APPLICATION OF THE BOUNDARY ELEMENT METHOD FOR SOIL ...

APPLICATION OF THE BOUNDARY ELEMENT METHOD FOR SOIL STRUCTURE INTERACTION PROBLEMS

by

JAYARAMAN SIVAKUMAR, B.E., M.E., M.S. in C.E.

A DISSERTATION

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

December, 1985

fol

l\Jd>' l l ^ ACKNOWLEDGMENTS

The author expresses h is deep sense of g r a t i t u d e to Dr. C. V.

G i r i j a Vallabhan for the keen supervision, encouragement and guidance

throughout the course of t h i s work. He also thanks Dr. K. C. Mehta,

Dr. W. P. Vann, and Dr. D. Gi l l iam fo r t he i r valuable suggestions as

members of the committee. Thanks are also due to Dr. R. P. Selvam for

f r u i t f u l d iscuss ions, Dr. J . R. McDonald f o r w i l l i n g n e s s to be an

examiner, and to Mr. D. Chou and Mr. K. Hong for a l l t h e i r help. The

author is indebted to Dr. E. W. K ies l ing, Chairman of the Department

o f C i v i l Eng ineer ing, f o r the f i n a n c i a l ass is tance throughout the

course of his study at Texas Tech Univers i ty .

The author thanks his wonderful parents f o r t h e i r a f f e c t i o n and

e x t r a o r d i n a r y moral support and the s a c r i f i c e s they have made, and

especia l ly thanks his wi fe, Shanthi, fo r her unl imited support.

The work presented here is part of the funded research p r o j e c t

sponsored by the U. S. Army Corps of Engineers, Waterways Experiment

Stat ion, Vicksburg, MS, whose f i nanc ia l ass is tance i s acknowledged.

F ina l l y , thanks are also due to Mrs. Sheryl Hensley fo r an outstanding

work in compi l ing and t yp i ng t h i s manuscr ip t , and to Mr. Richard

Dill ingham for his ed i t o r i a l assistance.

11

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ii

ABSTRACT v

LIST OF TABLES vii

LIST OF FIGURES viii

NOTATIONS xi

I. SOIL STRUCTURE INTERACTION 1

Introduction 1

Winkler Model 1

Alternate Methods 5

Boundary Element Method 5

Review of Literature 6

Scope of Research 7

II. BOUNDARY ELEMENT METHOD 9

Introduction 9

Basic Equations of Elasticity 9

Equations for Boundary Element Method 11

Fundamental Solution 16

III. NUMERICAL ANALYSIS 19

Introduction 19

Interpolation Functions 20

Numerical Analysis 20

Two-Dimensional Analysis 21

Three-Dimensional Analysis 24

IV. COUPLING OF BOUNDARY AND FINITE ELEMENT METHODS

IN TWO-DIMENSIONAL ELASTICITY PROBLEMS 31

Introduction 31

Equations Used in Coupling 31

Finite Element Equation 32

Boundary Element Equation 34

Soil Stiffness for Two Dimensional Problems 34

Boundary Element Model 35

iii

Compatible Boundary Stiffness Matrix 38

Method I 39

Method II 39

Properties of Boundary Element Stiffness Matrix 41

Coupling of Finite and Boundary Element Matrices 42

Example Problem 42

Discretization of the Example Problem 45

Presentation of Results 46

Comparison of Results 49

Application to a Layered Soil System 68

Previous Work Done 68

Boundary Element Technique for Layered Model 69

V. COUPLING OF BOUNDARY AND FINITE ELEMENT METHODS

IN THREE-DIMENSIONAL ELASTICITY 75

Introduction 75

Finite Element Equation 75

Boundary Element Equation 76

Coupling of Finite and Boundary Element Matrices 77

Example Problem 81

Comparison of Results 84

Summary 94

VI. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 98

Summary and Conclusions 98

Recommendations for Future Work 101

REFERENCES 102

TV

ABSTRACT

Soil-structure interaction problems are those in which the

behavior of the structure and the behavior of the soil surrounding it

are interdependent, and the solution requires the analysis of both the

structure and the soil in a compatible manner. Modeling of soil is

very complicated and approximations are made purely on the experience

and judgment of the engineer. So far, because of the simplicity of

the concept, Winkler's Model is being used extensively in soil-

structure interaction problems. Closed form solutions are available

only for simple geometry and loading conditions, thereby restricting

the analysts to idealize the problem. There are improved models

developed by Pasternak, Vlasov and Leontiev, adding complexities in

calculations. The drawback in these analyses is the nonunique value

of the coefficient of subgrade reaction of the soil. Recently, the

finite element method has been used to solve these problems. Here the

advantage of modeling the problem is offset by the tedium in the

preparation of input data for the analysis. It is in this context

that the boundary element method is used in this research for

application in some soil-structure interaction problems.

In this work, the structure is represented by finite elements and

the soil medium by boundary elements. The soil stiffness matrix is

developed and condensed up to the interface. This matrix is

efficiently transformed and coupled to the structure stiffness matrix

for complete analysis. Computer programs have been developed in two

and three dimensional elasticity for application to typical soil-

structure interaction problems. Also, a condensation procedure has

been suggested for analyzing layered soil media. The results compare

favorably with complete finite element analysis.

The elastic constants of the materials are sufficient for the

analysis, thereby totally avoiding the value of the coefficient of

subgrade reaction. Thus, the codes developed establish their

superiority for implementation in soil-structure interaction problems.

This procedure is a starting step in geotechnical problems for an

accurate and rational analysis to replace the semiempirical relations

presently in use.

VI

LIST OF TABLES

Table Page

4.1 Details of Example Problems 45

4.2 Comparison of Results with FEM 74

5.1 Comparison of Results with Other Methods 93

vn

LIST OF FIGURES

Figure Page

1.1 Beam on elastic foundation 4

2.1 An elastic domain with boundary conditions 10

2.2 Augmented surface for integration on the boundary 15

2.3 Kelvin's fundamental problem 17

3.1 Two-dimensional domain discretized into boundary

elements 22

3.2 Three-dimensional domain discretized into plane quadrilaterals 25

3.3 Quadrilateral boundary element 27

3.4 Quadrilateral element in dimensionless co-ordinate system 28

4.1 Domain with different properties 33

4.2 Boundary element model for a U-lock structure 36

4.3 Details of example problem 44

4.4 Discretization of the combined model 47

4.5 Discretization of the finite element analysis 48

4.6 Vertical displacement at the interface—case 1 and 2

of loading condition 1 49

4.7 Vertical displacement at.the interface—case 3 and 4 of loading condition 1 50

4.8 Vertical displacement at the interface—case 1 and 2 of loading condition 2 51




viii

Figure Page

4.12 Vertical tractions at the interface—case 1 and 2 of loading condition 1 55






4.18 Comparison of displacements at the interface— case 1 and 2 of loading condition 1 62





4.23 Comparison of displacements at the interface—

case 3 and 4 of loading condition 3 67

4.24 Boundary element model for layered media 71

5.1 Interface displacement relationship 78

5.2 Details of the pile example problem 82

5.3 Boundary element discretization of the top surface

of soil region 83 5.4 Vertical displacement of pile/soil interface

due to axial load 85

IX

Figure Page

5.5 Lateral displacement of pile/soil interface due to lateral load 86

5.6 Lateral displacement of pile/soil interface due to moment 87

5.7 Lateral traction of pile/soil interface due to

lateral load 88

5.8 Plan view of discretization of the finite element model 89

5.9 Comparison of vertical displacement of pile/soil

interface due to axial load (circular pile) 90 5.10 Comparison of lateral displacement of pile/soil

interface due to lateral load (circular pile) 91

5.11 Comparison of lateral displacement of pile/soil interface due to moment (circular pile) 92

5.12 Comparison of vertical displacement of pile/soil interface due to axial (square pile) 95

5.13 Comparison of lateral displacement of pile/soil interface due to lateral load (square pile) 96

5.14 Comparison of lateral displacement of pile/soil interface due to moment (square pile) 97

NOTATIONS

The following symbols are used in this dissertation:

A - System matrix

B - Boundary

b. - Body force

b - Vector at interface of region 'i'

c(s) - Constant of the diagonal term in 'H' matrix

C. . -, - Material property tensor

E - Modulus of elasticity

E, - Modulus of elasticity of the beam

E - Modulus of elasticity of the soil

F - Force vector

fg - Condensed force vector

G - Matrix

G - Shear modulus

||G|1 - Jacobian relating areas between two co-ordinate systems

g - Components of the Jacobian 'G'

gg - Interface of finite and boundary element regions

h - Depth of soil

H - Matrix

K - Coefficients of stiffness matrix

k - Subgrade modulus

kg - Soil stiffness matrix

L - Length of the beam

M - Number of boundary elements

xi

N. - Shape functions

n. - Unit normal vectors J

m - Modular ratio

p - Pressure

q - Field point

r - Distance between the field point 'q' and the source point 's'

R - Matrix for conversion of tractions into forces

s - Source point

S - Structure

T - Transformation matrix n ^h '

T. - Nodal traction on n boundary element J

t - Traction vector

t. - Thickness of the beam

t - Prescribed traction vector

U. - Nodal displacement on n boundary element

u - Displacement vector

u* - Weighting function

u - Prescribed displacement vector

V - Interface vector

V - Vector of prescribed forces and/or displacements

W. - Weighting factor

X - Vector of unknowns

Y - Vector of knowns

X. - Global cartesian co-ordinates

xn

a - Constant

3 - Constant

V - Poisson's ratio

V, - Poisson's ratio of beam

V - Poisson's ratio of soil s

a. . - Stress tensor

f2 - Volume of the body

fi* - infinite space

r - Boundary of the body

A., - Operator

6. . - Kronecker delta

4). . - Interpolation functions

Til. . -" Interpolation functions

^. - Co-ordinate of i integration point

n. - Co-ordinate

e - Radius of a sphere

TT - Constant of value 3.14

e. . - Linear strain tensor

xn 1

CHAPTER I

SOIL STRUCTURE INTERACTION

Introduct ion

S o i l - s t r u c t u r e i n t e r a c t i o n problems are those i n wh ich t h e

b e h a v i o r o f the s t r u c t u r e and t h a t of the surrounding s o i l are

interdependent, and the s o l u t i o n requ i res an ana lys i s of both the

s t r u c t u r e and the s o i l in a compatible manner. While structures are

often s a t i s f a c t o r i l y modeled as l i n e a r l y e l a s t i c , homogeneous and

i s o t r o p i c m a t e r i a l s , the modeling of so i l s is extremely complex. To

model the i n - s i t u b e h a v i o r o f s o i l s , one must make g r o s s

approximat ions using exper ience and judgment, and these evaluations

are mostly based on the r e l a t i v e importance of the p r o j e c t and the

des i red accuracy. The complexities in the cons t i tu t i ve re la t ions of

the so i l continuum are enhanced by the f a c t t h a t the s o i l has been

deposited in nature in a layered heterogeneous manner. Since, in most

ins tances , engineers are i n t e r e s t e d in the behavior of structures

only, they have assumed very s i m p l i f i e d p r o p e r t i e s of the s o i l i n

t he i r design considerations. One of the widely used models for s o i l -

st ructure in teract ion problems is the Winkler sp r ing model [31] f o r

ana lys is of beams on e las t i c foundation, mat foundations, pavements,

p i l e foundations, etc.

Winkler Model

Winkler proposed in 1867 that the def lect ion of the so i l surface

1

can be modeled by a simple equation,

p = kw (1.1)

where p is the pressure acting on the soil surface and k is the

proportionality constant, known as the subgrade modulus or the modulus

of subgrade reaction, and w is the deflection of the loaded region.

The unit of k is in pounds per cubic inch. Normally, plate bearing

tests are conducted to determine the value of k. This concept has

been widely used by engineers. For a given value of k, Hetenyi [17]

solved many problems of beams on elastic foundation, and Westergaard

[40] solved problems of slabs on elastic foundations.

When the question of the value of k for soil was raised, Terzaghi

[32] came up with some general guidelines. Matlock and Reese [24]

have widely used this technique for solving problems of laterally

loaded pile foundations. Terzaghi showed that even with linearly

elastic, isotropic and homogeneous properties for soil, the modulus of

subgrade reaction depends very much on the size of the loaded area.

