SOIL MECHANICS. Introduction to. Béla Bodó & Colin Jones SOIL MECHANICS. Introduction to INTRODUCTION TO SOIL MECHANICS. Béla Bodó & Colin Jones

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1 INTRODUCTION TO SOIL MECHANICS SOIL MECHANICS Béla Bodó & Colin Jones 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 wellillustrated text with diagrams, charts, graphs and tables. With many practical, worked examples and end-of-chapter problems (with fully worked solutions available at and coverage of Eurocode 7, Introduction to Soil Mechanics will be an ideal starting point for the study of soil mechanics and geotechnical engineering. About the Authors Béla Bodó 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. ebsite Bodó & Jones anion mp w free co 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 Béla 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 book s companion website is at and offers invaluable resources for both students and lecturers: M Supplementary problems M Solutions to supplementary problems Also Available Smith s Elements of Soil Mechanics 8th Edition Ian Smith Paperback: Béla Bodó & Colin Jones anion mp ebsite anion mp w ebsite SOIL MECHANICS w free co Fundamentals of Rock Mechanics 4th Edition J.C Jaeger, N.G.W Cook and R. Zimmerman Hardcover: Introduction to free co Introduction to

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3 Introduction to Soil Mechanics

4 About the companion website This book s companion website is at and offers invaluable resources for students and lecturers: Supplementary problems Solutions to supplementary problems

5 Introduction to Soil Mechanics Béla Bodó and Colin Jones

6 This edition first published by John Wiley & Sons, Ltd Registered Office John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom. Editorial Offices 9600 Garsington Road, Oxford, OX4 2DQ, United Kingdom. The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom. For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at The right of the author to be identified as the author of this work has been asserted in accordance with the UK Copyright, Designs and Patents Act 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. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. 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 (pbk. : alk. paper) ISBN (emobi) ISBN (epub) ISBN (epdf) 1. Soil mechanics. I. Title. TA710.B dc 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.com Cover design by Steve Thompson Set in 9/11.5pt Interstate-Light by SPi Publisher Services, Pondicherry, India

7 Contents Preface Dedication and Acknowledgments List of Symbols xii xiii xiv 1 Soil Structure olume relationships oids ratio (e) Porosity (n) Degree of saturation (S r ) Weight volume relationships Bulk densities Dry densities Saturated densities Submerged densities (g ) Density of solids (g s ) Specific gravity (G s ) Moisture content (m) Partially saturated soil Relative density (D r ) Alteration of soil structure by compaction Laboratory compaction tests Practical considerations Relative compaction (C r ) Compactive effort Under- and overcompaction Site tests of compaction California bearing ratio (CBR) test The pycnometer 35 Supplementary problems for Chapter Classification of Cohesive Soils Atterberg Limits Liquid Limit (LL) Plastic Limit Shrinkage Limit Swelling of cohesive soils Saturation Limit (Z%) Relationship between the limits Linear shrinkage and swelling Consistency indices Plasticity index (PI) 64 v

8 vi Contents Relative consistency index (RI) Liquidity index (LI) Classification of soils by particle size Sieve analysis Uniformity coefficient (U) Filter design Typical problems Combination of materials Sedimentation tests 85 Supplementary problems for Chapter Permeability and Seepage Coefficient of permeability (k) Seepage velocity (u s ) Determination of the value of k Constant head test Falling head test Field pumping tests Unconfined layer Radius of influence (R) Confined layer under artesian pressure (s A ) Permeability of stratified soil Flow nets Flow lines (FL) Head loss in a flow channel Equipotential lines (EPL) Flow net construction Application of flow nets Seepage flowrate (Q) Seepage pressure Seepage force (S) Erosion due to seepage Prevention of piping Flow net for earth dams 129 Supplementary problems for Chapter Pressure at Depth Due to Surface Loading Concentrated point load Concentrated line load Uniform strip loading (Michell s solution) Bulb of pressure diagrams ertical pressure under triangular strip load ertical pressure under circular area Rectangular footing Footings of irregular shape Pressure distribution under footings Influence of footing Influence of loading Linear dispersion of pressure 170 Supplementary problems for Chapter 4 173

