Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter 1. Introduction: Basic Concepts . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Soils and rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Engineering properties of soils . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3. Soils as an aggregation of particles . . . . . . . . . . . . . . . . . . . . . . 7
1.4. Interaction with pore water . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5. Transmission of the stress state in granular soil . . . . . . . . . . . . . . 10
1.6. Transmission of the stress state in the presence of a fluid . . . . . . . . 14
1.7. From discrete to continuum . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8. Stress and strain tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 2. Field Equations for a Porous Medium . . . . . . . . . . . . . . . . 27
2.1. Equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2. Compatibility equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3. Constitutive laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4. Geostatic stress state and over-consolidation . . . . . . . . . . . . . . . . 40
2.5. Continuity equation and Darcy’s law . . . . . . . . . . . . . . . . . . . . . 44
2.6. Particular cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6.1. Dry soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6.2. Saturated soil with still groundwater . . . . . . . . . . . . . . . . . . 50
2.6.3. Saturated soil with seepage: stationary conditions . . . . . . . . . . 50
2.6.4. Saturated soil with seepage: transient conditions . . . . . . . . . . . 51
2.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
vi Soil Mechanics
Chapter 3. Seepage: Stationary Conditions . . . . . . . . . . . . . . . . . . . . 57
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2. The finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3. Flow net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4. Excess pore pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5. Instability due to piping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6. Safety factor against piping . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.7. Anisotropic permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.8. Transition between soils characterized by different
permeability coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.9. Free surface problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.10. In situ methods for the permeability coefficient determination . . . . . 77
3.11. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Chapter 4. Seepage: Transient Conditions . . . . . . . . . . . . . . . . . . . . . 83
4.1. One-dimensional consolidation equation . . . . . . . . . . . . . . . . . . 83
4.2. Excess pore pressure isochrones . . . . . . . . . . . . . . . . . . . . . . . 86
4.3. Consolidation settlement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.4. Consolidation settlement: approximated solution . . . . . . . . . . . . . 93
4.5. Consolidation under different initial or boundary conditions . . . . . . . 97
4.6. Load linearly increasing over time: under consolidation . . . . . . . . . 101
4.7. Consolidation under axial symmetric conditions . . . . . . . . . . . . . . 104
4.8. Multidimensional consolidation: the Mandel-Cryer effect . . . . . . . . 106
4.9. Oedometer test and measure of cv . . . . . . . . . . . . . . . . . . . . . . . 114
4.10. Influence of the skeleton viscosity . . . . . . . . . . . . . . . . . . . . . 118
4.11. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Chapter 5. The Constitutive Relationship: Tests and
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2. Fundamental requirements of testing apparatus . . . . . . . . . . . . . . 127
5.3. Principal testing apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.3.1. The “true” triaxial test (TTA): Lamé’s ellipsoid and Mohr’s
sickle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.3.2. The (standard) triaxial apparatus . . . . . . . . . . . . . . . . . . . . . 135
5.3.3. The oedometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.3.4. The biaxial apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.3.5. Direct shear box and simple shear apparatus (SSA) . . . . . . . . . 147
5.3.6. Hollow cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Table of Contents vii
5.4. The stress path concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.5. Experimental results for isotropic tests on virgin soils . . . . . . . . . . 163
5.6. Experimental results for radial tests on virgin soils: stress,
dilatancy relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.7. Oedometric tests on virgin soil as a particular case of the
radial test: earth pressure coefficient at rest . . . . . . . . . . . . . . . . . . . 173
5.8. Drained triaxial tests on loose sands: Mohr-Coulomb
failure criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.9. Undrained triaxial tests on loose sands: instability line and static
liquefaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.10. Drained tests on dense and medium dense sand:
dilatancy and critical state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.11. Strain localization: shear band formation . . . . . . . . . . . . . . . . . 191
5.12. Undrained tests on dense and medium dense sands: phase
transformation line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.13. Sand behavior in tests in which the three principal stresses are
independently controled: failure in the deviatoric plane . . . . . . . . . . . . 198
5.14. Normally consolidated and over-consolidated clays: oedometric
tests with loading unloading cycles – extension failure . . . . . . . . . . . . . 201
5.15. Drained and undrained triaxial tests on normally consolidated
clays: normalization of the mechanical behavior . . . . . . . . . . . . . . . . 208
5.16. Over-consolidated clays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
