamental solutions. Matrices BR are independent

of 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, FE

if , to displacements FE

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

are presented symbolically as,

FE

i

FE

i ˜˜ f uK ˆˆ = (4)

where K ˆ is the dynamic stiffness matrix and ^ indicates quantities related to the Newmark’s process.

BEM-FEM Coupling

At the interface of the FE and BE models, the compatibility conditions of the

displacement vector, u, and force vector, f , need to be satisfied at all time instances t j,

( ) ( ) ( ) ( ) j

FE

j

BE

j

FE

j

BE t t t t intintintint f f uu −== (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 at

every time step, as depicted in Figure 1. In view of equation (3) the BEM solver evaluates displacements by a superposition of the B-Spline impulse responses without

solving any system of equations. The FEM solver, however, needs to solve for the

unknown 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

FEMSolver

Structure

j

BE

j

FE

t t intint uu =

( ) ( ) j

BE

j

FE t t intint f f −=

BEMSolver

Soil



Numerical Examples

Analysis of Rigid Sur face Massive Foundations

This example examines the through-the-soil interaction of two adjacent massive rigidfoundations. The “structural component” of this soil-structure system pertains only to

inertia 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 their

weights correspond to mass ratio M=10. The constants of the surrounding soil medium

are Poisson’s ratio ν = 1/3, mass density ρ=10.368 lb.sec2/ft

4 and modulus of elasticity

E=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 are

implemented according to the rigid surface element introduced by Rizos (1999). Thefrequency domain solutions are due to harmonic forcing functions of unit magnitude

applied on one foundation. The B-Spline impulse response matrices of the foundationsystem are obtained only once. Subsequently, for each excitation, the solution is obtained

in the time domain through the procedure outline above and the maximum amplitude of

the 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 foundations

as 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

A m p l i t u d e

Excited - Proposed Work Excited - Huang

Unloaded-Proposed Work Unloaded - Huang

Posin(ωt)

Half Space

2b

d



Hollow Rigid Sur face Foundation

This example examines a square footing of size 2a=5 ft with a centered square hole of size 2d for which d/a=0.75. The mass of the foundation varies so that the mass ratio M

takes 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 are

applied 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 the

dimensionless 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 have

been 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 the

Boundary Element Method for 3-D wave propagation and Soil-Structure InteractionAnalysis in the direct time domain. The method uses the newly developed B-Spline

BEM along with standard FEM processes. The coupling is obtained through a staggeredscheme, which satisfies the compatibility and equilibrium conditions at the interface boundary between the BEM and FEM domains. This article presented the first attempt to

implement the method and the problem of analysis of massive foundations was selected.In such problems, kinematic and inertial interaction effects are present. Although the

FEM domain does not contain elastic or damping forces of a real structure, this class of problems verifies the suitability of the B-Spline BEM method to such staggered solution

schemes. 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

A m p l i t u d e ( f t )

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

M=10

M=5

M=1

M=3Half Space

Posin(ωt)

2a2d



class of problems considered, and is very efficient. The present formulations will be

further developed to account for elastic and damping forces of a structure.

References

D. Aubry and D. Clouteau (1992): A subdomain approach to soil-structure interactionJ. Bielak, R.C. MacCamy and D.S. McGhee (1984): On the coupling of finite element

and boundary integral methods. In S.K. Datta (Ed.): Earthquake Source Modeling,Ground Motion, and Structural Response, AMD-Vol 60, pp. 115-132, ASME New York.

Z. Chuhan, J. Feng and W. Guanglun (1993): A method of FE-BE-IBE coupling for seismic interaction of arch dam-canyons. In M. Tanaka et al. (eds): Boundary Element

Methods. Elsevier.O. von Estorff (1990): Soil-structure interaction analysis by a combination of boundary

and finite elments. In: S.A. Savidis (ed.): Earthquake Resistant Constructions and Design.Balkem (Rotterdam).

O. von Estorff, and E. Kausel (1989): Coupling of boundary and finite elements for

soil-structure interaction problems, Earthquake Engineering and Structural Dynamics,18, 1065-1075.T. Fukui (1987): Time marching BE-FE method in 2-D elastodynamic problem.

International Conference BEM IX Stuttgart.A.P. Gaitanaros and D.L. Karabalis (1986): 3-D flexible embedded machine

foundations by BEM and FEM. In: D.L. Karabalis (ed.): Recent Applications inComputational Mechanics. ASCE (New York).

C-F Huang (1993 ) : Dynamic Soil-Foundation and Foundation-Soil-Foundation Interaction in 3-D. Ph.D. Dissertation, University of South Carolina, Columbia SC.

D.L. Karabalis and D.E. Beskos (1985): Dynamic response of 3-D flexible foundations by BEM and FEM. In D.L. Karabalis (ed.): Recent Applications in Computational

Mechanics. ASCE (New York).D.C. Rizos (1999): A Rigid Surface Element for the B-Spline direct time domain BEM.

Computational Mechanics. (To appear).D.C. Rizos (1993): An Advanced Time Domain Boundary Element Method For General

3-D Elastodynamic Problems. Ph.D. Dissertation, University of South Carolina,Columbia South Carolina .

D.C. Rizos, and D.L. Karabalis (1998): A time domain BEM for 3-D elastodynamicanalysis using the B-Spline fundamental solutions, Computational Mechanics, 22, No. 1,

pp 108-115.D.C. Rizos, and D.L. Karabalis (1994): An advanced direct time domain BEM

formulation for general 3-D elastodynamic problems. Computational Mechanics. 15,249-269.

C.C. Spyrakos and D.E. Beskos (1986): Dynamic response of flexible strip foundations by boundary and finite elements. Soil Dynamics and Earthquake Engineering , 5, 84-96.

O.C. Zienkiewicz, D.W. Kelly, and p. Bettess (1977): The coupling of the FiniteElement Method and Boundary Solution procedures, International Journal for Numerical

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