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Soil Mechanics
Soil Mechanics
Roberto Nova
Translated by Laura Gabrieli
First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc The translation of this book has been funded by SEPS (Segretariato Europeo per le Pubblicazioni Scientifiche) Via Val drsquoAposa 7 40123 Bologna Italy sepsalmauniboit ndash wwwsepsit Originally published in Italian under the title Fondamenti di meccanica delle terre ISBN 88-386-0894-6 copy The McGraw-Hill Compagnies S r l Milano 2002 Permission for this edition was arranged through The McGraw-Hill Compagnies Srl ndash Publishing Group Italia Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom copy ISTE Ltd 2010 The rights of Roberto Nova to be identified as the author of this work have been asserted by him in accordance with the Copyright Designs and Patents Act 1988
Library of Congress Cataloging-in-Publication Data Nova Roberto [Fondamenti di meccanica delle terre English] Soil mechanics Roberto Nova translated by Laura Gabrieli p cm Includes bibliographical references and index ISBN 978-1-84821-102-5 1 Soil mechanics I Title TA710N58713 2009 32415136--dc22
2009034935 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-102-5
Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne
Table of Contents
Preface ix
Chapter 1 Introduction Basic Concepts 1
11 Soils and rocks 1 12 Engineering properties of soils 3 13 Soils as an aggregation of particles 7 14 Interaction with pore water 9 15 Transmission of the stress state in granular soil 10 16 Transmission of the stress state in the presence of a fluid 14 17 From discrete to continuum 17 18 Stress and strain tensors 21 19 Bibliography 25
Chapter 2 Field Equations for a Porous Medium 27
21 Equilibrium equations 27 22 Compatibility equations 30 23 Constitutive laws 32 24 Geostatic stress state and over-consolidation 40 25 Continuity equation and Darcyrsquos law 44 26 Particular cases 48
261 Dry soil 49 262 Saturated soil with still groundwater 50 263 Saturated soil with seepage stationary conditions 50 264 Saturated soil with seepage transient conditions 51
27 Bibliography 55
vi Soil Mechanics
Chapter 3 Seepage Stationary Conditions 57
31 Introduction 57 32 The finite difference method 60 33 Flow net 63 34 Excess pore pressure 65 35 Instability due to piping 67 36 Safety factor against piping 68 37 Anisotropic permeability 70 38 Transition between soils characterized by different permeability coefficients 74 39 Free surface problems 75 310 In situ methods for the permeability coefficient determination 77 311 Bibliography 81
Chapter 4 Seepage Transient Conditions 83
41 One-dimensional consolidation equation 83 42 Excess pore pressure isochrones 86 43 Consolidation settlement 91 44 Consolidation settlement approximated solution 93 45 Consolidation under different initial or boundary conditions 97 46 Load linearly increasing over time under consolidation 101 47 Consolidation under axial symmetric conditions 104 48 Multidimensional consolidation the Mandel-Cryer effect 106 49 Oedometer test and measure of cv 114 410 Influence of the skeleton viscosity 118 411 Bibliography 123
Chapter 5 The Constitutive Relationship Tests and Experimental Results 125
51 Introduction 125 52 Fundamental requirements of testing apparatus 127 53 Principal testing apparatus 130
531 The ldquotruerdquo triaxial test (TTA) Lameacutersquos ellipsoid and Mohrrsquos sickle 130 532 The (standard) triaxial apparatus 135 533 The oedometer 144 534 The biaxial apparatus 144 535 Direct shear box and simple shear apparatus (SSA) 147 536 Hollow cylinder 152
Table of Contents vii
54 The stress path concept 157 55 Experimental results for isotropic tests on virgin soils 163 56 Experimental results for radial tests on virgin soils stress dilatancy relationship 167 57 Oedometric tests on virgin soil as a particular case of the radial test earth pressure coefficient at rest 173 58 Drained triaxial tests on loose sands Mohr-Coulomb failure criterion 174 59 Undrained triaxial tests on loose sands instability line and static liquefaction 179 510 Drained tests on dense and medium dense sand dilatancy and critical state 186 511 Strain localization shear band formation 191 512 Undrained tests on dense and medium dense sands phase transformation line 196 513 Sand behavior in tests in which the three principal stresses are independently controled failure in the deviatoric plane 198 514 Normally consolidated and over-consolidated clays oedometric tests with loading unloading cycles ndash extension failure 201 515 Drained and undrained triaxial tests on normally consolidated clays normalization of the mechanical behavior 208 516 Over-consolidated clays 214 517 The critical state Plasticity index 219 518 Natural soils apparent over-consolidation ndash yielding surface 226 519 Soil behavior under cyclic loading cyclic mobility and strength degradation 230 520 Bibliography 236
Chapter 6 The Constitutive Relationship Mathematical Modeling of the Experimental Behavior 241
61 Introduction 241 62 Nonlinear elasticity 242 63 Perfect elastic-plasticity 243 64 Yielding of metals 247 65 Taylor and Quinney experiments the normality postulate 251 66 Generalized variables of stress and strain 258 67 Plastic strains for a material behaving as described by the Mohr-Coulomb criterion 259 68 Drucker-Prager and Matsuoka-Nakai failure criteria 261 69 Dilatancy non-associated flow rule 267 610 Formulation of an elastic-perfectly plastic law 269
viii Soil Mechanics
611 Cam clay model 272 612 Reformulation of the Cam clay model as an elastic-plastic hardening model 282 613 Comparison between experimental behavior and mathematical modeling for normally consolidated clays 285 614 Lightly over-consolidated clays 290 615 Heavily over-consolidated clays 293 616 Subsequent developments and applications 298 617 Non-associated flow rule the Nova-Wood model 301 618 Sinfonietta classica a model for soils and soft rocks 309 619 Models for soils subjected to cyclic loading 315 620 Conceptual use of constitutive soil behavior models 318
6201 Oedometric test 318 6202 Unconfined undrained (UU) test 321 6203 Shear modulus ldquoanisotropyrdquo 324
621 Bibliography 325
Chapter 7 Numerical Solution to Boundary Value Problems 329
71 Introduction 329 72 The finite element method for plane strain problems 330 73 Earth pressures on retaining structures 344 74 Settlements and bearing capacity of shallow foundations 354 75 Numerical solution of boundary value problems for fully saturated soil 364 76 Undrained conditions short-term bearing capacity of a footing 371 77 Short- and long-term stability of an excavation 380 78 Bibliography 389
Postscript 391
Index 395
Preface
Atque neque uti docui solido cum corpore mundi naturast quoniam admixtumst in rebus inane hellip
Titus Lucretius Caro De Rerum Natura
According to the engineering nomenclature soil mechanics is concerned with the behavior of clastic rocks or ldquosoilsrdquo 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 ldquoΦαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητικήrdquo 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 structurersquos dimensions and safety assessment In fact it allows rational prediction of the structurersquos 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 ldquooverlappingrdquo two continua a solid continuum modeling the skeleton composed of the mineral particles the ldquosolid skeletonrdquo 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) ldquorapidrdquo 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 ldquopracticalrdquo 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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Soil Mechanics
Soil Mechanics
Roberto Nova
Translated by Laura Gabrieli
First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc The translation of this book has been funded by SEPS (Segretariato Europeo per le Pubblicazioni Scientifiche) Via Val drsquoAposa 7 40123 Bologna Italy sepsalmauniboit ndash wwwsepsit Originally published in Italian under the title Fondamenti di meccanica delle terre ISBN 88-386-0894-6 copy The McGraw-Hill Compagnies S r l Milano 2002 Permission for this edition was arranged through The McGraw-Hill Compagnies Srl ndash Publishing Group Italia Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom copy ISTE Ltd 2010 The rights of Roberto Nova to be identified as the author of this work have been asserted by him in accordance with the Copyright Designs and Patents Act 1988
Library of Congress Cataloging-in-Publication Data Nova Roberto [Fondamenti di meccanica delle terre English] Soil mechanics Roberto Nova translated by Laura Gabrieli p cm Includes bibliographical references and index ISBN 978-1-84821-102-5 1 Soil mechanics I Title TA710N58713 2009 32415136--dc22
2009034935 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-102-5
Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne
Table of Contents
Preface ix
Chapter 1 Introduction Basic Concepts 1
11 Soils and rocks 1 12 Engineering properties of soils 3 13 Soils as an aggregation of particles 7 14 Interaction with pore water 9 15 Transmission of the stress state in granular soil 10 16 Transmission of the stress state in the presence of a fluid 14 17 From discrete to continuum 17 18 Stress and strain tensors 21 19 Bibliography 25
Chapter 2 Field Equations for a Porous Medium 27
21 Equilibrium equations 27 22 Compatibility equations 30 23 Constitutive laws 32 24 Geostatic stress state and over-consolidation 40 25 Continuity equation and Darcyrsquos law 44 26 Particular cases 48
