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SOIL MECHANICS

Bla Bod & Colin Jones

Introduction to

INTR

OD

UC

TION

TO S

OIL M

ECH

ANIC

S B

od & Jones

SOIL MECHANICSBla Bod & Colin Jones

Introduction to

Introduction to Soil Mechanics covers the basic principles of soil mechanics, illustrating why the

properties of soil are important, the techniques used to understand and characterise soil behaviour

and how that knowledge is then applied in construction. The authors have endeavoured to define

and discuss the principles and concepts concisely, providing clear, detailed explanations, and a well-

illustrated text with diagrams, charts, graphs and tables. With many practical, worked examples and

end-of-chapter problems (with fully worked solutions available at www.wiley.com/go/bodo/soilmechanics)

and coverage of Eurocode 7, Introduction to Soil Mechanics will be an ideal starting point for the study

of soil mechanics and geotechnical engineering.

Also Available

Fundamentals of Rock Mechanics 4th EditionJ.C Jaeger, N.G.W Cook and R. Zimmerman Hardcover: 9780632057597

Smiths Elements of Soil Mechanics 8th EditionIan Smith Paperback: 9781405133708

About the Authors

Bla Bod B.Sc., B.A., C.Eng., M.I.C.E, was born in Hungary and studied at Budapest Technical University, the University of London and the Open University. He developed his expertise in Soil Mechanics during his employment with British Rail and British Coal.

Colin Jones B.Sc, C.Eng., M.I.C.E, P.G.C.E, studied at the University of Dundee, and worked at British Coal where he and Bla were colleagues. He has recently retired from the University of Wales, Newport where he was Programme Director for the Civil Engineering provision, specializing in Soil Mechanics and Geotechnics.

This books companion website is atwww.wiley.com/go/bodo/soilmechanics and offersinvaluable resources for both students and lecturers:

M Supplementary problems

M Solutions to supplementary problems

pg3913File Attachment9780470659434.jpg

Introduction to Soil Mechanics

About the companion website

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Supplementary problems Solutions to supplementary problems

www.wiley.com/go/bodo/soilmechanics

Introduction to Soil Mechanics

Bla Bod and Colin Jones

This edition first published 2013 2013 by John Wiley & Sons, Ltd

Registered OfficeJohn Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom.

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Library of Congress Cataloging-in-Publication Data

Bodo, Bela, (Engineer) Introduction to soil mechanics / Bela Bodo, Colin Jones. pages cm Includes bibliographical references and index. ISBN 978-0-470-65943-4 (pbk. : alk. paper) ISBN 978-1-118-55387-9 (emobi) ISBN 978-1-118-55388-6 (epub) ISBN 978-1-118-55389-3 (epdf) 1. Soil mechanics. I. Title. TA710.B617 2013 624.15136dc23

2012040913

A catalogue record for this book is available from the British Library.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Cover image courtesy of Shuttlestock.comCover design by Steve Thompson

Set in 9/11.5pt Interstate-Light by SPi Publisher Services, Pondicherry, India

1 2013

v

Contents

Preface xiiDedication and Acknowledgments xiiiList of Symbols xiv

1 Soil Structure 11.1 Volume relationships 1

1.1.1 Voids ratio (e) 21.1.2 Porosity (n) 31.1.3 Degree of saturation (S

r) 3

1.2 Weightvolume relationships 61.2.1 Bulk densities 71.2.2 Dry densities 81.2.3 Saturated densities 81.2.4 Submerged densities (g ) 91.2.5 Density of solids (g

s) 10

1.2.6 Specific gravity (Gs) 10

1.2.7 Moisture content (m) 111.2.8 Partially saturated soil 121.2.9 Relative density (D

r) 18

1.3 Alteration of soil structure by compaction 201.3.1 Laboratory compaction tests 211.3.2 Practical considerations 261.3.3 Relative compaction (C

r) 27

1.3.4 Compactive effort 271.3.5 Under- and overcompaction 281.3.6 Site tests of compaction 28

1.4 California bearing ratio (CBR) test 301.5 The pycnometer 35

Supplementary problems for Chapter 1 39

2 Classification of Cohesive Soils 432.1 Atterberg Limits 43

2.1.1 Liquid Limit (LL) 432.1.2 Plastic Limit 482.1.3 Shrinkage Limit 502.1.4 Swelling of cohesive soils 562.1.5 Saturation Limit (Z%) 562.1.6 Relationship between the limits 572.1.7 Linear shrinkage and swelling 59