5.17. The critical state. Plasticity index . . . . . . . . . . . . . . . . . . . . . . 219
5.18. Natural soils: apparent over-consolidation – yielding surface . . . . . 226
5.19. Soil behavior under cyclic loading: cyclic mobility and strength
degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
5.20. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Chapter 6. The Constitutive Relationship: Mathematical Modeling
of the Experimental Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
6.2. Nonlinear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
6.3. Perfect elastic-plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
6.4. Yielding of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
6.5. Taylor and Quinney experiments: the normality postulate . . . . . . . . 251
6.6. Generalized variables of stress and strain . . . . . . . . . . . . . . . . . . 258
6.7. Plastic strains for a material behaving as described by the
Mohr-Coulomb criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
6.8. Drucker-Prager and Matsuoka-Nakai failure criteria . . . . . . . . . . . 261
6.9. Dilatancy: non-associated flow rule . . . . . . . . . . . . . . . . . . . . . 267
6.10. Formulation of an elastic-perfectly plastic law . . . . . . . . . . . . . . 269
viii Soil Mechanics
6.11. Cam clay model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
6.12. Reformulation of the Cam clay model as an elastic-plastic
hardening model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
6.13. Comparison between experimental behavior and mathematical
modeling for normally consolidated clays . . . . . . . . . . . . . . . . . . . . 285
6.14. Lightly over-consolidated clays . . . . . . . . . . . . . . . . . . . . . . . 290
6.15. Heavily over-consolidated clays . . . . . . . . . . . . . . . . . . . . . . . 293
6.16. Subsequent developments and applications . . . . . . . . . . . . . . . . 298
6.17. Non-associated flow rule: the Nova-Wood model . . . . . . . . . . . . 301
6.18. Sinfonietta classica: a model for soils and soft rocks . . . . . . . . . . . 309
6.19. Models for soils subjected to cyclic loading . . . . . . . . . . . . . . . . 315
6.20. Conceptual use of constitutive soil behavior models . . . . . . . . . . . 318
6.20.1. Oedometric test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
6.20.2. Unconfined undrained (UU) test . . . . . . . . . . . . . . . . . . . . 321
6.20.3. Shear modulus “anisotropy” . . . . . . . . . . . . . . . . . . . . . . . 324
6.21. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
Chapter 7. Numerical Solution to Boundary Value Problems . . . . . . . . . 329
7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
7.2. The finite element method for plane strain problems . . . . . . . . . . . 330
7.3. Earth pressures on retaining structures . . . . . . . . . . . . . . . . . . . . 344
7.4. Settlements and bearing capacity of shallow foundations . . . . . . . . . 354
7.5. Numerical solution of boundary value problems for
fully saturated soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
7.6. Undrained conditions: short-term bearing capacity of a footing . . . . . 371
7.7. Short- and long-term stability of an excavation . . . . . . . . . . . . . . . 380
7.8. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
Preface
According to the engineering nomenclature, soil mechanics is concerned with the
behavior of clastic rocks, or “soils”, under different loading conditions: external
loading, such as that transmitted by the foundations of any structure, or generated by
the seepage of water, and also by its own weight as a consequence of geometric
changes, induced for instance by excavation or tunneling.
Knowledge of soil mechanical behavior is, in fact, an essential element for the
prediction of the displacements and internal actions of a structure founded on or
interacting with it. Soil Mechanics is, therefore, the fundamental subject of
geotechnical engineering, the branch of civil engineering concerned with soil and
with the interacting soil structure, dealing with the design and the construction of
civil and industrial structures and environment defense works against geological
hazards.
Aristotle said “Φαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητική”: any prediction
is based either on a rational calculation or on intuitive perception. Although the
latter has been for a long time the starting-point of any construction and still plays a
relevant role in design, it is the former that allows the definition of the structure’s
dimensions and safety assessment. In fact, it allows rational prediction of the
structure’s behavior in the different construction phases and during its life.
This calculation must be based on a mathematical model of the structure and the
soil. This should schematize the geometry of the problem, the mechanical behavior
both of materials and structures, as well as the loading. The definition of an overall
mathematical model of the structure and the soil is a very complex problem that is
x Soil Mechanics
beyond the scope of this book. In the following, only the bases upon which a
mathematical model of soil behavior can be formulated will be outlined.
Though limited in scope, soil modeling is rather complex and requires different
levels of abstract thinking. First, it is necessary to pass from the physical nature of
soil, composed of a discrete and innumerable number of solid mineral particles and
voids, into which fluids such as air, water or mineral oils can seep, to its
representation as a continuum. In fact, this allows a much more feasible
mathematical formulation. In order to achieve this goal, it is necessary to assume the
soil to be a special medium obtained by “overlapping” two continua: a solid
continuum, modeling the skeleton composed of the mineral particles, the “solid
skeleton”, and a fluid continuum, modeling the fluid, or the fluid mixture, seeping
through the voids.