Vesic [38] showed that the value of k is influenced by the stiffness

of the beam. For values of k, many empirical formulae arose, based on

experiments and theoretical concepts. These are always questioned by

structural engineers who often demand a value of k for the soil, for

the soil-structure interaction analysis.

The absence of a unique value of k for soil can be easily

realized from the following examples, even if one assumes idealized

properties such as linear, elastic, etc., for the soil continuum.

Consider a plane strain problem, where a strip of beam is resting on a

semi-infinite soil continuum and is loaded uniformly. If the beam has

very low rigidity, the deflections of the beam vary from a maximum

value at the center to a smaller value at the ends, as shown in Fig.

1.1(a). On the other hand, if the beam is very rigid compared to the

soil, the deflections along the beam are uniform, while the pressure

distribution varies from infinity at the edge to a finite value at the

center. This situation is illustrated in Fig. 1.1(b). In both cases,

the ratio of pressure to deflection is not a constant at the soil-

structure interface, hence the nonuniqueness of the value of k is

demonstrated.

Realizing these problems, Pasternak [28] and Vlasov and Leont'ev

[39] developed a two-parameter model to take into consideration the

end effects. But the evaluation of these parameters again becomes a

major problem confronting the geotechnical engineer. In addition, it

can be shown that the value of k depends on the depth of the soil

continuum, the stiffness of the structure and the distribution of the

loading. Many researchers [40] use Boussinesq equations assuming that

the soil continuum is semi-infinite. If a hard rock stratum occurs at

a finite depth, however, the use of a semi-infinite soil mass concept

can lead to substantial errors.

In spite of these limitations, for analysis of moderately simple

structures, the concept of the constant or even nonlinear k is used by

engineers. But when these concepts are applied to the analysis of

major structures such as a hydraulic U-lock, a better and more

accurate determination of the soil stiffness based on its constitutive

Uniform traction

mmmmmmmm\

Displacement

(a) Flexible beam

Uniform t rac t ion

Constant displacement

''Lt±±i^ Vertical traction

(b) Rigid beam

Fig. 1.1 Beam on elastic foundation

relations is warranted.

Alternate Methods

An alternate and better method of analysis is to model the

involved soil continuum in its entirety. Closed form mathematical

solutions to achieve this goal become too tedious even for relatively

simple problems, but electronic computers have made it possible to

analyze these problems numerically using methods such as finite

difference and finite element. Researchers [13, 37] have used finite

element methods for solving soil-structure interaction problems.

However, these methods become cumbersome as the number of unknowns

increases rapidly, and further when the discretizations of the

continuum are altered for more accurate representations of the soil

continuum. Normal discretization of a soil-structure interaction

problem results in a fairly coarse mesh of the structure. It is in

this context that the boundary element method is found to be

advantageous in the analysis of soil-structure interaction problems.

Boundary Element Method

The boundary element method (BEM) is gaining popularity and is

used extensively as a solution technique comparable to other numerical

methods such as finite difference and finite element. The method is

easily explained through a weighted residual procedure and is well

established [2, 4, 5]. In this method, two dimensional domains are

represented at the boundary as line elements, and three dimensional

domains are modeled at the boundary surface as surface elements. The

method offers a reduction in the dimensionality of the problem.

The boundary of the domain alone is discretized for analysis.

Therefore, the input data are highly simplified, reducing man hours in

preparation of data. Modeling of domains extending to large distances

is carried out very efficiently. The only disadvantage is that the

system matrix is fully populated and computational efficiency is not

great, as compared to solving half-banded matrices in the finite

element method. For slender regions, the domain can be divided into

regions to get a banded matrix. Overall, the accuracy and efficiency

of the method are much higher than those of the other prevailing

methods for soil-structure interaction problems. It is found to be an

excellent procedure for handling a large soil continuum with elements

required only on the boundaries; hence, it is attractive to the

analyst.

Review of Literature

The mathematical basis of this method is not recent. The

application of integral equations for elasticity problems was reported

by Muskhelishvili [26], Mikhlin [25], and Kupradze [20]. Kellog [19],

Jaswon and Symm [18], and Massonet [23] were involved in the indirect

formulation of the boundary integral method for potential problems.

For elastostatics problems, it was Cruse [10, 11] and Rizzo [30] who

gave an engineering touch to this mathematical technique. Brebbia has

done extensive work in boundary elements, and with his fellow

researchers has developed a computer software package called BEASY

[12]. Six international conferences dealing with the development of

the boundary element technique have been conducted. Banerjee and

Davies [1] have solved a pile soil interaction problem by coupling

finite difference and indirect boundary element methods. Nakaguma

[27] has developed a three dimensional boundary element program using

constant triangular elements for elastostatics, and has solved some

example problems. Georgiou [14] has coupled two-dimensional finite

element and boundary element programs for elastostatics problems.

Scope of Research

Engineers are seeking parameters to represent the so i l s t i f fness

underneath structures with simple and reasonable assumptions of the

so i l properties such as Young's modulus E and Poisson's ra t i o v of the

s o i l medium. Using the boundary element method, the s t i f f n e s s

parameters can be developed in a matrix form that is not dependent on

the s t i f f n e s s of the structure, size of the so i l - s t ruc tu re inter face

area or the d i s t r i b u t i o n of the l o a d i n g . The purpose o f t h i s

d i s s e r t a t i o n i s to i n v e s t i g a t e the boundary element technique to

represent a so i l medium in such a manner that the engineer can use i t

i n h i s s o i l - s t r u c t u r e i n t e r a c t i o n problems. For t h i s c lass of

problems, the f i n i t e element method i s used f o r represen t ing the

s t r u c t u r e , and the boundary element method is used for modeling the

s o i l . A condensed boundary element s t i f f n e s s equat ion represent ing

the s o i l s t i f f n e s s is added to the s t i f fness matrix of the structure

por t ion. The stresses and displacements of the s t r u c t u r e , i n c l u d i n g

the in ter face, are the resul ts of the analysis.

The a p p l i c a t i o n i s tes ted on (1) an ideal ized hydraulic U-lock

8

structure rest ing on an e las t i c so i l medium, and (2) a p i l e embedded

in an e las t i c so i l medium as a three-dimensional problem. The resul ts

are compared with f i n i t e element solut ions.

The resea rch performed i s presented here, h i g h l i g h t i n g the

fol lowing areas:

(a) d e r i v a t i o n of the boundary e lement e q u a t i o n based on

Kelvin 's fundamental so lu t ion ;

(b) numerical formulation of the boundary in tegra l equations;

(c) coup l ing of the displacement f i n i t e element method and the

d i rec t boundary element method;

(d) a combined model o f both methods f o r s o i l - s t r u c t u r e

in teract ion problems;

(e) a condensation procedure fo r analyzing layered so i l media;

( f ) appl icat ion of the developed computer programs to some s o i l -

st ructure in teract ion problems; and

(g) f i n a l l y , based on the s t u d y , f o r m u l a t i o n of general

conclusions and recommendations for future research.

CHAPTER II

BOUNDARY ELEMENT METHOD

Introduction

This chapter deals with the basic theory of the boundary element

method. Equations of linear theory of elasticity which are required

in the derivation of the boundary element equations are also presented

briefly. The sequence used in the derivation is for the direct method

wherein one uses a fundamental solution due to Kelvin [5]. Other

simplified boundary element techniques using Flamant equations [9, 33]

were also investigated; details of this method are given in a report

by Vallabhan and Sivakumar [35]. In this chapter the boundary element

equations are derived in such a form that they can be applied to two-

dimensional elasticity problems or three-dimensional elasticity

problems.

Basic Equations of Elasticity

The basic equations of elasticity [22] are presented here for a

three-dimensional case. Index notation is used for convenience. The

equilibrium equations at any point in the interior domain ^ of the

solid continuum shown in Fig. 2.1 are:

o . . . + b. = 0 in n (2.1)

where a is the stress tensor and b. the body force vector. The

comma after ij represents differentiation of the stress tensor a. .

10

u . = u- on To

V7////A t . = t . on r,

1 1 1

Fig. 2.1 An elastic domain with boundary conditions

n

with respect to the corresponding axis, represented by the subscript

following the comma.

The stress tensor has to match the prescribed tractions t. on the

boundary, i.e.,

a ..n. = t. (2.2)

where n. is the unit normal vector on the boundary. Displacements are

prescribed on boundary r :

u. = u. on Tp (2.3)

In this case, the total boundary r = r. + r . Eqs. 2.1, 2.2, and 2.3

are the fundamental equations in elasticity. However, to solve these

equations, one needs two additional sets of equations: one set to

represent strain-displacement relationships, i.e.,

e .. = i (u. . + u. .) (2.4)

where e.. is the linear strain tensor, and a second set to represent

the stress strain relations, i.e..

wh

a. . = C. ., e . (2.5) ij ijk k

ere C.., is a fourth order material property tensor.

Equations for Boundary Element Method

The derivation of the boundary element method can be achieved in

different ways. Brebbia [4] has used a weighted residual approach for

derivation of the boundary element equations. The approach used here

differs slightly and is more rational. If u* is assumed as a weighting

function, then the equilibrium equations can be written as

f (a.. . + b.) u*d$7= 0 (2.6)

12

Consider the integral shown below.

/(^..u*) ,dn (2.7)

Differentiating the integrand in the above equation, we get

/(<J.,.u*) .dn=J(a u*)dfi+/a u* .dfi (2.8)

By using the divergence theorem and substituting in Eq. 2.2, we have

/(a.ju*)^jd. = /(a.ju;)njdr

n r (2.9)

= /t.u*dr r

Substituting Eq. 2.9 and Eq. 2.6 into Eq. 2.8 and rearranging, we get

ft.u.dr - fa . .u. .d^ + (b.u.dJ2 = 0 (2.10) ' 1 1 ' 1 1 1 . 1 ''ii

By using symmetry of the stress tensor, i.e., a.. = a.., we have • J J '

J a. .u. .dn = )CT . .T(U. . + U . .)d n ?; TJ T.J A iJ T.J J.T n " (2.11)

c * = CT. .£. .df2

Substituting Eq. 2.11 into Eq. 2.10, we get

Jt.u.dr - /a. .£. .da + /b.u.dn = 0 (2.12)

Using the stress-strain relationship of Eq. 2.5, we find

i^J^J^'=i'^iJkl'k£V" * _ r * (2.13)

13

Following Eq. 2.8, we have

J(a.jU*)^.d.= |a..^jU>+|a.je;.d. (2.14)

Following Eq. 2.9, we have

/a*ju.^.d.= Ja^jU.n.dr " ^ (2.15)

= /t*u.dr

Substituting Eq. 2.15 into Eq. 2.14 and using Eq. 2.13, we have

/t*u.dr ==Ja* u.dfi + |a* £ dS (2.16) I I Q l J t J I I J I J

Eliminating the second i n teg ra l on the r igh t -hand side of Eq. 2.16,

using Eq. 2.12, we have

J t . u . d r - J t . u . d r •' - - - 1 1

+ ja . . .u.dn + Jb.u^d^ = 0 0. ""J'J "• 0. T 1

r •• ' r "• % (2 .17)

This is the governing equation for the domain under consideration, and

the first domain integral in the above equation can be removed by

assuming a solution of an equation such that

a*.^. + A.^ (s,q) = 0 inf^ (2.18)

where s is the source point where a unit load is applied in the £

direction, and q is the field point where the displacements and

tractions are calculated due to the unit load. Mathematically,

if s = q, A = 0;

if s ? q, and i = £, J A „dn = 1 ,

and if i f I; A.^= 0

14

The solutions to the above equation u** and tl'. for 1 = 1,2,3 represent

the displacements and tractions in the i-direction due to a unit

concentrated load at 's' in the '£' direction. Substituting Eq. 2.18

into Eq. 2.17, for any source point s, we get

-U£(s) +/u^,^t^dr - /t i uj dr +/u , b| dn = 0 (2.19)

If we omit body forces, we have

u^(s) +/t*^u,^dr =/u*^t,^dr (2.20)

Eq. 2.20 is for a source point inside the domain. For the boundary

element method, the source point has to be moved onto the boundary.