261 Dry soil 49 262 Saturated soil with still groundwater 50 263 Saturated soil with seepage stationary conditions 50 264 Saturated soil with seepage transient conditions 51
27 Bibliography 55
vi Soil Mechanics
Chapter 3 Seepage Stationary Conditions 57
31 Introduction 57 32 The finite difference method 60 33 Flow net 63 34 Excess pore pressure 65 35 Instability due to piping 67 36 Safety factor against piping 68 37 Anisotropic permeability 70 38 Transition between soils characterized by different permeability coefficients 74 39 Free surface problems 75 310 In situ methods for the permeability coefficient determination 77 311 Bibliography 81
Chapter 4 Seepage Transient Conditions 83
41 One-dimensional consolidation equation 83 42 Excess pore pressure isochrones 86 43 Consolidation settlement 91 44 Consolidation settlement approximated solution 93 45 Consolidation under different initial or boundary conditions 97 46 Load linearly increasing over time under consolidation 101 47 Consolidation under axial symmetric conditions 104 48 Multidimensional consolidation the Mandel-Cryer effect 106 49 Oedometer test and measure of cv 114 410 Influence of the skeleton viscosity 118 411 Bibliography 123
Chapter 5 The Constitutive Relationship Tests and Experimental Results 125
51 Introduction 125 52 Fundamental requirements of testing apparatus 127 53 Principal testing apparatus 130
531 The ldquotruerdquo triaxial test (TTA) Lameacutersquos ellipsoid and Mohrrsquos sickle 130 532 The (standard) triaxial apparatus 135 533 The oedometer 144 534 The biaxial apparatus 144 535 Direct shear box and simple shear apparatus (SSA) 147 536 Hollow cylinder 152
Table of Contents vii
54 The stress path concept 157 55 Experimental results for isotropic tests on virgin soils 163 56 Experimental results for radial tests on virgin soils stress dilatancy relationship 167 57 Oedometric tests on virgin soil as a particular case of the radial test earth pressure coefficient at rest 173 58 Drained triaxial tests on loose sands Mohr-Coulomb failure criterion 174 59 Undrained triaxial tests on loose sands instability line and static liquefaction 179 510 Drained tests on dense and medium dense sand dilatancy and critical state 186 511 Strain localization shear band formation 191 512 Undrained tests on dense and medium dense sands phase transformation line 196 513 Sand behavior in tests in which the three principal stresses are independently controled failure in the deviatoric plane 198 514 Normally consolidated and over-consolidated clays oedometric tests with loading unloading cycles ndash extension failure 201 515 Drained and undrained triaxial tests on normally consolidated clays normalization of the mechanical behavior 208 516 Over-consolidated clays 214 517 The critical state Plasticity index 219 518 Natural soils apparent over-consolidation ndash yielding surface 226 519 Soil behavior under cyclic loading cyclic mobility and strength degradation 230 520 Bibliography 236
Chapter 6 The Constitutive Relationship Mathematical Modeling of the Experimental Behavior 241
61 Introduction 241 62 Nonlinear elasticity 242 63 Perfect elastic-plasticity 243 64 Yielding of metals 247 65 Taylor and Quinney experiments the normality postulate 251 66 Generalized variables of stress and strain 258 67 Plastic strains for a material behaving as described by the Mohr-Coulomb criterion 259 68 Drucker-Prager and Matsuoka-Nakai failure criteria 261 69 Dilatancy non-associated flow rule 267 610 Formulation of an elastic-perfectly plastic law 269
viii Soil Mechanics
611 Cam clay model 272 612 Reformulation of the Cam clay model as an elastic-plastic hardening model 282 613 Comparison between experimental behavior and mathematical modeling for normally consolidated clays 285 614 Lightly over-consolidated clays 290 615 Heavily over-consolidated clays 293 616 Subsequent developments and applications 298 617 Non-associated flow rule the Nova-Wood model 301 618 Sinfonietta classica a model for soils and soft rocks 309 619 Models for soils subjected to cyclic loading 315 620 Conceptual use of constitutive soil behavior models 318
6201 Oedometric test 318 6202 Unconfined undrained (UU) test 321 6203 Shear modulus ldquoanisotropyrdquo 324
621 Bibliography 325
Chapter 7 Numerical Solution to Boundary Value Problems 329
71 Introduction 329 72 The finite element method for plane strain problems 330 73 Earth pressures on retaining structures 344 74 Settlements and bearing capacity of shallow foundations 354 75 Numerical solution of boundary value problems for fully saturated soil 364 76 Undrained conditions short-term bearing capacity of a footing 371 77 Short- and long-term stability of an excavation 380 78 Bibliography 389
Postscript 391
Index 395
Preface
Atque neque uti docui solido cum corpore mundi naturast quoniam admixtumst in rebus inane hellip
Titus Lucretius Caro De Rerum Natura
According to the engineering nomenclature soil mechanics is concerned with the behavior of clastic rocks or ldquosoilsrdquo 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 ldquoΦαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητικήrdquo 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 structurersquos dimensions and safety assessment In fact it allows rational prediction of the structurersquos 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 ldquooverlappingrdquo two continua a solid continuum modeling the skeleton composed of the mineral particles the ldquosolid skeletonrdquo 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) ldquorapidrdquo 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 ldquopracticalrdquo 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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Soil Mechanics
Roberto Nova
Translated by Laura Gabrieli
First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc The translation of this book has been funded by SEPS (Segretariato Europeo per le Pubblicazioni Scientifiche) Via Val drsquoAposa 7 40123 Bologna Italy sepsalmauniboit ndash wwwsepsit Originally published in Italian under the title Fondamenti di meccanica delle terre ISBN 88-386-0894-6 copy The McGraw-Hill Compagnies S r l Milano 2002 Permission for this edition was arranged through The McGraw-Hill Compagnies Srl ndash Publishing Group Italia Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom copy ISTE Ltd 2010 The rights of Roberto Nova to be identified as the author of this work have been asserted by him in accordance with the Copyright Designs and Patents Act 1988
Library of Congress Cataloging-in-Publication Data Nova Roberto [Fondamenti di meccanica delle terre English] Soil mechanics Roberto Nova translated by Laura Gabrieli p cm Includes bibliographical references and index ISBN 978-1-84821-102-5 1 Soil mechanics I Title TA710N58713 2009 32415136--dc22
2009034935 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-102-5
Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne
Table of Contents
Preface ix
Chapter 1 Introduction Basic Concepts 1
11 Soils and rocks 1 12 Engineering properties of soils 3 13 Soils as an aggregation of particles 7 14 Interaction with pore water 9 15 Transmission of the stress state in granular soil 10 16 Transmission of the stress state in the presence of a fluid 14 17 From discrete to continuum 17 18 Stress and strain tensors 21 19 Bibliography 25
Chapter 2 Field Equations for a Porous Medium 27
21 Equilibrium equations 27 22 Compatibility equations 30 23 Constitutive laws 32 24 Geostatic stress state and over-consolidation 40 25 Continuity equation and Darcyrsquos law 44 26 Particular cases 48
261 Dry soil 49 262 Saturated soil with still groundwater 50 263 Saturated soil with seepage stationary conditions 50 264 Saturated soil with seepage transient conditions 51
27 Bibliography 55
vi Soil Mechanics
Chapter 3 Seepage Stationary Conditions 57
31 Introduction 57 32 The finite difference method 60 33 Flow net 63 34 Excess pore pressure 65 35 Instability due to piping 67 36 Safety factor against piping 68 37 Anisotropic permeability 70 38 Transition between soils characterized by different permeability coefficients 74 39 Free surface problems 75 310 In situ methods for the permeability coefficient determination 77 311 Bibliography 81
Chapter 4 Seepage Transient Conditions 83
41 One-dimensional consolidation equation 83 42 Excess pore pressure isochrones 86 43 Consolidation settlement 91 44 Consolidation settlement approximated solution 93 45 Consolidation under different initial or boundary conditions 97 46 Load linearly increasing over time under consolidation 101 47 Consolidation under axial symmetric conditions 104 48 Multidimensional consolidation the Mandel-Cryer effect 106 49 Oedometer test and measure of cv 114 410 Influence of the skeleton viscosity 118 411 Bibliography 123
Chapter 5 The Constitutive Relationship Tests and Experimental Results 125
51 Introduction 125 52 Fundamental requirements of testing apparatus 127 53 Principal testing apparatus 130
531 The ldquotruerdquo triaxial test (TTA) Lameacutersquos ellipsoid and Mohrrsquos sickle 130 532 The (standard) triaxial apparatus 135 533 The oedometer 144 534 The biaxial apparatus 144 535 Direct shear box and simple shear apparatus (SSA) 147 536 Hollow cylinder 152
Table of Contents vii