9 Contents vii 5 Effective Pressure (s ) Unloaded state Loaded state Flooded state Types of problem Effect of seepage on shallow footings Ground water lowering (at atmospheric pressure) Reduction of artesian pressure Capillary movement of water Equilibrium moisture content (m E ) Soil suction (S s ) 208 Supplementary problems for Chapter Shear Strength of Soils Coulomb Mohr Theory Stresses on the plane of failure Friction and cohesion Apparent cohesion Stress path Stress path failure envelope ariation of stress path Effect of saturation Effective Mohr s circle Effective stress path (ESP) Measurement of shear strength Triaxial tests ariation of pore pressure Total excess pore pressure Unconsolidated undrained tests Quick-undrained test Consolidated undrained (CU) test Consolidated drained (CD) test Unconfined compression strength of clays Standard shear box test The ane shear test Residual shear strength Thixotropy of clay Undrained cohesion and overburden pressure 263 Supplementary problems for Chapter Consolidation and Settlement Consolidation The pressure voids ratio curve Analytical solution Equation of the s e curve Alternative conventional procedure Graphical solution Forms of the s e curve Normally consolidated clay 280

10 viii Contents Overconsolidated clays Coefficient of compressibility (a v ) Coefficient of volume change (m v ) oids ratio method Direct method Estimation of settlement oids ratio method Method using m v Direct method Rate of consolidation ariation of excess pore pressure with time Typical pore pressure distributions Estimation of time Coefficient of consolidation (c v ) Pore pressure isochrones Average percentage consolidation Coefficient of permeability (k) Time from similarity Total settlement Initial compression Primary consolidation Secondary consolidation 312 Supplementary problems for Chapter Lateral Earth Pressure Resistance to active expansion The value of K Stress path representation Rankine s theory of cohesionless soil Stress path representation (Lambe) Rankine-Bell theory for c f soil Tension cracks Effect of surcharge (q kn/m) on z Water in the cracks only Rankine Bell theory for c soil Pressure force and its line of action Triangular diagram for uniform soil Triangular diagram for water Rectangular diagram for surcharge only Wall supporting sloping surface General formulae for c f soil Active case Passive case (with surcharge) Formulae for pure clay (f = 0) Height of unsupported clay Wedge theories Procedure for cohesionless soil 351

11 Contents ix Procedure for cohesive soil Point of application of P a (x) Effect of static water table Stability of retaining walls Gravity walls Cantilever walls Buttress and counterfort walls Stability check Sheet piles Cantilever sheet pile walls Factor of safety Bending of sheet piles Sheet pile in cohesive soils Anchored sheet pile walls Free-earth support method Fixed-earth support method Anchorage Length of tie rod (L) Stability of anchors Effect of ground water Stability of deep trenches Horizontal bracing Bentonite slurry support Trench in clay Trench in sand 408 Supplementary problems for Chapter Bearing Capacity of Soils Terminology Foundation pressure (s) Net foundation pressure (s n ) Effective overburden pressure (s 0 ) Ultimate bearing capacity (q u ) Net ultimate bearing capacity (q n ) Safe net bearing capacity (q sn ) Safe bearing capacity (q s ) Allowable foundation pressure (s a ) Presumed bearing values Shallow strip footing Terzaghi s equation for q u Effect of static water table Influence of footing shape Shallow rectangular footing Method of Fellenius Deep foundations Moderately deep foundations Standard penetration test (SPT) 443

12 x Contents 9.7 Pile foundations z > B Types of pile Some reasons for choosing piles Some reasons for not choosing piles Effects necessitating caution Negative skin friction Stress distribution around piles Load-carrying capacity of piles Static formulae End-bearing resistance (Q e ) Shaft resistance (Q s ) Ultimate carrying capacity of pile Allowable carrying capacity of piles (Q a ) Negative skin friction (Q f ) End bearing resistance and SPT Influence of pile section on Q u Group of piles Eccentrically loaded pile group Settlement of pile groups Raking piles 472 Supplementary problems for Chapter Stability of Slopes Short-term and long-term stability Total stress analysis (cohesive soils) Homogeneous, pure clay (f u = 0) Increasing the value of F s Minimum value of F s Potential slip surface Determination of the factor of safety Homogeneous c f soil (total stress analysis) Stratified slopes Slopes under water Taylor s stability numbers Effective stress analysis (cohesive soils) Method of slices (radial procedure) Bishop s conventional method Bishop s rigorous iterative method Stability of infinite slopes 523 Supplementary problems for Chapter Eurocode Introduction Recommended units Limit states Design procedures erification procedures Application of partial factors 534

13 Contents xi Appendices Appendix A Mass and Weight 552 Appendix B Units, Conversion Factors and Unity Brackets 556 Appendix C Simpson s Rule 562 Appendix D Resultant Force and Its Eccentricity 567 Appendix E References 570 Index 572 About the companion website This book s companion website is at and offers invaluable resources for students and lecturers: Supplementary problems Solutions to supplementary problems

14 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 concisely, 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 problems. 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. Béla Bodó and Colin Jones xii

