## Contents

Preface page ix

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction 1

1.2 Soil mechanics 2

1.3 Range of problems/applications 2

1.4 Scope of this book 10

1.5 Mind maps 11

2 Stress in soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Introduction 12

2.2 Equilibrium 12

2.3 Gravity 13

2.4 Stress 16

2.5 Exercises: Stress 18

2.6 Vertical stress profile 19

2.6.1 Worked examples 21

2.7 Water in the ground: Introduction to hydrostatics 23

2.7.1 Worked example: Archimedes uplift on spherical object 26

2.8 Total and effective stresses 28

2.8.1 Worked examples 32

2.9 Summary 37

2.10 Exercises: Profiles of total stress, effective stress, pore pressure 37

3 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

3.1 Introduction 40

3.2 Units 40

3.3 Descriptions of packing and density 41

3.3.1 Volumetric ratios 43

3.3.2 Water content 44

3.3.3 Densities 44

3.3.4 Unit weights 46

3.3.5 Typical values 46

3.4 Measurement of packing 47

3.4.1 Compaction 50

3.5 Soil particles 52

3.6 Laboratory exercise: particle size distribution and other

classification tests 56

3.6.1 Sieving 56

3.6.2 Sedimentation 57

3.6.3 Particle shape 61

3.6.4 Sand: relative density 61

3.7 Summary 62

3.8 Exercises: Density 64

3.8.1 Multiple choice questions 64

3.8.2 Calculation exercises 65

4 Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Introduction 67

4.2 Linear elasticity 67

4.3 Natural and true strain 70

4.4 One-dimensional testing of soils 70

4.4.1 Hooke’s Law: confined one-dimensional stiffness ♣ 72

4.5 One-dimensional (confined) stiffness of soils 74

4.6 Calculation of strains 78

4.6.1 Worked examples: Calculation of settlement 79

4.7 Overconsolidation 82

4.7.1 Worked examples: Overconsolidation 84

4.8 Summary 87

4.9 Exercises: Stiffness 87

5 Seepage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.1 Introduction 90

5.2 Total head: Bernoulli’s equation 90

5.3 Poiseuille’s equation 96

5.4 Permeability 99

5.4.1 Darcy or Forchheimer? 102

5.5 Measurement of permeability 104

5.6 Permeability of layered soil 106

5.7 Seepage forces 108

5.8 Radial flow to vertical drain 111

5.9 Radial flow to point drain 112

5.10 Worked examples: Seepage 113

5.10.1 Example: flow through soil column 113

5.10.2 Example: effect of changing reference datum 116

5.10.3 Example: pumping from aquifer 117

5.10.4 Example: flow into excavation 119

5.11 Summary 121

5.12 Exercises: Seepage 123

6 Change in stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.1 Introduction 127

6.2 Stress change and soil permeability 127

6.3 Worked examples 130

6.3.1 Example 1 130

6.3.2 Example 2 131

6.3.3 Example 3 133

6.4 Summary 134

6.5 Exercises: Change in stress 136

7 Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.1 Introduction 138

7.2 Describing the problem 140

7.3 Parabolic isochrones 142

7.4 Worked examples 149

7.4.1 Example 1: Determination of coefficient of consolidation 149

7.4.2 Example 2 152

7.4.3 Example 3 154

7.4.4 Example 4 155

7.5 Consolidation: exact analysis ♣ 155

7.5.1 Semi-infinite layer 159

7.5.2 Finite layer 161

7.6 Summary 165

7.7 Exercises: Consolidation 167

8 Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

8.1 Introduction 169

8.2 Failure mechanisms 169

8.3 Shear box and strength of soils 171

8.4 Strength model 173

8.5 Dilatancy 174

8.6 Drained and undrained strength 177

8.7 Clay: overconsolidation and undrained strength 179

8.8 Pile load capacity 181

8.9 Infinite slope 185

8.9.1 Laboratory exercise: Angle of repose 191

8.10 Undrained strength of clay: fall-cone test 193

8.11 Simple model of shearing ♣ 195

8.11.1 Stiffness 196

8.11.2 Strength 197

8.11.3 Mobilisation of strength 197

8.11.4 Dilatancy 198

8.11.5 Complete stress:strain relationship 199

8.11.6 Drained and undrained response 200

8.11.7 Model: summary 203

8.12 Summary 203

8.13 Exercises: Strength 205

9 Soil-structure interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

9.1 Introduction 208

9.2 Pile under axial loading ♣ 211

9.2.1 Examples 215

9.3 Bending of an elastic beam ♣ 216

9.4 Elastic beam on elastic foundation ♣ 220

9.5 Pile under lateral loading ♣ 224

9.6 Soil-structure interaction: next steps 226

9.7 Summary 227

9.8 Exercises: Soil-structure interaction 227

10 Envoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

10.1 Summary 230

10.2 Beyond the single dimension 231