2.2 Consistency indices 642.2.1 Plasticity index (PI) 64

vi Contents

2.2.2 Relative consistency index (RI) 642.2.3 Liquidity index (LI) 64

2.3 Classification of soils by particle size 692.3.1 Sieve analysis 692.3.2 Uniformity coefficient (U) 732.3.3 Filter design 742.3.4 Typical problems 772.3.5 Combination of materials 782.3.6 Sedimentation tests 85

Supplementary problems for Chapter 2 91

3 Permeability and Seepage 923.1 Coefficient of permeability (k) 933.2 Seepage velocity (u

s) 94

3.3 Determination of the value of k 963.3.1 Constant head test 963.3.2 Falling head test 98

3.4 Field pumping tests 1023.4.1 Unconfined layer 1023.4.2 Radius of influence (R) 1043.4.3 Confined layer under artesian pressure (s

A) 106

3.5 Permeability of stratified soil 1073.6 Flow nets 108

3.6.1 Flow lines (FL) 1093.6.2 Head loss in a flow channel 1113.6.3 Equipotential lines (EPL) 1113.6.4 Flow net construction 1133.6.5 Application of flow nets 1143.6.6 Seepage flowrate (Q) 1143.6.7 Seepage pressure 1153.6.8 Seepage force (S) 119

3.7 Erosion due to seepage 1213.8 Prevention of piping 1283.9 Flow net for earth dams 129Supplementary problems for Chapter 3 135

4 Pressure at Depth Due to Surface Loading 1394.1 Concentrated point load 1404.2 Concentrated line load 1424.3 Uniform strip loading (Michells solution) 1444.4 Bulb of pressure diagrams 1474.5 Vertical pressure under triangular strip load 1514.6 Vertical pressure under circular area 1564.7 Rectangular footing 1594.8 Footings of irregular shape 1634.9 Pressure distribution under footings 167

4.9.1 Influence of footing 1674.9.2 Influence of loading 170

4.10 Linear dispersion of pressure 170Supplementary problems for Chapter 4 173

Contents vii

5 Effective Pressure (s ) 1755.1 Unloaded state 1755.2 Loaded state 1775.3 Flooded state 1805.4 Types of problem 1825.5 Effect of seepage on shallow footings 1945.6 Ground water lowering (at atmospheric pressure) 1955.7 Reduction of artesian pressure 1965.8 Capillary movement of water 199

5.8.1 Equilibrium moisture content (mE) 204

5.8.2 Soil suction (Ss) 208

Supplementary problems for Chapter 5 214

6 Shear Strength of Soils 2196.1 CoulombMohr Theory 220

6.1.1 Stresses on the plane of failure 2216.1.2 Friction and cohesion 2236.1.3 Apparent cohesion 224

6.2 Stress path 2246.2.1 Stress path failure envelope 2256.2.2 Variation of stress path 231

6.3 Effect of saturation 2346.3.1 Effective Mohrs circle 2346.3.2 Effective stress path (ESP) 234

6.4 Measurement of shear strength 2386.4.1 Triaxial tests 2386.4.2 Variation of pore pressure 2406.4.3 Total excess pore pressure 2416.4.4 Unconsolidatedundrained tests 2426.4.5 Quick-undrained test 2486.4.6 Consolidatedundrained (CU) test 2506.4.7 Consolidateddrained (CD) test 2526.4.8 Unconfined compression strength of clays 2536.4.9 Standard shear box test 2566.4.10 The Vane shear test 2596.4.11 Residual shear strength 261

6.5 Thixotropy of clay 2636.6 Undrained cohesion and overburden pressure 263Supplementary problems for Chapter 6 265

7 Consolidation and Settlement 2687.1 Consolidation 2687.2 The pressurevoids ratio curve 270

7.2.1 Analytical solution 2707.2.2 Equation of the s e curve 2717.2.3 Alternative conventional procedure 2747.2.4 Graphical solution 276

