Numerical Methods in Soil Mechanics BEM FEM soil structure

Numerical Methods in Soil Mechanics BEM FEM soil structure The chapter reviews aspects of the soil–structure interface behavior at the element level and the numerical integration of the corresponding interface constitutive models. The design of structures subjected to soil–structure interaction and to contact with friction should be tackled using soil–structure interface constitutive equations. These laws differ from the laws for soils because of three main features: the size of the relative displacements and of relative rotations between grains, the high level of dilatancy and contraction under shearing, and the presence of an intense degradation effect resulting from localization in the pattern of a shear band. The elastoplastic interface constitutive equations are easy to use but do not modelize all these effects. The incrementally non-linear interface constitutive equations are versatile for modeling all these interdependent phenomena. In addition, applications to piles under tension loading are presented to illustrate the results of these procedures.

Applications of Staggered BEM-FEM Solutions to Soil-Structure

Interaction

D.C Rizos, M ASCE and Z.Y Wang

University 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 family

of fundamental solutions computes the dynamic response of the soil domain through a superposition of the characteristic B-Spline impulse responses Standard direct

integration 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 demonstrate the 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 organized

formulation 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 a Boundary 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 Staggered

BEM-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-BEM solutions 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 staggered solution 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 case

of 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 excessive computational effort and implicitly satisfy the continuity conditions of the Stokes

fundamental solutions A detailed formulation and integration of the B-Spline

fundamental 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 the Love integral identity form as,

( ) ( )=∫ { [ ( )( ) ]− [ ( ) ] }

S

i B

ij i

B ij i

c î î, x, ;î n x, x, ;î x, (1)

where S is the bounding surface of the elastodynamic domain, î, and x represent the

receiver and source points, respectively, and the tensor c ij is known as the “jump” term that depends on the geometric characteristics of the domain in the neighborhood of the receiver point The tensors u i( )x,t , and t( )ni( )x,t are the displacement and traction fields

of the actual elastodynamic state, and the tensors U , and ij B T are the B-Spline ij B

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 is derived as

RN f

G RN f

L G u

soil N

soil

*

(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 and

vector 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 form

that 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 appropriate

superposition of the B-Spline impulse responses as

1

and , , )

−

+ + +

=

∈

−

=

k

t t

t t

t t t

t

i j

j i

i

L

τ τ

f BR

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 marching scheme using Newmark’s algorithm The FEM equations relate incremental forces,

FE

i

˜f , to displacements ˜u i FE on discrete nodes in the FE model at time interval i, and

are presented symbolically as,

FE i FE

Kˆ = ˆ (4)

where Kˆ 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,

FE j

BE j

FE j BE

t t

t

int u f f

The coupling of the two models is achieved through a staggered solution approach according 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 increases the efficiency of the solution by reducing the computing time The proposed scheme has shown superior accuracy and stability for the examples examined so far

Figure 1 Staggered Solution Scheme

uBE

FEM

Solver Structure

( ) ( )j BE j FE

t

uFE

( ) ( )j

BE j

FE t int t

int f

BEM

Solver Soil

uBE

Numerical Examples

Analysis of Rigid Surface Massive Foundations

This example examines the through-the-soil interaction of two adjacent massive rigid foundations 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 BE solver 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/ft4 and modulus of elasticity E=2.5898x109 lb/ft2 The surface of the half space is modeled by 8 node quadrilateral elements and each footing covers an area of 4x4 elements The rigid conditions are implemented according to the rigid surface element introduced by Rizos (1999) The frequency domain solutions are due to harmonic forcing functions of unit magnitude applied on one foundation The B-Spline impulse response matrices of the foundation system 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 to

a mere superposition of pre-computed quantities, as implied by Equation (3) Figure 2 shows 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 a distance 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 have shown 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

Excited - Proposed Work Excited - Huang Unloaded-Proposed Work Unloaded - Huang

P o sin( ω t)

Half Space

2b d

Hollow Rigid Surface 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 described

in 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 The amplitudes 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 solution reported 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 compared

favorably 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 Interaction Analysis 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 staggered scheme, 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

Dimensionless Frequency a 0

M=10

M=5

M=1

P o sin( ω t)

2a 2d

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

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