15 Dedication I dedicate this book to my late wife Dorie. Béla Bodó Acknowledgments 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. xiii

16 List of Symbols Chapter 1 CBR California bearing ratio C r Relative compaction D r Relative Density e oids ratio G s Specific gravity k CBR Load-ring factor M Total Mass of sample m Moisture (water) content m o Optimum moisture content M s Mass of solids M w Mass of water n Porosity P CBR applied force P a Percentage of air voids Q CBR Load gauge reading S r Degree of saturation Total volume of sample a olume of air c olume of calibrating cylinder s olume of solids v olume of voids w olume of water W Total weight of sample W s Weight of solids W w Weight of water d CBR Penetration distance (delta) g Bulk weight density (Gamma) g Submerged weight density g d Dry Weight density g d Dry Unit weight to be achieved by compaction g s Weight density of solids g sat Saturated weight density r Bulk mass density r d Dry mass density r sat Saturated mass density r Submerged mass density r s Mass density of solids xiv

17 List of Symbols xv Chapter 2 C d Correction for dispersing agent C m Meniscus correction D Equivalent particle diameter D 10 Effective size of a particle f Specific olume change H Height from the top of the bulb to surface h b Length of bulb H R Height of centre of bulb to surface LI Liquidity index LL Liquid limit M p Mass passing the n th sieve M r Mass retained on the n th sieve m T Temperature correction N Number of blows PI Plasticity index PL Plastic limit P n Percentage of soil passing the n th sieve R Mixing ratio R h Recorded hydrometer reading R h Corrected hydrometer reading RI Relative consistence index SL Shrinkage limit T Temperature t Time U Uniformity coefficient u elocity of sedimentation b olume of hydrometer bulb o olume of over-dried specimen olume at SL x Magnitude of linear shrinkage or swelling Z Saturation limit h Dynamic viscosity <eta> Chapter 3 A a A s A v EPL FL F s GL GWL Cross-sectional area of specimen Cross-sectional area of standpipe Cross-sectional area of solids in specimen Cross-sectional area of voids in specimen Equipotential line Flow Line Factor of safety Ground level Groundwater level (Water Table)

18 xvi List of Symbols h H T H x h x i i av i c i e k L N e N f N x P Q q R r r o S u x Δh u u s Head loss Total head at x Head loss to point x Pressure head at x Hydraulic gradient Average hydraulic gradient Critical hydraulic gradient Exit gradient Coefficient of permeability Length of flow path Number of squares (head drops) Number of flow channels Number of head drops to point x Hydrostatic force Flowrate Quantity of flow in time (t) Radius of influence Radius to observation well Radius of central well Seepage force Seepage pore pressure at x Head Loss between equipotential line Discharge velocity Seepage velocity Chapter 4 I n Q q r z s s v t Influence factor Number of elements on the Newmark chart Concentrated point load Uniformly distributed load (UDL) Radius Depth Horizontal pressure ertical pressure Shear stress Chapter 5 dh h A h c h s i c m E m o pf PI S r Total deformation of specimen of thickness h Artesian pressure head Capillary head Seepage pressure head Critical hydraulic gradient Equilibrium moisture content Optimum moisture content Soil suction index Plasticity index Degree of saturation

19 List of Symbols xvii S s Soil suction T Surface tension u Pore pressure u cs Pore pressure in the capillary fringe u h Static pore pressure at depth h u s Seepage pore pressure z c Critical depth for piping Δu Small change in u Δg Change in unit weight Δs Small change in s Ds Small change in s d Deformation of specimen at time t s Total pressure s Effective pressure Artesian pressure s A Chapter 6 A A B c c u CD CU ESP NCC n OCC p&q p f &q f QU r x TSP UU x Δu d Δu c Δs c Δs d e f s n s x s u t t f t p t m Pore pressure coefficient Pore pressure coefficient Pore pressure coefficient Cohesion Undrained shear strength Consolidated-drained test Consolidated-undrained test Effective stress path Normally consolidated clay Proving ring constant Over consolidated clay Stress path coordinates Stress path coordinates at failure Quick-undrained test Force dial reading at x Total stress path Unconsolidated-undrained test Strain gauge reading Change in pore pressure due to Δs d Change in pore pressure due to Δs c Change in cell pressure Change in the deviator stress Strain at x Angle of friction Normal pressure Deviator stress at x Unconfined compression strength Shear stress Shear stress at failure Shear stress on a plain Maximum shear stress

