Table Of ContentCOMPUTATIONAL GEOMECHANICS
Theory and Applications of Transport in Porous Media
Series Editor:
Jacob Bear, Technion -Israel Institute ofTechnology, Haifa, Israel
Volume?
The titles published in this series are listed at the end of this volume.
Computational
Geomechanics
by
Arnold Verruijt
Department of Civil Engineering,
Delft University of Technology,
The Netherlands
'~'
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4532-4 ISBN 978-94-017-1112-8 (eBook)
DOI 10.1007/978-94-017-1112-8
Printed on acid-free paper
All Rights Reserved
© 1995 Springer Science+B usiness Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint of the hardcover 1st edition 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Soil Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Theory of Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Sea Bed Response to Cyclic Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Beams on Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Sheet Pile Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6 Axially Loaded Piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7 Development of Pile Plug . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Laterally Loaded Piles ........................................ 122
9 Pile in Layered Elastic Material ................................. 151
10 Waves in Piles ............................................. 167
11 Gravity Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
12 Slope Stability ............................................. 204
13 Finite Elements for Plane Groundwater Flow ........................ 219
14 Finite Elements for Plane Strain Elastostatics ........................ 244
15 Finite Elements for Elasto-Plastic Deformations ...................... 266
16 Finite Elements for One-dimensional Consolidation .................... 280
17 Finite Elements for Two-dimensional Consolidation ................... 291
18 Transport in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
19 Adsorption and Dispersion ..................................... 337
Appendix A. Integral Transforms .................................. 356
Appendix B. Solution of Linear Equations ............................ 365
References .................................................. 375
Index ...................................................... 381
PREFACE
In modern geomechanics computational methods are widely used to predict the be
haviour of foundation structures and soil bodies. Because of the complex non-linear
mechanical behaviour of soils simple analytical methods often fail to describe the
detailed behaviour of foundation structures, although for many practical problems
effective approximate methods have been developed by geotechnical engineers.
The advent of computers in recent decades has made it possible to develop models
using realistic non-linear relations to describe the mechanical behaviour of soils.
This has lead to an enormous increase in the number of available computer models,
varying from simple calculation procedures in BASIC to complex packages that
require special courses to obtain familiarity with their use.
This book attempts to present a systematic development of computer models
for most problems of geotechnical engineering, emphasizing the physical (mechani
cal) background of the basic assumptions used in the models. The book has grown
out of a collection of lecture notes, used at the Department of Civil Engineering of
the Delft University of Technology, especially for final year courses on Numerical
Geomechanics and Offshore Soil Mechanics.
The book is mainly concerned with the development, implementation, and
application of geotechnical models, presenting some analytical methods, but con
centrating on numerical methods for the solution of problems of geotechnical en
gineering. The type of problem ranges from the calculation of the generation and
the dissipation of pore pressures in soils to the analysis of the stability of slopes
of embankments and the analysis of the deformation of sheet pile walls, and the
analysis of the flexibility of axially and laterally loaded piles. Both static problems
of load transfer are considered as well as dynamic problems of wave propagation
during the installation of foundation piles. The physics of stress transfer in soils
due to surface loading is considered in some detail. This leads to problems of
elasticity and elasto-plasticity, and problems of consolidation for fluid-saturated
media.
For the numerical mathematical solution of most one-dimensional problems
the finite difference method is used, because of its mathematical and compu
tational simplicity. For two-dimensional problems the finite element method is
generally preferred, because of the possibilities it offers to solve problems for non
homogeneous materials. For all these methods the basic principles are presented
as well as the details needed to develop real numerical models and computer pro
grams.
Finite element solutions are presented for problems such as the flow and trans
port of groundwater, elastic and elasto-plastic deformation problems, and prob
lems of consolidation or poro-elasticity. Problems of transport of pollutants in
moving groundwater are considered in some detail, taking into account the three
dimensional aspects of this transport, and secondary phenomena such as adsorp-
vii
viii
tion and dispersion.
A basic knowledge of the principles of engineering mathematics and mechanics
is assumed, including an understanding of basic soil mechanics. This requires
familiarity with the basic concepts of continuum mechanics, such as the analysis
of stress and strain, and the basic concepts of soil mechanics such as the principle
of effective stress, and the parameters to describe deformations and strength of
granular materials.
For many problems elementary computer programs that execute the procedures
developed in the book are given. Complete listings of 23 programs are presented
in the book, and on the diskette accompanying it. The language used is Turbo
Pascal (as defined ami distributed by Borland International, Inc.), which is a well
defined and widely available language. In order to extend the applicability of these
programs certain extensions and generalizations are suggested in the exercises.