54 The stress path concept 157 55 Experimental results for isotropic tests on virgin soils 163 56 Experimental results for radial tests on virgin soils stress dilatancy relationship 167 57 Oedometric tests on virgin soil as a particular case of the radial test earth pressure coefficient at rest 173 58 Drained triaxial tests on loose sands Mohr-Coulomb failure criterion 174 59 Undrained triaxial tests on loose sands instability line and static liquefaction 179 510 Drained tests on dense and medium dense sand dilatancy and critical state 186 511 Strain localization shear band formation 191 512 Undrained tests on dense and medium dense sands phase transformation line 196 513 Sand behavior in tests in which the three principal stresses are independently controled failure in the deviatoric plane 198 514 Normally consolidated and over-consolidated clays oedometric tests with loading unloading cycles ndash extension failure 201 515 Drained and undrained triaxial tests on normally consolidated clays normalization of the mechanical behavior 208 516 Over-consolidated clays 214 517 The critical state Plasticity index 219 518 Natural soils apparent over-consolidation ndash yielding surface 226 519 Soil behavior under cyclic loading cyclic mobility and strength degradation 230 520 Bibliography 236
Chapter 6 The Constitutive Relationship Mathematical Modeling of the Experimental Behavior 241
61 Introduction 241 62 Nonlinear elasticity 242 63 Perfect elastic-plasticity 243 64 Yielding of metals 247 65 Taylor and Quinney experiments the normality postulate 251 66 Generalized variables of stress and strain 258 67 Plastic strains for a material behaving as described by the Mohr-Coulomb criterion 259 68 Drucker-Prager and Matsuoka-Nakai failure criteria 261 69 Dilatancy non-associated flow rule 267 610 Formulation of an elastic-perfectly plastic law 269
viii Soil Mechanics
611 Cam clay model 272 612 Reformulation of the Cam clay model as an elastic-plastic hardening model 282 613 Comparison between experimental behavior and mathematical modeling for normally consolidated clays 285 614 Lightly over-consolidated clays 290 615 Heavily over-consolidated clays 293 616 Subsequent developments and applications 298 617 Non-associated flow rule the Nova-Wood model 301 618 Sinfonietta classica a model for soils and soft rocks 309 619 Models for soils subjected to cyclic loading 315 620 Conceptual use of constitutive soil behavior models 318
6201 Oedometric test 318 6202 Unconfined undrained (UU) test 321 6203 Shear modulus ldquoanisotropyrdquo 324
621 Bibliography 325
Chapter 7 Numerical Solution to Boundary Value Problems 329
71 Introduction 329 72 The finite element method for plane strain problems 330 73 Earth pressures on retaining structures 344 74 Settlements and bearing capacity of shallow foundations 354 75 Numerical solution of boundary value problems for fully saturated soil 364 76 Undrained conditions short-term bearing capacity of a footing 371 77 Short- and long-term stability of an excavation 380 78 Bibliography 389
Postscript 391
Index 395
Preface
Atque neque uti docui solido cum corpore mundi naturast quoniam admixtumst in rebus inane hellip
Titus Lucretius Caro De Rerum Natura
According to the engineering nomenclature soil mechanics is concerned with the behavior of clastic rocks or ldquosoilsrdquo 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 ldquoΦαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητικήrdquo 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 structurersquos dimensions and safety assessment In fact it allows rational prediction of the structurersquos 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 ldquooverlappingrdquo two continua a solid continuum modeling the skeleton composed of the mineral particles the ldquosolid skeletonrdquo 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) ldquorapidrdquo 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 ldquopracticalrdquo 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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc The translation of this book has been funded by SEPS (Segretariato Europeo per le Pubblicazioni Scientifiche) Via Val drsquoAposa 7 40123 Bologna Italy sepsalmauniboit ndash wwwsepsit Originally published in Italian under the title Fondamenti di meccanica delle terre ISBN 88-386-0894-6 copy The McGraw-Hill Compagnies S r l Milano 2002 Permission for this edition was arranged through The McGraw-Hill Compagnies Srl ndash Publishing Group Italia Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom copy ISTE Ltd 2010 The rights of Roberto Nova to be identified as the author of this work have been asserted by him in accordance with the Copyright Designs and Patents Act 1988
Library of Congress Cataloging-in-Publication Data Nova Roberto [Fondamenti di meccanica delle terre English] Soil mechanics Roberto Nova translated by Laura Gabrieli p cm Includes bibliographical references and index ISBN 978-1-84821-102-5 1 Soil mechanics I Title TA710N58713 2009 32415136--dc22
2009034935 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-102-5
Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne
Table of Contents
Preface ix
Chapter 1 Introduction Basic Concepts 1
11 Soils and rocks 1 12 Engineering properties of soils 3 13 Soils as an aggregation of particles 7 14 Interaction with pore water 9 15 Transmission of the stress state in granular soil 10 16 Transmission of the stress state in the presence of a fluid 14 17 From discrete to continuum 17 18 Stress and strain tensors 21 19 Bibliography 25
Chapter 2 Field Equations for a Porous Medium 27
21 Equilibrium equations 27 22 Compatibility equations 30 23 Constitutive laws 32 24 Geostatic stress state and over-consolidation 40 25 Continuity equation and Darcyrsquos law 44 26 Particular cases 48
261 Dry soil 49 262 Saturated soil with still groundwater 50 263 Saturated soil with seepage stationary conditions 50 264 Saturated soil with seepage transient conditions 51
27 Bibliography 55
vi Soil Mechanics
Chapter 3 Seepage Stationary Conditions 57
31 Introduction 57 32 The finite difference method 60 33 Flow net 63 34 Excess pore pressure 65 35 Instability due to piping 67 36 Safety factor against piping 68 37 Anisotropic permeability 70 38 Transition between soils characterized by different permeability coefficients 74 39 Free surface problems 75 310 In situ methods for the permeability coefficient determination 77 311 Bibliography 81
Chapter 4 Seepage Transient Conditions 83
41 One-dimensional consolidation equation 83 42 Excess pore pressure isochrones 86 43 Consolidation settlement 91 44 Consolidation settlement approximated solution 93 45 Consolidation under different initial or boundary conditions 97 46 Load linearly increasing over time under consolidation 101 47 Consolidation under axial symmetric conditions 104 48 Multidimensional consolidation the Mandel-Cryer effect 106 49 Oedometer test and measure of cv 114 410 Influence of the skeleton viscosity 118 411 Bibliography 123
Chapter 5 The Constitutive Relationship Tests and Experimental Results 125
51 Introduction 125 52 Fundamental requirements of testing apparatus 127 53 Principal testing apparatus 130
531 The ldquotruerdquo triaxial test (TTA) Lameacutersquos ellipsoid and Mohrrsquos sickle 130 532 The (standard) triaxial apparatus 135 533 The oedometer 144 534 The biaxial apparatus 144 535 Direct shear box and simple shear apparatus (SSA) 147 536 Hollow cylinder 152
Table of Contents vii
54 The stress path concept 157 55 Experimental results for isotropic tests on virgin soils 163 56 Experimental results for radial tests on virgin soils stress dilatancy relationship 167 57 Oedometric tests on virgin soil as a particular case of the radial test earth pressure coefficient at rest 173 58 Drained triaxial tests on loose sands Mohr-Coulomb failure criterion 174 59 Undrained triaxial tests on loose sands instability line and static liquefaction 179 510 Drained tests on dense and medium dense sand dilatancy and critical state 186 511 Strain localization shear band formation 191 512 Undrained tests on dense and medium dense sands phase transformation line 196 513 Sand behavior in tests in which the three principal stresses are independently controled failure in the deviatoric plane 198 514 Normally consolidated and over-consolidated clays oedometric tests with loading unloading cycles ndash extension failure 201 515 Drained and undrained triaxial tests on normally consolidated clays normalization of the mechanical behavior 208 516 Over-consolidated clays 214 517 The critical state Plasticity index 219 518 Natural soils apparent over-consolidation ndash yielding surface 226 519 Soil behavior under cyclic loading cyclic mobility and strength degradation 230 520 Bibliography 236
Chapter 6 The Constitutive Relationship Mathematical Modeling of the Experimental Behavior 241
61 Introduction 241 62 Nonlinear elasticity 242 63 Perfect elastic-plasticity 243 64 Yielding of metals 247 65 Taylor and Quinney experiments the normality postulate 251 66 Generalized variables of stress and strain 258 67 Plastic strains for a material behaving as described by the Mohr-Coulomb criterion 259 68 Drucker-Prager and Matsuoka-Nakai failure criteria 261 69 Dilatancy non-associated flow rule 267 610 Formulation of an elastic-perfectly plastic law 269
viii Soil Mechanics
611 Cam clay model 272 612 Reformulation of the Cam clay model as an elastic-plastic hardening model 282 613 Comparison between experimental behavior and mathematical modeling for normally consolidated clays 285 614 Lightly over-consolidated clays 290 615 Heavily over-consolidated clays 293 616 Subsequent developments and applications 298 617 Non-associated flow rule the Nova-Wood model 301 618 Sinfonietta classica a model for soils and soft rocks 309 619 Models for soils subjected to cyclic loading 315 620 Conceptual use of constitutive soil behavior models 318
