This book focuses on the unified solutions and analysis for the problems in soil mechanics based on the unified strength

*406*
*152*
*10MB*

*English*
*Pages 222
[223]*
*Year 2023*

*Table of contents : PrefaceContents1 Introduction 1.1 Backgrounds 1.2 Discussions of the Mohr–Coulomb Strength Theory 1.2.1 On the Intermediate Principal Stress 1.2.2 On the Stress Circle 1.2.3 On the Frication Angle 1.2.4 Comparison Between Strength and Deformation Obtained by Plane Strain and Triaxial Tests 1.2.5 Strength Index of Soils: Friction Angle φ (True Triaxial Test) 1.2.6 Shear Strength Index C 1.2.7 Peak Strength σ01 of Soils 1.2.8 On the Analysis Results of Structural 1.2.9 On Axisymmetric Triaxial Tests Method 1.3 New Theoretical Basis of Soil Strength 1.4 Soil Mechanics Based on Unified Strength Theory References2 Stress State and Element 2.1 Introduction 2.2 Stresses on the Oblique Plane 2.2.1 Stresses on the Oblique Plane 2.2.2 Principal Shear Stresses 2.2.3 Octahedral Shear Stress 2.3 Hexahedron, Octahedron, and Dodecahedron 2.3.1 Principal Stress Element 2.3.2 Isoclinal Octahedron Element 2.3.3 Single-Shear Element 2.3.4 Twin-Shear Element 2.3.5 Dodecahedral Principal Shear Stress Element 2.3.6 Twin-Shear Stress State and Twin-Shear Element 2.4 Stress Space 2.4.1 Relation Between ( σ1 ,σ2 ,σ3 ) and (x, y, z) 2.4.2 Relation Between ( σ1 ,σ2 ,σ3 ) and (ξ, r, θ) or 2.5 Summary Reference3 Stresses in Soil Masses 3.1 Introduction 3.2 Effective Stress of Soils 3.3 Self-weight Stress 3.4 Contact Pressure 3.4.1 Vertical Centric Load 3.4.2 Vertical Eccentric Load 3.5 Additional Stress in Soils 3.5.1 Additional Stress Due to Vertical Concentrated Load 3.5.2 Additional Stress Due to Distributed Load 3.5.3 Stress Analysis in Spatial Problems 3.6 Additional Stresses in Plane Strain State 3.6.1 Vertical Line Load 3.6.2 Vertical Uniform Strip Load 3.7 Stresses Induced by Uniform Circular Load 3.8 A Brief Introduction of Soil Elasto-plastic Stress Analysis References4 Strength Characteristics of Soil 4.1 Introduction 4.2 Strength Different Effect (SD Effect) 4.3 Shear Strength and Effect of Normal Stress 4.4 Normal Stress Effect of Twin-Shear Strength 4.5 Effect of Hydrostatic Stress 4.6 Effect of Intermediate Principal Stress 4.7 Effect of Stress Angle 4.8 Convexity and Inter- and Outer Boundary of the Failure Limit Surface of Soils References5 Yu Unified Strength Theory 5.1 Introduction 5.2 Voigt-Timoshenko Conundrum 5.3 Mechanical Model of the Yu Unified Strength Theory 5.4 Mathematical Modeling of the Yu Unified Strength Theory 5.5 Experimental Determination of Material Parameters 5.6 Mathematical Expression of the Yu Unified Strength Theory 5.7 Other Formulations of the Yu Unified Strength Theory 5.7.1 Yu Unified Strength Theory Cohesion C and Friction Angle Φ Expression 5.7.2 Unified Theoretical Expression of Strength with Positive Compressive Stress 5.8 Relation Among the Parameters of the UST 5.9 Special Cases of the UST for Different Parameter b 5.10 Special Cases of the UST by Varying Parameter a 5.11 Limit Loci of the UST by Varying Parameter b in the π-Plane 5.12 Limit Loci of the Yu Unified Strength Theory in Plane Stress State 5.13 Limit Surface of the Yu Unified Strength Theory 5.14 Limit Surfaces of the Yu Unified Strength Theory Drawed by Kolupave-Altenbach 5.15 Comparison Between the UST and Experimental Results 5.16 Significance of the Yu Unified Strength Theory 5.17 Summary References6 Compression and Settl 6.1 Introduction 6.2 Compression Test and Compression Index 6.2.1 Compression Test and Compression Curve 6.2.2 Compressibility Index 6.2.3 Rebound Curve and Recompression Curve 6.3 Influence of Stress History on Settlement 6.3.1 Stress History of Natural Soil Layers 6.3.2 Determination of Pre-consolidation Pressure 6.3.3 Final Settlement of Foundation Considering the Effect of Stress History 6.4 Relationship Between Deformation and Time 6.4.1 Seepage Deformation of Saturated Soils 6.4.2 Terzaghi’s One-Dimensional Consolidation Theory 6.4.3 One-Dimensional Consolidation Differential Equation 6.4.4 Calculation of Consolidation Degree 6.4.5 In Late-Stage Settlement Calculated by Settlement Observation Data 6.5 Settlement Analysis for the City Wall of East Gate in Xi’an 6.6 Analysis of Unidirectional Compression Consolidation in Saturated Soft-Soil 6.7 Analysis of Shear Consolidation in Saturated Soft-Soil 6.8 Consolidation Analysis of in Saturated Soft-Soil Foundation Under Uniform Load 6.9 Settlement Analysis of the Foundation of Big Goose Pagoda References7 Earth Pressure Theory 7.1 Introduction 7.2 Theoretical Solution of Rankine’s Earth Pressure 7.2.1 Theoretical Analysis Model 7.2.2 Formula Derivation When the Intermediate Principal Stress Is Larger 7.2.3 Formula Derivation When the Intermediate Principal Stress Is Smaller 7.3 Expression of UST with Shear Strength 7.4 Unified Solution of Theory on Sliding Wedge for Earth Pressure 7.5 Examples for Unified Solution of Rankine's Earth Pressure 7.5.1 Example 1 (Zhang J, Hu RL, et al.) 7.5.2 Example 2 (Zhang J, Hu RL, Yu WL, et al.) 7.5.3 Example 3 (Fan W, Shen ZJ, et al.) 7.5.4 Example 4 (Yuan JL) 7.5.5 Example 5 7.5.6 Example 6 (Ying J, Liao HJ, et al.) 7.6 Study on J. Karstedt Space Earth Pressure of Reinforced Retaining Wall Based on the UST 7.7 Unified Solution of Space Earth Pressure Computing Theory 7.8 Summary References8 Ultimate Bearing Capacity of Strip Footings 8.1 Introduction 8.2 Bearing Capacity of Strip Footings 8.2.1 Formula of Ultimate Bearing Capacity 8.2.2 The Influence of Various Parameters on Ultimate Bearing Capacity 8.3 Ultimate Bearing Capacity of Footings Caused by Cohesion and Overloading 8.3.1 Formula Deduction of Ultimate Bearing Capacity of Foundation 8.3.2 The Influence of Parameters on the Ultimate Bearing Capacity 8.4 Examples 8.4.1 Compared with the Calculated Results of Terzaghi Formula 8.4.2 Comparison Between Unified Strength Formula and Meyerhof Formula 8.5 Slip Line Unified Solution of Ultimate Bearing Capacity for Strip Footings 8.6 Summary References9 Slope Stability 9.1 Introduction 9.2 Theoretical Derivation of Bearing Capacity 9.3 Examples 9.4 Unified Solution of Bearing Capacity of a Trapezoid Structure When It Is Subjected to Uniform Load 9.5 Slip Line Unified Solution of Trapezoidal Structures 9.6 Unified Solution of Slope Bearing Capacity When It Is Subjected to Uniformly Distributed Load 9.7 Slip Line Unified Solution of Slope References*

Mao-Hong Yu

Soil Mechanics New Concept and Theory

Soil Mechanics

Mao-Hong Yu

Soil Mechanics New Concept and Theory

Mao-Hong Yu School of Aerospace Xi’an Jiaotong University Xi’an, Shanxi, China

ISBN 978-981-99-2780-7 ISBN 978-981-99-2781-4 (eBook) https://doi.org/10.1007/978-981-99-2781-4 Jointly published with Zhejiang University Press. The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Zhejiang University Press. © Springer Nature Singapore Pte Ltd. and Zhejiang University Press 2023 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Professor Karl Terzaghi published his book Erdbaumechanik in 1925 which is acknowledged as the foundation of soil mechanics. Nowadays, soil mechanics has already been a required course for most civil, hydraulic, and geotechnical engineering undergraduate students. One of the most important concepts in soil mechanics is the Mohr-Coulomb strength theory: σ1 − ασ3 = σt , in which σ1 and σ3 are, respectively, the first and third principal stress, σt the strength in tension, and α = σt /σc the tension-compression strength ratio. It was clear that there are three principal stresses, i.e., σ1 , σ2 , and σ3 , acted on the soil mass. The Mohr-Coulomb strength theory, however, does not take into account the influence of the intermediate principal stress. There have been a number of attempts on developing modified failure criteria to include the effect of the intermediate principal stress. Most of these failure criteria are nonlinear equations and can hardly be used in the analytical analysis of the practical engineering problems. In most monographs in this field, the analytical solutions are still based on the MohrCoulomb strength theory. The traditional soil mechanics is consequently considered as the soil mechanics without considering the effect of the intermediate principal stress. The Soil Mechanics: New Concept and Theory is the first part of the Trilogy of Geomechanics. The main difference between the failure criterion adopted in this book and in the traditional soil mechanics lies only in two variables, i.e., the intermediate principal stress σ2 and the parameter b which reflects the influence of the intermediate principal stress on the failure of materials. These two variables are not included in the failure criterion of the traditional soil mechanics. As these two variables are added by a linear expression, readers will not encounter the mathematical difficulties in understanding the new failure criterion. A Unified Strength Theory which can be used for most kinds of materials had been considered impossible. In 1901, Prof. Voigt at Göttingen University, Germany, concluded that it is impossible to formulate a single strength criterion which can be applied to various materials. In 1953, Prof. Timoshenko at Stanford University, USA, v