7.3 Forms of the s e curve 2797.3.1 Normally consolidated clay 280

viii Contents

7.3.2 Overconsolidated clays 2807.4 Coefficient of compressibility (a

v) 281

7.5 Coefficient of volume change (mv) 282

7.5.1 Voids ratio method 2827.5.2 Direct method 282

7.6 Estimation of settlement 2847.6.1 Voids ratio method 2867.6.2 Method using m

v 288

7.6.3 Direct method 2897.7 Rate of consolidation 291

7.7.1 Variation of excess pore pressure with time 292

7.7.2 Typical pore pressure distributions 2937.7.3 Estimation of time 2947.7.4 Coefficient of consolidation (c

v) 295

7.8 Pore pressure isochrones 3017.8.1 Average percentage consolidation 302

7.9 Coefficient of permeability (k) 3107.10 Time from similarity 3107.11 Total settlement 311

7.11.1 Initial compression 3117.11.2 Primary consolidation 3117.11.3 Secondary consolidation 312

Supplementary problems for Chapter 7 314

8 Lateral Earth Pressure 3198.1 Resistance to active expansion 3208.2 The value of K

0 321

8.3 Stress path representation 3228.4 Rankines theory of cohesionless soil 324

8.4.1 Stress path representation (Lambe) 3308.5 Rankine-Bell theory for c f soil 334

8.5.1 Tension cracks 3358.5.2 Effect of surcharge (q kN/m) on z

0 336

8.5.3 Water in the cracks only 3368.6 RankineBell theory for csoil 3368.7 Pressureforce and its line of action 336

8.7.1 Triangular diagram for uniform soil 3378.7.2 Triangular diagram for water 3378.7.3 Rectangular diagram for surcharge only 338

8.8 Wall supporting sloping surface 3428.9 General formulae for c f soil 342

8.9.1 Active case 3438.9.2 Passive case (with surcharge) 345

8.10 Formulae for pure clay (f = 0) 3498.11 Height of unsupported clay 3508.12 Wedge theories 350

8.12.1 Procedure for cohesionless soil 351

Contents ix

8.12.2 Procedure for cohesive soil 3558.12.3 Point of application of P

a(x) 359

8.12.4 Effect of static water table 3608.13 Stability of retaining walls 360

8.13.1 Gravity walls 3608.13.2 Cantilever walls 3618.13.3 Buttress and counterfort walls 3618.13.4 Stability check 362

8.14 Sheet piles 3688.14.1 Cantilever sheet pile walls 3698.14.2 Factor of safety 3708.14.3 Bending of sheet piles 3748.14.4 Sheet pile in cohesive soils 375

8.15 Anchored sheet pile walls 3758.15.1 Free-earth support method 3768.15.2 Fixed-earth support method 3848.15.3 Anchorage 3908.15.4 Length of tie rod (L) 3908.15.5 Stability of anchors 390

8.16 Effect of ground water 3938.17 Stability of deep trenches 400

8.17.1 Horizontal bracing 4008.18 Bentonite slurry support 406

8.18.1 Trench in clay 4078.18.2 Trench in sand 408

Supplementary problems for Chapter 8 413

9 Bearing Capacity of Soils 4209.1 Terminology 420

9.1.1 Foundation pressure (s) 4209.1.2 Net foundation pressure (s

n) 421

9.1.3 Effective overburden pressure (s 0) 421

9.1.4 Ultimate bearing capacity (qu) 421

9.1.5 Net ultimate bearing capacity (qn) 421

9.1.6 Safe net bearing capacity (qsn

) 4229.1.7 Safe bearing capacity (q

s) 422

9.1.8 Allowable foundation pressure (sa) 422

9.1.9 Presumed bearing values 4249.2 Shallow strip footing 424

9.2.1 Terzaghis equation for qu 425

9.2.2 Effect of static water table 4289.3 Influence of footing shape 4359.4 Shallow rectangular footing 436

9.4.1 Method of Fellenius 4389.5 Deep foundations 439

9.5.1 Moderately deep foundations 4399.6 Standard penetration test (SPT) 443

x Contents

9.7 Pile foundations >

5zB

445

9.7.1 Types of pile 4469.8 Some reasons for choosing piles 4499.9 Some reasons for not choosing piles 4519.10 Effects necessitating caution 4519.11 Negative skin friction 4539.12 Stress distribution around piles 4559.13 Load-carrying capacity of piles 455