When the source point is on the boundary, a singularity occurs in the

fundamental solution and Eq. 2.20 has to be integrated in a special

manner. In Fig. 2.2, two boundaries are considered: r for r = E at the

e source point andr which is equal t o r - r . In a two-dimensional case, r e

these boundaries are l ines, and for a three-dimensional case, they are

sur faces. The i n t e g r a t i o n has to be per fo rmed as e->-o. For an

explanation of the in tegrat ion, l e t us consider

|u^t*^dr=/u^t*^dr^+/u^t;^dr^ (2.21)

e r " k

The first part of the integral in Eq. 2.21 can be shown to be - -j-,

after substitution of the fundamental traction into the integral sign

and noting that e= r. It can also be shown that the second part does

not introduce any new term when e^o and when Eq. 2.20 is used on the

boundary. Thus, the boundary integral equation on the boundary is

15

Fig. 2.2 Augmented surface for integration on the boundary

16

written as

c(s)u^(s) +It^kU,^dr =/u^^t,^dr (2.22)

where c(s) = i when the boundary is smooth, and c(s) =1 if s is inside

the domain. The value of c(s) for higher order elements can be derived

separately or calculated from rigid body motion criteria [4, 5].

Fundamental Solution

The fundamental solution is an analytical point load solution in

the domain, and this solution is used to convert the domain integral

into a boundary integral. The solution of Eq. 2.18 is the fundamental

solution of displacements and tractions for elastostatics problems.

For three- and two-dimensional cases, Kelvin [4] developed the

fundamental solution due to a point load in an infinite continuum.

Kelvin's fundamental problem is shown in Fig. 2.3. Several others [2,

5] produced solutions for different domains and loads. The use of a

particular fundamental solution is a matter of choice and each choice

has its own advantages in application. Kelvin's solution is adopted in

this work for two reasons: first, it can be used for bounded domains;

second, it is convenient and required to solve problems in layered

media. The expressions for the fundamental solution of Kelvin for

displacements and tractions for a unit load in an infinite continuum

are given below:

"*j(^'^) = 16.(1 -v)Gr ^(3 - 4v)6.j + r .r .} (2.23)

for three-dimension problems, and

17

Components of u* and t*

n* s q

Infinite space Load point Field point

Fig. 2.3 Kelvin's fundamental problem

18

" i j ^ ^ ' ^ ) = 8 . ( 1 " ! v) G (3 - 4v) n(r)6. j - r .r^ .} (2.24)

for two-dimension plane stra in problems. Corresponding t r a c t i o n

functions are

t l j ( = - ' ) = 4 . ( r l v ) r . ( [ ( l -2v )« i j +6r_, r_ j ] | i ^^ ^^^

-(1-2.)(r_,„.-r_,„.)>

where a = 2,1 and g = 3,2 for three- and two-dimensional plane strain,

respectively. Also, r = r(s,q) represents the distance between the

load point s and the field point q, and its derivatives are taken with

reference to the coordinates of q, i.e.,

r = (r.r.)^=||s-q||,

r^ = x^.(q) - x..(s),

- 9 r _ ^ ^ 1 ~ 3irTqy ~ r

where j| x || is equal to / x - x . .

CHAPTER III

NUMERICAL ANALYSIS

Introduction

The boundary integral equation required for the boundary element

method was described in Chapter 2. This equation is reproduced here

for continuity of discussion as Eq. 3.1:

c(s)u (s) + / t * u. d = / u* . t ,dr (3.1) I p IJ J p IJ J

where t | . and u^ . are the fundamental t r a c t i o n s and displacements,

respect ively, and t . and u. are the tract ions and displacements on the J J

boundary.

An analyt ical in tegrat ion of the boundary i n t e g r a l equat ion as

seen in Eq. 3.1 is extremely tedious and not p rac t i ca l for solving

engineering problems. These integrations are performed in a piecewise

manner on the boundary using discrete boundary elements. Interpolat ion

f unc t i ons are used to represent the var ia t ions of displacements and

t ract ions over each element. Using Gaussian quadrature formulae, the

i n t e g r a t i o n s are performed over the boundary elements one by one, with

the source po in t in one element. The source is appl ied over each

element and th i s procedure produces a system of equations of the type,

A X = Y, a f ter applying the boundary condi t ions of the problem. The

set of simultaneous equations i s then solved to f i n d the unknown

displacements and tract ions on the boundary.

19

20

Interpolation Functions

The variations of the unknown displacements and tractions on the

boundary are described by interpolation functions, namely,

u. = (j)..U." T *ij J

(3.2)

t. =,i;..T." 1 ^TJ J

where U. and T. are the nodal values of the displacement and traction J J

J.L.

vectors on the n boundary element, respectively, and <() .. and ij;. . are 'J 'J

the in te rpo la t ion funct ions. The s implest of these i s the constant

element, where the displacements and t rac t ions are considered to be

constants w i th in the element. In the case of l i n e a r elements, the

displacements and t rac t ions vary l i near ly wi th in the element. Higher

order elements can also be formulated [ 5 ] .

Numerical Analysis

The boundary i s d i s c r e t i z e d i n t o l i n e e lements i n a t w o -

d i m e n s i o n a l problem and surface elements in a three-d imens iona l

problem. In t h i s work, constant elements are used in the computer

program; hence, explanat ion of the numerical analysis is l imi ted to

t h i s type of element.

The boundary is discret ized in to a number of constant elements.

Then Eq. 3.1 i s appl ied on the boundary in a d i s c r e t e form. The

corresponding boundary element equation in the constant elements would

be of the form.

21

M M c(s)u.(s) + E { / t*,dr} U," = E { r u*,dr} T," (3.3)

m=1 r J ^ m=l r - ^ n n

where

M is the total number of boundary elements,

the range of 'j' is equal to the number of degrees of freedom, J.L.

r^ is the m boundary element, and

U. and T. are the displacements and tractions in element m. J J

J. L.

Eq. 3.3 represents the assembled equation for the i node (source

point 's'). Using Gaussian quadrature formulae, the integration is

carried out throughout the boundary on the M elements sequentially by

numerical integration.

Two-Dimensional Analysis

The domain is discretized as shown in Fig. 3.1. Each element has

a node at its midpoint. The values of U and T are assumed constant on

each element and are equal to the values at its midpoint.

The source point on each element is chosen to be the midpoint of

the element. Hence, Eq. 3.3 is applied at the midpoint of every

element to form the system equations. The integrals are represented as

/ t»dr = H = P IJ

nd

/ "tid^ = ij P IJ

Hence, Eq. 3.3 becomes

c(s)U.(s) + I H .U '" =Z G T T m=l - m=1 '"

M m . n T m (3^4)

22

Element

Nodes

*• X

Fig. 3.1 Two-dimensional domain discretized into boundary elements

23

Eq. 3.4 relates the value of the displacement 'U.' at the midpoint of

element i (namely, the source point) to the displacements and tractions

at all the elements on the boundary, including the source point. Eq.

3.4 can be written as

M M E H. .U. = E G. J. , i = 1, 2M j=l '' J j=1 J J ^

where H. . = H. . for i j 'J 'J

H.. = H.. + c. for i = j

When i = j in Eq. 3.4, the integrat ion becomes s ingu la r and has to be

evaluated a n a l y t i c a l l y or by other means. I t is easier to calculate

H . . using r i g i d body cons idera t ions , and f o r a constant element i t

works out to be 0.5 [ 4 ] . G.. can be calculated ana ly t i ca l l y or by a

logar i thmica l ly weighted numerical integration formula.

Numerical Integrat ion

The elements are converted into dimensionless coordinates and the

numerical integrat ion is performed using Gaussian quadrature. The l ine

e lement i s i n t e g r a t e d using one-dimensional Gaussian quadrature

formulae,

+1 N I = /F (Od = Ew, f (? , ) (3.5)

-1 i=1 ^ •f- h

where w. i s the weight ing f a c t o r , ? . i s the coordinate of the i 1 3 3 T

integration point, and N is the number of integration points.

The system equations for the M boundary nodes can now be written

as

[H]{U}= [G] (T) (3.6)

24

w i t h known d i sp lacements and t r a c t i o n s s p e c i f i e d as boundary

condit ions. H and G are square matrices of the order 2xM.

Solution of System Equations

Whenever displacements are prescr ibed, the tract ions cannot be

prescr ibed and v ice versa. Rearranging the set of equations w i t h

unknown d i sp lacemen ts and t r a c t i o n s on the le f t -hand side and

corresponding knowns on the right-hand side, Eq. 3.6 becomes a set o f

l inear simultaneous equations,

[ A ] { X } = { Y } (3.7)

Solving Eq. 3.7. we get a l l the unknown displacements and t ract ions on

the boundary.

Three-Dimensional Analysis

In a three-dimensional problem, the boundary is a surface and it

is discretized into a number of surface elements. In this work, the

elements employed are plane quadrilaterals, as shown in Fig. 3.2. For

a constant element, the values of displacements and tractions at the

center are assumed to be constant over the element surface. The

numbering of the nodes of a particular element is such that the normal

to its surface is always outward. For M boundary elements on the

surface, Eq. 3.1 is applied on each of the elements in order to form

the system equations. The integrals are represented as

j t* d r = H.. and r Ij J n

I u* .d r = G,, r TJ J n

25

Element Nodes

H^X^

Fig. 3.2 Three-dimensional domain discretized into plane quadrilaterals

26

In matrix form, Eq. 3.4 can be written as

[H] {U}= [G]{T} (3.6)

where H and G are square matrices of the order 3Mx3M. By use of the

prescribed boundary conditions, the equations are solved for unknown

displacements and tractions on the surface as explained in the previous

section.

Numerical Integration

The plane quadrilateral element is transformed into a unit square

on a dimensionless coordinate system (ni.lo) as shown in Figs. 3.3 and

3.4. The nodal coordinates are interpolated using the shape functions.

X. = N.X.. ^ J Ji

for

where

i = 1,2,3 j = 1,2,3,4

(3.8)

1. = kO + 6. B) fore = ±1, 6 = n^, n^ and

X.. = cartesian coordinate of node 'j' in the 'i' direction. Ji

Since the interpolation functions are expressed in terms of rii, no

coordinates, it is necessary to transform the elements of area dr from

Cartesian coordinates to the n-i, no coordinates. A differential of

(3.9)

area dr w i l l b€

dr =

=

J given

II — 8n^

i

3x,

Srvj"

•—

CM

X

cr

CO

C

O

by

^ IT^ " ^

j t oXrt dXo

8n-] 9ni

oXo oXo

3 ^ 3 ^

II II dn^dn2 = Ii g-]i + g2J + 93k II dn^dn^

= jjGjI drii dno

27

'2

i 'x^]^ 'X42'^43'

^^ll'^12'^13^

3l^^3l'^32'^33^

\Xpi >'^po ' 2 ?

Fig. 3.3 Quadrilateral boundary element

28

(-1,1) (1,1)

- ^ n i

(-1,-1) (1,-1)

Fig. 3.4 Quadrilateral element in dimensionless co-ordinate system

29

where

3 X 0

3n,

3x3

3ni

3x,

9n,

!!3 3n2

.—

CM

X

(T

C

O

CO

3X2

3n2

! ! ! 3ni

3x^

3n-|

3x2

3x2

9n2

3x3

3n2 <—

C

M

X

cr C

O

CO

91 =

99 =

| |G | |= (g2 + g2 + g2) i

II G II i s the Jacobian re lat ing the elements of area in the two systems of

coordinates.