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also repeated this conclusion as “Voigt came to the conclusion that the question of strength is too complicated and that it is impossible to devise a single theory for successful application to all kinds of structural materials.” In 1985, the same thought was expressed in Encyclopedia of China (1985) as “it is impossible to establish a Unified Strength Theory for various materials.” It can be referred to as “VoigtTimoshenko” conundrum. The solution to the “Voigt-Timoshenko Conundrum” is the Unified Strength Theory. The development of the UST can be summarized as three failure equations: 1. Twin-shear stress yield criterion 1 σ1 + σ3 F = σ1 − (σ2 + σ3 ) = σt , when σ2 ≤ 2 2 F' =

1 σ1 + σ3 (σ1 + σ2 ) − σ3 = σt , when σ2 ≥ 2 2

2. Generalized twin-shear strength theory F = σ1 − F' =

α σ1 + σ3 (σ2 + σ3 ) = σt , when σ2 ≤ 2 2

1 σ1 + σ3 (σ1 + σ2 ) − ασ3 = σt , when σ2 ≥ 2 2

3. Unified Strength Theory F = σ1 − F' =

α σ1 + ασ3 (bσ2 + σ3 ) = σt , when σ2 ≤ 1+b 1+α

1 σ1 + ασ3 (σ1 + bσ2 ) − ασ3 = σt , when σ2 ≥ 1+b 1+α

It can be found that there are three stages in the development of the Unified Strength Theory. The Unified Strength Theory is easy to understand and simple to use. Since the Unified Strength Theory was proposed in 1991, it has been extensively applied to many fields. I am immensely grateful to Xi’an Jiaotong University Alumni Association of Hong Kong, Xi’an Jiaotong University Alumni Association of Civil Engineering Department, and State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, China, for their support. I would also like to express my sincere thanks to my research assistants Xia-Xia Wu and Jia-Yu Liang for the help of writing this book. Xi’an, China Spring 2022

Mao-Hong Yu

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Discussions of the Mohr–Coulomb Strength Theory . . . . . . . . . . . . 1.2.1 On the Intermediate Principal Stress . . . . . . . . . . . . . . . . . . . 1.2.2 On the Stress Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 On the Frication Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Comparison Between Strength and Deformation Obtained by Plane Strain and Triaxial Tests . . . . . . . . . . . . . 1.2.5 Strength Index of Soils: Friction Angle ϕ (True Triaxial Test) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Shear Strength Index C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Peak Strength σ 0 1 of Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.8 On the Analysis Results of Structural . . . . . . . . . . . . . . . . . . 1.2.9 On Axisymmetric Triaxial Tests Method . . . . . . . . . . . . . . . 1.3 New Theoretical Basis of Soil Strength . . . . . . . . . . . . . . . . . . . . . . . 1.4 Soil Mechanics Based on Unified Strength Theory . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 5 5 7

10 10 11 12 15 19 20 21

2 Stress State and Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stresses on the Oblique Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Stresses on the Oblique Plane . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Principal Shear Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Octahedral Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hexahedron, Octahedron, and Dodecahedron . . . . . . . . . . . . . . . . . . 2.3.1 Principal Stress Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Isoclinal Octahedron Element . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Single-Shear Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Twin-Shear Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Dodecahedral Principal Shear Stress Element . . . . . . . . . . . 2.3.6 Twin-Shear Stress State and Twin-Shear Element . . . . . . . .

23 23 24 24 25 25 26 28 28 29 30 30 30

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2.4

Stress Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Relation Between (σ1 , σ2 , σ3 ) and (x, y, z) . . . . . . . . . . . . . . 2.4.2 Relation Between (σ1 , σ2 , σ3 ) and (ξ, r, θ ) or . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 33 35 36 36

3 Stresses in Soil Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Effective Stress of Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Self-weight Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Contact Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Vertical Centric Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Vertical Eccentric Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Additional Stress in Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Additional Stress Due to Vertical Concentrated Load . . . . . 3.5.2 Additional Stress Due to Distributed Load . . . . . . . . . . . . . . 3.5.3 Stress Analysis in Spatial Problems . . . . . . . . . . . . . . . . . . . . 3.6 Additional Stresses in Plane Strain State . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Vertical Line Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Vertical Uniform Strip Load . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Stresses Induced by Uniform Circular Load . . . . . . . . . . . . . . . . . . . 3.8 A Brief Introduction of Soil Elasto-plastic Stress Analysis . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 39 40 42 42 42 44 44 46 47 49 49 50 54 56 57

4 Strength Characteristics of Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Strength Different Effect (SD Effect) . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Shear Strength and Effect of Normal Stress . . . . . . . . . . . . . . . . . . . 4.4 Normal Stress Effect of Twin-Shear Strength . . . . . . . . . . . . . . . . . . 4.5 Effect of Hydrostatic Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Effect of Intermediate Principal Stress . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Effect of Stress Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Convexity and Inter- and Outer Boundary of the Failure Limit Surface of Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 61 61 63 63 65 66

5 Yu Unified Strength Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Voigt-Timoshenko Conundrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Mechanical Model of the Yu Unified Strength Theory . . . . . . . . . . 5.4 Mathematical Modeling of the Yu Unified Strength Theory . . . . . . 5.5 Experimental Determination of Material Parameters . . . . . . . . . . . . 5.6 Mathematical Expression of the Yu Unified Strength Theory . . . . . 5.7 Other Formulations of the Yu Unified Strength Theory . . . . . . . . . .

79 79 80 81 83 84 84 85

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5.7.1 Yu Unified Strength Theory Cohesion C and Friction Angle ϕ Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.7.2 Unified Theoretical Expression of Strength with Positive Compressive Stress . . . . . . . . . . . . . . . . . . . . . . 85 5.8 Relation Among the Parameters of the UST . . . . . . . . . . . . . . . . . . . 86 5.9 Special Cases of the UST for Different Parameter b . . . . . . . . . . . . 87 5.10 Special Cases of the UST by Varying Parameter a . . . . . . . . . . . . . . 88 5.11 Limit Loci of the UST by Varying Parameter b in the π-Plane . . . 89 5.12 Limit Loci of the Yu Unified Strength Theory in Plane Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.13 Limit Surface of the Yu Unified Strength Theory . . . . . . . . . . . . . . . 94 5.14 Limit Surfaces of the Yu Unified Strength Theory Drawed by Kolupave-Altenbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.15 Comparison Between the UST and Experimental Results . . . . . . . . 99 5.16 Significance of the Yu Unified Strength Theory . . . . . . . . . . . . . . . . 101 5.17 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6 Compression and Settl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Compression Test and Compression Index . . . . . . . . . . . . . . . . . . . . 6.2.1 Compression Test and Compression Curve . . . . . . . . . . . . . 6.2.2 Compressibility Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Rebound Curve and Recompression Curve . . . . . . . . . . . . . 6.3 Influence of Stress History on Settlement . . . . . . . . . . . . . . . . . . . . . 6.3.1 Stress History of Natural Soil Layers . . . . . . . . . . . . . . . . . . 6.3.2 Determination of Pre-consolidation Pressure . . . . . . . . . . . . 6.3.3 Final Settlement of Foundation Considering the Effect of Stress History . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Relationship Between Deformation and Time . . . . . . . . . . . . . . . . . . 6.4.1 Seepage Deformation of Saturated Soils . . . . . . . . . . . . . . . . 6.4.2 Terzaghi’s One-Dimensional Consolidation Theory . . . . . . 6.4.3 One-Dimensional Consolidation Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Calculation of Consolidation Degree . . . . . . . . . . . . . . . . . . . 6.4.5 In Late-Stage Settlement Calculated by Settlement Observation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Settlement Analysis for the City Wall of East Gate in Xi’an . . . . . 6.6 Analysis of Unidirectional Compression Consolidation in Saturated Soft-Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Analysis of Shear Consolidation in Saturated Soft-Soil . . . . . . . . . . 6.8 Consolidation Analysis of in Saturated Soft-Soil Foundation Under Uniform Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Settlement Analysis of the Foundation of Big Goose Pagoda . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 108 108 110 112 113 113 113 114 117 117 119 119 121 122 125 130 131 131 133 138

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7 Earth Pressure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Theoretical Solution of Rankine’s Earth Pressure . . . . . . . . . . . . . . . 7.2.1 Theoretical Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Formula Derivation When the Intermediate Principal Stress Is Larger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Formula Derivation When the Intermediate Principal Stress Is Smaller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Expression of UST with Shear Strength . . . . . . . . . . . . . . . . . . . . . . . 7.4 Unified Solution of Theory on Sliding Wedge for Earth Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Examples for Unified Solution of Rankine’s Earth Pressure . . . . . . 7.5.1 Example 1 (Zhang J, Hu RL, et al.) . . . . . . . . . . . . . . . . . . . . 7.5.2 Example 2 (Zhang J, Hu RL, Yu WL, et al.) . . . . . . . . . . . . . 7.5.3 Example 3 (Fan W, Shen ZJ, et al.) . . . . . . . . . . . . . . . . . . . . 7.5.4 Example 4 (Yuan JL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.5 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.6 Example 6 (Ying J, Liao HJ, et al.) . . . . . . . . . . . . . . . . . . . . 7.6 Study on J. Karstedt Space Earth Pressure of Reinforced Retaining Wall Based on the UST . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Unified Solution of Space Earth Pressure Computing Theory . . . . 7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 143 144

