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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
This books companion website is at www.wiley.com/go/bodo/soilmechanics and offers invaluable resources for students and lecturers:
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.
Editorial Offices9600 Garsington Road, Oxford, OX4 2DQ, United Kingdom.The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom.
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All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.
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Limit of Liability/Disclaimer of Warranty: While the publisher and author(s) have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought.
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
About the companion website
This books companion website is at www.wiley.com/go/bodo/soilmechanics and offers invaluable resources for students and lecturers:
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