For the quadri lateral element,

3x^ g 3N^ 3N2 3N3 3N4

grrj" = sTTj" ( ^ i ^ i l ^ " 37^ ^11 ^ ^ h ^ " ^ 3 ^ ^31 " sTTj" ^ 41

3x2 3 ^^1 ^^2 ^^3 ^"^4 sTTj" = gTTj" C ^ i ^ z ^ = sTT^ 12 "*• 3 ^ ^22 " sTTj" ^32 + sTTj" ^42

3x3 g 3N^ 3N2 3N3 3N4

STT = 9 ^ (^•^•3^ = 9T:^ ^13 + 9 ~ ^23 + 9 ~ h3 " 3— ^43

3 ^ = 3-^ ( " ^ i ^ l ^ = 9"^ ^11 "• 9T^ ^21 + 9n2 ^31 + 9n2 ^41

3x, . 3Ni 3Np 9N3 9 „ , — Y + — - X + — -I2 21 ^9n2 31 ^ 9

9Xo 3 8N^ 9N2 9N3 3J^

97^ = 9 -1^ ( ^ . ^2^ = 9 l ^ X l 2 + 9 - 7 ^ X 2 2 + 9 n 2 ^ 3 2 ^ 9 n 2 ^42

3 . g 3N^ 3N2 9N3 9J^

3T^ = 9T^ ('^iXi3) = 9"^ ^13 + 3-^ ^23 - 9 1 ^ X33 + 3^ ^ X43

In matrix form we can wri te

30

3x, 3X2 ^^3

3n-] 3ni 3ni

9x, 3xp 9 Xo

9n2 9n2 SrTT

3N^ 3N2 3N3 sN^

9ni 9ni 9ni 3n'|

9N., 9N2 9N3 3N4

9n2 9n2 9nT SnT

^ 1 1 ^12 ^13

^21 ^22 ^23

^31 ^32 ^33

^41 ^42 ^43

(3.10)

Subst i tut ing Eq. 3.10 into Eq. 3.9, we can calculate the Jacobian

||G||. Then from Eq. 3 .1 ,

M c (s ) U (s) + E { | t * l lGl ldn, dnp } u""

^ m=1 Tffl IJ I ^ J

M = 2 { J u* |G| dn, dn2} T" '

m=1 m ^J 1 ^ J

Replacing the integrals by summations again, we get

M N

(3.11)

c( ) U. (s) + E { E t* I G| W } U" ' m=l n=1 IJ n n J

M N = S { E U * . | G L W l T!" (3.12)

1 I 1 1J 1 n J m=1 k=1 m " ''

where N is the number of Gaussian integral points,

W_ is the weighting factor, and

IGII^ is the Jacobian of the m element.

Eq. 3.12 is applied to all M boundary elements, thereby producing a set

of 3M equations.

CHAPTER IV

COUPLING OF BOUNDARY AND FINITE ELEMENT METHODS FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS

Introduction

Since the boundary element method has been found to be a very

powerfu l technique f o r so lv ing stresses and deformations of large

continua where the boundaries are at large d is tances , i t i s used to

represent large so i l media in so i l -s t ruc ture interact ion problems. The

f i n i t e element method i s used to represent the st ructure; propert ies

such as complex geometry, heterogeneity, non l i near i t y , re in fo rcement ,

e t c . , are better modeled by the f i n i t e element method. For a l i nea r l y

e las t i c continuum, the boundary element method y i e l d s be t t e r r e s u l t s

with a r e l a t i v e l y small number of unknowns and with easy preparation of

i npu t da ta . I f one needs to use nonlinear properties of the so i l in

the immediate v i c i n i t y of the s t ruc ture , i t i s convenient to use the

f i n i t e element method f o r the nonl inear port ion of the so i l medium.

The use of these two methods makes the analys is computa t iona l l y and

economically more e f f i c i e n t .

Equations Used in Coupling

There are two basic procedures for combining the f i n i t e element

and boundary element methods [2 , 5 ] . One is to convert the f i n i t e

e lement e q u a t i o n s to s u i t the boundary element equat ions. This

p rocedu re has been the main techn ique used by many p r e v i o u s

31

32

researchers; however, this procedure has serious efficiency problems,

especially when the finite element model has nonlinear behavior. The

procedure adopted here is in the reverse order; i.e., the boundary

element equations are transformed into an equivalent stiffness matrix

and added on to the finite element half banded stiffness matrix

equation. This procedure has many advantages in solving soil-structure

interaction problems.

Consider two regions J^ and S 2 w' ' different material properties

and geometries as shown in Fig. 4.1. Region Q, is divided into finite

elements and region p "•"'to boundary elements. Coupling of the two

models is done using equilibrium and compatibility on the interface gg,

namely:

1. compatibility of displacements; i.e., displacements on the gg

interface should be equal for regions , and p; and

2. equilibrium of tractions; i.e., the tractions on gg interface

should be equal and opposite for region fi, andfip.

Finite Element Equation

The structure is discretized into plane strain finite elements for

the analysis. The structure stiffness matrix is developed using the

computer program [35], including the soil-structure interface, denoted

by gg in Fig. 4.1. The finite element stiffness matrix equation is of

the form.

K SS

'BS

K SB

'BB

•

' ^ s "

%

(4.1)

33

lA

+J S-O) a. o Q .

C O)

rtJ

o

en

34

where the square matrix is the global s t i f f n e s s mat r ix o f the f i n i t e

element p o r t i o n , Ug and Fg are the displacement and force vectors of

the boundary element i n t e r f a c e , and U^ and F^ are the r ema in i ng

displacement and force vectors of the f i n i t e element system. The

f i n i t e element s t i f fness matrix shown in Eq. 4.1 is developed as a hal f

banded matrix fo r computational economy.

Boundary Element Equation

By use of the boundary e lement method, a s t i f f n e s s m a t r i x

rep resen t ing the s o i l medium has to be developed on the in ter face.

This should be in such a form tha t i t i s compatible w i th the f i n i t e

element system. Therefore, the boundary element s t i f f n e s s matr ix

equation should be of the type:

[ kg ] { Ug}= - { F g } + { f g } (4.2)

where fg is a condensed force vector representing the prescribed forces

and displacements on the boundary of the so i l medium.

Soil St i f fness for Two-Dimensional Problems

The condensed equation (Eq. 4.2) representing the so i l medium has

to be developed for coupling with the f i n i t e element s t i f fness matrix.

This development is explained in r e l a t i o n to an i dea l i zed U-lock

s t r u c t u r e . U-lock s t ruc tu res are very massive, about 600-1000 f t in

l e n g t h , 80-120 f t in w id th , and 60-100 f t in he igh t . Hence, the

s t r u c t u r e r e s t i n g on the s o i l a l toge ther can be assumed as a two-

dimensional problem of plane s t ra in in sol id mechanics.

35

Boundary Element Model

To obtain a condensed stiffness matrix [kg] to represent the soil,

Kelvin's fundamental solution as shown in Eqs. 2.23-2.25 is used for

the formulation of the boundary element equations. Fig. 4.2 shows the

boundary element model for a U-lock structure. The advantage in

symmetry of the problem is taken care of in the computer code.

The boundary element equations are developed using Eq. 2.25, and

the final set of matrices is of the form:

M

{1 ••

where U and T are the nodal displacement and traction vectors on the

four boundaries gg, g,, g2. and g3 as shown in Fig. 4.2. The

subscripts on U and T denote the respective boundaries. The following

boundary conditions are prescribed for the problem:

on g.|, T.| = 0,

T, = h, a prescribed traction, ly

on g2, U2^ = 0,

on g3, U3y = 0,

on gg, both Ug and Tg are unknowns.

36

> UpT 2'2 on g^

Fig. 4.2 Boundary element model for a U-lock structure

37

Considering the prescribed and the unknown components of displacements

and tractions, we can reorganize the H and G matrices such that the new

H and G matrices are:

H 11

71

H 12

22

U B '11

'21

'12

'22 • (4.3)

where Vp and V^ are vectors representing the prescribed and the unknown

components, respectively, of all displacements and tractions on g,, g^

and g3. The sizes of the submatrices of H and G correspond to the

orders of the interface vectors Ug and Vy, respectively. From Eq. 4.3,

^U = "22"^ (' 21 8 + S22VP - 21 8)

Substituting this result in the upper part of Eq. 4.3,

11 ~ ^ 12 22 21 * 8 ~ ( 11 ~ ^12^22 21 ''"B

+ (G^2 ~ ^12^22 ^22^^P

(4.4)

(4.5)

(4.6)

we can reduce this equation to

[kg] {Ug} = - ^ ' ' B ^ + ^^B^

where kg is the required stiffness matrix of the soil medium, Fg is the

vector of equivalent nodal forces from the traction vector Tg, and fg

is the vector of equivalent forces representing the prescribed forces

and displacements on g,, g2 and g3. The development of kg and fg

appears to be quite cumbersome. However, the computations can be

simplified considerably by using a static condensation procedure [42].

Using the Gauss/Jordan elimination procedure, if one transforms both

the H and G matrices such that H22 is made into an identity matrix and

H,p is made into a null matrix, then the new transformed matrices

38

H,,, G.,, and G,p are

"ti % = h h^^h^p (4.7)

i.e., the matrices within the respective brackets in Eq. 4.4. Again,

by use of the elimination procedure, Eq. 4.7 is transformed such that

G. l is made into an identity matrix, i.e.,

"ri ^B = Tg + G*| Vp (4.8)

The coup l ing i s done by t ransforming the nodal t rac t ion vector Tg to

the equ i va len t nodal fo rce vec tor -Fg in Eq. 4.8 and adding t h i s

transformation to Eq. 4 . 1 . This resu l t is achieved by mul t ip ly ing both

s ides of Eq. 4.8 by a d i s t r i bu t i on matrix R to convert t ract ions in to

nodal forces. For two-dimensional problems w i t h constant elements,

ma t r i x R i s a diagonal mat r ix w i t h the lengths of the respec t i ve

elements as coe f f i c ien ts . Thus, we have

[R H**] Ug = R Tg + R G*| Vp = -(Fg) + ( fg ) (4.9)

i . e . , [kg] { U g } = - ^ F g } + { f g } (4.2)

Compatible Boundary St i f fness Matrix

The boundary elements representing the s o i l are assumed to have

constant var ia t ions of displacements and t ract ions over t he i r lengths.

For coupling, the condensed e f f e c t i v e s t i f f n e s s of the s o i l p o r t i o n

obta ined by Eq. 4.2 has to be made compatible with the f i n i t e element

s t i f fness matrix representing the st ructure.

The coupling of s t i f fnesses from the constant boundary element

f o r m u l a t i o n of the s o i l t o the f i n i t e element s t i f f n e s s of the

39

structure is effected in two ways, as explained below.

Method I

In this method, the midpoints of the boundary elements are

positioned so that they lie on the finite element nodes of the

structure at the interface. The stiffnesses of the boundary elements

at the ends of the interface are reduced by half.

Method II

In this method, the end points of the boundary elements coincide

with the finite element nodes. Therefore, the stiffness matrix of the

subgrade from the boundary element region has 2 less rows and 2 less

columns. The condensed stiffness matrix at the interface, representing

the subgrade, is transformed by premultiplying it by a transformation

matrix and then postmultiplying it by the transpose of the

transformation matrix. This transformation is possible by use of the

contragradient law [43] and satisfies energy principles. Let Ug and Fi

be the nodal displacements and forces at the midpoints of the boundary

elements at the interface. Let Ug and Fg be the displacements and

forces at the finite element nodes at the interface. Let T be the

transformation matrix which converts the finite element displacements

of the interface nodes into the boundary element displacement at the

midpoints. T will be of the form.

40

5

0

0

0

0

0

0

.5

0

0

0

0

.5

0

.5

0

0

0

0

.5

0

.5

0

0

0

0

.5

0

.5

0

0

0

0

.5

0

.5

0

0

0

0

.5

0

0

0

0

0

0

.5

.5 0 .5 0 0

0 .5 0 .5 0

0 0 .5 0 .5

The order in this matrix is (2M, 2M + 2) where M is the number of

constant boundary elements at the interface. Therefore,

{Ug}= [T]{U* } (4.10)

By the contragradient law, we have

(4.11)

Reverting back to our notation, the condensed stiffness matrix of

the subgrade is

{ F j } = [ T ] * { F g }

(4.12)

[ k g ] { U g } = { F g }

[kg] [T] [Ug*] ={Fg }

[T"^] [kg] [T] {U*}= [T"^] {Fg}={F*}

[k*]{Ug*}={Fg*}

The matrix kg will be the required kg for use in Eq. 4.9. The Ug and

Fg vectors represent the corresponding new displacement and force

vectors on the boundary interface.

41

This method is more convenient than Method I and yields a better

solution.