8 Ultimate Bearing Capacity of Strip Footings . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Bearing Capacity of Strip Footings . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Formula of Ultimate Bearing Capacity . . . . . . . . . . . . . . . . . 8.2.2 The Influence of Various Parameters on Ultimate Bearing Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Ultimate Bearing Capacity of Footings Caused by Cohesion and Overloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Formula Deduction of Ultimate Bearing Capacity of Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 The Influence of Parameters on the Ultimate Bearing Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Compared with the Calculated Results of Terzaghi Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Comparison Between Unified Strength Formula and Meyerhof Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Slip Line Unified Solution of Ultimate Bearing Capacity for Strip Footings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 169 171 172

146 148 150 152 155 155 157 158 159 160 161 162 163 166 167

176 182 183 191 194 194 195 195 196 198

Contents

9 Slope Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Theoretical Derivation of Bearing Capacity . . . . . . . . . . . . . . . . . . . 9.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Unified Solution of Bearing Capacity of a Trapezoid Structure When It Is Subjected to Uniform Load . . . . . . . . . . . . . . . 9.5 Slip Line Unified Solution of Trapezoidal Structures . . . . . . . . . . . . 9.6 Unified Solution of Slope Bearing Capacity When It Is Subjected to Uniformly Distributed Load . . . . . . . . . . . . . . . . . . . . . 9.7 Slip Line Unified Solution of Slope . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

199 199 202 204 207 209 211 212 214

Chapter 1

Introduction

1.1 Backgrounds Soil mechanics is a branch of engineering that describes the behavior of soils. The research object of soil mechanics is the soil on the earth’s surface, including large stone, clay, silt, sand, gravel soil, soft and highly compressed peat organic sediments, and so on. In the field of construction, the soil is the most widely distributed and the most complex engineering material on earth. Most kinds of structures in civil engineering are built on the soil, except for a few built directly on the rock. Generally, the soil is composed of three phases: solid, air, and water, as illustrated in Fig. 1.1. The problems related to soil mechanics have already existed in ancient times. One of the most famous examples related to the soil bearing capacity and foundations is the Leaning Tower of Pisa in Italy, as shown in Fig. 1.2a. It is the third-oldest structure in Pisa’s Cathedral Square. Construction of the tower started in 1173 A.D. and was completed in 1370 A.D. The tower has also been used for experiments by the famous Italian scientist called Galileo Galilei who threw two balls of different masses from the tower to prove that their descending time was not related to their masses. Since 1174 A.D., the tower has been settling differentially for over 800 years. By 1990, the tilt had reached 5.5 degrees. Figure 1.2b shows the foundation soil layer of the Leaning Tower of Pisa. The tilt problem is still an issue that scientists are constantly exploring. There are also many leaning towers in China, of which the largest one is the Care Pearl pylon, as shown in Fig. 1.3a. The pylon is a masonry-wood mixed structure with seven layers and eight corners, which was built in the Northern Song Dynasty (1079). It is 18.82 m high and is located in Songjiang District, Shanghai. The Care Pearl pylon has tilted to the east by 2.28 m and has a tilt of 7.1 degrees, of which the inclination has surpassed that of the famous Leaning Tower of Pisa in Italy. Figure 1.3b is a structure diagram of the Care Pearl pylon. It can be found that the pylon is constructed on a hillside from the structural diagram, and the uneven settlement is caused by the different depths of foundation soils (Fig. 1.4).

© Springer Nature Singapore Pte Ltd. and Zhejiang University Press 2023 M.-H. Yu, Soil Mechanics, https://doi.org/10.1007/978-981-99-2781-4_1

1

2

1 Introduction

Fig. 1.1 Three components of soil

Fig. 1.2 Learning Tower of Pisa, Italy. a Leaning Tower of Pisa (www.towerofpisa.org), b structure diagram (Krynine 1941)

The stability analysis of the foundations of Chinese ancient structures is also an interesting topic. The Big Goose Pagoda is one of the most famous Buddhist pagodas in China. It has seven stories and is 64.7 m in height, as shown in Fig. 1.5. From the 1980s to the 1990s, Yu carried out a series of situ experiments on the vibration characteristics of the Big Goose Pagoda and Small Goose Pagoda. The

1.1 Backgrounds

3

Fig. 1.3 Care Pearl Tower at Shanghai. a Care Pearl Tower, b structure diagram Fig. 1.4 Big Goose Pagoda

bearing capacity of the foundations of these structures was also analyzed. The detailed discussion on the foundation of the Big Goose Pagoda will be found in Chap. 6. Some geological disasters are also related to soil mechanics. In 2005, a major landslide occurred after the culmination of an extremely wet two-week period in La Conchita, California (Horton and Mancini, 2008). It can be found that the construction of roads has destroyed the stress equilibrium system of the slope, as shown in Fig. 1.5a. Gu-Zhao Highway (see Fig. 1.5b), which is widely regarded as “the most

4

1 Introduction

Fig. 1.5 Landslide La Conchita and Guzhao Highway. a Landslide La Conchita (Horton and Mancini 2008), b Guzhao Highway, China

beautiful overwater highway,” is an amazing route linking Gufu Town and Zhaojun Bridge in Xingshan County of China’s Hubei province. This scheme not only avoids deforestation and protects the environment, but also keeps the original stress balance of the slope. It is found that engineers have to deal with many challenges in soil mechanics. There are three typical engineering problems in soil mechanics, i.e., bearing capacity of soil foundations, stability of retaining walls, and stability of slopes, as illustrated in Fig. 1.6.

Fig. 1.6 Typical engineering problems of soil mechanics

1.2 Discussions of the Mohr–Coulomb Strength Theory

5

1.2 Discussions of the Mohr–Coulomb Strength Theory With the rise of the industrial revolution in the nineteenth century, the construction of large cities, water conservancy, roads, and railways has encountered many problems related to soil mechanics. Based on the experiment results, Coulomb proposed the shear strength of soil and developed the wedge theory for the determination of lateral earth pressure on a retaining wall. He is considered to be the ancestor of soil mechanics. After that, Mohr (1900) developed a generalized form of the theory for the failure of materials, which is now referred to as the Mohr–Coulomb strength theory. The Mohr–Coulomb strength theory is still by far the most widely used model in most soil mechanics textbooks. The traditional soil mechanics can also be referred to as the soil mechanics based on the Mohr–Coulomb strength theory. Some limitations of the Mohr–Coulomb strength theory, however, will be discussed in this section.

1.2.1 On the Intermediate Principal Stress In most applications in geotechnical engineering, such as the foundations, slopes, and retaining walls, the soil is under a three-dimensional stress state. It is demonstrated that there exist three planes of zero shear stresses. These planes are mutually perpendicular, and the normal stresses on these planes have maximum or minimum values. These normal stresses are referred to as principal stresses. The maximum principal stress, intermediate principal stress, and the minimum principal stress can be denoted as σ1 , σ2 , σ3 , respectively. The Mohr–Coulomb strength theory can be expressed in terms of principal stresses as 1 1 (σ1 − σ3 ) − (σ1 + σ3 ) sin ϕ = c cos ϕ, 2 2

(1.1)

where ϕ denotes the internal friction angle and c the cohesion. It is found that only two principal stresses, i.e., maximum and minimum principal stresses, are taken into account in the Mohr–Coulomb strength theory. The influence of the intermediate principal stress on the failure of material is ignored.

1.2.2 On the Stress Circle The Mohr–Coulomb strength theory can be expressed in terms of maximum shear stress τ13 and the corresponding normal stress σ13 acting on the same plane as τ13 − σ13 sin ϕ = c cos ϕ

(1.2)

6

1 Introduction

Fig. 1.7 Mohr stress circle

or in terms of shear stress τ and normal stress σ in the triaxial stress state as τ − σ tan ϕ = c.

(1.3)

The Mohr–Coulomb strength theory only takes the maximum shear stress τ13 and the corresponding normal stress σ13 acting on the same plane into account in the material failure. In engineering practice, the material strength test is based on the Mohr stress circle, as shown in Fig. 1.7. In a strength test, the confining pressure σ 3 is applied to the specimen and remains constant, the vertical pressure σ 1 increases at a certain rate until the specimen fails. The Mohr stress circle can be obtained as the small circle A at the initial state, and the ultimate stress circle of the material can be finally obtained as the circle B in Fig. 1.7. It is found that the circle C in Fig. 1.7 cannot be obtained in the test. The strength test and Mohr stress circle are very effective for understanding the Mohr–Coulomb strength theory. However, it is found that the Mohr–Coulomb strength theory is only related to the maximum principal stress σ 1 and the minimum principal stress σ 3 , or only related to the maximum shear stress τ13 and the maximum normal stress σ13 , as shown in Fig. 1.8. The intermediate principal stress σ 2 is not taken into account. In general, there are three principal shear stresses and three stress circles in any stress element. The maximum shear stress circle and other two small stress circles are shown in Fig. 1.9. It is found that the relationship between the two small stress circles is uncertain, as shown in Fig. 1.9. All the three-shear stresses should have an influence on the failure of the material.

1.2 Discussions of the Mohr–Coulomb Strength Theory

7

Fig. 1.8 Limit stress circle and strength envelope

Fig. 1.9 Two small stress circles and their variations. a τ 12 ≤ τ 23 , b τ 12 ≥ τ 23

1.2.3 On the Frication Angle The material parameters can be determined by the triaxial test or the plane strain test. It is found that the friction angle ϕ obtained by the plane strain test is different from that by the triaxial test. Kjellman (1936) has pointed out that the results obtained by plane strain test was 8° higher than those by the triaxial confining test. In Table 1.1, a comparison between the plane strain and triaxial tests on sand was given by Lee (1970). It is obvious that the friction angle obtained by the plane strain test (ϕ ps ) and triaxial confining test (ϕ tri ) is different.