20 xviii List of Symbols Chapter 7 A c A t a v C α C c C v D x Area indicating completed consolidation Area under an isochrone Coefficient of compressibility Coefficient of Secondary settlement () to consolidation Compression index Coefficient of consolidation Dial reading at stage x dh i Initial settlement E Modulus of elasticity e 0 Initial voids ratio e f Final voids ratio e s oids ratio after swelling e x oids ratio at stage x H Layer thickness H 0 Flow path h x Height of specimen at stage x I p Influence factor k Coefficient of permeability m v Coefficient of volume change OCR Overconsolidation ratio q Bearing pressure T v Time factor t Time U Average degree of consolidation U z Degree of consolidation u Pore pressure at time t u 0 Initial pore pressure ΔH Long-term consolidation settlement Δs Effective consolidating pressure d Depth factor (Delta) μ Poisson s ratio (My) s x Effective pressure at stage x Chapter 8 c u c W e F f f max f min F s H H 0 K Unconfined compression strength Adhesion between soil and wall Eccentricity Factor of safety in terms of friction angle Maximum compressive stress Minimum compressive stress Factor of safety Height of wall Height of unsupported clay Coefficient of lateral pressure

21 List of Symbols xix K 0 Coefficient of earth pressure at rest K a Coefficient of active earth pressure K f Coefficient of earth pressure at failure K p Coefficient of passive earth pressure L Length of slip surface M max Maximum bending moment M 0 Overturning moment M R Resisting moment P a Active force P p Passive force P W Force of water in tension crack R Force on wedge T Tension force in tie rod z c Pile penetration z 0 Depth of tension crack d Angle of wall friction f m Mobilised friction m Coefficient of friction s a Active earth pressure s c Cell pressure in triaxial test s d Deviator stress in triaxial test s p Passive earth pressure s a Effective active earth pressure s p Effective passive earth pressure σ Average pressure Shear stress at failure t f Chapter 9 cu A e A s B c F 0 F s K s l N n N N c N q N g P Q Q a Q ag Average undrained shear strength End bearing area Surface area of pile Width of footing Cohesion Overall factor of safety Factor of safety Average coefficient of earth pressure Length of pile Number of SPT blows Number of piles Corrected value of N Bearing capacity factors Failure load on pile Design working load Allowable carrying capacity of pile Allowable carrying capacity of pile group

22 xx List of Symbols Q e End bearing resistance Q f Negative skin friction Q S Shaft resistance Q u Ultimate carrying capacity of pile Q ug Ultimate carrying capacity of pile group q n Net ultimate bearing capacity q s Safe bearing capacity q sn Safe net bearing capacity q u Ultimate bearing capacity SPT Standard penetration test W P Weight of pile a Adhesion factor (Alpha) d Angle of friction between soil and pile (Delta) h Efficiency of pile group (Eta) f Angle of friction s Safe bearing pressure of footing s n Net bearing pressure of footing σ o Average effective overburden pressure s o Effective overburden pressure Chapter 10 c u F F C F S F f L M D M R N N C R r u S T W Shear strength Friction force Factor of safety with respect to cohesion Factor of safety Factor of safety with respect to friction Length of slip surface Disturbing moment Resisting moment Normal (or radial) component of W Stability number Radius of slip circle Pore pressure ratio Shear force Tangential component of W 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: E d E dst;d E stb;d F d F rep F s Design value of the effect of actions Design value of the effect of destabilizing action Design value of the effect of stabilizing action Design value of an action Representative value of an action Factor of safety

23 List of Symbols xxi G dst;d G stb;d Q dst;d R d S dst;d T d U dst;d dst;d X d X k g G g G;dist g G;stb g m g Q g R;h Design value of destabilising permanent action Design value of stabilising permanent action Design value of destabilising variable action Design value of resistance action Design value of destabilising seepage force Design value of total shear resistance Design value of destabilising pore water pressure Design value of destabilising vertical action Design value of a material property Characteristics value of a material property Partial factor for a permanent action Partial factor for a destabilising action Partial factor for a stabilising action Partial factor for soil parameters (material property) Partial factor for a variable action Partial factor for sliding resistance

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25 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. (A) Air (B) Water (C) Solid Solid Air Solid Water Figure 1.1 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. olume occupied by each constituent. 2. Mass (or weight) of the constituents. 1.1 olume 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 Introduction to Soil Mechanics, First Edition. Béla Bodó and Colin Jones John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 1