9.13.1 Static formulae 4569.13.2 End-bearing resistance (Q

e) 456

9.13.3 Shaft resistance (Qs) 457

9.13.4 Ultimate carrying capacity of pile 4589.13.5 Allowable carrying capacity of piles (Q

a) 458

9.13.6 Negative skin friction (Qf) 458

9.14 End bearing resistance and SPT 4649.15 Influence of pile section on Q

u 465

9.16 Group of piles 4659.16.1 Eccentrically loaded pile group 4689.16.2 Settlement of pile groups 4719.16.3 Raking piles 472

Supplementary problems for Chapter 9 474

10 Stability of Slopes 47910.1 Short-term and long-term stability 47910.2 Total stress analysis (cohesive soils) 480

10.2.1 Homogeneous, pure clay (fu = 0) 480

10.2.2 Increasing the value of Fs 481

10.2.3 Minimum value of Fs 482

10.2.4 Potential slip surface 48210.2.5 Determination of the factor of safety 48310.2.6 Homogeneous c f soil (total stress analysis) 49710.2.7 Stratified slopes 50010.2.8 Slopes under water 50110.2.9 Taylors stability numbers 505

10.3 Effective stress analysis (cohesive soils) 51310.3.1 Method of slices (radial procedure) 51310.3.2 Bishops conventional method 51810.3.3 Bishops rigorous iterative method 519

10.4 Stability of infinite slopes 523Supplementary problems for Chapter 10 528

11 Eurocode 7 53011.1 Introduction 53011.2 Recommended units 53011.3 Limit states 53111.4 Design procedures 53111.5 Verification procedures 53211.6 Application of partial factors 534

Contents xi

Appendices Appendix A Mass and Weight 552Appendix B Units, Conversion Factors and Unity Brackets 556Appendix C Simpsons Rule 562Appendix D Resultant Force and Its Eccentricity 567Appendix E References 570

Index 572

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Supplementary problems Solutions to supplementary problems

www.wiley.com/go/bodo/soilmechanics

Preface

This book is intended to introduce the subject to students studying for BTEC Higher National Certificate/Diploma in Civil Engineering and Building Studies or for a Degree in Civil Engineering. It should also be practical reference to Architects, Geologists, Structural and Geotechnical Technicians.

The primary aim is to provide a clear understanding of the basic concepts of Soil Mechanics. We endeavoured to avoid the temptation of over-elaboration by providing excessively detailed text, unnecessary at this early stage of technical studies.

The purpose of this publication is threefold:

1. To introduce the student to the basics of soil mechanics.2. To facilitate further advanced study.3. To provide reference Information.

In order to satisfy the above requirements, the concepts of the subject are defined con-cisely, aided by diagrams, charts, graphs, tables and worked examples as necessary.

The text may appear to be excessively analytical at first sight, but all formulas are derived in terms of basic mathematics, except for a few requiring complicated theory, for those interested in working from first principles. They can be applied however, without reference to the derivation. The expressions are numbered and referred to throughout the text.

There are numerous worked examples on each topic as well as supplementary prob-lems. All examples and problems are solved, many of them interrelated so that solutions can be compared and verified by means of several methods.

Some soil testing procedures are outlined only, as there are a number of excellent, detailed, specialized books and laboratory manuals available to cover this part of the subject.

There is some emphasis on the units employed and on the difference between mass and weight. This subject is discussed in Appendix A.

Bla Bod and Colin Jones

xii

xiii

Acknowledgments

Dedication

I dedicate this book to my late wife Dorie.Bla Bod

We wish to express our appreciation to Mr. Norman Seward, Senior Lecturer in Civil Engineering at the University of Wales College, Newport for his technical advice as to the presentation of the subject.We are also grateful to Mr. Gregory Williams for his help in the production of this book.We would like to thank ELE International for their support in providing product images.

