Post on 17-May-2022
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
4.9 Vertical displacement at the interface—case 3 and 4 of loading condition 2 52
4.10 Vertical displacement at the interface—case 1 and 2 of loading condition 3 53
4.11 Vertical displacement at the interface—case 3 and 4 of loading condition 3 54
viii
Figure Page
4.12 Vertical tractions at the interface—case 1 and 2 of loading condition 1 55
4.13 Vertical tractions at the interface—case 3 and 4 of loading condition 1 56
4.14 Vertical tractions at the interface—case 1 and 2 of loading condition 2 57
4.15 Vertical tractions at the interface—case 3 and 4 of loading condition 2 58
4.16 Vertical tractions at the interface—case 1 and 2 of loading condition 3 59
4.17 Vertical tractions at the interface—case 3 and 4 of loading condition 3 60
4.18 Comparison of displacements at the interface— case 1 and 2 of loading condition 1 62
4.19 Comparison of displacements at the interface— case 3 and 4 of loading condition 1 63
4.20 Comparison of displacements at the interface— case 1 and 2 of loading condition 2 64
4.21 Comparison of displacements at the interface— case 3 and 4 of loading condition 2 65
4.22 Comparison of displacements at the interface— case 1 and 2 of loading condition 3 66
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)
Fig. 4.7 Vertical displacement at the interface—case 3 and 4 of loading condition 1
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 )
Fig. 4.8 Vertical displacement at the interface—case 1 and 2 of loading condition 2
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
FE/BE Interface -> X (ft)
Fig. 4.9 Vertical displacement at the interface—case 3 and 4 of loading condition 2
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
Fig. 4.10 Vertical displacement at the interface—case 1 and 2 of loading condition 3
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
Fig. 4.11 Vertical displacement at the interface—case 3 and 4 of loading condition 3
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)
Fig. 4.13 Vertical tractions at the interface—case 3 and 4 of loading condition 1
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
Fig. 4.15 Vertical tractions at the interface—case 3 and 4 of loading condition 2
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-
Fig. 4.16 Vertical tractions at the interface—case 1 and 2 of loading condition 3
60
M=5000ft.k A
c ;^-Lb h/L L/t, V^s
Fig. 4.17 Vertical tractions at the interface—case 3 and 4 of loading condition 3
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
FE/BE Interface -> X (ft)
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
FE/BE Interface -> X (ft)
Fig. 4.19 Comparison of displacements at the interface-case 3 and 4 of loading condition 1
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)
Fig. 4.20 Comparison of displacements at the interface-case 1 and 2 of loading condition 2
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
FE/BE Interface -> X (ft)
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
Fig. 4.22 Comparison of displacements at the interface— case 1 and 2 of loading condition 3
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
Fig. 4.23 Comparison of displacements at the interface— case 3 and 4 of loading condition 3
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.
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