Properties of Boundary Element Stiffness Matrix

The stiffness matrix kg formed from the boundary element domain is

generally asymmetric. This poses problems for addition into the

symmetric stiffness matrix of the finite element region. The asymmetry

of the boundary element is not new and has been reported in the

literature before [2, 5, 14, 15]. This asymmetry has been attributed

to three factors, namely, discretization of the boundary element

domain, the collocation process and the nature of the fundamental

solution. To alleviate this anomaly, the boundary element stiffness

matrix is symmetrized, discarding the unsymmetric part. This

symmetrization has been justified by Georgiou [14] by showing results

of examples which are reasonably accurate. A detailed mathematical

procedure to derive a symmetric stiffness matrix from an asymmetric one

is explained by Hartmann [15], He says that if G H u = t is solved

with Galerkin's method or by minimizing a potential <!> (u), a symmetric

stiffness matrix is guaranteed.

In this present work, the boundary element stiffness matrix is

symmetrized by averaging the off-diagonal terms, by the principle of

least squares as suggested by Brebbia [4]. It is found that the error

involved is negligible. Also, the stiffness matrix derived from the

constant elements are superior to the higher order formulation in

relation to its symmetric properties [14].

42

Coupling of Finite and Boundary Element Matrices

Having developed the stiffness matrices of the structure and the

soil as shown in Eqs. 4.1 and 4.2, the coupling is done ensuring

compatibility and equilibrium at the interface. Now the combined

FEM/BEM model can be written as

• SS " SB

• BS " BB "•" "B

U<

U B B

(4.13)

This equation is readily solved to get the displacements and stresses

of the structure.

Example Problem

In this section the results of an example problem solved using the

two-dimensional computer program are presented. The example shown here

is to illustrate the coupling of the boundary element method and the

finite element method for soil-structure interaction problems. The

example problem was run on the IBM 3033 computer at Texas Tech

University. The results are compared with finite element solutions for

displacements to validate the code developed and to determine the level

of accuracy obtained by solving, using the coupling technique.

A typical problem which concerns a hydraulic U-lock structure was

selected for this purpose. The problem deals with a strip of plate of

unit width resting on a finite soil medium. The problem is considered

as one of the plane strain type. The strip is modeled as a beam of

length L equal to 100 ft. Two cases of depth of the beam are

considered, one for a depth equal to 5 ft and the other for a depth

43

equal to 10 ft. It is further assumed that the soil is resting on a

smooth, unyielding rock surface at a depth of 200 ft. The effect of

the smoothness of the bottom boundary on the accuracy of the results is

found to be negligible for the above geometry. It is assumed that the

vertical boundaries 200 ft away from the center line are smooth. All

these boundary conditions are selected only from a practical design

point of view, and they do not reflect any limitations of the

technique. (The loads, geometry and the boundary conditions of the

problem are illustrated in Fig. 4.3.)

The modulus of elasticity of the concrete beam E, is taken as 3 x

10 psi. The modulus of elasticity of the soil E is varied such that

two values of the modular ratio m = E,/E are 10 and 100. For

convenience in this analysis, the Poisson's ratio of the concrete and

the soil are both equal to 0.2, even though this setting is not a

limitation on the methodology. The load conditions considered are:

(1) a uniformly distributed load of 5 k/ft on the top of the beam,

(2) a concentrated load of 250 k at each end of the beam, and

(3) a concentrated moment of 5,000 ft-k at each end of the beam.

For each loading condition there are four cases, with nomenclature as

shown in Table 4.1.

44

+-> 4-

O o o in

in CM

f LT)

v.

o_ i n CM f

V-

LU p

O

o

o CM

n

o t—1

II

o o 1—1

n

o 1—1

II

^ 1 CO LU | L U

II

-Q S 4->

" ^ _ 1

E (U ,— J 3

o S-Q .

(U

Q .

ra X 0)

4 -O

->.

^

^

• 1 - CM t/1 • ^ o O II o O I/) CO ?

II II

j a X I LU ?

4 -

o o CVJ

II

x:

cu Q

• CO

• «d-

•

Li_

45

TABLE 4.1

DETAILS OF EXAMPLE PROBLEMS

Case Foundation Depth (ft)

200

200

200

200

Beam Depth (ft)

10

10

5

5

Modular Ra m = E,/E^

10

100

10

100

1

2

3

4

Ano the r q u e s t i o n which comes i n t o t h e a n a l y s i s i s t h e

compat ib i l i t y of horizontal displacements between the st ructure and the

s o i l at the i n t e r f a c e . I t i s a common p r a c t i c e in beam on e las t i c

f ounda t i on s t u d i e s t o i g n o r e the c o m p a t i b i l i t y o f h o r i z o n t a l

d isplacements at the inter face [36 ] . I f we ignore t h i s condi t ion, we

are essent ia l l y assuming that the so i l in ter face i s p e r f e c t l y smooth.

Thus, the ana lys i s i s performed in two c a t e g o r i e s . In the f i r s t

category, the compat ib i l i t y of horizontal displacements i s neg lec ted ,

making the s o i l a smooth boundary. In the second case, compat ib i l i t y

of hor izontal displacements on the inter face is enforced.

Discret izat ion of the Example Problem

Due to the symmetry o f the problem o n l y h a l f t he domain i s

d isc re t i zed . Rectangular f i n i t e elements with 2 degrees of freedom per

node are employed. The beam is d iscret ized in to f i v e layers wi th ten

elements in each l aye r . This d i s c r e t i z a t i o n i s found to model the

46

bending of beams for displacements and stresses accurately. The

boundary element discretization has to be made compatible with the

finite element discretization at the interface. The sizes of the

boundary elements are varied such that wherever the displacements and

stresses are small or uniform, larger elements are used as shown in

Fig. 4.4. A convergence study was made on the number of boundary

elements in the combined model, and it was found that the

discretization with 40 elements, shown in Fig. 4.4, gave essentially

the same results as with a larger number of boundary elements.

In order to verify the accuracy of the finite element/boundary

element model, a complete finite element study was also made for the

four cases of Table 4.1 for each loading condition. The finite element

discretization for the problem and the loading and boundary conditions

are shown in Fig. 4.5.

Presentation of Results

The results are seen in a clear perspective in graphical form.

The results mainly pertain to the vertical displacements and vertical

tractions at the interface. In all figures, the displacements are

shown from the line of symmetry at 5 ft intervals, at the soil-

structure interface. In the case of vertical tractions at the soil-

structure interface, the values are constant in each boundary element

and are shown at the midpoint of the element.

Figs. 4.6-4.11 show the plots of vertical displacements at the

interface for each loading condition. Figs. 4.12-4.17 show the plots

of corresponding vertical tractions at the interface. These graphs are

47

250 k

r: 5k/ f t

II .1 u

^ SOOOkft

1/2

Es,y S r ^ S

C|

<^

c |

4

9 I • — I — • — I — • — I • ' • I • I

^77 V77 7 ^ V77

•\ • — * • -

Fig. 4.4 Discret izat ion of the combined model

48

o in CM

ftK

00

0

i n

s:

C^ -.rrr ^ : : :

f *..

" H I ?

'\X

Mt^

iJL

kr

TJ ^

k \

T 1

c 1

^ 1

I. \

% 2

I ••• ^

^

.4 ^

•^ Hi

«

• \ • «

k

ro

(O

(U

+->

O) x : +->

s-o

(*-c o

to

IVJ

+->

S -

o U1

i n

49

5k/ft.^^ , ^ , j - t u

• H i i^ lib N ^ >l t ^

'

,

h

f

h/L L/t. m = E./E D b s

Case 1

Case 2

2

2

10

10

10

100

>iB

FE/BE Interface -> x (ft)

Fig. 4.6 Vertical displacement at the interface—case 1 and 2 of loading condition 1

50

5 k/ft.

frVrTii?'^ h/L L/ t , m = Ej /E^

>is

FE/BE Interface -> X (ft)


51

I o

250K 250K

L

\jlb t ,

'

1

h

h/L L/tj j m = Ejj/E^

Case 1

Case 2

2

2

10

10

10

100

FE/BE Interface -> X ( f t )


52

I o

>^e 0) o

10

0) -p c

c (1) E o <0

CL V)

10 <J

4-> i-0)

250K 250K

L

'Jlh t ,

'

h

h/L L/t. m = E./E b b s

Case 3

Case 4

2

2

20

20

10

100



53

-p ^-

I o

M=5000ft.k A

h/L L/t. m = E,/E b b s

Case 1 10

Case 2 2 10

10

100


54

M=5000ft.k A

I

o

>:(S

<U <J (O M -S-0) -p c:

-P 10

-p c <u E 0) o ItJ

CL I/)

(O (J

0)

• 1 • ! •

1. ^ J t

'

.

h

•

h/L L/t. m . E./E^

Case 3

Case 4

2

2

20

20

10

100


5 k/ft.-^

• H U 1 lib 1 . ^ -1 t i

'

,

h Case 1

Case 2

55

h/L L/t, m = E./E b b s

2

2

10

10

10

100

c o -p o (O

.4:1

-5--

-6-

-7--

10 o -8 •r-+->

.9..

•10

O - Case 1

• - Case 2

+ I 20 30

FE/BE Interface

50 10 40

-> X (ft)

Fig. 4.12 Vertical tractions at the interface—case 1 and 2 of loading condition 1

56

5 k/ft. pVn- | f b

Case 3

Case 4

h/L L/t, m = E./E b b s

2

2

20

20

10

100

- 4 . .

^ - 5 ' • c o

(J 10 i-

10

o •r-i-(V

4 10 20 30

FE/BE Interface

40 50

X (ft)


57

250K 250K ' [ t ^'^

r\ • ' * L '

1

L

h

'

Case 1

Case 2

h/L L/tj j m = E^/E^

2

2

10

10

10

100

. ^ -5

• 1 0 • •

•2 -15 -p '"^ o 10

• i : - 2 0

10

o -25 -P

-S -30 +

-35 • •

•40-

•45

FE/BE Interface

t

•> X ( f t )

Fiq 4.14 Vertical tractions at the interface-case 1 and 2 of loading condition 2

58

250 K 250K

L

[ilb t ,

'

1

h

o 10

(O

o -p L. 0)

1 0 -

0

-10 ••

•20 •

- 3 0 ••

•40 ••

- 5 0 ••

- 6 0 ;

h/L L/t. V^s Case 3

Case 4

2

2

10 FE/BE Interface

20

-> X (ft) O - Case 3

• - Case 4

20

20

10

100


59

i=;)a X) ft.k A rr ^ ^ 1. ^ -1 t

h Case 1

Case 2

h/L L/t,

2

2

10

10

m = E,/E^

10

100

•p

.:

Tra

cti

on

ert

icc

A so-so '

40-

X -

20-

10-


60

M=5000ft.k A

c ;^-Lb h/L L/t, V^s


61

for the first category analyzed, i.e., where the compatibility of

horizontal displacements is not enforced at the interface.

Comparison of Results

The comparison of interface displacements obtained from the finite

element analysis for the four cases for each loading condition is made

with the corresponding cases in the coupled model for the second

category, i.e., where the horizontal displacements of the soil

interface are also included in the analysis. The plots are shown in

Figs. 4.18 through 4.23. It was found that the differences ranged from

3 to 8 percent in all of the cases, except at the end of the beam in

the moment loading condition.

A difference as high as 50 percent at the very end of the beam was

seen in case 3. The BE/FE solution yielded a higher displacement at

this loading point, suggesting the fact that it can predict more

accurately at localized regions. The shape of the displacement curve

is very smooth, indicating no discontinuity to question this single

value. Here the modular ratio was 10 and the L/t, ratio was 20. This

may be attributed to the lack of flexibility of the finite element

model at the loaded point. In reality, there is a stress concentration

at the loading point which can lead to large displacements at that

point. At all other points, the solutions matched very well.

The difference in computer time was insignificant in the example ,

problem. The main advantage lies in preparation of data for the

problem. The finite element model had 892 degrees of freedom as

opposed to 224 degrees of freedom in the combined model. If the mesh

62

I o

H.OO-

-1.02 •

-1.04 •

5;B-L06-

"" -1.08-•

-I.IO-

-LI2 •

-1.14 ••

-1.16 •

-LI8 •>

u <0

0) -p c

-p 10

ai -1.20 f

i -1.22 + lO

-^ -1.24 +

;:; -1.26

5 k/ ''ft.^

' M i i j f b U-j-—1 '

'

h

'

h/L Ut^ n, = E,/E^

Case 1

Case 2

2

2

10

10

10

100

m

10

100

Legend

o BE/FE

0 FE

• BE/FE

A FE


Fig. 4.18 Comparison of displacements at the interface-case 1 and 2 of loading condition 1

63

5 k/ft.-.