8

1 Introduction

Table 1.1 Comparison of plane strain and triaxial tests (Lee 1970) Soil

ϕ ps − ϕ tri (°)

Plane strain apparatus

References

Sand

+8

Cube-vary all three stresses as desired

Kjellman (1936)

Dense sand

+4

Loose sand

−1

Direct shear; critical void radio is highly in plane strain

Sand

+5

Direct shear

Dense sand

+4

Direct shear

Loose sand

−2

Direct shear

Sand, gravel, lead shot

+ 2 to + 7 Direct shear

Sand

+8

Direct shear

Sand

+2

Hollow cylinder

Compacted clay

+ 2 to + 4 Plane strain apparatus

Dense sand

+4

Plane strain apparatus

Loose sand

+0

Drained test

Sand

+ 4 to + 5 Active earth pressure on model retaining wall

Ottawa sand

+ 2 to + 5 Bearing capacity of model strip footings

Sand

+ 3 to + 4 Vacuum compression on long rectangular specimens

Ottawa sand

+6

Hollow cylinders

Ottawa sand

+5

Hollow cylinders failed by increasing outside

Ottawa sand

− 4 to − 6 Torsion tests on very thin annular rings of soil

Compacted clay

+ 2 to + 4 Rectangular plane strain apparatus

Glass spheres

ϕ ps > ϕ tri

Rectangular plane strain apparatus

Dense sand

+5

Loose sand

+3

Bishop plane strain apparatus; plane strain gives lower strain to failure

Saturated NC silty clay

+ 3.5

Cornforth (1964)

Plane strain apparatus (continued)

1.2.4 Comparison Between Strength and Deformation Obtained by Plane Strain and Triaxial Tests In soil mechanics and soil tests, the conventional triaxial test (axisymmetric triaxial test) is generally used for determining of friction angle of soil. However, Cornforth (1964) pointed out that the friction angle ϕ ps obtained a plane strain test which is

1.2 Discussions of the Mohr–Coulomb Strength Theory

9

Table 1.1 (continued) Soil

ϕ ps − ϕ tri (°)

Plane strain apparatus

References

Saturated clay

+1

Plane strain apparatus

Henkel, Wade (1966)

Ottawa dense sand

+3

Vacuum plane strain and vacuum triaxial

Ottawa loose sand

+1

Monterey dense sand

+3

Monterey loose sand

+ 0.5

Dense fine sand low pressure

+2

Dense fine sand elevated pressure

+0

Direct shear

Lee (1970)

generally greater than that by the conventional triaxial test ϕ tri . Henkel and Wade (1966), Hussaini (1973), as well as Vaid and Campanella (1974) conducted a series of plane strain tests and triaxial tests and obtained almost the same results. The differences between the results of plane strain and conventional triaxial tests on sand obtained by Cornforth (1964) and Al-Hussaini (1973) are illustrated in Fig. 1.10.

Fig. 1.10 Change curve of friction angle under different tests. a Cornforth (1964), b Al-Hussaini (1973)

10

1 Introduction

1.2.5 Strength Index of Soils: Friction Angle ϕ (True Triaxial Test) A special triaxial stress state can be produced, i.e., the stress state of σ 1 ≥ σ 2 = σ 3 or σ 1 = σ 2 ≥ σ 3 for axisymmetric triaxial experiments can be obtained. Although a triaxial stress state, i.e., σ 1 ≥ σ 2 /= σ 3 can be produced by plane strain experiments, the size of the intermediate principal stress should not be changed arbitrarily. The value of intermediate principal stress produced by the plane strain experiment is σ 2 ≤ (σ 1 + σ 3 )/2. In order to study the strength of soil under other stress conditions, great efforts were dedicated to the development of true triaxial testing facilities, which then were used to test engineering materials. Figure 1.11a is the variation of internal friction angle of the two kinds of sand obtained by Proster and Benden (1969) using the true triaxial test, in which the ordinate is the friction angle and the abscissa is the stress state parameter τμ = τ23 /τ13 = (σ2 − σ3 )/(σ1 − σ3 ). It can be seen that the friction angle of soil under different stress state is larger than that get by the Mohr–Coulomb theory.. When the stress state parameter is equal to τ23 /τ13 = 0, the intermediate principal stress is σ2 = σ3 , that is the stress state in the axisymmetric triaxial test. The stress state parameter τ23 /τ13 increases with the increase of the intermediate principal stress; when the stress state parameter is equal to τ23 /τ13 = 0.5, the intermediate principal stress is σ2 = (σ1 + σ3 )/2 ; when the stress state parameter is equal to τ23 /τ13 = 1 , the intermediate principal stress is σ2 = σ1 ; the results predicted by Mohr–Coulomb strength theory are its corresponding to the horizontal line. From result, it can be noticed that the friction angle obtained in true triaxial test is greater than that in axisymmetric triaxial test, except in case of τ23 /τ13 = 0 (equivalent to axisymmetric triaxial test). The friction angle obtained by Proster and Benden is 5° greater than that calculated by the Mohr–Coulomb strength theory. It is obvious that the Mohr–Coulomb theory does not agree well with the experimental results.

1.2.6 Shear Strength Index C In the previous section, we analyze the difference of friction angle between the conventional triaxial and plane strain tests. Mohr–Coulomb theory cannot account for this difference. Not only that, the test results also show that the difference between the shear strength index Ctri or (σ1 − σ3 )tri in conventional triaxial test and the shear strength index Cps or (σ1 − σ3 )ps is obtained by the plane strain test. Combined with project practice and the core wall of earth dam in Xiaolangdi, a series of consolidated drained conventional triaxial and plane strain tests were carried out. The experimental results of stress–strain relationship of soil at three levels of confining pressures of 100, 200, and 400 kPa are given by Fig. 1.12a. The comparative data of shear strength of sand are obtained. It can be found that the shear strength of soil in plane strain test is larger than that measured by the conventional

1.2 Discussions of the Mohr–Coulomb Strength Theory

11

Fig. 1.11 Results of sand obtained by true triaxial test. 1. Proctor and Barden (dense sand, 1969); 2. Sutherl and Mesdary (dense sand,1969); 3. Sutherl and Mesdary (loose sand, 1969); 4. Lade (dense sand, 1972); 5. Lade (Loose sand, 1972); 6. Lomiza (sand, 1969); 7. Rawat and Ramamurty (dense sand, l973); 8. A1-Ani, Quasi (dense sand, l975); 9. Ergun (dense sand, l976); 10. Ergun (Loose sand, l976)

triaxial test, and the greater the confining pressure is, the higher the shear strength is. Comparison results of shear strength of sand measured by this two different tests are shown in Fig. 1.12b. It is obvious that the experimental results are inconsistent with the Mohr–Coulomb theory. When the material enters the plastic state, the axial strain of the peak value in the axisymmetric triaxial test is much larger than that obtained by plane strain test under the same load condition, as given in Table 1.2 and Fig. 1.12a.

1.2.7 Peak Strength σ 0 1 of Soils Test results indicate that the difference of the internal friction angle or shear strength of soil is obtained by conventional triaxial test and plane strain test. Similarly, the difference of peak strength obtained by conventional triaxial and plane strain tests can be observed. Figure 1.13 shows the test results of middle dense sand and cement soil obtained by Li (1982) and Song and Xu (2011), respectively. It can be seen that the peak strength under the plane strain test is obviously larger than that measured by the conventional triaxial test. A systematic study of rockfill materials was carried out by the Yangtze River Academy of Science and Southeast University, and the theoretical and experimental

12

1 Introduction

Fig. 1.12 Comparison of triaxial and plane strain tests on a clay and b sand

results of the peak strength of three kinds of rockfill materials between plane strain and conventional triaxial tests were compared (Shi and Cheng 2011). The regularity of the three kinds of rockfill is consistent. Figure 1.14 shows the relationship of radio of peak strength to confining pressure (σ1 /σ3 ) and confining pressure σ3 of rockfill material between theoretical and test results. It is found that the ratio of peak strength to confining pressure (σ1 /σ3 )ps under the plane strain test is significantly larger than that measured by the conventional triaxial test (σ1 /σ3 )tri . At the same time, the strength of the materials with different tensile and compressive strengths was calculated under the complex stress condition. The theoretical and experimental results are shown in Fig. 1.14.

1.2.8 On the Analysis Results of Structural Figure 1.15 shows a plane stress problem of the trapezoidal structure, the upper part is subjected to uniform load q, and the ultimate load of trapezoidal structure is obtained by using different strength theory, as shown in Fig. 1.15b. The q0 is the ultimate load obtained by Mohr–Coulomb strength theory; q is the ultimate load of various strength theory; b is the parameter of strength theory; α = σt /σc is the tension–compression strength ratio. The result is the Mohr–Coulomb theory when b = 0. It can be seen that the results obtained by Mohr–Coulomb theory cannot reflect the different tensile and compressive strength of materials. The result is obviously unreasonable.