6201 Oedometric test 318 6202 Unconfined undrained (UU) test 321 6203 Shear modulus ldquoanisotropyrdquo 324
621 Bibliography 325
Chapter 7 Numerical Solution to Boundary Value Problems 329
71 Introduction 329 72 The finite element method for plane strain problems 330 73 Earth pressures on retaining structures 344 74 Settlements and bearing capacity of shallow foundations 354 75 Numerical solution of boundary value problems for fully saturated soil 364 76 Undrained conditions short-term bearing capacity of a footing 371 77 Short- and long-term stability of an excavation 380 78 Bibliography 389
Postscript 391
Index 395
Preface
Atque neque uti docui solido cum corpore mundi naturast quoniam admixtumst in rebus inane hellip
Titus Lucretius Caro De Rerum Natura
According to the engineering nomenclature soil mechanics is concerned with the behavior of clastic rocks or ldquosoilsrdquo 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 ldquoΦαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητικήrdquo 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 structurersquos dimensions and safety assessment In fact it allows rational prediction of the structurersquos 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 ldquooverlappingrdquo two continua a solid continuum modeling the skeleton composed of the mineral particles the ldquosolid skeletonrdquo 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) ldquorapidrdquo 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 ldquopracticalrdquo 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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Table of Contents
Preface ix
Chapter 1 Introduction Basic Concepts 1
11 Soils and rocks 1 12 Engineering properties of soils 3 13 Soils as an aggregation of particles 7 14 Interaction with pore water 9 15 Transmission of the stress state in granular soil 10 16 Transmission of the stress state in the presence of a fluid 14 17 From discrete to continuum 17 18 Stress and strain tensors 21 19 Bibliography 25
Chapter 2 Field Equations for a Porous Medium 27
21 Equilibrium equations 27 22 Compatibility equations 30 23 Constitutive laws 32 24 Geostatic stress state and over-consolidation 40 25 Continuity equation and Darcyrsquos law 44 26 Particular cases 48
261 Dry soil 49 262 Saturated soil with still groundwater 50 263 Saturated soil with seepage stationary conditions 50 264 Saturated soil with seepage transient conditions 51
27 Bibliography 55
vi Soil Mechanics
Chapter 3 Seepage Stationary Conditions 57
31 Introduction 57 32 The finite difference method 60 33 Flow net 63 34 Excess pore pressure 65 35 Instability due to piping 67 36 Safety factor against piping 68 37 Anisotropic permeability 70 38 Transition between soils characterized by different permeability coefficients 74 39 Free surface problems 75 310 In situ methods for the permeability coefficient determination 77 311 Bibliography 81
Chapter 4 Seepage Transient Conditions 83
41 One-dimensional consolidation equation 83 42 Excess pore pressure isochrones 86 43 Consolidation settlement 91 44 Consolidation settlement approximated solution 93 45 Consolidation under different initial or boundary conditions 97 46 Load linearly increasing over time under consolidation 101 47 Consolidation under axial symmetric conditions 104 48 Multidimensional consolidation the Mandel-Cryer effect 106 49 Oedometer test and measure of cv 114 410 Influence of the skeleton viscosity 118 411 Bibliography 123
Chapter 5 The Constitutive Relationship Tests and Experimental Results 125
51 Introduction 125 52 Fundamental requirements of testing apparatus 127 53 Principal testing apparatus 130
531 The ldquotruerdquo triaxial test (TTA) Lameacutersquos ellipsoid and Mohrrsquos sickle 130 532 The (standard) triaxial apparatus 135 533 The oedometer 144 534 The biaxial apparatus 144 535 Direct shear box and simple shear apparatus (SSA) 147 536 Hollow cylinder 152
Table of Contents vii
54 The stress path concept 157 55 Experimental results for isotropic tests on virgin soils 163 56 Experimental results for radial tests on virgin soils stress dilatancy relationship 167 57 Oedometric tests on virgin soil as a particular case of the radial test earth pressure coefficient at rest 173 58 Drained triaxial tests on loose sands Mohr-Coulomb failure criterion 174 59 Undrained triaxial tests on loose sands instability line and static liquefaction 179 510 Drained tests on dense and medium dense sand dilatancy and critical state 186 511 Strain localization shear band formation 191 512 Undrained tests on dense and medium dense sands phase transformation line 196 513 Sand behavior in tests in which the three principal stresses are independently controled failure in the deviatoric plane 198 514 Normally consolidated and over-consolidated clays oedometric tests with loading unloading cycles ndash extension failure 201 515 Drained and undrained triaxial tests on normally consolidated clays normalization of the mechanical behavior 208 516 Over-consolidated clays 214 517 The critical state Plasticity index 219 518 Natural soils apparent over-consolidation ndash yielding surface 226 519 Soil behavior under cyclic loading cyclic mobility and strength degradation 230 520 Bibliography 236
Chapter 6 The Constitutive Relationship Mathematical Modeling of the Experimental Behavior 241
61 Introduction 241 62 Nonlinear elasticity 242 63 Perfect elastic-plasticity 243 64 Yielding of metals 247 65 Taylor and Quinney experiments the normality postulate 251 66 Generalized variables of stress and strain 258 67 Plastic strains for a material behaving as described by the Mohr-Coulomb criterion 259 68 Drucker-Prager and Matsuoka-Nakai failure criteria 261 69 Dilatancy non-associated flow rule 267 610 Formulation of an elastic-perfectly plastic law 269
viii Soil Mechanics
611 Cam clay model 272 612 Reformulation of the Cam clay model as an elastic-plastic hardening model 282 613 Comparison between experimental behavior and mathematical modeling for normally consolidated clays 285 614 Lightly over-consolidated clays 290 615 Heavily over-consolidated clays 293 616 Subsequent developments and applications 298 617 Non-associated flow rule the Nova-Wood model 301 618 Sinfonietta classica a model for soils and soft rocks 309 619 Models for soils subjected to cyclic loading 315 620 Conceptual use of constitutive soil behavior models 318
6201 Oedometric test 318 6202 Unconfined undrained (UU) test 321 6203 Shear modulus ldquoanisotropyrdquo 324
621 Bibliography 325
Chapter 7 Numerical Solution to Boundary Value Problems 329
71 Introduction 329 72 The finite element method for plane strain problems 330 73 Earth pressures on retaining structures 344 74 Settlements and bearing capacity of shallow foundations 354 75 Numerical solution of boundary value problems for fully saturated soil 364 76 Undrained conditions short-term bearing capacity of a footing 371 77 Short- and long-term stability of an excavation 380 78 Bibliography 389
Postscript 391
Index 395
Preface
Atque neque uti docui solido cum corpore mundi naturast quoniam admixtumst in rebus inane hellip
Titus Lucretius Caro De Rerum Natura
According to the engineering nomenclature soil mechanics is concerned with the behavior of clastic rocks or ldquosoilsrdquo 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 ldquoΦαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητικήrdquo 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 structurersquos dimensions and safety assessment In fact it allows rational prediction of the structurersquos 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 ldquooverlappingrdquo two continua a solid continuum modeling the skeleton composed of the mineral particles the ldquosolid skeletonrdquo 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) ldquorapidrdquo 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 ldquopracticalrdquo 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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
vi Soil Mechanics
Chapter 3 Seepage Stationary Conditions 57
31 Introduction 57 32 The finite difference method 60 33 Flow net 63 34 Excess pore pressure 65 35 Instability due to piping 67 36 Safety factor against piping 68 37 Anisotropic permeability 70 38 Transition between soils characterized by different permeability coefficients 74 39 Free surface problems 75 310 In situ methods for the permeability coefficient determination 77 311 Bibliography 81
Chapter 4 Seepage Transient Conditions 83
41 One-dimensional consolidation equation 83 42 Excess pore pressure isochrones 86 43 Consolidation settlement 91 44 Consolidation settlement approximated solution 93 45 Consolidation under different initial or boundary conditions 97 46 Load linearly increasing over time under consolidation 101 47 Consolidation under axial symmetric conditions 104 48 Multidimensional consolidation the Mandel-Cryer effect 106 49 Oedometer test and measure of cv 114 410 Influence of the skeleton viscosity 118 411 Bibliography 123
Chapter 5 The Constitutive Relationship Tests and Experimental Results 125
51 Introduction 125 52 Fundamental requirements of testing apparatus 127 53 Principal testing apparatus 130
531 The ldquotruerdquo triaxial test (TTA) Lameacutersquos ellipsoid and Mohrrsquos sickle 130 532 The (standard) triaxial apparatus 135 533 The oedometer 144 534 The biaxial apparatus 144 535 Direct shear box and simple shear apparatus (SSA) 147 536 Hollow cylinder 152
Table of Contents vii