xiv

List of Symbols

Chapter 1

CBR California bearing ratioC

rRelative compaction

Dr

Relative Densitye Voids ratioG

sSpecific gravity

k CBR Load-ring factorM Total Mass of samplem Moisture (water) contentm

oOptimum moisture content

Ms

Mass of solidsM

wMass of water

n PorosityP CBR applied forceP

aPercentage of air voids

Q CBR Load gauge readingS

rDegree of saturation

V Total volume of sampleV

aVolume of air

Vc

Volume of calibrating cylinderV

sVolume of solids

Vv

Volume of voidsV

wVolume of water

W Total weight of sampleW

sWeight of solids

Ww

Weight of waterd CBR Penetration distance (delta)g Bulk weight density (Gamma)g Submerged weight densityg

dDry Weight density

gd

Dry Unit weight to be achieved by compactiong

sWeight density of solids

gsat

Saturated weight densityr Bulk mass densityr

dDry mass density

rsat

Saturated mass densityr Submerged mass densityr

sMass density of solids

List of Symbols xv

Chapter 2

Cd

Correction for dispersing agentC

mMeniscus correction

D Equivalent particle diameterD

10Effective size of a particle

f Specific Volume changeH Height from the top of the bulb to surfaceh

bLength of bulb

HR

Height of centre of bulb to surfaceLI Liquidity indexLL Liquid limitM

pMass passing the nth sieve

Mr

Mass retained on the nth sievem

TTemperature correction

N Number of blowsPI Plasticity indexPL Plastic limitP

nPercentage of soil passing the nth sieve

R Mixing ratioR

hRecorded hydrometer reading

Rh

Corrected hydrometer readingRI Relative consistence indexSL Shrinkage limitT Temperaturet TimeU Uniformity coefficientu Velocity of sedimentationV

bVolume of hydrometer bulb

Vo

Volume of over-dried specimenVolume at SL

x Magnitude of linear shrinkage or swellingZ Saturation limith Dynamic viscosity

Chapter 3

A Cross-sectional area of specimena Cross-sectional area of standpipeA

sCross-sectional area of solids in specimen

Av

Cross-sectional area of voids in specimenEPL Equipotential lineFL Flow LineF

sFactor of safety

GL Ground levelGWL Groundwater level (Water Table)

xvi List of Symbols

h Head lossH

TTotal head at x

Hx

Head loss to point xh

xPressure head at x

i Hydraulic gradientiav

Average hydraulic gradientic

Critical hydraulic gradientie

Exit gradientk Coefficient of permeabilityL Length of flow pathN

eNumber of squares (head drops)

Nf

Number of flow channelsN

xNumber of head drops to point x

P Hydrostatic forceQ Flowrateq Quantity of flow in time (t)R Radius of influencer Radius to observation wellr

oRadius of central well

S Seepage forceu

xSeepage pore pressure at x

h Head Loss between equipotential lineu Discharge velocityu

sSeepage velocity

Chapter 4

I Influence factorn Number of elements on the Newmark chartQ Concentrated point loadq Uniformly distributed load (UDL)r Radiusz Depths Horizontal pressures

vVertical pressure

t Shear stress

Chapter 5

dh Total deformation of specimen of thickness hh

AArtesian pressure head

hc

Capillary headh

sSeepage pressure head

ic

Critical hydraulic gradientm

EEquilibrium moisture content

mo

Optimum moisture contentpF Soil suction indexPI Plasticity indexS

rDegree of saturation

List of Symbols xvii

Ss

Soil suctionT Surface tensionu Pore pressureu

csPore pressure in the capillary fringe

uh

Static pore pressure at depth hu

sSeepage pore pressure

zc

Critical depth for pipingu Small change in ug Change in unit weights Small change in ss Small change in s d Deformation of specimen at time ts Total pressures Effective pressures

AArtesian pressure

Chapter 6

A Pore pressure coefficient

A Pore pressure coefficient

B Pore pressure coefficientc Cohesionc

uUndrained shear strength

CD Consolidated-drained testCU Consolidated-undrained testESP Effective stress pathNCC Normally consolidated clayn Proving ring constantOCC Over consolidated clayp&q Stress path coordinatesp