Medium K 0 sand

Loose sand

Mihe sand

Cornforth (1964)

Cornforth (1964)

Green (1972)

K0 σ3 = 1

K0 σ3 = 1

Lee (1970) Loose sand

Lee (1970) Dense sand

Isotropic

K0

Medium K 0 dense sand

Cornforth (1964)

K0

Dense sand

Peak strength ϕ (°) Axial strain at peak value under the same load εf , %

3.41

3.00

7.40

6.65

7.87

9.6

11.1 3.00 (31.3%)

2.83 (25.6%)

5.75 2.34 (69%)

3.95 0.95 (32%0

9.46 2.06 (28%)

7.44 0.79 (12%)

9.05 1.18 (15%)

12.6

13.9

40.0º

38.0º

39.0º

33.5º

35.9º

39.0º

41.7º

48.0º 8.0º (20%)

45.0º 7.0º (18.4%)

44.0º 5.0º (12.8%)

34.0º 0.5º (1.5%)

38.0º 2.1º (5.9%)

43.7º 4.7º (12%)

46.0º 4.3º (10.3%)

7.11

14.7

6.36

11.1

6.82

4.00

3.55

3.33

5.30

3.56

3.30

2.07

1.51

1.32

(continued)

3.78 (53.2%)

9.40 (63.9%)

2.80 (44%)

7.76 (69.9%)

4.75 (69.6%)

2.49 (62.2%)

2.23 (62.8%)

Axisymmetric Plane Difference Axisymmetric Plane Difference Axisymmetric Plane Difference triaxial strain (increase%) triaxial strain (increase%) triaxial strain (reduce %)

Consolidation (σ 1 -σ 3 )max Kg/cm2 mode

Cornforth (1964)

References Soil

Table 1.2 Difference of strength and deformation in plane strain test of sand relative to triaxial compression

1.2 Discussions of the Mohr–Coulomb Strength Theory 13

Peak strength ϕ (°) Axial strain at peak value under the same load εf , %

10.6

2.89

8.06

13.24

K0

Dense fine sand

Medium Isotropic σ 3 chengde = 1 sand

Medium Isotropic σ 3 chengde = 3 sand

Medium Isotropic σ 3 chengde = 5 sand

Wade (1963)

Tsinghua University Li Shuqin (1982)

13.8

K0 σ3 = 5 1.3 (12.3%)

0.80 (5.8%)

17.74 4.5 (34%)

10.93 2.87 (35.6%)

3.89 1.0 (34.6%)

11.9

14.6

34.7º

35.0º

36.3º

40.7º

35.4º

39.8º 5.1º (14.7%)

40.2º 5.2º (14.9%)

41.3º 5º (13.8%)

42.7º 2.0º (4.9%)

36.4º 1.0º (2.8%)

6.06

5.30

4.04

4.00

21.7

3.59

2.91

2.03

1.87

9.60

2.47 (40.8%)

2.39 (45.1%)

2.01 (49.8%)

2.13 (53.3%)

12.10 (55.8%)

Axisymmetric Plane Difference Axisymmetric Plane Difference Axisymmetric Plane Difference triaxial strain (increase%) triaxial strain (increase%) triaxial strain (reduce %)

Consolidation (σ 1 -σ 3 )max Kg/cm2 mode

Lee (1970) Dense sand

References Soil

Table 1.2 (continued)

14 1 Introduction

1.2 Discussions of the Mohr–Coulomb Strength Theory

15

Fig. 1.13 Comparison of peak strength on a sand and b cement soil

Fig. 1.14 Relations of (σ1 /σ3 ) ∼ σ3 between theoretical and test results (Shi 2011). a Misong, b Shuibuya

Hence, the Mohr–Coulomb theory is inconsistent with the test results of soil material, and applying this theory to soil structure may also lead to unreasonable results.

1.2.9 On Axisymmetric Triaxial Tests Method In geotechnical engineering, the triaxial experiment is one of the most reliable methods available for determining shear strength parameters. Triaxial compression

16

1 Introduction

Fig. 1.15 Plane stress problem: the ultimate load of trapezoid structure

apparatus is composed of a pressure cell, axial loading system, ambient pressure system and pore water pressure measuring system, and so on, as shown in Fig. 1.16. The circular base has a central pedestal on which the specimen is placed, there being access through the pedestal for drainage or for the measurement of pore water pressure. A perspex cylinder sealed between a ring and a which the loading ram passes. The main steps of conventional test methods are described as follows: The specimen is placed on either a porous or a solid disk on the pedestal of the apparatus.

Fig. 1.16 Triaxial apparatus (Craig 1978)

1.2 Discussions of the Mohr–Coulomb Strength Theory

17

A loading cap is placed on top of the specimen, and the specimen is then sealed in a rubber membrane, O-rings under tension being used to seal the membrane to the pedestal and the loading cap. In the cases of soils, the specimen must be prepared in a rubber membrane inside a rigid former which fits around the pedestal; a small negative pressure is applied to the pore water to maintain the stability of the specimen, while the former is removed prior to the application of the all-round pressure. At this time, the three principal stresses in the specimen are equal, so there are no shear stress existing on any plane. And then an all-round vertical pressure is applied to the specimen through the transfer rod, in this way, the vertical principal stress is greater than the horizontal principal stress, and the horizontal principal stress is kept constant, then the vertical principal stress is gradually increased by the application of compressive load until failure of the specimen takes place, usually on a diagonal plane. The system for applying the all-round pressure must be capable of compensating for pressure changes due to cell leakage or specimen volume change. Let vertical compressive stress be taken as Δσ1 when failure takes place in the specimen, the maximum principal stress of the specimen is σ1 = σ3 + Δσ1 , and the minimum principal stress is σ3 , a limit stress circle can be drawn with a diameter of (σ1 − σ3 ), such as circle A in figure. The same type of soil samples (more than three groups) is tested by above methods, each specimen is subjected to different confining pressures σ3 , and then the maximum principal stress σ 1 of shear failure can be obtained, respectively. Finally, these results will be coming up with a set of limit stress circles, such as circle A, B, and C in Fig. 1.17. According to the limit stress circle, make a set of common tangents, that is, the shear strength envelope of soil, which can be usually approximate to a straight line. The angle between the line and the lateral axis is the internal friction angle ϕ of soil, and the intercept of the line and the ordinate is the cohesion C of soil. Fig. 1.17 Envelope of triaxial compression test

18

1 Introduction

According to the degree of consolidation and drainage conditions before and after shearing, the triaxial compression test can be divided into the following three test methods: (1) Unconsolidated-undrained triaxial test (UU), undrained test. The specimen is subjected to a specified all-round pressure, and then the principle stress different is applied immediately, with no drainage being permitted at any stage of the test. (2) Consolidated undrained triaxial test (CU). Drainage of the specimen is permitted under a specified all-round pressure σ3 until consolidation is complete; the principle stress different is then applied with no drainage being permitted, and until specimen has caused shear damage. (3) Consolidated drained triaxial test, (CD). Drainage of the specimen is permitted under a specified all-round pressure σ 3 until consolidation is complete; the principle stress different is then applied with drainage still being permitted in the whole process of shear failure. It is necessary to point out that the triaxial compression test (axisymmetric triaxial test) can produce a especial complex stress σ 1 , σ 2, and σ 3 , which are all in a special plane, as shown in Fig. 1.18. However, in the confining pressure triaxial test, whether it is isotropic consolidation, K 0 consolidation, triaxial compression shear, or the triaxial extension shear, the two principal stresses in them are always equal, that is σ 2 = σ 3 , or σ 1 = σ 2 . The results of the test can be used to obtain the strength parameters of material, but they cannot distinguish the difference of different strength theories.

Fig. 1.18 Stress system in triaxial test

1.3 New Theoretical Basis of Soil Strength

19

Fig. 1.19 Comparison of friction angle between plane strain and triaxial tests (Li GX)

Many studies have shown that the Mohr–Coulomb strength theory is unreasonable. Figure 1.19 shows a series of results obtained from the two tests of Montary sand under different confining pressure. It can be seen that, in the range of the test, the results of the plane strain test are higher than those obtained by triaxial test no matter what the compactness of the sand.

1.3 New Theoretical Basis of Soil Strength Due to the various limitations of the Mohr–Coulomb strength theory, a new strength theory is needed. According to the discussions, the following characteristics of the new strength theory should be required: (1) (2) (3) (4) (5) (6)

The mathematical expression should be simple. The physical meaning should be clear. Three principal stresses should be taken into account in the failure of material. The new strength theory should be easy to use in engineering practice. The new strength theory should be easy for analytical analysis. The new strength theory should agree well with the experimental results.

Many strength theories were proposed in the past 100 years. Among them, the Drucker-Prager criterion, the Matsuoka-Nakai criterion, and the Lade-Duncan criterion may be the most famous criteria. The intermediate principal stress effect is taken into account in these criteria. Based on these famous criteria, some other researchers have put forward various combinations of criteria, such as the DruckerPrager criterion combined with the Matsuoka-Nakai criterion, Matsuoka-Nakai criterion combined with the Lade-Duncan criterion, and the Matsuoka-Nakai criterion combined with the Mohr–Coulomb criterion. These criteria can flexibly change a number of different failure criteria between the two criteria, which extend the limits

20

1 Introduction

Fig. 1.20 Bounds and regions of the Mohr–Coulomb theory and twin-shear theory

of a single criterion. However, these combinations are generally not up to the entire region of the Drucker convexity. In 1951, a famous postulate was proposed by Drucker. According to Drucker’s postulate, all kinds of strength theory must be convex and also situated between the inner bound and the outer bound, as shown in Fig. 1.20a. In fact, the Mohr–Coulomb strength theory can be thought of as the lower bound of all convex limit surfaces, the twin-shear strength theory can be thought of as the upper bound of all convex limit surfaces. The Unified Strength Theory (UST) gives a series of new yield and failure criteria, establishes a relationship among various failure criteria, and encompasses previous yield criteria and failure criteria as its special cases or linear approximations. It covers all areas within the domain, as shown in Fig. 1.20b. Unified Strength Theory was proposed by Yu in 1991, which is suitable for more kinds of materials and structures. The theory and its application in soil mechanics is a completely new system; we will systematically discuss it further in Chap. 5.