54 The stress path concept 157 55 Experimental results for isotropic tests on virgin soils 163 56 Experimental results for radial tests on virgin soils stress dilatancy relationship 167 57 Oedometric tests on virgin soil as a particular case of the radial test earth pressure coefficient at rest 173 58 Drained triaxial tests on loose sands Mohr-Coulomb failure criterion 174 59 Undrained triaxial tests on loose sands instability line and static liquefaction 179 510 Drained tests on dense and medium dense sand dilatancy and critical state 186 511 Strain localization shear band formation 191 512 Undrained tests on dense and medium dense sands phase transformation line 196 513 Sand behavior in tests in which the three principal stresses are independently controled failure in the deviatoric plane 198 514 Normally consolidated and over-consolidated clays oedometric tests with loading unloading cycles ndash extension failure 201 515 Drained and undrained triaxial tests on normally consolidated clays normalization of the mechanical behavior 208 516 Over-consolidated clays 214 517 The critical state Plasticity index 219 518 Natural soils apparent over-consolidation ndash yielding surface 226 519 Soil behavior under cyclic loading cyclic mobility and strength degradation 230 520 Bibliography 236
Chapter 6 The Constitutive Relationship Mathematical Modeling of the Experimental Behavior 241
61 Introduction 241 62 Nonlinear elasticity 242 63 Perfect elastic-plasticity 243 64 Yielding of metals 247 65 Taylor and Quinney experiments the normality postulate 251 66 Generalized variables of stress and strain 258 67 Plastic strains for a material behaving as described by the Mohr-Coulomb criterion 259 68 Drucker-Prager and Matsuoka-Nakai failure criteria 261 69 Dilatancy non-associated flow rule 267 610 Formulation of an elastic-perfectly plastic law 269
viii Soil Mechanics
611 Cam clay model 272 612 Reformulation of the Cam clay model as an elastic-plastic hardening model 282 613 Comparison between experimental behavior and mathematical modeling for normally consolidated clays 285 614 Lightly over-consolidated clays 290 615 Heavily over-consolidated clays 293 616 Subsequent developments and applications 298 617 Non-associated flow rule the Nova-Wood model 301 618 Sinfonietta classica a model for soils and soft rocks 309 619 Models for soils subjected to cyclic loading 315 620 Conceptual use of constitutive soil behavior models 318
6201 Oedometric test 318 6202 Unconfined undrained (UU) test 321 6203 Shear modulus ldquoanisotropyrdquo 324
621 Bibliography 325
Chapter 7 Numerical Solution to Boundary Value Problems 329
71 Introduction 329 72 The finite element method for plane strain problems 330 73 Earth pressures on retaining structures 344 74 Settlements and bearing capacity of shallow foundations 354 75 Numerical solution of boundary value problems for fully saturated soil 364 76 Undrained conditions short-term bearing capacity of a footing 371 77 Short- and long-term stability of an excavation 380 78 Bibliography 389
Postscript 391
Index 395
Preface
Atque neque uti docui solido cum corpore mundi naturast quoniam admixtumst in rebus inane hellip
Titus Lucretius Caro De Rerum Natura
According to the engineering nomenclature soil mechanics is concerned with the behavior of clastic rocks or ldquosoilsrdquo 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 ldquoΦαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητικήrdquo 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 structurersquos dimensions and safety assessment In fact it allows rational prediction of the structurersquos 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 ldquooverlappingrdquo two continua a solid continuum modeling the skeleton composed of the mineral particles the ldquosolid skeletonrdquo 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) ldquorapidrdquo 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 ldquopracticalrdquo 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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Table of Contents vii
54 The stress path concept 157 55 Experimental results for isotropic tests on virgin soils 163 56 Experimental results for radial tests on virgin soils stress dilatancy relationship 167 57 Oedometric tests on virgin soil as a particular case of the radial test earth pressure coefficient at rest 173 58 Drained triaxial tests on loose sands Mohr-Coulomb failure criterion 174 59 Undrained triaxial tests on loose sands instability line and static liquefaction 179 510 Drained tests on dense and medium dense sand dilatancy and critical state 186 511 Strain localization shear band formation 191 512 Undrained tests on dense and medium dense sands phase transformation line 196 513 Sand behavior in tests in which the three principal stresses are independently controled failure in the deviatoric plane 198 514 Normally consolidated and over-consolidated clays oedometric tests with loading unloading cycles ndash extension failure 201 515 Drained and undrained triaxial tests on normally consolidated clays normalization of the mechanical behavior 208 516 Over-consolidated clays 214 517 The critical state Plasticity index 219 518 Natural soils apparent over-consolidation ndash yielding surface 226 519 Soil behavior under cyclic loading cyclic mobility and strength degradation 230 520 Bibliography 236
Chapter 6 The Constitutive Relationship Mathematical Modeling of the Experimental Behavior 241
61 Introduction 241 62 Nonlinear elasticity 242 63 Perfect elastic-plasticity 243 64 Yielding of metals 247 65 Taylor and Quinney experiments the normality postulate 251 66 Generalized variables of stress and strain 258 67 Plastic strains for a material behaving as described by the Mohr-Coulomb criterion 259 68 Drucker-Prager and Matsuoka-Nakai failure criteria 261 69 Dilatancy non-associated flow rule 267 610 Formulation of an elastic-perfectly plastic law 269
viii Soil Mechanics
611 Cam clay model 272 612 Reformulation of the Cam clay model as an elastic-plastic hardening model 282 613 Comparison between experimental behavior and mathematical modeling for normally consolidated clays 285 614 Lightly over-consolidated clays 290 615 Heavily over-consolidated clays 293 616 Subsequent developments and applications 298 617 Non-associated flow rule the Nova-Wood model 301 618 Sinfonietta classica a model for soils and soft rocks 309 619 Models for soils subjected to cyclic loading 315 620 Conceptual use of constitutive soil behavior models 318
6201 Oedometric test 318 6202 Unconfined undrained (UU) test 321 6203 Shear modulus ldquoanisotropyrdquo 324
621 Bibliography 325
Chapter 7 Numerical Solution to Boundary Value Problems 329
71 Introduction 329 72 The finite element method for plane strain problems 330 73 Earth pressures on retaining structures 344 74 Settlements and bearing capacity of shallow foundations 354 75 Numerical solution of boundary value problems for fully saturated soil 364 76 Undrained conditions short-term bearing capacity of a footing 371 77 Short- and long-term stability of an excavation 380 78 Bibliography 389
Postscript 391
Index 395
Preface
Atque neque uti docui solido cum corpore mundi naturast quoniam admixtumst in rebus inane hellip
Titus Lucretius Caro De Rerum Natura
According to the engineering nomenclature soil mechanics is concerned with the behavior of clastic rocks or ldquosoilsrdquo 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 ldquoΦαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητικήrdquo 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 structurersquos dimensions and safety assessment In fact it allows rational prediction of the structurersquos 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 ldquooverlappingrdquo two continua a solid continuum modeling the skeleton composed of the mineral particles the ldquosolid skeletonrdquo 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) ldquorapidrdquo 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 ldquopracticalrdquo 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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
viii Soil Mechanics
611 Cam clay model 272 612 Reformulation of the Cam clay model as an elastic-plastic hardening model 282 613 Comparison between experimental behavior and mathematical modeling for normally consolidated clays 285 614 Lightly over-consolidated clays 290 615 Heavily over-consolidated clays 293 616 Subsequent developments and applications 298 617 Non-associated flow rule the Nova-Wood model 301 618 Sinfonietta classica a model for soils and soft rocks 309 619 Models for soils subjected to cyclic loading 315 620 Conceptual use of constitutive soil behavior models 318
6201 Oedometric test 318 6202 Unconfined undrained (UU) test 321 6203 Shear modulus ldquoanisotropyrdquo 324
621 Bibliography 325
Chapter 7 Numerical Solution to Boundary Value Problems 329
71 Introduction 329 72 The finite element method for plane strain problems 330 73 Earth pressures on retaining structures 344 74 Settlements and bearing capacity of shallow foundations 354 75 Numerical solution of boundary value problems for fully saturated soil 364 76 Undrained conditions short-term bearing capacity of a footing 371 77 Short- and long-term stability of an excavation 380 78 Bibliography 389
Postscript 391
Index 395
Preface
Atque neque uti docui solido cum corpore mundi naturast quoniam admixtumst in rebus inane hellip
Titus Lucretius Caro De Rerum Natura