f&q

fStress path coordinates at failure

QU Quick-undrained testr

xForce dial reading at x

TSP Total stress pathUU Unconsolidated-undrained testx Strain gauge readingu

dChange in pore pressure due to s

d

uc

Change in pore pressure due to sc

sc

Change in cell pressures

dChange in the deviator stress

e Strain at xf Angle of frictions

nNormal pressure

sx

Deviator stress at xs

uUnconfined compression strength

t Shear stresst

fShear stress at failure

tp

Shear stress on a plaint

mMaximum shear stress

xviii List of Symbols

Chapter 7

Ac

Area indicating completed consolidationA

tArea under an isochrone

av

Coefficient of compressibilityC Coefficient of Secondary settlement () to consolidation C

cCompression index

Cv

Coefficient of consolidationD

xDial reading at stage x

dHi

Initial settlementE Modulus of elasticitye

0Initial voids ratio

ef

Final voids ratioe

sVoids ratio after swelling

ex

Voids ratio at stage xH Layer thicknessH

0Flow path

hx

Height of specimen at stage xIp

Influence factork Coefficient of permeabilitym

vCoefficient of volume change

OCR Overconsolidation ratioq Bearing pressureT

vTime factor

t TimeU Average degree of consolidationU

zDegree of consolidation

u Pore pressure at time tu

0Initial pore pressure

H Long-term consolidation settlements Effective consolidating pressured Depth factor (Delta) Poissons ratio (My)s

xEffective pressure at stage x

Chapter 8

cu

Unconfined compression strengthc

WAdhesion between soil and wall

e EccentricityFf Factor of safety in terms of friction anglef

maxMaximum compressive stress

fmin

Minimum compressive stressF

sFactor of safety

H Height of wallH

0Height of unsupported clay

K Coefficient of lateral pressure

List of Symbols xix

K0

Coefficient of earth pressure at restK

aCoefficient of active earth pressure

Kf

Coefficient of earth pressure at failureK

pCoefficient of passive earth pressure

L Length of slip surfaceM

maxMaximum bending moment

M0

Overturning momentM

RResisting moment

Pa

Active forceP

pPassive force

PW

Force of water in tension crackR Force on wedgeT Tension force in tie rodz

cPile penetration

z0

Depth of tension crackd Angle of wall frictionf

mMobilised friction

m Coefficient of frictions

aActive earth pressure

sc

Cell pressure in triaxial tests

dDeviator stress in triaxial test

sp

Passive earth pressures

aEffective active earth pressure

s p

Effective passive earth pressure Average pressuret

fShear stress at failure

Chapter 9

uc Average undrained shear strengthA

eEnd bearing area

As

Surface area of pileB Width of footingc CohesionF

0Overall factor of safety

Fs

Factor of safetyK

sAverage coefficient of earth pressure

l Length of pileN Number of SPT blowsn Number of pilesN Corrected value of NN

c

Nq

Bearing capacity factorsNgP Failure load on pileQ Design working loadQ

aAllowable carrying capacity of pile

Qag

Allowable carrying capacity of pile group

xx List of Symbols

Qe

End bearing resistanceQ

fNegative skin friction

QS

Shaft resistanceQ

uUltimate carrying capacity of pile

Qug

Ultimate carrying capacity of pile groupq

nNet ultimate bearing capacity

qs

Safe bearing capacityq

snSafe net bearing capacity

qu

Ultimate bearing capacitySPT Standard penetration testW

PWeight of pile

a Adhesion factor (Alpha)d Angle of friction between soil and pile (Delta)h Efficiency of pile group (Eta)f Angle of frictions Safe bearing pressure of footings

nNet bearing pressure of footing

o Average effective overburden pressures

oEffective overburden pressure

Chapter 10

cu

Shear strengthF Friction forceF

CFactor of safety with respect to cohesion

FS

Factor of safetyFf Factor of safety with respect to frictionL Length of slip surfaceM

DDisturbing moment

MR

Resisting momentN Normal (or radial) component of WN

CStability number

R Radius of slip circler

uPore pressure ratio

S Shear forceT Tangential component of WW Weight

Chapter 11

The comprehensive list of symbols for EC7 is given in Eurocode 7. Geotechnical design Part 1: General rule. Only some of the symbols, applied in this book, are reproduced here:

Ed

Design value of the effect of actionsE

dst;dDesign value of the effect of destabilizing action

Estb;d

Design value of the effect of stabilizing actionF

dDesign value of an action

Frep

Representative value of an actionF

sFactor of safety

List of Symbols xxi

Gdst;d

Design value of destabilising permanent actionG

stb;dDesign value of stabilising permanent action

Qdst;d

Design value of destabilising variable actionR

dDesign value of resistance action

Sdst;d

Design value of destabilising seepage forceT

dDesign value of total shear resistance

Udst;d

Design value of destabilising pore water pressureV

dst;dDesign value of destabilising vertical action

Xd

Design value of a material propertyX

kCharacteristics value of a material property

gG

Partial factor for a permanent actiong

G;distPartial factor for a destabilising action

gG;stb

Partial factor for a stabilising actiong

mPartial factor for soil parameters (material property)

gQ

Partial factor for a variable actiong

R;hPartial factor for sliding resistance

Introduction to Soil Mechanics, First Edition. Bla Bod and Colin Jones. 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd.

1

Chapter 1

Soil Structure

Soils consist of solid particles, enclosing voids or pores. The voids may be filled with air or water or both. These three soil states (or phases) can be visualized by the enlargement of three small samples of soil.

Sample A: The soil is oven-dry, that is there is only air in the voids.Sample B: The soil is saturated, that is the voids are full of water.Sample C: The soil is partially saturated, that is the voids are partially filled with water.

The above three soil states can be described mathematically by considering:

1. Volume occupied by each constituent.2. Mass (or weight) of the constituents.

1.1 Volume relationships

The expressions derived in this section will answer two questions:

1. How much voids and solids are contained in the soil sample?2. How much water is contained in the voids?

In order to obtain these answers, the partially saturated sample (C) is examined. It is assumed, for the purpose of analysis, that the soil particles are lumped together into a homogeneous mass. Similarly, the voids are combined into a single volume, which is

Figure 1.1

Air(A)

Solid

(B)Water

Solid

(C)

Water

Air

Solid

2 Introduction to Soil Mechanics

partly occupied by a volume of water. The idealisation of the sample, indicating the volumes occupied by the constituents, is shown diagrammatically in Figure 1.2b.

Idealized representation of sample C.Where: V = Total volume of the sample

Vv = Volume of voids in the sample

Vs = Volume of soil in the sample

Vw = Volume of water in the sample

Va = Volume of air in the sample

The basic relationships between the volumes can be seen in the diagram.

Total volume: s vV V V= + (1.1)

Volume of voids: v w aV V V= + (1.2)

Hence: s w aV V V V= + + (1.3)

Three important relationships are derived from the basic ones. These are:

e = voids ratio (or void ratio)n = porosityS

r = degree of saturation

1.1.1 Voids ratio (e)

This shows the percentage of voids present in the sample, compared to the volume of solids. Thus, if V

s is considered to be 100%, then V

v is e%.

Hence: v

s

100 %V

eV

= (1.4)

For example: if Vs = 60 cm3

and Vv = 15 cm3

then 15

100 25%60

e = =

That is, the volume of voids is 25% of the volume of solids, in this particular sample. Alternatively, the voids ratio maybe expressed as a decimal e.g. e = 0.25.

Formula (1.4) now becomes: v

s

Ve

V= (1.5)

Figure 1.2

Solids

Water

Air

(a)

Air

Water

Solids

Va

VvVw

Vs

V

(b)

Soil Structure 3

The ratio of voids to solids in a sample is represented by Figure 1.3.

1.1.2 Porosity (n)

This shows how many percent of voids are present in the sample, compared to the total volume V. Thus, if V is considered to be 100%, then V

v is n%.

V100 %

Vn

V=

(1.6)

For example: if V = 75 cm3 and V

v = 15 cm3

then 15

100 20%75

n = =

That is, the volume of voids is 20% of the total volume of the sample of soil. Again, n maybe expressed as a decimal number n = 0.2.

Formula (1.6) now becomes: vV

nV

= (1.7)

The diagrammatic representation of porosity is:

1.1.3 Degree of saturation (Sr)

This shows the percentage of voids filled with water. Thus, if Vv is considered to be 100%,

then Vw is S

r%.