1.4 Soil Mechanics Based on Unified Strength Theory The strength theory is one of the most important bases of the soil mechanics. As the traditional soil mechanics only takes the maximum shear stress and the corresponding normal stress acting on the same plane into account on the failure of materials, it can be referred to as single-shear soml Mechanics or soil mechanics based on Mohr– Coulomb strength theory. As the intermediate principal stress effect is not taken

References

21

into account, the soil mechanics based on Mohr–Coulomb strength theory should be improved. Unlike the Mohr–Coulomb strength theory, the Unified Strength Theory takes all three principal stresses into account and formed a series of new criteria. The intermediate principal stress effect is taken into account in the UST by introducing a parameter b. Thus, it can be applied to a wide range of materials and structures. In order to distinguish it from the traditional soil mechanics, it can be referred to as soil mechanics based on Unified Strength Theory. The soil mechanics based on the Mohr–Coulomb strength theory is included in soil mechanics based on Unified Strength Theory as a special case. Compared with the traditional soil mechanics, the new characteristics of soil mechanics based on Unified Strength Theory are summarized as follows: (1) The Unified Strength Theory is introduced in the monograph. The effect of intermediate is taken into account in the Unified Strength Theory by the introduction of parameter b. The Mohr–Coulomb criterion is a special case of Unified Strength Theory. The Unified Strength Theory is well suitable for the analytical analysis. (2) The Unified Strength Theory has been widely adopted in the analysis of strip foundation stability analysis, underground structure stability analysis, slope stability analysis, and wellbore stability analysis. The unified solutions are provided for these problems in this book. It can be found that the solution obtained based on Mohr–Coulomb criterion is a special case of the unified solutions.

References Al Hussaini MM (1973) Influence of relative density on the strength and deformation of sand under plane strain conditions. ASTM 523:332–347 Cornforth DH (1964) Some experoments on the influence of strain conditions on the strength of sand. Geotechnique 14(2):143–167 Craig RF (1978, 1983) Soil mechanics, 2nd edn. Van Nostrand Reinhold Co Henkel DJ, Wade NH (1966) Plane strain tests on a saturated remolded clay. J Soil Mech Found Div ASCE 92(SM6):67–80 Horton J, Mancini M (2008) How landslides work. HowStuffWorks.com. https://science.howstu ffworks.com/environmental/earth/geology/landslide.htm Kjellman W (1936) Report on an apparatus for consummate investigation of the mechanical properties of soils. In: Proceedings of the 1st international conference on soil mechanics and foundation engineering, vol 2 Lee KL (1970) Comparison of plane strain and triaxial tests on sand. J Soil Mech Found Div ASCE 96(3):901–923 Li SQ (1982) Experimental study on constitutive relation of sand under plane strain condition, master thesis. Tsinghua University, Beijing (in Chinese) Mohr O (1900) Welche umstande bedingen die elastizitatsgrenze und den bruch eines materials? Z Ver Dtsch Ing 44(1524–1530):1572–1577

22

1 Introduction

Proctor DC, Barden L (1969) Correspondence on a note on the drained strength of sand under generalized stain conditions by G.E. Green and A.W. Bishop. Geotech. 19(3):424–426 Shi XS, Chen ZL (2011) Unified strength theory paramaters of rockfill materials in plane strain state. Chin J Rock Mech Eng 30(11):2244–2253 (in Chinese) Song XJ, Xu HB (2011) Experimental study of strength characteristics of cement-soil under plane strain condition. Rock Soil Mech 32(8):2325–2330 Vaid YP, Campanella RG (1974) Triaxial and plane strain behaviour of natural clay. J Geotech Eng ASCE 100(3):207–224

Chapter 2

Stress State and Element

2.1 Introduction Stress is an important concept in soil mechanics. A structure will have stresses in various parts of the material (points, elements) under load. The stresses in different parts are often different. And for the same element, the stresses vary from section to section. The analysis of stresses in the element and their relation to the strength of the soil is a fundamental problem in soil mechanics and an important problem in engineering. In this chapter, we will discuss the basic concepts of stresses and stress states. The study is the same as that of solid mechanics. However, we will introduce some new concepts such as the twin-shear mechanics model, twin-shear stress circle, and twin-shear stress parameter. This approach is not only useful for the future study of the Unified Strength Theory but for understanding the engineering applications in other chapters. When the stress on a point is determined, the stresses on the sections passing through this point in different directions are different, but they all refer to the stress state at the same point. In general, the stress state of a point is represented by three groups of stresses on three mutually perpendicular sections of a hexahedron, i.e., nine stress components. If the shear stress on a section of the element is equal to zero, this section is referred to as the principal plane. The normal stress in the principal plane is referred to as the principal stress. For a stress state at any point, the three principal planes are perpendicular to each other. The principal stresses acting on the principal plane are written as σ1 , σ2 , σ3 , which satisfy σ1 ≥ σ2 ≥ σ3 . According to the number that the

© Springer Nature Singapore Pte Ltd. and Zhejiang University Press 2023 M.-H. Yu, Soil Mechanics, https://doi.org/10.1007/978-981-99-2781-4_2

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24

2 Stress State and Element

principal stress is not equal to zero, the stress states of a point are classed into three categories: (1) One-dimensional stress state: Two principal stresses of the element are equal to zero. (2) Two-dimensional stress state: One principal stress of the element is equal to zero. (3) Three-dimensional stress state: All principal stresses of the element are not equal to zero.

2.2 Stresses on the Oblique Plane If the three principal stresses σ 1 , σ 2 , σ 3 acting on three principal planes, respectively, at a point are given, we can determine the stresses acting on any plane through this point. This can be done by consideration of the static equilibrium of an infinitesimal tetrahedron formed by this plane and the principal planes, as shown in Fig. 2.1. In this figure, we have shown the principal stresses acting on the three principal planes. These stresses are assumed to be known. We wish to find the stresses σ α , τ α acting on the oblique plane whose normal has direction cosines l, m, and n.

2.2.1 Stresses on the Oblique Plane The normal stress σ α and shear stress τ α acting on this plane can be determined as follows:

Fig. 2.1 Stress on an infinitesimal tetrahedron

2.2 Stresses on the Oblique Plane

25

σα = σ1l 2 + σ2 m 2 + σ3 n 2

(2.1)

τα = σ12 l 2 + σ22 m 2 + σ32 n 2 − (σ1l 2 + σ2 m 2 + σ3 n 2 )

(2.2)

p→α = σ→α + τ→α .

(2.3)

2.2.2 Principal Shear Stresses The three principal shear stresses τ13 , τ12 , and τ23 can be obtained as follows: τ13 =

1 1 1 (σ1 − σ3 ), τ12 = (σ1 − σ2 ), τ23 = (σ2 − σ3 ). 2 2 2

(2.4)

The maximum shear stress acts on the plane bisecting the angle between the largest and smallest principal stresses and is equal to half of the difference between these principal stresses τmax = τ13 =

1 (σ1 − σ3 ). 2

(2.5)

The corresponding normal stresses σ13 , σ12 , and σ23 acting on the sections where τ 13 , τ 12, and τ 23 are acting, respectively, are σ13 =

1 1 1 (σ1 + σ3 ), σ12 = (σ1 + σ2 ), σ23 = (σ2 + σ3 ). 2 2 2

(2.6)

It is seen from Eq. (2.14) that the maximum principal shear stress τ13 equals to the sum of the other two (τ12 + τ23 ), i.e., τ13 = τ12 + τ23 .

(2.7)

2.2.3 Octahedral Shear Stress If the normal of the oblique plane makes equal angles with all the principal axes, 1 l = m = n = ±√ . 3

(2.8)

26

2 Stress State and Element

Table 2.1 Direction cosines of the principal planes and the principal shear stress planes, etc Principal plane

Principal shear stress plane

Octa. plane

l=

±1

0

0

± √1

± √1

0

√1 3

m=

0

±1

0

± √1

0

± √1

n=

0

0

±1

0

± √1

√1 3 √1 3

σ=

σ1

σ2

σ3

σ12 =

τ=

0

0

0

τ12 =

2

2

2

2 ± √1 2

2

σ1 +σ2 2 σ1 +σ2 2

σ13 = τ13 =

σ1 +σ3 2 σ1 +σ3 2

σ23 = τ23 =

σ2 +σ3 2 σ2 +σ3 2

σoct τoct

These planes are called the octahedral plane, and the shear stresses acting on it are called the octahedral shear stresses. The normal stress, called the octahedral normal stress σ 8 (or σ oct ), acting on this plane equals the mean stress σoct =

1 (σ1 + σ2 + σ3 ) = σm . 3

(2.9)

A tetrahedron similar to this one can be constructed in each of the four quadrants above the x–y plane and in each of the four quadrants below the x–y plane. On the oblique face of each of these eight tetrahedral, the condition l2 = m2 = n2 = 1/3 will apply. The difference between the tetrahedra will be in the signs attached to l, m, and n. The eight tetrahedra together form an octahedra as shown in Fig. 2.6 and on each of the eight planes form the faces of this octahedron. The octahedral normal stress is given by Eq. (2.9), and the octahedral shear stress τ oct acting on the octahedral plane is 1 [(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 ]1/2 3 1 = √ [(σ1 − σm )2 + (σ2 − σm )2 + (σ3 − σm )2 ]1/2 . 3

τoct =

(2.10)

The direction cosines l, m, and n of principal plane, principal shear stress plane, and the octahedral plane, as well as the principal shear stresses and corresponding normal stresses are listed in Table 2.1.