For each computer program at least one example is given in the text, in many
cases described by a dataset defining the parameters of the problem. The usual
naming convention is that the Pascal version of the program is called (for instance)
SLOPE.PAS, the compiled (and executable) version is called SLOPE.EXE, and
the dataset is called SLOPE!. The listing of SLOPE.PAS is given in the book, as
well as the dataset SLOPE!. The diskette also contains the executables, and in
some cases an additional dataset, say SLOPE2. The computer programs may be
used, copied and distributed without restriction. No responsibility for any loss or
damage resulting from use of these programs is accepted, however.
It is a pleasure to acknowledge that many of the numerical models presented
in this book have been developed and tested with the help of undergraduate or
graduate students. A few of them have published their work, and can be found
in the list of references. Others have found their way in the world of applied civil
engineering, where the actual design or realization of engineering works is of more
importance than the publication of papers in academic journals. They all deserve
many thanks.
The book has been prepared using the U.TEJX version (Lamport, 1986) of the
program 'lEX (Knuth, 1986). Most of the figures have been prepared using the
PJClEX macros (Wichura, 1987). It has been a constant delight to discover the
various possibilities of these programs.
Delft, January 1995 Arnold Verruijt
CHAPTER 1
SOIL PROPERTIES
In this chapter the mechanical properties of soils are briefly discussed, emphasizing
the properties of soft soils, for which the variability is the largest.
Soil mechanics is the branch of science concerned with the equilibrium and
movement of soil bodies, where soil is understood to be the weathered natural
material in the upper layers (say the upper 20 to 100 m) of the earth's crust.
This material may be gravel, sand, clay, peat, or some other rather soft and loose
granular medium. The nature of these materials is quite different from artificial
man-made materials such as steel, concrete, etc. These materials usually are
much more consistent than soils, and exhibit relatively simple, linear, mechanical
behaviour, at least if the deformations are not too large.
The mechanical properties of soils are usually strongly non-linear, with the ma
terial exhibiting irreversible plastic deformations when loaded and unloaded, even
at low stress levels, and often showing anisotropic behaviour, creep and sometimes
even such typical effects as dilatancy (a volume increase during shear). Another
difficulty is that the structure of a soil often is highly inhomogeneous, due to its
geological history, and it is often not possible to determine the detailed behaviour
of each individual layer or element in the soil by tests in the laboratory or in situ.
The behaviour of soils may be further complicated by the presence of water in the
pores. This relatively stiff fluid in the pores may prevent or at least retard volume
deformations. Although there have been many attempts to describe the relation
between stresses and strains of a soil by one single relation, this relation then is
so complicated, and involves so many parameters, that it is practically impossible
to determine all these parameters, and the complex relation is very difficult to
implement in practical calculations.
For all these reasons the characterization of the mechanical behaviour of soils
is often done in a schematic way only, and itfl form is adapted to the particular
type of problem under consideration. Thus for a problem of long term settlements
of an embankment the relevant soil properties are quite different from those for
a problem of stability of the slopes of the same embankment. Even though these
problems may seem to be closely related, the methods of analysis involve different
concepts, and different parameters. In the case of stability of slopes the strength of
the soil layers is the dominating effect, whereas the settlement of an embankment
is mainly governed by the deformation properties of the soil, such as a creep law.
Thus in soil mechanics the range of applicability of a certain parameter is often
restricted to a limited class of problems. Many properties can not be used outside
their intended field of application. Nevertheless, various properties may all derive
from such common basic phenomena as interparticle friction, or the structure
1
2 CHAPTER 1
of a granular medium, so that there may well exist good correlations between
certain properties. In this chapter some of these properties are reviewed, and
some correlations are discussed. It should be noted that in engineering practice
nothing can beat the results of experimental determination of the soil parameters
in situ, or in the laboratory. A correlation may at best give a first estimate of the
order of magnitude of a parameter.
1.1 Stiffness
In many problems of soil-structure interaction the soil is modeled by a subgrade
modulus,
p
w= -, (1.1)
c
where w is the settlement of a foundation element, pis the pressure upon it, and c
is the subgrade modulus. The soil is modeled by a linear spring in this case. One of
w
Figure 1.1. Linear spring.
the main reasons for the use of this relation may well be the simplicity of a linear
relationship for analytical or numerical calculations. It should be remembered,
however, that this is a very simplistic way of modeling real soil behaviour, taking
into account that in reality soils exhibit various forms of non-linear and irreversible
behaviour. In some cases it may well be justified, however, to use a linear spring
model, as a first approximation.