According to the engineering nomenclature soil mechanics is concerned with the behavior of clastic rocks or ldquosoilsrdquo 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 ldquoΦαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητικήrdquo 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 structurersquos dimensions and safety assessment In fact it allows rational prediction of the structurersquos 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 ldquooverlappingrdquo two continua a solid continuum modeling the skeleton composed of the mineral particles the ldquosolid skeletonrdquo 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) ldquorapidrdquo 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 ldquopracticalrdquo 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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Preface
Atque neque uti docui solido cum corpore mundi naturast quoniam admixtumst in rebus inane hellip
Titus Lucretius Caro De Rerum Natura
According to the engineering nomenclature soil mechanics is concerned with the behavior of clastic rocks or ldquosoilsrdquo 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 ldquoΦαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητικήrdquo 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 structurersquos dimensions and safety assessment In fact it allows rational prediction of the structurersquos 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 ldquooverlappingrdquo two continua a solid continuum modeling the skeleton composed of the mineral particles the ldquosolid skeletonrdquo 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) ldquorapidrdquo 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 ldquopracticalrdquo 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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
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 ldquooverlappingrdquo two continua a solid continuum modeling the skeleton composed of the mineral particles the ldquosolid skeletonrdquo 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) ldquorapidrdquo 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 ldquopracticalrdquo 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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
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 ldquopracticalrdquo 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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
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 Pisanograve 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 ldquomyrdquo Maddalena Tommaso and Tobia who patiently bore the consequences of its writing
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Chapter 1
Introduction Basic Concepts
11 Soils and rocks
The term ldquosoilrdquo is used in civil engineering to describe a material composed of a natural accumulation of mineral particles whose sizes range between specified limits according to a conventional classification system
Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering) rocks being the primary element that constitutes the Earthrsquos crust Soil particles can be completely uncemented or weakly cemented depending on the degree of alteration of the parent rock On the other hand soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis)
Since the processes of weathering and diagenesis are gradual the distinction between soil and rock is to a certain extent arbitrary To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds such that the stress levels typical of civil engineering applications can easily exceed their strength On the other hand rock is defined as a material with strong chemical bonds The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself
Several geological materials (eg tuff clay stone marble limestone etc) have an intermediate behavior These materials behave as rocks if subjected to relatively
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
2 Soil Mechanics
low stresses and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles
Soil grains are mainly composed of silica minerals (eg silicon dioxide and other silica-based minerals) which are more resistant to chemical-physical attack by weathering than other minerals Quartz (SiO2) is almost insoluble in water is relatively acid proof and is a very stable mineral It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands followed by feldspars
Feldspars are chemically altered by water oxygen and carbon dioxide The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin Similarly phillosillicates existing in large quantities in igneous rocks delaminate along their basal plane due to their mineralogical foil structure and form illite and smectite Kaolin illite and smectite are the primary minerals appearing in clay they are characterised by plate-like particles with length and width in the order of a micron
Soil particles are also composed of calcite and gypsum as well as of minerals of volcanic origin (pyroclasts) Particles formed by these types of minerals are usually weaker than those formed by silica minerals therefore they have a greater influence on the strain behavior of these materials
The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history
For example on the one hand rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation On the other hand sand grains that remain in their original location where weathering of the parent rock took place are angular and have an irregular shape
The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways If clay particles align in the same direction (face-to-face orientation Figure 11a) it is referred to as dispersed structure while a structure similar to a card house (edge-to-face or edge-to-edge orientation) is referred to as flocculated structure (Figure 11b) and is much more unstable than the former With the change in the deposit chemical conditions the structure can pass from dispersed to flocculated and vice versa
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Introduction Basic Concepts 3
Figure 11 a) Dispersed clay structure b) flocculated clay structure (ldquocardhouserdquo)
12 Engineering properties of soils
As seen in the previous section several types of minerals compose a soil its ldquosolid skeletonrdquo and its fabric are influenced by its geological history and by the chemical environment However for the majority of engineering aims different types of soil can be initially classified according to the size of the constituent particles The classification of the different types of soils is somewhat arbitrary Examples of classifications adopted by British Standards (BS) Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 11
Cobbles Gravel (G) Sand (S) Silt (M) Clay (C) BS 60 2 006 0002
AGI 2 002 0002
AASHO 75 2 0015 0002
006
0020015
0002
0002
0002
Table 11 Classification of different types of soils Sizes are in mm
Note that in the proposed classifications there is no direct reference to the grain chemical composition to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration) This type of classification has two advantages Firstly it is a quantitative classification and hence it is almost free from the subjectivity of the operator Secondly it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior The range of possible particle sizes is enormous Soil particle sizes range from sub microscopic clay particles discernible only by a scanning electron microscope to rounded sand grains with a diameter a thousand times larger to cobbles with a diameter a hundred times larger
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
4 Soil Mechanics
On a single particle both body forces (weight) and surface forces (electrostatic forces) have effect The former are proportional to the volume of the particle while the latter are proportional to the external surface An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface An indicator of the relative role played by the two types of forces is the specific surface Ss defined as the ratio of the area of the surface of the particle to the mass of the particle ρV
=sASVρ
[11]
where ρ is the density and V is the volume of the particle
In the case of a rounded particle of silica sand the specific surface is inversely proportional to the diameter of the grain dg
6s
g
Sdρ
= [12]
Quartz density is equal to 265 gcm3 hence for a rounded particle of diameter 1 mm the specific surface will be 000226 m2g A clay particle of plate-like shape has instead a specific surface equal to
2sS
sρ= [13]
where s is the thickness of the particle The particle thickness largely depends on the type of clay For kaolin it can be of the order of a tenth of a micron while it can be of the order of only 10 Aring (10-3 microm) for the smallest particles this is typical of montmorillonite
For kaolin the specific surface is of the order of 10 m2g (more than 3000 times the value of the sand considered) For montmorillonite the specific surface is of the order of 1000 m2g Electrostatic forces are then negligible in sand however they become relevant when dealing with clay In the presence of water clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes This layer is referred to as adsorbed water The water forming this layer has very different mechanical properties in comparison with those of free water for instance adsorbed water is capable of transferring shear stresses In practice adsorbed water can be considered as a first approximation as an integral part of the mineral clay particle Unless otherwise specified the mineral particle is assumed to be coated by a layer of
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Introduction Basic Concepts 5