Exercises: numerical answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Index 237

## Preface

This book has emerged from a number of stimuli.

There is a view that soils are special: that their characteristics are so extraordinary

that they can only be understood by a small band of specialists. Obviously,

soils do have some special properties: the central importance of density and change

of density merits particular attention. However, in the context of teaching principles

of soil mechanics to undergraduates in the early years of their civil engineering

degree programmes, I believe that there is advantage to be gained in trying to integrate

this teaching with other teaching of properties of engineering materials to

which the students are being exposed at the same time.

It is a fundamental tenet of critical state soil mechanics – with which I grew up

in my undergraduate days – that consideration of the mechanical behaviour of soils

requires us to consider density alongside effective stresses, thus permitting the unification

of deformation and strength characteristics. This can be seen as a broad

interpretation of the phrase critical state soil mechanics. I believe that such a unification

can aid the teaching and understanding of soil mechanics.

There is an elegant book by A. J. Roberts1 which demonstrates in a unified

way how a common mathematical framework can be applied to problems of solid

mechanics, fluid mechanics, traffic flow and so on. While I cannot hope to emulate

this elegance, the title prompted me to explore a similar one-dimensional theme

for the presentation of many of the key concepts of soil mechanics: density, stress,

stiffness, strength and fluid flow.

This one-dimensional approach to soil mechanics has formed the basis for an

introductory course of ten one-hour lectures with ten one-hour problem classes and

one three-hour laboratory afternoon for first-year civil engineering undergraduates

at Bristol University. The material of that course is contained in this book. I have

added a chapter on the analysis of one-dimensional consolidation, which fits neatly

with the theme of the book. I have also included a model of the shearing of soil and

some examples of soil-structure interaction which are capable of theoretical analysis

using essentially one-dimensional governing equations.

1 Roberts, A. J. (1994). A one-dimensional introduction to continuum mechanics. World Scientific.

Simplification of more or less realistic problems leads to differential equations

which can be readily solved: this is the essence of modelling with which engineers

need to engage (and to realise that they are engaging) all the time. A few of these

topics require some modest mathematical ability – a bit of integration, solution of

ordinary and partial differential equations – but nothing beyond the eventual expectations

of an undergraduate engineering degree programme. Sections that might, as

a consequence, be omitted on a first reading, or until the classes in mathematics have

caught up, are indicated by the symbol ♣.

Exercises are given for private study at the end of all chapters and some suggestions

for laboratory demonstrations that could accompany such an introductory

course are sprinkled through the book.

I am grateful to colleagues at Bristol and elsewhere – especially Danuta

Lesniewska, Erdin Ibraim and Dick Clements – who have provided advice and comments

on drafts of this book to which I have tried to respond. Erdin in particular has

helped enormously by using material and examples from a draft of this book in his

own teaching and has made many useful suggestions for clarification. However, the

blame for any remaining errors must remain with me.

I am grateful to Christopher Bambridge, Ross Boulanger, Sarah Dagostino,

David Eastaff, David Nash and Alan Powderham for their advice and help in locating

and giving permission to reproduce suitable pictures.

I thank Bristol University for awarding me a University Research Fellowship

for the academic year 2007–8 which gave me some breathing space after a particularly

heavy four years of administrative duty.

I am particularly grateful to Peter Gordon for his editorial guidance and wisdom

and his intervention at times of stress.

I acknowledge with gratitude Helen’s indulgence and support while I have been

preparing and revising this book.