' U H j fb ^ t ^

'

,

h

'

Case 3

Case 4

/ L/t ^ >"= V E ^ 2

2

20

20

10

100

I o

.00:

>:)E

0) u lO M-1. 0) -p

c <o -1.20

c 0) E 0)

u lO

^ -L30 10

m

10

100

Legend

o BE/FE

0 FE

• BE/FE

A FE



64

I o

250K •

250K

L

'jilb * i

'

h

^B - 9

S -10 + (O

t -11 + • p

^ -12 + 4->

« -13 +

•14 ••

•15-

-16-

^ -17 • • (O

• -18 f

c 0) E 0) u lO

Q.

^ -19

h/L L/t. m = E./E jb b ;

Case 1 2 10 10

Case 2

m

10

TOO

Legend

o BE/FE

0 FE

• BE/FE

A FE

10

10 - 1 — 20

100

30 40

FE/BE Interface

50

-> X (ft)


65

250K 250K ,'

1 . r ' L

f ^^b + .

'

,

h

'

>4E

0) CJ to

4 -i . (U

- P

c

(O

-p c 0) E 0) u ro Q. I/)

10

•r— •P i . 0)

h/L L/t , m = E./E b s

Case 3

Case 4

2

2

20

20

20 30


10

100

Fig. 4.21 Comparison of displacements at the interface— case 3 and 4 of loading condition 2

66

M=5000 ft.k

^ (f

t)

'o

- ^ l i b

u ^ f h Case 1

Case 2

h/L CM

CM

L/t,

10

10

m = E,/E^

10

100

0) o lO

L.

-P c -p (O

•p

c 0) E (1) O (O Q . I/)

lO

o •r— + j

s-0)

11-

10 •

9

8

7

6

5

4

3

2

I

0

-I

-2

-3

-4

m

10

100

Legend

o BE/FE

0 FE

• BE/FE

A FE


67

M=5000 ft.k t.k t .

< •

. >iB —' (U

10 M-

0) - p c

+J (0

+J c 0) E 0) o (0

C3.

(/» •r-Q

lO O

- P i. 0)

o r ^ X

24

22

20

18

16

14

12 10

8

6

4

2

0

-2

-4

-6

• 1 1 •

1. _ J t

'

h

•

m

10

100

Legend

o BE/FE

0 FE

• BE/FE

^ FE

h/L jA_r=ib/^s Case 3

Case 4

2

2

20

20

10

100


68

has to be refined for the problem, the combined model has greater

flexibility than the finite element model.

This clearly demonstrates the ease in input data preparation and

the quality of results with lesser degrees of freedom in the combined

model. On the whole, the combined model is efficient, if not superior

to the finite element model.

Application to a Layered Soil System

In this section the coupling of the finite element and boundary

element methods when the soil has layers of different properties is

presented. The solution to a Boussinesq problem is normally employed

for a linearly elastic, single homogeneous material. In reality, the

soil can be nonhomogeneous with anisotropic layers of different

constitutive relations. It is not possible to get meaningful results

out of an idealized elastic region for problems of this kind. A brief

review of the work done in this area for elastic, isotropic soil layers

is discussed in the following paragraph.

Previous Work

A solution to two or three layers of different elastic properties

was obtained by Burmister [6]. Due to the tedium of this solution, it

is not in common practice among engineers. A numerical approach to

this problem would be to use the finite element method. Once again

this poses problems of modeling and preparation of input data. Brebbia

[4] has outlined the basic concept of analyzing regions of different

properties, using the direct boundary element approach. He has solved

69

a potential problem to highlight this method. Computer implementation

of the above procedure to examples has been given by Nakaguma [27].

Butterfield [7] has suggested a method for solving extensive, thin

layered soil media. In this model, adjacent layers abut immediately

above and below each other. Each layer is divided into top and bottom

elements, with the demarcation in the middle of the layer. The direct

boundary element formulation is employed, and equations are transposed

in such a way that the displacements and tractions of the top elements

are related to the displacements and tractions of the bottom elements.

By use of compatibility and equilibrium, a recurrence relation is set

up so that a system of transfer matrices incorporating the material

properties of the different layers will relate displacements and

tractions of any top elements of one layer to those of the top elements

of the layer above it. In this method, the assumed boundary conditions

of the top and bottom elements of the extreme layers are propagated

unreal istical ly to the ends of the intermediate layers, according to

the author. Numerical results using the above algorithm are not

reported, to the knowledge of the author.

Boundary Element Technique for Layered Model

In the approach suggested here, boundary elements are used to

represent, separately, each of the homogeneous soil layers. The

boundary element equations of the various soil layers are statically

condensed very efficiently in succession through a series of

Gauss/Jordan elimination procedures up to the soil-structure interface.

A stiffness matrix for the soil system is thereby formed for addition

70

to the structure s t i f fness matrix fo r f i n a l a n a l y s i s . The p r i n c i p a l

advantages of th i s procedure are:

(a) d i f f e ren t homogeneous so i l layers are numbered independently,

i n a l o c a l sense; t h i s p rocedure w i l l avo id renumber ing f o r

elements/nodes to form system equations g l o b a l l y , a f t e r f o r m i n g

boundary element equations for each subregion, and th i s renumbering can

be ted ious and complex as the order of i n t e r p o l a t i o n f unc t i ons is

increased;

(b) memory requirements for the variables are on ly to the order

of the maximum number of degrees of freedom of any par t icu lar layer;

( c ) so l v i ng f o r unknowns in th i s procedure is on a smaller size

matr ix; and

( d ) da ta p r e p a r a t i o n i s ve ry c o n v e n i e n t as i t i s done

independently for each layer.

The Ke lv in fundamental s o l u t i o n i s used f o r the layered s o i l

system of the boundary element domain. The s t a t i c condensation t o

ob ta in s o i l s t i f f n e s s in the case of a single homogeneous so i l system

has already been explained in the previous sect ion. When there are two

or more so i l layers, the f o l l o w i n g scheme i s s y s t e m a t i c a l l y c a r r i e d

ou t . To i l l u s t r a t e the method, a s o i l region w i t h three layers is

chosen as shown in Fig. 4.24. Start ing from Region I , the degrees of

freedom with subscript ' 1 ' are condensed:

Region I

The basic direct boundary element formulation for Region I can be

written as

71

m^^Mf^:M:m^m ' i i i i i i i i i i i i I •, M

^B Tg II T ^ 1 'l

U ^ T

Fig. 4.24 Boundary element model for layered media

72

"n

"21

"12

" 2 2 _

« "1' A.

• —

"« l i

_ ^21

^ 2

G22_

•

• ^ l '

A. (4.14)

With the prescribed boundary conditions, the [H] and [G] matrices are

reorganized and the equations are condensed up to the elements denoted

by subscript '1' to get

[H22] { U } = { b" } + { TJ } (4.15)

1 1 where U2 and Tp are the unknown displacements and tractions of layer

one on the the interface of layer two. b is a vector arising out of

nonzero boundary conditions in Region I.

Region II

The boundary element equations f o r Region I I are independently

formed as

x2 "11 "12

H2 H2 L 21 "22 J

' " ?

A>

r p 2 p2 Si S2 2 2

.' 21 ^22 _

'1

'I, (4.16)

Eq. 4.15 i s combined w i t h Eq. 4 .16 by us ing c o m p a t i b i l i t y and

equi l ibr ium at the in ter face, i . e . .

U = U^2 U2 - u T = T 2 '2 '1

(4.17)

where U 12 12 1

and T^ are the unknown displacements and tractions of

Region II up to its interface with Region I.

The system equations before condensation will be of the form.

I I (H T + G T H-) H^2

II L^"21 + 21 22) 22

2 1 1

2

n 2 21

'12

,2 '22

r,iii

73

(4.18)

2 II The addition of G ^ H22 is only to the interface degrees of freedom

connecting Region I and Region II. Thus, giving a new notation, the

equations up to Region II are written as:

H2

"11 H2 "12

"21 "22

> =

>2 p2 "

Si S2 p2 p2 •21 ^ 22

•

>?•

,i. (4.19)

Once again, these equations are condensed up to the interface connected

to Region III as

[H22] { U^ } = { b^^} + {T^ } (4.20)

2 2

where Up and Tp are the unknown displacements and tractions at the

interface between layers two and three.

The above algorithm is carried in succession from the bottom most

layer to the top, condensing step-by-step to represent boundary element

equations at the interface of the soil and the structure in the form [H,J {U.} = {b.} + {T,} "BB ' "B B 'B

(4.21)

Now the t r a c t i o n s are converted in to equivalent nodal forces by post

mul t ip ly ing Eq. 4.21 by a transformation matrix R to get the s t i f fness

of the layered so i l s t i f fness , namely,

[kg] {Ug } = { f g } + { - F g } (4.22)

This equation is the same as Eq. 4.2 for a single homogeneous layer.

The a l g o r i t h m developed f o r layered media was coded and the

74

program was tes ted using the example problem solved be fo re . The

u n i f o r m l y loaded case was run f o r modular ra t ios of 10, 100 and 1000

f o r the case of a beam depth of 10 f t . As the number o f l a y e r s

inc reased , i t was found t h a t the resul ts were not sa t i s fac to ry . The

percentage differences from the f i n i t e element s o l u t i o n are given in

Table 4 . 2 . I t i s concluded t h a t the e r ro r s caused in the layered

system are due to errors in numerical s ta t i c condensation and numerical

in tegra t ion used in the boundary element method. More research i s

necessary to supplement these f ind ings.

TABLE 4.2

COMPARISON OF RESULTS WITH FEM

Number of Layers % D i f f . in Max Disp. in Beam in Soil Region wi th the f i n i t e el em. method

2

3

for m = E,/E b s

10 100

23.5 11

40 23

1000

10

23

CHAPTER V

COUPLING OF BOUNDARY AND FINITE ELEMENT METHODS IN THREE-DIMENSIONAL ELASTICITY

Introduct ion

This chapter presents the theory of coupl ing of the two methods

fo r solving three-dimensional problems. The theory for coupling of the

two methods i s exac t l y the same as i n a two-dimensional problem,

exp la ined i n Chapter 4. The expansion o f the boundary e lement

s t i f f n e s s mat r i x as ou t l i ned in Eq. 4 .12, i f adopted, becomes very

cumbersome. Therefore, th is operation i s ca r r i ed out i n a d i f f e r e n t

manner f o r three-dimensional problems for e f f i c iency. Results fo r the

problem of a c i r c u l a r p i l e embedded in an e l a s t i c domain are a lso

inc luded here. Some of the resul ts are compared with other solut ions

to determine the degree of accuracy of the suggested technique.

F in i te Element Equation

The structure is represented by three-dimensional f i n i t e elements.

An isoparametric fo rmu la t ion i s used in the c a l c u l a t i o n of element

s t i f f n e s s m a t r i c e s . E i g h t noded b r i c k e lements are used f o r

d i s c r e t i z i n g the s t r u c t u r e , as is recommended f o r the ana lys is of

general e l a s t i c so l i ds [ 8 ] . Moreover, the use of a higher order

element increases the number o f degrees of freedom and makes the

c o n n e c t i v i t y much more complex. The d e t a i l s of the i n t e r p o l a t i o n

functions and the der ivat ion of the element s t i f f n e s s mat r ix can be

75

76

found in standard books on the finite element method [3, 43]. A

computer program has been developed to formulate the stiffness latrix

of the structure using eight noded brick elements. The global finite

element matrix is obtained for the structure in the form of Eq. 4.".

Boundary Element Equation

The soil domain is discretized by constant surface elements,

consisting of plane quadrilaterals. The fundamental solution of Kelvin

for an infinite elastic continuum, as shown in Eqs. 2.23 through Eq.

2.25, is used to develop the H and G matrices. The details of the

theory and numerical analysis were explained in Chapter III. The

columns of the H and G matrices are rearranged as shown in Eqs. -1.3

through 4.8. By use of the Gauss elimination procedure, the equations

are statically condensed up to the interface in the form of Eq. 4.8.