adsorbed water From the mechanical point of view interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains
Moving away from the surface of the particle the attractive force decreases and progressively water starts behaving as free water which can be gradually removed from a sample of soil for example by applying compression stresses
From an engineering point of view the most relevant aspect related to the particle size distribution is the ability of water or other fluids such as oil to seep through the soil pores
Figure 12 A soil element as an aggregation of particles
A soil element can be visualized as an aggregation of solid particles weakly cemented or uncemented the void space between the particles containing one or more fluids principally air andor water (Figure 12) A fluid can seep through a soil more or less easily depending on the width of the flow channel section The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius which is of the same order of magnitude of the soil particle size The size of a clay particle is approximately a thousand times smaller than the one of a sand grain Thereafter water discharge velocity in a clay layer must be a million times lower than the one in a sand layer all other conditions being equal As will be observed in the following this difference has relevant practical consequences
A load applied on a sample of soil provokes the rearrangement of its structure Since soil grains are principally composed of extremely resistant and rigid minerals the deformability of a soil element is mainly associated with a change in the configuration among grains which is related to a change in the volume occupied by voids In fact grain deformability is negligible with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads Water is also considered under the
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
6 Soil Mechanics
stress levels typical of civil engineering applications to be an incompressible fluid If soil is fully saturated by water a change in volume can take place only if water is free to drain throughout the soil If soil is coarsely grained drainage is instantaneous and the particles are free to change their configuration while loads are applied On the contrary in fine grained soils water flow is subjected to a higher resistance The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (eg the construction of a building a road embankment or an excavation) In the initial phases of the load referred to as ldquoshort termrdquo the possible configurations are only the ones that maintain the total volume constant which means that the soil has an internal kinematic constraint With time water gradually drains through the soil and at ldquolong termrdquo also fine grained soils can freely change their configuration without any internal constraints
The first and main difference between coarsely and fine grained soils is then apparent Fine grained soils change their configuration after a change in load even though initially without a change in volume Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles Therefore the strain process continues also after the stabilization of the load
It is worth noting however that in relatively coarsely grained soils such as fine sands there can be kinematic constraints preventing changes in volume after rapid variations in load as in the case of earthquakes Moreover a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles
Another important difference between fine and coarsely grained soils is represented by capillary rise Let Ts be the surface tension of water α the angle between the tangent to the meniscus and the wall of the capillary tube γw the unit weight of water and d the diameter of the tube (see Figure 13) Equilibrium in the vertical direction implies that the capillary height hc of the liquid column is
4 cos= s
cw
Th
dα
γ [14]
Surface tension of water in standard conditions is equal to 0075 Nm therefore in a capillary of 1 mm diameter the rise is of the order of 2 cm If the diameter of the tube is instead 1 microm the capillary rise is 20 meters In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry Conversely fine grained soils are saturated up to several tens of meters over the ground-water table
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Introduction Basic Concepts 7
Ts
hc
d α
Figure 13 Rise in a capillary tube
13 Soils as an aggregation of particles
As a first approximation the structural arrangement of an elementary volume of soil can be schematized as in Figure 12 Solid particles occupy only a portion of the space relative to an element of soil The remaining portion called ldquovolume of voidsrdquo is occupied by a fluid usually air andor water
The ratio of the volume occupied by voids Vv to the volume occupied by solids Vs is called void ratio e
equiv v
s
Ve
V [15]
Alternatively porosity n is defined as the ratio of the volume of voids to the total volume V of a soil element
1equiv =
+vV en
V e [16]
It is clear that the higher the porosity the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads On the other hand a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
8 Soil Mechanics
In order to define the range of soil porosities an ideal material composed of rigid spheres of equal radius is considered A simple cubic structure in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below is characterized by a porosity equal to 0476 This configuration is highly unstable A small external perturbation is sufficient to reduce its porosity Conversely in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral in contact among them) porosity is much lower and equal to 0259 In this case the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated an external perturbation will cause a decrease in volume in the case of loose sand while it will cause an increase in volume in the case of dense sand (dilatancy)
The proposed model is only an example Firstly a soil is composed of particles of different sizes and non-rounded shapes Moreover smaller particles have a greater possibility of occupying a minor total volume the volume of solids being equal For instance a sphere of radius R occupies a cube of radius 2R with a porosity of 0476 On the other hand eight spheres of radius R2 and hence occupying the same volume of the spheres just considered would fill the same cube only if disposed in the most unstable configuration previously described However the eight small spheres can dispose in several ways for example in the tetrahedral configuration that is characterized by a much lower porosity A sample of sand composed of several particle sizes will be characterized in general by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size) To take into account the effect of grading in sands it is more appropriate to refer other than to porosity to the relative density (density index) which is traditionally defined as
max
max minR
e eD
e eminus
equivminus
[17]
where emax and emin are two void ratios conventionally determined (refer to ASTM D2216-66) which define the loosest and the densest state for a criterion of sand
Moreover particles are not rigid Calcareous and volcanic sands are composed of fragile grains that can crush under loading As stated before on the effect of particle size porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves
Finally when particles develop cohesive bonds configurations with very high void ratios are possible For example loess deposits and cohesive silts deposited by
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Introduction Basic Concepts 9
wind can be characterized by porosities higher than 60 (e gt 15) These configurations are stable only for tensional levels that are lower than the bond strength For higher tensional levels the bonds break and the soil assumes a much more compact configuration The collapse of these kinds of soils usually causes big problems from an engineering viewpoint
14 Interaction with pore water
Inter-granular voids can be partially or totally filled with water The degree of saturation Sr is defined as the percent ratio of the volume occupied by water to the volume of voids
wr
v
VS
Vequiv [18]
A soil is referred to as saturated when Sr = 100 and as dry when Sr = 0 In general soil is not in its limit conditions A certain percentage of moisture is always present in the soil above the groundwater table due to the humidity of air and to the capillary rise of groundwater However full saturation is not reached even under the groundwater table due to small air bubbles trapped within the soil voids
The soil water content w is defined as the ratio of the mass of water within the sample Ww to the mass of the solid part Ws this being the dry weight of the considered sample
w
s
Ww
Wequiv [19]
Let γw be the unit weight of water and γs the unit weight of the material composing the grains The water content is then linked to the degree of saturation and to the void ratio by the relationship
w w w r v r
s s s s s
V S V Sw
V V G eγ γγ γ
= = = [110]
where
ss
w
Gγγ
equiv [111]
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
10 Soil Mechanics
The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 25 to 29 The Gs value of quartz is 265 of calcite is 271 with 27 the typical average value of clayey minerals
It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content Let γ be the total unit weight of a certain sample of soil
( (1 ) )s w s w s w r vw s r
W W G V S VW G n nSV V V
γ γγ γ+ += = = = minus + [112]
In particular for a dry sample the dry unit weight γd is equal to
(1 )d w sG nγ γ= minus [113]