2.3 Hexahedron, Octahedron, and Dodecahedron According to the stress state, various polyhedral elements can be illustrated as shown in Figs. 2.2, 2.3, 2.4 and 2.5.

2.3 Hexahedron, Octahedron, and Dodecahedron

Fig. 2.2 Principal stress element

Fig. 2.3 σ 8 and τ 8 element

Fig. 2.4 Single-shear elements

27

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2 Stress State and Element

Fig. 2.5 Twin-shear elements

2.3.1 Principal Stress Element Principal stress element is a cubic element; the three principal stresses σ 1 , σ 2 , and σ 3 act on this element. The principal stress element and three principal stresses (σ 1 , σ 2 , σ 3 ) are shown in Fig. 2.2.

2.3.2 Isoclinal Octahedron Element Isoclinal octahedral element subjected to the octahedral normal stresses σ oct and octahedral shear stresses τ oct , as shown in Fig. 2.3, is a regular octahedron.

2.3 Hexahedron, Octahedron, and Dodecahedron

29

Fig. 2.6 From principal stress element to twin-shear element

2.3.3 Single-Shear Element The maximum shear stress element (τ 13 , σ 13 , σ 2 ) is a quadrangular prism element, which the maximum shear stress τ 13 , corresponding normal stress σ 13 , as well as the intermediate principal stress σ 2 act on. This kind of element, as shown in Fig. 2.4a, may be referred to as the single-shear element because only one shear stress and corresponding normal stress act on the element. Quadrangular prism element (τ 12 , σ 12 , σ 3 ) is shown in Fig. 2.4b, acted by the intermediate principal shear stress element (when τ 12 ≥ τ 23 ), the intermediate principal shear stress τ 12 and the corresponding normal stress σ 12 , as well as the minimum principal stress σ 3 . Quadrangular prism element (τ 23 , σ 23 , σ 1 ) is shown in Fig. 2.4c,

30

2 Stress State and Element

acted by the minimum principal shear stress element (when τ 12 ≤ τ 23 ), the minimum principal shear stress τ 23, and the corresponding normal stress σ 23 , as well as the maximum principal stress σ 1 .

2.3.4 Twin-Shear Element Figure 2.5a shows an orthogonal octahedron (τ 13 , τ 12 ; σ 13 , σ 12 ), in which the principal shear stresses τ 13 , τ 12 and the corresponding normal stresses σ 13 , σ 12 act on this element. Yu proposed this new element in 1988 and 1989 (Yu 2004). It can be referred to as the twin-shear element. The principal shear stresses τ 13 , τ 23 and the corresponding normal stresses σ 13 , σ 23 act on an orthogonal octahedron element (τ 13 , τ 23 ; σ 13 , σ 23 ), as shown in Fig. 2.5b. This element can also be referred to as the twin-shear element. They are available to use for the mechanical model of strength theory. How to get the twin-shear element from principal stress element to single-shear element and then from single-shear element to twin-shear element? The process is illustrated in Fig. 2.6.

2.3.5 Dodecahedral Principal Shear Stress Element The three principal shear stresses τ 13 , τ 12 , and τ 23 and the corresponding normal stresses σ 13 , σ 12 , and σ 23 acting on a element are shown in Fig. 2.6. The first presentation of the dodecahedron element may be by Walczak at Krakov, Poland in 1951. This element of dodecahedron can be referred to as the three-shear element. It is interesting that the three principal shear stresses τ 13 , τ 12 , and τ 23 only have two independent variations because the maximum principal shear stress equals the sum of the other two, i.e., τ 13 = τ 12 + τ 23 . The formation of the three-shear element is illustrated in Fig. 2.7.

2.3.6 Twin-Shear Stress State and Twin-Shear Element The stress state at a point can be determined by the combination of the three principal stresses (σ 1 , σ 2 , σ 3 ). It is expressed by f (σ 1 , σ 2 , σ 3 ). The principal stress state f (σ 1 , σ 2 , σ 3 ) can be converted to principal shear stress state f (τ 13 , τ 12 , τ 23 ). However, only two principal shear stresses of the three are dependent variables because the maximum principal shear stress τ 13 equals the sum of the other two shear stresses. This relationship can be expressed as follows: τ13 ≡ τ12 + τ23 .

(2.11)

2.4 Stress Space

31

Fig. 2.7 Formation of the three-shear element

The Mohr stress circle can be used to visually reflect the three principal stresses σ 1 , σ 2 , σ 3 and three principal shear stressesτ 13, τ 12 , τ 23 , as well as their relationships, as shown in Fig. 2.8. Like the twin-shear element, there are two types of twin-shear stress circle, as shown in Fig. 2.9a, b.

2.4 Stress Space The stress point P (σ 1 , σ 2 , σ 3 ) in stress space can be expressed by other forms, such as P(x, y, z), P(r, θ, ξ ), or P (J 2 , θ, ξ ). The geometrical representation of these transfers can be seen in Fig. 2.10 and Fig. 2.11. For the straight line OZ passing through the origin and making the same angle with each of the coordinate axes, the equation is σ1 = σ2 = σ3 .

(2.12)

σ1 + σ2 + σ3 = 0.

(2.13)

The equation of the π0 -plane is

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2 Stress State and Element

Fig. 2.8 Concept of twin-shear can be illustrated by Mohr’s stress circle

The stress vector σ can also be divided into two parts: the hydrostatic stress vector σm and the mean shear stress vector τm . σ = σm = τm .

(2.14)

1 ξ = √ (σ1 + σ2 + σ3 ) 3

(2.15)

] √ 1 [ √ (σ1 + σ2 )2 + (σ2 + σ3 )2 + (σ3 + σ1 )2 = 3τoct = 2τm 3

(2.16)

Their magnitudes are given by

/ r= in which σ / r=

oct

is the octahedral normal stress and τoct is the octahedral shear stresses.

2 2 2 τ13 + τ12 + τ23 = 3

/

] 1[ (σ1 + σ2 )2 + (σ2 + σ3 )2 + (σ3 + σ1 )2 12

(2.17)

The π-plane is parallel to the π0 -plane and is given by σ1 + σ2 + σ3 = C.

(2.18)

The projections of the three principal stress axes in stress space σ 1 , σ 2 , and σ 3 are σ1' , σ2' , σ3' . The relationship between them is

2.4 Stress Space

33

Fig. 2.9 Two types of twin-shear stress circle. a τ 12 > τ 23 , b τ 12 < τ 23

/ σ1' = σ1 cos β =

2 σ1 , σ2' = σ2 cos β = 3

/

2 σ2 , σ3' = σ3 cos β = 3

/

2 σ3 (2.19) 3

in which β is the angle between O ' A, O ' B, O ' C and the three coordinates as shown in Fig. 2.12.

2.4.1 Relation Between (σ1 , σ2 , σ3 ) and (x, y, z) The relations between the coordinates of the deviatoric plane and the principal stresses are

34

2 Stress State and Element

Fig. 2.10 Cylindrical coordinates

Fig. 2.11 Stress state in the π-plane

1 1 1 x = √ (σ3 − σ2 ), y = √ (2σ1 − σ2 − σ3 ), z = √ (σ1 + σ2 + σ3 ) 2 6 3 ( ( ) √ √ ) √ 1 √ 1 √ 2 3z − 6y − 3 2x , σ1 = 6y + 3z , σ2 = 3 6

(2.20)

2.4 Stress Space

35

Fig. 2.12 Deviatoric plane

σ3 =

√ √ ) 1( √ 3 2x − 6y + 2 3z . 6

(2.21)

2.4.2 Relation Between (σ1 , σ2 , σ3 ) and (ξ, r, θ) or The relations between the cylindrical coordinates (ξ, r, θ ) and the principal stresses (σ1 , σ2 , σ3 ) are √ 1 ξ = |O N | = √ (σ1 + σ2 + σ3 ) = 3σm 3

(2.22)

]1 1 [ r = |N P| = √ (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 2 3 √ )1 1 ( 2 = √ S1 + S22 + S32 2 = 3τoct = 2τm 3 ( ) x . θ = tan−1 y

(2.23) (2.24)

36

2 Stress State and Element

From Eqs. (2.20) and (2.23), we can obtain cos θ =

y . r

(2.25)

These relations are suitable to the conditions σ 1 ≥ σ 2 ≥ σ 3 and 0 ≤ θ ≤ π /3. The limit loci in the π-plane have threefold symmetry, so if the limit loci in the range of ◦ 60 are given, then the limit loci in π-plane can be obtained. The three principal stresses can be expressed as follows: / 2 1 r cos θ σ1 = √ ξ + 3 3 / 2 1 π σ2 = √ ξ + r cos(θ + 2π/3)0 ≤ θ ≤ 3 3 3 / 2 1 σ3 = √ ξ + r cos(θ + 2π/3). 3 3

(2.26)

2.5 Summary Elements and stress states are described briefly in this chapter. Stress state theory is studied in many courses, such as mechanics of materials, elasticity, plasticity, mechanics of solids, rock mechanics, and soil mechanics. Only the basic formulas are given here. The twin-shear stresses, the twin-shear element, and the twin-shear stress parameter are new concepts. These new concepts will be used in the following chapters.