"ft ^B = ^B ^t! Vp

This equat ion i s modif ied to a s t i f fness form by premult iply ing i t by

the matrix R. For constant elements, a diagonal mat r ix i s used w i th

the areas of the respective elements as coef f i c ien ts .

Hong [16] has developed a three-d imensional boundary element

program to solve e l a s t o s t a t i c s problems. Th is program has been

mod i f ied to solve problems w i t h the xy plane as a plane Q-^ symmetry.

The modeling of the symmetric c o n d i t i o n by the use o f boundary

c o n d i t i o n s , as performed in f i n i t e element models, i s not found to be

successful in boundary element models. Errors can be introduced in the

c a l c u l a t e d t r a c t i o n s and displacements a t the corners , where the

77

quantities vary substantially. One way to overcome this problem is to

consider elements with reflections on the plane of symmetry, thus

avoiding major discontinuities in the variables on the boundary. The

elements are reflected on the other side of the plane of symmetry and

are numerically integrated by the usual procedure. In this way, there

are no boundary elements on the plane of symmetry, which substantially

reduces the total number of unknowns. The implementation of this

technique for a two-dimensional problem is explained by Crouch [9].

Coupling of Finite and Boundary Element Matrices

In the boundary element analysis, the unknown displacements and

tractions are considered at the midpoint of the element. In the finite

element analysis, the unknown displacements and forces are considered

at the corners of the element. For this formulation, it is not

possible to ensure compatibility on a one-to-one basis. To achieve

this compatibility, the boundary element stiffness matrix is

transformed to represent the displacements at the corners of the finite

elements and to be compatible with the finite element degrees of

freedom at the interface. It is assumed that the displacements at the

midpoint of a boundary element matching with the face of a finite

element are the averages of the displacements at the four corners of

the finite element. This is shown in Fig. 5.1.

The transformation matrix T for the expansion of the boundary

element stiffness matrix for a single element would be of the form

78

ment node

^ 1

(u.)g,, = %(u|.u?H.u3.up,,, = 1,3

Fig. 5.1 Interface displacement relationship

79

T =

;25 .25 .25 .25 0 0 0 0 0 0 0 0 '

0 0 0 0 .25 .25 .25 .25 0 0 0 0

0 0 0 0 0 0 0 0 .25 .25 .25 .25

(5.1)

Theoretically, the size of the matrix T would be (3n x 12n), where n is

the number of boundary elements to be coupled. Upon transformation, by

use of Eqs. 4.10 through Eq. 4.12, the rows and columns of the global

finite element stiffness matrix must be adjusted by addition, because

of the contributions of stiffness coefficients at a common finite

element node between the boundary elements. This procedure can be

laborious and leads to inefficient multiplication of huge matrices,

requiring large storage on the computer. This transformation is

efficiently done by taking the stiffness coefficients of the boundary

element stiffness matrix, one by one, and identifying their positions

in the finite element stiffness matrix, through a connectivity matrix.

The computer implementation of the above procedure is given below.

DO 90 II = I,NC0MB 13 = 3 * II 12 = 13 - I n = 12 - I DO 70 J = 1,4 N3 = 3 * NC0N(II,J) N2 = N3 - 1 NI = N3 - 2 DO 80 JJ = 1, NCOMB J3 = 3 * JJ J2 = J3 - 1 J1 = J2 - 1 DO 40 KK = 1,4 L3 = 3 * NC0N(JJ,KK) L2 = L3 - 1 Ll = L2 - 1

IF(L3.EQ.N3) GO TO 60 IF(L3.LT.N3) GO TO 40 A(N2,L1-N2+1) = A(N2,L1-N2+1) + BS(I2,J1)

80

A(N3,L1-N3-H) = A(N3,L1-N3-H) + BS(I3,J1) A(N3,L2-N3-H) = A(N3,L2-N3+1) + BS(I3,J2)

60 A(N1,L1-N1-H) = A(N1,L1-N1+1) + BS(I1,J1) A(N1,L2-N1+1) = A(N1,L2-N1+1) + BS(I1,J2) A(N1,L3-N1-H) = A(N1,L3-N1+1) + BS(I1,J3) A(N2,L2-N2+1) = A(N2,L2-N2+1) + BS(I2,J2) A(N2,L3-N2+1) = A(N2,L3-N2+1) + BS(I2,J3) A(N3,L3-N3+1) = A(N3,L3-N3+1) + BS(I3,J3)

40 CONTINUE 80 CONTINUE 70 CONTINUE

RETURN

END

where

NCOMB = Number of boundary elements to be combined;

NCON ( I I , J ) = Connectivity of boundary element I I with f i n i t e element

nodes (J = 1,4);

A (N,L) = F in i te element s t i f fness matrix; and

BS ( I , J ) = Boundary element s t i f fness matrix.

The s t i f f n e s s coe f f i c i en t s of the boundary element matrix are to

be divided by 16 for addit ion into the f i n i t e element s t i f fness matrix

i n the above a l g o r i t h m . The c o n s t a n t 16 comes from the T kgT

o p e r a t i o n , which i s equ iva lent to the o r i g i n a l p rocedure o f t he

t rans fo rmat ion of the boundary element s t i f fness matrix as required by

Eq. 4.12.

F ina l l y , the combined BE/FE model i s developed in the form as

shown in Eq. 4.13. This equation is solved to obtain displacements and

s t r e s s e s o f the s t r u c t u r e i nc lud ing the i n t e r f a c e . Knowing the

displacements of the f i n i t e element nodes at the in ter face, we compute

the c o n s t a n t d isp lacements of the boundary elements. These are

subst i tuted in Eq. 4.8 to get the t r a c t i o n s at the i n t e r f a c e , which

81

represent the boundary forces on the soil medium.

Example Problem

The response of a concrete pile in an elastic homogeneous soil

medium was studied using the three-dimensional coupled BE/FE computer

program. Three types of loading were considered on the top of the

pile: (1) axial load; (2) lateral load; and (3) moment. These basic

loading types signify the general load a pile would be subjected to in

reality. Moreover, these can be represented on a plane of symmetry to

avoid extensive modeling of the whole problem with very little or no

deviations from actual conditions.

A circular pile 2 feet in diameter and 20 feet in length was

embedded in a bounded soil mass. The geometry, boundary conditions,

and loading of the example problem are shown in Fig. 5.2. A smooth

boundary condition was assumed at the bottom and the sides of the soil

region with zero displacements in the normal to the surface direction.

The top surface of the soil region was assumed to be traction free.

Due to the symmetry of the geometry and loading conditions, only one

half of the problem was analyzed. The soil domain was discretized by

102 constant surface boundary elements and the pile by 20 eight-noded

isoparametric finite elements. The discretization of the top surface

of the soil domain is shown in Fig. 5.3.

The following elastic constants were used in the problem:

E ., = 432000 ksf pile

V ., = 0.2; v^^., = 0.2 pile soil

In each loading condit ion, the modular ra t i o m (=Ep/E^) was var ied as

82

P =160 K

M =71.2 Kft

H = 40 K

Fig. 5.2 Details of p i le example problem

# — *

^ -

%D-

^ -

^ -

^ -

* -

% "C

J: 101 JL

^ C"

^ 83

ft

• : ^

- *

10'

- *

:*

=^

^

10'

1

:%

: • ^

Fig. 5.3 Boundary element discretization of the top surface of soil region

84

100, 1000, and 10,000 to h i g h l i g h t the e f f e c t o f s o i l s t r u c t u r e

i n t e r a c t i o n due to v a r i a t i o n s o f modular r a t i o . The v e r t i c a l

displacements of the p i l e along the p i l e / s o i l i n te r face due to the

axial load fo r modular r a t i os of 100, 1000, and 10,000 are shown in

F i g . 5 .4 . The l a t e r a l displacements of the p i l e along the p i l e / s o i l

in ter face for modular rat ios of 100, 1000 and 10,000 due to the la tera l

load and moment on the top of the p i l e are shown in Figs. 5.5 and 5.6,

respect ively. Fig. 5.7 shows the var ia t ion of la tera l t rac t ion due to

l a t e r a l loading along the p i l e / s o i l i n t e r f a c e . These graphs very

c l e a r l y ind ica te the capabi l i ty of the combined model to analyze s o i l -

s t ructure in teract ion problems.

Comparison of Results

The same problem was solved with the finite element program for

displacements along the length of the pile. The plan view of the

discretization of the finite element model consisting of 400 eight-

noded elements is shown in Fig. 5.8.

The displacements along the length of the pile for the combined

BE/FE model are compared with those of the complete finite element

model for the modular ratio 1000 and are shown individually in Figs.

5.9 through 5.11.

The displacement profiles of all the cases in the two methods are

fairly close to each other. In the axial loading condition, the

combined model yielded smaller values to the order of 3.5% than the

finite element solution. In the other two loading conditions, a

difference of 16% for the maximum displacement was seen. The BE/FE

85

o ro i p i-

O I/)

O-

I

I I

4 --

6 ' I

8 r

10

12+

14

16--

18-

20--

Vertical displacement {^) ^ m

xlO'^ft

om = 100

am = 1000

Am = 10,000

Fig. 5.4 Vertical displacement of pile/soil interface due to axial load

86

Lateral displacement ( ) ^ _ ^m'

o m = 100

D m = 1000

A m = 10,000

2.5 xlO"^(ft)

Fig. 5.5 Lateral displacement of p i l e / s o i l interface due to la tera l load

87

Lateral displacement (^)

OJ

o M-i-

. t

o m = 100 n m = 1000 A m = 10,000

xlO"''(ft) •—

Fig. 5.6 Lateral displacement of pile/soil interface due to moment

88

Lateral t ract ion (k/f t )

(U o (O

<+-5-0)

c

o

(U

o m = 100 D m = 1000 Am = 10,000

Fig. 5 7 Lateral t ract ion of Pi le /soi l interface due to la tera l load

1 2 10'

^ "Tt

10'

*-

10'

Fig. 5.8 Plan view of discretization of the finite element model

90

0 BE/FE

• FE

m = 1000

Vertical displacement (ft)

0

2

4 -I-

p i 6

u 4H 8 S-

^ 10

o 12

£ 14

16

18

20

4 xlO

Fig. 5.9 Comparison of vertical displacement of pile/soil interface due to axial load (circular pile)

91

0 BE/FE

• FE

m = 1000

Lateral displacement (ft)

1.5 xlO

Fig. 5.10 Comparison of la tera l displacement of p i l e / so i l interface due to la tera l load (c i rcu lar p i le)

92

0 BE/FE

• FE

m = 1000

Lateral displacement(f t) XlO

Fig. 5.11 Comparison of la tera l displacement of p i l e / so i l interface due to moment (c i rcu lar p i le )

93

combined model predicted higher values than the finite element model.

It is known that a very fine mesh is necessary for finite elements to

achieve good results, especially for three-dimensional problems.

Moreover, eight-noded isoparametric elements are not very efficient in

bending, unless several layers are used in the model [8].

The displacements at the top of the pile for the three loading

conditions are compared with solutions given by Poulos and Davis [29]

in Table 5. Of the three modular ratios analyzed, the ratio of m =

1000 was used for comparison.

TABLE 5

COMPARISON OF RESULTS WITH OTHER METHODS

Displacement (ft)

Load

BE/FE

FE

Poulos & Davis (semi-infini soil mass)

Axial

0.298

0.309

0.277

te

Lateral

0.142

0.122

0.208

Moment

.0329

.0288

.0416

The response of a pile under lateral loading is very much

dependent upon the geometry of the soil medium for a specified pile

geometry and nature of loading. The use of a semi-infinite medium for

the soil region in the combined model is possible through the use of an

appropriate fundamental solution. This is seldom used, as it is not

very realistic. Moreover, the program is developed exclusively for a

bounded soil domain. A limited convergence study was made to determine

the horizontal distance up to which the soil domain has to be modeled

94

for a given finite layer depth. In the example problem solved, the

lateral displacement at the top of the pile converged at 80 ft. from

the center line of the pile to a value of 0.0157 ft. in the laterally

loaded case. In the case of the moment loading, convergence was seen

at 40 ft., to a value of 0.0320ft. It should also be noted that the

convergence study was made for a particular value of modular ratio of

1000. This study gives a general rule for modeling the soil domain for

pile analysis.