The flexibility of a plate foundation may be described a little better by mod
eling the soil as a linear elastic continuum, having a modulus of elasticity E, and
Poisson's ration v. For a rigid circular plate the following relation between pressure
and settlement can be derived, on the basis of Boussinesq's elementary solution
for a point load on an elastic half plane (Timoshenko & Goodier, 1951),
= -1-r-"(-1----v'-2.:)..p.._D-
w (1.2)
4E
SOIL PROPERTIES 3
!U!UU
..................... ! ........... ! .................... .
.............. .. .. .. ..... . .. ........ .... ... .. . . ... ... . . . .. ......... .. ..... . ... .. .... .. . . . ........... ........ . . . ....... .... . . ........... .......... .. . .......... . ..... ..... .. . ............ ...... . . . .... . .... . . . . ... ...... ........ ....... . . ... . .... .. . .. .... ..... . ...... ......... ..... ....... .. ... . .. ..... .... .. .. .... ...... .... ....... . .. ....... . .. . .. . ........... ........ .. . . ....... .. . . .. . .. . ....·... ........... . .. .. ...... ... . ........ ... . .. . . .. ........ ............ . ......... ... . . . ....................... . .......... ... . . ..... ...... .............. ............ .. .. ..... ..... . .. ... ...... ............ . . . ...... ... . ............ .......... . . ...... .... .. . . ... ...·... ............ .. ... ...... . . . . ............ ....... .. . ...... ......... ....... . . . . . .. ........... ...... ....· ......... .... .. . ..·....... ... .......... .... . . · .... . . . ......... .. .·...... . .... ..... .... ... .. ·... . . . . ............. ..·....... . . . ......... ... . . . .· .......... ............. . . ......·. . . . .. . . ............. ...·........ . ............. . . ... .. · ...... . . . .. ...... ...... ...·........ . . . ....... .. .. . ... · . .... ..... ......... ... . . ....·.... . . .. .. . .. . ........ ....·..... . . .. .. .... . . .. . .... ... .... .. . .. .. . . . ......... ........... .. ... ....... . . .. ... ... . ...... ....... .. . ... ......... . .. ... .......... .......... .. .. ........ . . ... ......... ...... ..... . ......... ...... ....... . ... . . .......... . ........ .... ...... .. .. ............. ... . .... . ... .. . .... .........
Figure 1.2. Foundation on elastic soil.
where Dis the diameter of the plate. A linear relationship between the load p and
the displacement w is obtained, which is of course a consequence of the assumption
that the soil behaves as a linear elastic material. The relation appears to depend
upon the diameter of the plate, however, and it is therefore a little more refined
than the simple subgrade modulus model.
It is interesting to note that a similar relation can be obtained by starting
from the assumption that in the soil the stresses are distributed uniformly over an
area of gradually increasing magnitude, over an angle of say 45° with the vertical
.. ..... . . . . ... ... ... .. .. . .. .... . . .. ... ... .. . .... . . . .. ... ... . . . .... .. ... . . . .... .. . ..... . ... . . . ... .. .. . .. ........ ...... ... .. .. . . .
.. .. .. ... ... .... ... ... . .. . .... .. ..... ... ....... ... . ... .... ... .... ....... .............. ....... .. . ........... ....... ... .... .......... . .......... ........... .............. ...... ..... . ........ ... . . .. . ... .... .... ... ... . .. . ... ... . . . . . .. .. . . . .
.. .. . ... .. .. .. .. .... .. ... ... .... ... ... . ... ....... ... . .......... ........... ...... . ... ... . ... .. .. . .. ... . ...... . ... ... ........ ........ . . .... ... ... . ... ... ... . ... . .. ... . .... . .... . ... . . ..... ...... ...... ...... . .... .. . ...... . .... .. . ...... . .... .. . .. ..... ... .. ..... ........ . ...... .... ... . .. .. .
Figure 1.3. Foundation on elastic soil, approximation.
direction, see figure 1.3. This means that at a depth z below the surface the area
carrying the load is
(1.3)
The vertical normal stress at that depth is the total force F = ~7r D2p, divided by
the area A,
p
ffv = (1 + 2z/D)2 · (1.4)
If the medium is assumed to be linear elastic, the vertical strain at this depth
is, ignoring the influence of horizontal stresses (or, in other words, assuming that
=
v 0),