= wrv

100 %V

SV

(1.8)

Vv

Vs

V

Solids

Voids

Figure 1.3

Solids

VvVoids

V

Figure 1.4

4 Introduction to Soil Mechanics

For example, if Vw = 6 cm3

and Vv = 15 cm3

then r6

100 40%15

S = =

That is, water fills 40% of the volume of voids. In decimal form Sr = 0.4 and formula

(1.8) becomes:

wrv

VS

V= (1.9)

Diagrammatically,

Note: For oven-dry soil (Sample A, Figure 1.1):

w r0, hence 0V S= =

For fully saturated soil (Sample B, Figure 1.1):

w v r, hence 1V V S= =

For partially saturated soil therefore: 0 < Sr < 1

Combined formulaeThe quantities defined by formulae (1.1) to (1.9) can be interrelated:

( )s s ss v

vv sv

eith( )

( )

er 1From 1.1 :

From 1.5 : or

V V eV V e VV V V

VV eV V Ve

= + = += += = +

(1.10)

v v v

1 11

eV V V

e e+ = + =

(1.11)

( )

vs

ss

From 1.7 :

1 1From 1

( )

.10 :( )) (1

Vn eV e

n nVe V e

V e V

== =

+ += + (1.12)

From 1.12 :1

((1 )

)

1

n ne ee

n ne n e n e

n

+ ==

+ = =

(1.13)

Air

Water

Solids

VwVv

Figure 1.5

Soil Structure 5

=+ = =

= ++

wr

w wvr r

v

From 1.9 :1

From 1.11 :1

(

) 1

)

(

VS

V VV eS S

eV e VeVV ee

(1.14)

= =+w

rFrom 1.12 : 1( ) or

Ven S

e nV (1.15)

Example 1.1

Given: V = 946 cm3 Calculate: Vv, V

a, e, n and S

r

3

s 533cmV =

3

w 303cmV =

3v sFrom 1.1 : 946 533 413c( m) V V V= = =

3a v wFrom 1.2 : 413 303 110cm( ) V V V= = =

v

s

413From 1.5 : 0.775( )

533

Ve

V= = = , that is the volume of voids is 77.5% that of

solids.

v 413From 1.7 : 0.437That is, the volume of voids is 43.7%946of the sample.0.775

or From 1.12 : 0.4371 1

( )

).7 5

(7

Vn

Ve

ne

= = =

= = =+

wr

v

wr

( )

( )( )

303From 1.9 : 0.73

413 That is, water fills 73% of voids.

The sample is partially saturated.303or From 1.15 : 0.73

0.437 946

VS

V

VS

nV

= = =

= = =

Example 1.2

A sample of sand was taken from below the ground water table. The volumes measured were:

V = 1000 cm3 Calculate: Vv, V

a, V

s, e and n

Vw = 400 cm3

Note: Assume sand samples taken from above the water table as partially satu-rated (S

r < 1) and saturated (S

r = 1) if taken from below.

In this example, therefore, Sr = 1 V

a = 0.

From (1.8) wr w vV

3v

1

400cm

VS V V

V

V

= = =

=

(1.16)

6 Introduction to Soil Mechanics

From (1.2) : Va = V

v-V

w = 400 -400 = 0 The voids are full of water

From (1.1): 3s v 1000 400 600cmV V V= = =

From (1.5): v v ss

4000.67 is 67% of

600

Ve V V

V= = =

From (1.7): v v400

0.4 is 40% of1000

Vn V V

V= = =

1.2 Weightvolume relations

As the title implies, the formulae derived in this section take into account the weights of V

s and V

w. It is assumed that air is weightless. The weight volume relations are shown

diagrammatically:

s

w s w

Where : Weight of solids

Weight of water From Figure 1.6

Totalweight

W

W W W W

W

== = +

=

(1.17)

Note: The concepts of mass and weight are defined in Appendix A. Suffice to say here, that if mass (M) is given in kilograms, then weight (W) is calculated from:

39.81 mass ( ) 9.81 10 kNW M N W M= = (1.18)

Several important relationships are derived below in terms of mass, weight and volume. These are:

r = bulk mass densityg = bulk weight density (unit weight)

rd =dry mass density

gd = dry weight density

rsat

=saturated mass densityg

sat =saturated weight density

Air

Vv

Va

Ww

Ws

Vw

Vs

Water

Solids

VW

Figure 1.6