26 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. (a) (b) Air a Water Air Solids Water Solids w s v Figure 1.2 Idealized representation of sample C. Where: = Total volume of the sample v = olume of voids in the sample s = olume of soil in the sample w = olume of water in the sample a = olume of air in the sample The basic relationships between the volumes can be seen in the diagram. Total volume: = s + v (1.1) olume of voids: v = w + a (1.2) Hence: = s + w + a (1.3) Three important relationships are derived from the basic ones. These are: e = voids ratio (or void ratio) n = porosity S r = degree of saturation oids ratio (e) This shows the percentage of voids present in the sample, compared to the volume of solids. Thus, if s is considered to be 100%, then v is e%. Hence: e v = 100 % (1.4) s For example: if s = 60 cm 3 and v = 15 cm 3 then 15 e = 100 = 25% 60 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 = Formula (1.4) now becomes: v e = (1.5) s

27 Soil Structure 3 The ratio of voids to solids in a sample is represented by Figure 1.3. oids v Solids s Figure Porosity (n) This shows how many percent of voids are present in the sample, compared to the total volume. Thus, if is considered to be 100%, then v is n%. n = 100 % (1.6) For example: if = 75 cm 3 and v = 15 cm 3 then 15 n = 100 = 20% 75 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: n The diagrammatic representation of porosity is: v = (1.7) oids v Solids Degree of saturation (S r ) Figure 1.4 This shows the percentage of voids filled with water. Thus, if v is considered to be 100%, then w is S r %. S 100 % w r = v (1.8)

28 4 Introduction to Soil Mechanics For example, if w = 6 cm 3 and v = 15 cm 3 then 6 S r = 100 = 40% 15 That is, water fills 40% of the volume of voids. In decimal form S r = 0.4 and formula (1.8) becomes: w Sr = (1.9) v Diagrammatically, Air Water w v Solids Figure 1.5 Note: For oven-dry soil (Sample A, Figure 1.1): w = 0, hence Sr = 0 For fully saturated soil (Sample B, Figure 1.1): =, hence S = 1 w v r For partially saturated soil therefore: 0 < S r < 1 Combined formulae The quantities defined by formulae (1.1) to (1.9) can be interrelated: From ( 1.1 ): = s + v From ( 1.5 ): = e v s ( ) either = s + es = 1 + e s v or = + v e 1 1+ e = + 1 = e e (1.10) v v v (1.11) From ( 1.7 ): = From ( 1.10 ) : = (1 + e ) v n es e s n = n = 1+ e 1+ e ( ) s (1.12) n + ne = e e From ( 1.12 ) : n = n 1 + e n = e(1 n) e = (1.13) 1 n

29 Soil Structure 5 w From ( 1.9 ) : Sr = e From ( 1.11 ) : v = 1 + e v w 1 + e w Sr = Sr = e e 1 + e e w From ( 1.12 ): n = or Sr = 1 + e n (1.14) (1.15) Example 1.1 Given: = 946 cm 3 Calculate: v, a, e, n and S r = 533 cm s w 3 = 303 cm v 3 From ( 1.1 ) : = = = 413 cm s From ( 1.2 ): = = = 110cm a v w v From ( 1.5 ): e = = =, that is the volume of voids is 77.5% that of s 533 solids. v 413 From ( 1.7 ): n = = = That is, the volume of voids is 43.7% e of the sample. or From ( 1.12 ) : n = = = e ( ) 303 S = = = 413 That is, water fills 73% of voids. w From 1.9 : r 0.73 v 303 ( ) S = = = n ( ) w or From 1.15 : r 0.73 The sample is partially saturated. Example 1.2 A sample of sand was taken from below the ground water table. The volumes measured were: = 1000 cm 3 Calculate: v, a, s, e and n w = 400 cm 3 Note: Assume sand samples taken from above the water table as partially saturated (s r < 1) and saturated (S r = 1) if taken from below. In this example, therefore, S r = 1 a = 0. From (1.8) w Sr = = 1 w = v (1.16) 3 = 400cm v

30 6 Introduction to Soil Mechanics From (1.2): From (1.1): From (1.5): From (1.7): a = v - w = = 0 The voids are full of water s = v = = 600cm is 67% of v e = = = v s s is 40% of 1000 v n = = = v Weight volume relations As the title implies, the formulae derived in this section take into account the weights of s and w. It is assumed that air is weightless. The weight volume relations are shown diagrammatically: a Air v w Water W w W s Solids W s Figure 1.6 Where : W = Weight of solids s W = Weight of water From Figure 1.6 W = W + W w s w W = Totalweight (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: W MN W M 3 = 9.81 mass ( ) = kn (1.18) Several important relationships are derived below in terms of mass, weight and volume. These are: r = bulk mass density g = bulk weight density (unit weight) r d = dry mass density g d = dry weight density r sat = saturated mass density g sat = saturated weight density