Critical State Soil Mechanics Andrew Schofield and Peter Wroth Lecturers in Engineering at Cambridge University Preface This book is about the mechanical properties of saturated remoulded soil. It is written at the level of understanding of a final-year undergraduate student of civil engineering; it should also be of direct interest to post-graduate students and to practising civil engineers who are concerned with testing soil specimens or designing works that involve soil. Our purpose is to focus attention on the critical state concept and demonstrate what we believe to be its importance in a proper understanding of the mechanical behaviour of soils. We have tried to achieve this by means of various simple mechanical models that represent (with varying degrees of accuracy) the laboratory behaviour of remoulded soils. We have not written a standard text on soil mechanics, and, as a consequence, we have purposely not considered partly saturated, structured, anisotropic, sensitive, or stabilized soil. We have not discussed dynamic, seismic, or damping properties of soils; we have deliberately omitted such topics as the prediction of settlement based on Boussinesq’s functions for elastic stress distributions as they are not directly relevant to our purpose. The material presented in this book is largely drawn from the courses of lectures and associated laboratory classes that we offered to our final year civil engineering undergraduates and advanced students in 1965/6 and 1966/7. Their courses also included material covered by standard textbooks such as Soil Mechanics in Engineering Practice by K. Terzaghi and R. B. Peck (Wiley 1948), Fundamentals of Soil Mechanics by D. W. Taylor (Wiley 1948) or Principles of Soil Mechanics by R. F. Scott (Addison-Wesley 1963). In order to create a proper background for the critical state concept we have felt it necessary to emphasize certain aspects of continuum mechanics related to stress and strain in chapter 2 and to develop the well-established theories of seepage and one-dimensional consolidation in chapters 3 and 4. We have discussed the theoretical treatment of these two topics only in relation to the routine experiments conducted in the laboratory by our students, where they obtained close experimental confirmation of the relevance of these theories to saturated remoulded soil samples. Modifications of these theories, application to field problems, three-dimensional consolidation, and consideration of secondary effects, etc., are beyond the scope of this book. In chapters 5 and 6, we develop two models for the yielding of soil as isotropic plastic materials. These models were given the names Granta-gravel and Cam-clay from that river that runs past our laboratory, which is called the Granta in its upper reaches and the Cam in its lower reaches. These names have the advantage that each relates to one specific artificial material with a certain distinct stress – strain character. Granta-gravel is an ideal rigid/plastic material leading directly to Cam-clay which is an ideal elastic/plastic material. It was not intended that Granta-gravel should be a model for the yielding of dense sand at some early stage of stressing before failure: at that stage, where Rowe’s concept of stress dilatancy offers a better interpretation of actual test data, the simple Granta-gravel model remains quite rigid. However, at peak stress, when Granta-gravel does yield, the model fits our purpose and it serves to introduce Taylor’s dilatancy calculation towards the end of chapter 5. Chapter 6 ends with a radical interpretation of the index tests that are widely used for soil classification, and chapter 7 includes a suggested computation of ‘triaxial’ test data that allows students to interpret much significant data which are neglected in normal methods of analysis. The remainder of chapter 7 and chapter 8 are devoted to testing the relevance of the two models, and to suggesting criteria based on the critical state concept for choice of strength parameters in design problems. Chapter 9 begins by drawing attention to the actual work of Coulomb – which is often inaccurately reported – and its development at Gothenberg; and then introduces Sokolovski’s calculations of two-dimensional fields of limiting stress into which we consider it appropriate to introduce critical state strength parameters. We conclude in chapter 10 by demonstrating the place that the critical state concept has in our understanding of the mechanical behaviour of soils. We wish to acknowledge the continual encouragement and very necessary support given by Professor Sir John Baker, O.B.E., Sc.D., F.R.S., of all the work in the soil mechanics group within his Department. We are very conscious that this book represents only part of the output of the research group that our teacher, colleague, and friend, Ken Roscoe, has built up over the past twenty years, and