The most relevant aspect lies in the fact that both these continua completely
occupy the same region of space. They interact by parting the stress state in a way
that directly derives from the conditions of conservation of energy and mass, and
that is a function of how the behavior of the solid continuum under loading and the
fluid seepage in the soil are independently modeled. Hence, it is necessary to
mathematically formulate models for the description of the mechanical behavior of
the solid skeleton (stress-strain relationship) and a conceptually equivalent law
ruling the motion of fluid with respect to the solid skeleton.
Once the model is defined, in order to mathematically reproduce with the best
approximation possible the experimental results obtained by elementary tests, the
parameters describing the soil (or the different soil layers) behavior have to be
specified for the case under examination.
Finally, a further step in modeling is necessary to transform the system of
differential equations and boundary conditions ruling any soil mechanics problem in
the light of continuum mechanics into a system of algebraic equations that can be
solved by means of a computer.
This book will be developed in logical sequence according to what has been
previously outlined.
Chapter 1 presents some elementary concepts necessary to pass from the discrete
nature of soil to its continuum representation. Differential and boundary equations
for a generic soil mechanics problem will then be presented in Chapter 2. Special
cases will be analyzed, such as stationary seepage conditions (Chapter 3), “rapid”
loading conditions (undrained conditions), and transient seepage conditions (Chapter
4). In this last case, under constant loading, the stress state is transferred from the
water to the solid skeleton, inducing soil deformations and structure assessments
Preface xi
over time (consolidation). For the sake of simplicity, in this case soil will be
assumed to be characterized by an incrementally linear behavior.
Nevertheless, the mechanical behavior of the solid skeleton is much more
complex. In fact, it is nonlinear, irreversible, and highly influenced by the average
pressure to which it is subjected. These aspects will be detailed in Chapter 5, which
is dedicated to the study of the response of elementary soil samples in laboratory
tests. In Chapter 6, mathematical models of increasing complexity describing the
behavior outlined in the previous chapter will be formulated. Finally, in Chapter 7,
methods of discretizing the continuum and integration procedures will be
mentioned. A few examples, referring to archetypes of geotechnical problems
(foundations, sheet piles, slopes), will illustrate the results that can be obtained in
this way.
This book is not intended to be exhaustive on all the geotechnical issues or to
give “practical” suggestions. For these purposes several good and topical books
already exist and there is no reason to write another. On the contrary, the goal of this
work is to tackle the fundamental aspects of a very complex subject at a deeper level
than current works. These aspects can have remarkable consequences on the choices
that the engineer has to make in order to build the geotechnical model of the soil that
is appropriate for the particular case under examination (geometry of the problem,
type of model to describe soil behavior, parameters to be assumed, type of numerical
solution) and thus, as a consequence, on the design.
Having worked in the field of soil mechanics for many years, I know that there is
some confusion concerning the fundamental principles which this subject is based
on. Frequently, even people working in the geotechnical engineering field do not
completely understand the formulae that they use, especially the computer methods,
whose bases they do not have knowledge of. The dialog between the several actors
involved in a geotechnical project (civil and environmental engineers, geologists,
architects) risks becoming a dialog between deaf people, in which not even the
specific role of each of them is clear.
As any good geotechnical engineer knows, a safe structure has to be based on
solid foundations. The book is therefore intended to give, to those who will have the
patience to read it, the bases necessary to understand the fundamentals of soil
mechanics. It is my firm belief that only through the thorough understanding of such
fundamentals can appropriate geotechnical characterization and soil modeling be
carried out. Though the main point of this book is undoubtedly theoretical, its final
goal is very practical: to give adequate means for a correct framing of geotechnical
design.
xii Soil Mechanics
In writing this book, I was privileged to collaborate with some young colleagues:
Claudio di Prisco, Roberta Matiotti, Silvia Imposimato, Riccardo Castellanza,
Francesco Calvetti, Cristina Jommi, Rocco Lagioia, Claudio Tamagnini, Stefano
Utili, Giuseppe Buscarnera, Matteo Oryem Ciantia, Giuseppe Dattola and Federico
Pisanò. They helped me to clarify the text (in addition to taking care of the
graphics). To them and to all those who have been so kind as to highlight mistakes
and omissions or simply been willing to discuss the non-traditional approach
followed in this book, my most sincere thanks.
This book is dedicated to “my” Maddalena, Tommaso and Tobia, who patiently
bore the consequences of its writing.