The program was also tested for a different pile geometry, namely,

a square pile in the above problem. Comparisons with finite element

solutions for displacements along the pile soil interface for a modular

ratio of 1000 are shown in Figs. 5.12 through 5.14. The displacements

of the combined model are comparable to the finite element solutions

and consistent like the circular pile problem.

Summary

The comparison of displacements of the pile/soil interface of the

combined model with the complete finite element solution is seen to be

close. This comparison has been tested for different modular ratios

and pile shapes. The number of degrees of freedom in the combined

model was much less and modeling was much easier. The computer time

for the combined model was 60 seconds more than for the complete finite

element model. The extra computer time is due to the numerical

integration of the boundary elements to form the H and G matrices.

However, the man hours spent in preparation of the data were fewer.

Moreover, the results indicate that the combined model yields a better

95

o ro

i p S-(U

• p

cu I

0

2 +

4

6 ..

8 -.

10

12.

14t

16

18

20-(-

Vertical displacement ( )

xlO -2 2

H Of

• FE

® BE/FE

m = 1000

Fig. 5.12 Comparison of vertical displacement of pile/soil interface due to axial (square pile)

96

Lateral displacement (^) ( f t )

o ro ^-S -

+->

o CO

• r -Q -

xlO -2

Fig. 5.13 Comparison of la tera l displacement of p i l e / so i l interface due to la te ra l load (square p i le)

97

Lateral displacement (^) (ft)

o ro

< 4 -

o to

<u

a.

xlO"

o BE/FE

• FEM m = 1000

Fig. 5.14 Comparison of lateral displacement of pile/soil interface due to moment (square pile)

98

solution. This clearly demonstrates the super ior i ty of the program

over conventional f i n i t e element programs for analyzing problems of

this category.

CHAPTER VI

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

FOR FUTURE WORK

Summary and Conclusions

The basic principles involved in the formulation of the boundary

element method have been presented. This work is primarily concerned

with the development of a stiffness matrix from the boundary element

equations for coupling to a finite element stiffness matrix. This

approach has been used to solve soil-structure interaction problems

with linearly elastic behavior.

Computer programs were developed in two- and three-dimensional

elasticity for the coupling of the boundary element and finite element

methods. Constant boundary elements were used in the program as this

avoided the problem caused by discontinuities in the geometry and

loading.

The complete programs were used to solve a beam on an elastic

foundation and a pile in an elastic soil domain. The displacements of

the soi 1-structure interface were accurate and compared favorably with

complete finite element solutions, indicating that the developed

computer programs are suitable and reliable for analyzing soil-

structure interaction problems. They are also superior to other codes

by way of easy preparation of input data and greater flexibility in

refining the mesh for further analysis.

99

100

In the development of these codes, a simple transformation

procedure has been adopted for the soil stiffness matrix for coupling

to the structure stiffness matrix. This procedure is necessary due to

the implementation of constant boundary elements in the analysis. This

procedure does not introduce any appreciable errors in the analysis as

seen from the results. In fact, this procedure in three-dimensional

analysis is efficiently carried out by reducing the matrix operations

as explained in Chapter V. As such, this procedure is unique in

application for coupling problems.

A convergence study on a limited basis has thrown light on

satisfactory modeling of the soil domain when using these programs for

a bounded soil region. It is recommended that a convergence study be

made for modeling purposes in high cost structures when the true

boundary conditions of the soil region are not exactly known.

A new approach has been suggested to solve layered soil media in

coupling problems using the condensation technique. Although the

results are not very satisfactory, this approach paves the way for

future work in this area.

The coupling methodology gives a rational approach to determine

the load-displacement relationship at the soil interface. The computer

programs developed would be handy for engineers to incorporate the

effects of soil-structure interaction in their design. The elastic

constants of the materials are sufficient as data for an accurate

linear elastic analysis. The use of a subgrade modulus in the analysis

is therefore totally avoided. Even though soil is nonlinear, a

101

realistic linear analysis would supply a great deal of information as

to its behavior for small deformations. This procedure is a starting

step in geotechnical problems for a reasonably accurate analysis to

replace semi-empirical relations presently in use.

Recommendations for Future Work

Higher order boundary elements could be tried in the two-

dimensional layered model for evaluating the soil stiffness.

The nonl inearities of the soil region surrounding the structure

should be taken into consideration in the computer programs as a next

step. It is felt that the finite element method would be easier and

more economical for a nonlinear analysis. Hence, a portion of the soil

close to the structure could be modeled by finite elements along with

the structure. The elastic stiffness matrix of the remaining soil

region would be calculated only once, using the boundary element

method. This analysis is likely to be superior to a conventional

nonlinear finite element analysis.

REFERENCES

1. Banerjee, P. K., and Davies, T. G., "Analysis of some reported case histories of laterally loaded pile groups," Int. Conf. Num. Meth. in Offshore Piling, Institute of Civil Engineers, London, 1979.

2. Banerjee, P. K., and Butterfield, R., "Boundary Element Methods in Engineering Science," McGraw-Hill Book Co., Ltd., London, 1981.

3. Bathe, K. J. "Finite Element Procedures in Engineering Analysis," Prentice-Hall, New Jersey, 1982.

4. Brebbia, C. A., and Walker, S., "Introduction to Boundary Element Methods," Pentech Press, 1972.

5. Brebbia, C. A., Telles, J. C. F., and Wrobel, L. C , "Boundary Element Techniques Theory and Application in Engineering," Springer-Verlag, New York, 1984.

6. Burmister, D. M., "Stress and Displacement Characteristics of a Two Layer Rigid Base Soil System: Influence Diagrams and Practical Applications," Proceedings Highway Research Board, 35, 773 (1956).

7. Butterfield, R. B.E.M. in Geomechanics, Lecture notes. Boundary Element Techniques in Computer Aided Engineering, NATO, Advanced Study Institute, Southampton, Sep. 1983.

8. Clough, R. W., "Comparison of Three Dimensional Finite Elements," Proc. of Symposium on Application of Finite Element Methods in Civil Engineering, Tennessee, 1969.

9. Crouch, S. L., and Starfield, A. M., "Boundary Element Methods in Solid Mechanics," George Allen and Unwin, London, 1983.

10. Cruse, T. A., Numerical Solutions in Three Dimensional Elastostatics, Int. J. Solids Structure, vol. 5, 1969.

11. Cruse, T. A., Application of the Boundary Integral Equation Method to Three Dimensional Stress Analysis, Computer and Structures, vol. 3, 1973.

102

103

12. Danson, D. J., "Beasy A Boundary Element Analysis System," Boundary Element Methods in Engineering. Proc. Fourth International Seminar, Southampton, England, 1982.

13. Duncan, J. M., and Clough, G. W., "Finite Element Analysis of Port Allen Lock," J. of the Soil Mechanics and Foundations Div., ASCE, Vol. 77 No. SM8, 1971.

14. Georgiou, P., "The Coupling of the Direct Boundary Element Method with the Finite Element Displacement Technique in Elastostatics," Ph.D. Thesis, Southampton University, 1981.

15. Hartmann, F., "The Derivation of Stiffness Matrices from Integral Equations," jji Boundary Element Methods (C. A. Brebbia, Ed.), Springer-Verlag, Berlin, 1981.

16. Hetenyi, M., "Beams on Elastic Foundation," Ann Arbor, University of Michigan Press, 1946.

17. Hong, K., "Application of Boundary Element Method for Some 3-D Elasticity Problems," M.S. Thesis, Texas Tech University, Lubbock, 1984.

18. Jaswon, M. A., and Symm, G. T., "Integral Equation Methods in Potential Theory and Elastostatics," Academic Press, 1977.

19. Kellog, 0. D., "Foundations of Potential Theory," Frederick Ungar Pub. Co., New York, 1929.

20. Kupradze, V. D., "Potent ial Methods in the Theory of E l a s t i c i t y , " Frederick Ungar Pub. Co., New York, 1965.

21. Lachat, J . C , and Watson, J . 0 . , "A Second Generation Boundary I n t e g r a l Equation Program f o r Three-Dimensional E las t i c Analysis," Appl. Mech. Div. , ASME 11, 1975.

22. Love, A. E. H. , " T r e a t i s e on the Ma themat i ca l Theory o f E l a s t i c i t y , " Dover, 1964.

23. Massonet, C. E., "Numerical Use of Integral Procedures," Stress Analysis, ed. by Z ienk iewicz , 0. C , and H o l i s t e r , G. S. John Wiley and Sons, New York, 1965.

24. Matlock, H. , and Reese, L. C , "Genera l i zed So lu t ions f o r Laterally-Loaded P i l es , " Proc. ASCE, 86, SMS, 63-91 , D e c , 1960.

25. M ikh l in , S. G., " I n t e g r a l Equations," Pergamon Press, Elmsford, New York, 1957.

104

26. Muskhelishvili, N. I., "Some Basic Problems of the Mathematical Theory of Elasticity," P. Noordhoff, 1953.

27. Nakaguma, R. K., "Three Dimensional Elastostatics Using the Boundary Element Method," Ph.D. Dissertation, Southampton University, 1979.

28. Pasternak, P. L., "On a New Method of Analysis of an Elastic Foundation by Means of Two Foundation Constants," Moscow: Gos. Izd. Lit. Postroiti Arkh., 1954 (in Russian).

29. Poulos, H. G., and Davis, E. H., "Elastic Solutions for Soil and Rock Mechanics," John Wiley and Sons, New York, 1974.

30. Rizzo, F. J., "An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics," Quart. Appl. Math., 25, 83, 1967.

31. Scott, R. F., "Foundation Analysis," Prentice-Hall, New Jersey, 1981.

32. Terazaghi, K., "Evaluation of Coeff icients of Subgrade Reaction," Geotechnique, 5, 297, 1955.

33. Timoshenko, S. , and Goodier, J . N., "Theory of E l a s t i c i t y , " McGraw H i l l , New York, 1971.

34. Vallabhan, C. V. G., "A Feas ib i l i t y Study of Boundary Element for Soi l -Structure Interact ion Problems," Report for U. S. Army Corps of Engineers, WES, Vicksburg, MS, Sept. 1983.

35. Vallabhan, C. V. G., and Sivakumar, J . , "The A p p l i c a t i o n of Boundary Element Techniques f o r Some S o i l S t ruc tu re I n t e r a c t i o n Problems," Report f o r U. S. Army Corps o f Engineers, WES, Vicksburg, MS, Dec. 1983.

36. Val labhan, C. V. G. , Sivakumar, J . , and Radhakrishnan, N., "Appl icat ion of Boundary Element Method f o r S o i l - S t r u c t u r e Interact ion Problems," Proc. of Sixth Internat ional Conf. on Boundary Element Methods, July, 1984.

37. Vallabhan, C. V. G., and Jain, R. K., "Octahedral Stress Approach t o A n a l y s i s o f Water Resources S t r u c t u r e s , " Proc. of Symposium on A p p l i c a t i o n of F i n i t e Element Method i n Geotechnical Engineering, Vicksburg, MS, May, 1972.

38. Vesic, A. S., "Bending of Beams Resting on I s o t r o p i c E las t i c S o l i d , " Proc. ASCE, 87, Jour. Engg. Mech. Div. EM2, 35, A p r i l , 1961.

105

39. Vlasov, V. Z., and Leont'ev, N. N., "Beams, Plates and Shells on Elastic Foundations." Translated from Russian. Israel Program for Scientific Translations, Jerusalem, 1966.

40. Wang, S. K., Sargious, M., and Cheung, Y. K., "Advanced Analysis of Rigid Pavements," Transportation Engineering Journal of ASCE, Vol. 98, No. TE1, Proc. Paper 8699, Feb. 1972.

41. Westergaard, H. M., "Stresses on Concrete Pavements Computed by Theoretical Analysis," Public Roads, 7, 25, 1926.

42. Wilson, E. L., "The Static Condensation Algorithm," Int. Journal of Numerical Methods in Eng. Vol. 8, pp. 199-203, 1974.

43. Zienkiewicz, 0. C , "The Finite Element Method," McGraw-Hill Book Co., 1977.

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