we owe him our unbounded gratitude. We are indebted to E. C. Hambly who kindly read the manuscript and made many valuable comments and criticisms, and we thank Mrs Holt-Smith for typing the manuscript and helping us in the final effort of completing this text. A. N. Schofield and C. P. Wroth To K. H. Roscoe Table of contents Contents Preface Glossary of Symbols Table of Conversions for S.L Units Chapter 1 Basic Concepts 1 1.1 Introduction 1 1.2 Sedimentation and Sieving in Determination of Particle Sizes 2 1.3 Index Tests 4 1.4 Soil Classification 5 1.5 Water Content and Density of Saturated Soil Specimen 7 1.6 The Effective Stress Concept 8 1.7 Some Effects that are ‘Mathematical’ rather than ‘Physical’ 10 1.8 The Critical State Concept 12 1.9 Summary 14 Chapter 2 Stresses, Strains, Elasticity, and Plasticity 16 2.1 Introduction 16 2.2 Stress 16 2.3 Stress-increment 18 2.4 Strain-increment 19 2.5 Scalars, Vectors, and Tensors 21 2.6 Spherical and Deviatoric Tensors 22 2.7 Two Elastic Constants for an Isotropic Continuum 23 2.8 Principal Stress Space 25 2.9 Two Alternative Yield Functions 28 2.10 The Plastic Potential Function and the Normality Condition 29 2.11 Isotropic Hardening and the Stability Criterion 30 2.12 Summary 32 Chapter 3 Seepage 34 3.1 Excess Pore-pressure 34 3.2 Hydraulic Gradient 35 3.3 Darcy’s Law 35 3.4 Three-dimensional Seepage 37 3.5 Two-dimensional Seepage 38 3.6 Seepage Under a Long Sheet Pile Wall: an Extended Example 39 3.7 Approximate Mathematical Solution for the Sheet Pile Wall 40 3.8 Control of Seepage 44 Chapter 4 One-dimensional Consolidation 46 4.1 Spring Analogy 46 4.2 Equilibrium States 49 4.3 Rate of Settlement 50 4.4 Approximate Solution for Consolidometer 52 4.5 Exact Solution for Consolidometer 55 4.6 The Consolidation Problem 57 Chapter 5 Granta-gravel 61 5.1 Introduction 61 5.2 A Simple Axial-test System 62 5.3 Probing 64 5.4 Stability and Instability 66 5.5 Stress, Stress-increment, and Strain-increment 68 5.6 Power 70 5.7 Power in Granta-gravel 71 5.8 Responses to Probes which cause Yield 72 5.9 Critical States 73 5.10 Yielding of Granta-gravel 74 5.11 Family of Yield Curves 76 5.12 Hardening and Softening 78 5.13 Comparison with Real Granular Materials 81 5.14 Taylor’s Results on Ottawa Sand 85 5.15 Undrained Tests 87 5.16 Summary 91 Chapter 6 Cam-clay and the Critical State Concept 93 6.1 Introduction 93 6.2 Power in Cam-clay 95 6.3 Plastic Volume Change 96 6.4 Critical States and Yielding of Cam-clay 97 6.5 Yield Curves and Stable-state Boundary Surface 98 6.6 Compression of Cam-clay 100 6.7 Undrained Tests on Cam-clay 102 6.8 The Critical State Model 104 6.9 Plastic Compressibility and the Index Tests 105 6.10 The Unconfined Compression Strength 111 6.11 Summary 114 Chapter 7 Interpretation of Data from Axial Tests on Saturated Clays 116 7.1 One Real Axial-test Apparatus 116 7.2 Test Procedure 118 7.3 Data Processing and Presentation 119 7.4 Interpretation of Data on the Plots of v versus ln p 120 7.5 Applied Loading Planes 123 7.6 Interpretation of Test Data in (p, v, q) Space 125 7.7 Interpretation of Shear Strain Data 127 7.8 Interpretation of Data of ε& and Derivation of Cam-clay Constants 130 7.9 Rendulic’s Generalized Principle of Effective Stress 135 7.10 Interpretation of Pore-pressure Changes 137 7.11 Summary 142 Chapter 8 Coulomb’s Failure Equation and the Choice of Strength Parameters 144 8.1 Coulomb’s Failure Equation 144 8.2 Hvorslev’s Experiments on the Strength of Clay at Failure 145 8.3 Principal Stress Ratio in Soil About to Fail 149 8.4 Data of States of Failure 152 8.5 A Failure Mechanism and the Residual Strength on Sliding Surfaces 154 8.6 Design Calculations 158 8.7 An Example of an Immediate Problem of Limiting Equilibrium 160 8.8 An Example of the Long-term Problem of Limiting Equilibrium 161 8.9 Summary 163 Chapter 9 Two-dimensional Fields of Limiting Stress 165 9.1 Coulomb’s Analysis of Active Pressure using a Plane Surface of Slip 165 9.2 Coulomb’s Analysis of Passive Pressure 167 9.3 Coulomb’s Friction Circle and its Development in Gothenberg 169 9.4 Stability due to Cohesion Alone 172 9.5 Discontinuity Conditions in a Limiting-stress Field 174 9.6 Discontinuous Limiting-stress Field Solutions to the Bearing Capacity Problem 180 9.7 Upper and Lower Bounds to a Plastic Collapse Load 186 9.8 Lateral Pressure of Horizontal Strata with Self Weight (γ>0, ρ>0) 188 9.9 The Basic Equations and their Characteristics for a Purely Cohesive Material 191 910 The General Numerical Solution 195 9.11 Sokolovski’s Shapes for Limiting Slope of a Cohesive Soil 197 9.12 Summary 199 Chapter 10 Conclusion 201 10.1 Scope 201 10.2 Granta-gravel Reviewed 201 10.3 Test Equipment 204 10.4 Soil Deformation and Flow 204 Appendix A 206 Appendix B 209 Appendix C 216 Glossary of symbols The list given below is not exhaustive, but includes all the most important symbols used in this book.