while for a saturated sample the total unit weight is equal to
( (1 ) )sat w sG n nγ γ= minus + [114]
Finally a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced The buoyant unit weight of a soil is hence equal to
( 1)(1 )sat w w sG nγ γ γ γprime = minus = minus minus [115]
Notice that this result is equally achieved by considering the soil sample as composed of a unique material or by considering the solid part namely the single particles independently of the fluid
15 Transmission of the stress state in granular soil
Imagine applying a load to a volume of soil composed of rigid particles The stress state will be transmitted from the boundaries within the sample through the contacts between the particles To study this phenomenon tests on discs made of CR-39 co-polymer can be run and exploiting photo-elasticity techniques the average stress and strain tensors can be determined
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Introduction Basic Concepts 11
Figure 14 Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong 1972)
Exposed to rays of polarized light the transparent ldquograinsrdquo develop isochromatic patterns that are a function of the intensity of the stress state Figure 14 (Drescher and De Josselin de Jong 1972) shows the chains of aligned contact points along which forces are transmitted The thicker the black line in the figure the greater the intensity of the transmitted forces Analogously Figure 15 from Calvetti (1998) illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon
Figure 15 Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti 1998)
Some grains are intensely stressed others less and others do not even carry any load The most interesting aspect observed both in the real test and in the numerical
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
12 Soil Mechanics
simulation is that chains are not stable For a certain value of load two grains initially in contact slide over one another interrupting the contact Hence the column is no longer able to transmit the load Grains that were previously loaded suddenly are unloaded the load is immediately redistributed and a new chain is formed If the number of grains is large enough this intense internal assessment is not externally visible that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure
Due to column instability the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts more than the stiffness of the grains themselves This resistance is due principally to the friction among grains and partly to their cementation (often absent) The law governing the shear strength is known as Coulombrsquos equation (even if it was firstly formulated by Amonton a century before) This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact T reaches a proportion of the normal force at contact N
=T Nmicro [116]
where micro is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact
T
N
Figure 16 Sliding of a brick on a rough plane
Tabor (1959) provided a simple explanation of [116] For simplicity refer to Figure 16 where a brick on a rough plane is illustrated The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A The normal strength σy that can by transmitted from one body to the other through the contact points is limited In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value Ac such that
cy
NAσ
= [117]
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Introduction Basic Concepts 13
Under this stress state the two surfaces develop a sort of cold welding Suppose we now apply a shear force T If τy is the shear strength of the weld sliding will take place at
yy c
y
T A Nτ
τσ
= = [118]
Since τy and σy are constants that are characteristic of the material that composes the grains it follows from Amontonrsquos law that
y
y
τmicro
σequiv [119]
which is independent of the apparent contact area
As the normal stresses at the contacts determine the resistance to sliding it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains
Moreover smaller or larger strains will take place in the sample according to the grains structure more or less dense (the external load increment being equal) When a column collapses a new one will form more easily in a dense sample than in a loose one Also the shape of the particles is relevant For instance rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones As a consequence a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles
If the particle configuration is compact that is if the material is in a dense state and if the particles are rigid a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample This phenomenon is referred to as dilatancy
To visualize this phenomenon consider Figure 17 in which two rough rigid plates are in contact Assume the lower plate to be fixed The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate Since voids increase the overall volume between the two plates increases
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
14 Soil Mechanics
Note from the figure that in order for displacement to take place the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry Geometry in its turn controls dilatancy It is therefore evident that a strong relationship exists between dilatancy and strength
Notice that grains cannot always be assumed to be rigid If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 17 the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate However the effective contact area between the grains is much smaller than the total area of the plates This implies that the stress state at contacts is at least three orders of magnitude higher Small average stresses can therefore generate very high stress states at the contacts among grains In calcareous and volcanic sands this can lead to the breaking of grains In contrast in quartz sands this can occur only at very high average stresses (higher than 1 MPa) as for instance at the base of foundations piles
N
T
N
Figure 17 The phenomenon of dilatancy
16 Transmission of the stress state in the presence of a fluid
In a saturated soil under hydrostatic conditions the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above For porous enough materials such as soils voids are continuous and water can seep through the pores driven by differences in the hydraulic head
The pressure of water tends to separate the grains one from each other facilitating their sliding Imagine filling the void space between the brick and the rough plane considered in section 15 (Figure 16) with a fluid under pressure the fluid having a pressure equal to u
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
Introduction Basic Concepts 15
According to vertical equilibrium under the assumption of a contact area much smaller than the total one
1⎛ ⎞= + minus cong +⎜ ⎟⎝ ⎠
cy c y c
AN A u A uA
Aσ σ [120]
Hence
minus=cy
N uAAσ
[121]
and then sliding takes place at
( )= minusT N uAmicro [122]
The higher the pressure of the water the lower the force T necessary for sliding to occur
Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane equation [122] then gives
( ) prime= minus =uτ micro σ micro σ [123]
where primeσ is referred to as effective stress
The overall deformability of a volume of soil is governed by the relative sliding among grains It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones Therefore it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes
If soil is not fully saturated water tends to concentrate around the contact areas (Figure 18) and its surface tension gives rise to adhesion among the grains pressed together Thereafter since shear strength is provided by the frictional strength generated by capillary pressure the grains remain in contact even in the absence of external compression loads This strength often referred to as apparent cohesion vanishes if soil is submerged in water and thus the water pressure becomes positive
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
16 Soil Mechanics
Figure 18 Formation of water menisci around grain contacts in unsaturated soil
In fine grained soils the role of porosity is played by water content According to [110] in the case of a fully saturated soil
= Se G w [124]
in which void ratio and water content are synonymous except for a multiplicative non dimensional factor
Nevertheless fine grained soils behave according to this relationship only in a limited range In fact for high water contents fine grained soils will tend to behave more as a viscous fluid than as a solid even though porous and saturated The liquid limit wL is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer Obviously a similar level is arbitrary and therefore it is defined by a standard procedure
The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface If the water content of a soil sample decreases (for example under a compression load) the particles will tend to get closer and the strength to increase Below a certain value the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils Soil then becomes brittle and strength no longer increases with the decrease in water content or at least grows at a lower rate This limit of plasticity wP is arbitrarily determined by means of a procedure that is reported in ASTM standards The difference between wL and wP is referred to as the plasticity index (PI) Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties In fact although in a qualitative manner Atterberg limits and in particular wL and PI are related to some important mechanical characteristics such as oedometric compressibility and short term shear strength Moreover since they can be determined by employing simple equipment