Reference Yu MH (2004) Unified strength theory and its applications. Springer, Berlin

Chapter 3

Stresses in Soil Masses

3.1 Introduction Stress is an important concept in soil mechanics. The basic concepts of stress, element, and stress state were introduced in Chap. 2. In this chapter, we will study the stress caused by the self-weight of soil above a certain point and the externally applied loads. The stress in the soil is very complex, and it is mainly studied in the framework of continuum mechanics; then the results of solid mechanics are applied to the problem of soil mechanics. The soil or rock formation supporting every artificial structure is called the ground. The stress analysis of general soil structure is more complex. The stress distribution in soil is determined by the shape of the foundation. Plane stress, plane strain, and spatial axisymmetric problems are three important problems in plasticity and engineering, as shown in Fig. 3.1. However, the soil cannot be subjected to loading in the plane stress, so the soil mechanics generally studies the plane strain and spatial axisymmetric problems. Figure 3.1a is a plane stress structure with a uniform thickness thin lamina deformed under the action of force which lie in its median plane. Figure 3.1b is a plane strain problem with zero strain at z direction (length direction) in a very large thickness structure. Figure 3.1c is an axisymmetric problem which is symmetrical in terms of geometry, boundary conditions, and external loading about an axis. In addition, there are square and rectangular foundations, which belong to the general three-dimensional space problem. Their stress analyses are more complex. In 1885, Boussinesq (1987), a French mathematician and physicist, derived the analytical solution of the stress in an elastic body subjected to a vertical concentrated load P on the surface. This is a spatial axisymmetric problem; the axis of symmetry is the action line that passes through the vertical concentrated load P, taking the point P as the origin, the point M coordinates are (x, y, z), as shown in Fig. 3.2.

© Springer Nature Singapore Pte Ltd. and Zhejiang University Press 2023 M.-H. Yu, Soil Mechanics, https://doi.org/10.1007/978-981-99-2781-4_3

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38

3 Stresses in Soil Masses

Fig. 3.1 Three kinds of engineering structures. a Plane stress problem, b Plane strain problem, c Spatial axisymmetric problem Fig. 3.2 Boussinesq project

The expression of the six stress components and the three displacement components of the M point was obtained by Boussinesq, which is called the Boussinesq project. Since then, many scholars have had the basis to do a lot of research about this problem and obtained various solutions. For simple problems, we can find the analytical solution of the stress distribution in the soil. For complex problems, where the analytical solution is more difficult to solve, the computer numerical analysis method can be used. For example, Fig. 3.3 shows the stress distribution of a strip foundation subjected a uniform load. The two mutually perpendicular line segments in Fig. 3.3a indicate the magnitude and direction of the principal stress, and Fig. 3.3b shows the stress contour map (Han et al.2012). This ellipsoidal spherical shape of the lines is also called stress bubble. The study of stress problems in this chapter focuses on effective stresses, selfweight stresses, and additional stresses under foundation loads. These are the same as in general geomechanics. In addition, it should be noted that when the soil enters the plastic state, the distribution of stresses in the soil is related to the chosen strength theory. Several studies in recent years have shown that there are significant differences in the stress distributions derived using different strength theories.

3.2 Effective Stress of Soils

39

Fig. 3.3 Distribution of stresses along strip foundation under vertical load

3.2 Effective Stress of Soils The soil consists of solid particles, water, and gas. The stress distribution inside the soil subjected to a load is complicated. In 1913–1915, Fillunger carried out a large number of experiments and pointed out “It is less the actual individual stress problems, than the complete agreement of the final results, which lead straight to the conviction that a pressure-carrying liquid penetrating the masonry construction, creates a pressure in the material equal in all directions.” Subsequently, he defined his statement more precisely regarding the causation of effective pressure by the liquid in the pores of the masonry: “one can assume that the uniform internal pressure cannot cause a significant reduction of the strength.” This was the first elaboration of the concept that pore water pressure does not have any effect on the strength of porous solids. In 1915, Fillunger again concluded that the pore water pressure did not affect the material behavior of the porous solid at all based on the experimental results (de Boer 1988). In 1923, von Terzaghi proposed the principle of effective stress for saturated soils, in which effective stress was defined as the difference between the total stress and the pore water pressure. He proposed the equation σ = σ ' + u and pointed out that the total stress of the soil consists of two parts. One part is u, the strength acting on water and solids is equal in all directions, and this part is regarded as the pore water pressure (Fig. 3.4); another part is the effective stress, which is defined as the difference between the total stress and the pore water pressure, i.e., σ ’ = σ − u, it occurs only in the solid phase of soil (The volume does not vary with changes of pore water pressure, and pore water pressure has nothing to do with the cracking of soil subjected to a load). This is called the principle of effective stress.

40

3 Stresses in Soil Masses

Fig. 3.4 Geostatic vertical stress in homogeneous soil

The principle of effective stress mainly includes the following two points: (1) The total stress acting on soil is the sum of effective stress and pore water pressure, namely σ ' = σ − u.

(3.1)

(2) The strength and deformation of soil are determined only by the effective stress.

3.3 Self-weight Stress Before a building is constructed, there are already stresses in the soil from the weight of the soil itself. Soil is a discontinuous medium made up of soil particles, water, and gas. If the soil is assumed as a continuum and the continuum theory is applied to study the stress distribution in the soil, only the average stress per unit area in the soil is considered (Qian ea al.1988, Lambe et al. 1979, Pearson 1959). When calculating the stress of soil, it is assumed that the natural ground is an infinite horizontal plane. If the soil underground is homogeneous and the natural weight-specific density is γ , the vertical gravity stress at any depth z is σcz = γ × z.

(3.2)

The vertical self-weight stress varies linearly with depth z, and the distribution of vertical self-weight stress σ cz along the depth is shown in Fig. 3.4. In addition to the vertical self-weight stress acting on the horizontal plane, there is also a lateral self-weight stress in the vertical plane. According to the elastic

3.3 Self-weight Stress

41

mechanics, σ cx and σ cy should be proportional to σ cz , and the shear stress is zero, i.e., σcx = σcy = k0 σcz , τx y = τ yz = τzx = 0,

(3.3)

where the coefficient k 0 is known as the coefficient of earth pressure at rest. Soils are often layered, and thus, each layer has different weights. If the water table is located in the same soil layer, the water table surface should be used as a layered interface when calculating the self-weight stress, as shown in Fig. 3.5. Soil layers below the water table must have an effective weight γ ' instead of the natural weight γ . This gives the formula for calculating the self-weight stress of a layered soil: σcz =

n

γi h i ,

(3.4)

i=1

where n is the total number of soil layers in the range of depth z; hi is the thickness of ith layer; γ i is the natural weight-specific density. If the layer is below the water level, the effective unit weight should be used. The distribution of vertical self-weight stresses in the soil along the depth shows that (1) The line of distribution of the self-weight stress in the soil is a folding line with the folding point at the junction of the soil layers and at the water table, with abrupt changes in the line of distribution at the impermeable level. (2) Self-weight stresses in the same layer of soil vary in a straight line.

Fig. 3.5 Distribution of geostatic vertical stress along with depth

42

3 Stresses in Soil Masses

(3) Self-weight stresses become greater with increasing depth. (4) Self-weight stresses are equal at all points in the same plane. The natural soil layer in nature has been formed for a long time, and the compression deformation of the soil caused by its own self-weight has long been completed, so the self-weight stress does not generally cause the settlement of building foundations. However, the deformation caused by the self-weight stress should be taken into account for recently deposited or stockpiled soils.

3.4 Contact Pressure The major function of the foundation of structure is to transmit the load of the structure to the supporting ground. Contact pressure is the load transmitted from the foundation to the ground soil. The magnitude and the distribution of the contact pressure have an important impact on the stress in the ground. Therefore, the analysis of the interaction between a structural foundation and the supporting ground soil is of primary importance to both structural and geological engineering (Fan et al. 2017; Chen et al. 1994).

3.4.1 Vertical Centric Load The length and width of a rectangular foundation are l and b, respectively, as shown in Fig. 3.6a. A vertical centric load F is applied to the foundation. According to the linear distribution assumption, the value of the contact pressure is p=

F +G , A

(3.5)

where lowercase p is the contact pressure (kPa); F is the vertical load on the underside on the foundation (kN); G is the total load self-weight and the refill soil (kN), G = γ G Ad, γ G is the average weight-specific density of soil, generally, it is assumed as 20 kN/m3 , but the parts underwater level should be considered as 10 kN/m3 ; A = l × b represents the area of the foundation (m2 ).

3.4.2 Vertical Eccentric Load Generally, when a one-way eccentric load is applied to a rectangular foundation, the contact pressure at any arbitrary point can be calculated using the formula of eccentric compression in mechanics of materials, pmax and pmin are the maximum

3.4 Contact Pressure

43

Fig. 3.6 Contact pressure distribution due to vertical centric load. a Centric load, b eccentric load (e < l/6), c eccentric load (e > l/6)

and minimum contact pressures on both sides of the underside of the foundation, as given by max = pmin

F+G 6e 1± , lb l

(3.6)

where l and b represent the length and width of a rectangular foundation and e is the offsetting of eccentric load line to Y –Y axis. According to the e and Eq. (3.6), we can obtain the distribution of contact pressure may appear three cases: (1) When e < 1/6, pmin > 0, the distribution curve of the contact pressure is trapezoidal (Fig. 3.6b). (2) When e = 1/6, pmin = 0, the distribution curve of the contact pressure is triangular. (3) When e > 1/6, pmin < 0, i.e., the tension force would appear on one side of the underside of the foundation. According to the balance condition of eccentric load and contact pressure, the load force should pass through the centroid of triangle distribution (Fig. 3.6c).

44

3 Stresses in Soil Masses

3.5 Additional Stress in Soils For a natural soil layer before construction, the displacement due to self-weight stress has been finished. Only the contact pressure increase of underside may cause the stress increase and the displacement of the ground. The distribution schematic diagram of self-weight pressure and additional stress of soil under construction is shown in Fig. 3.7. The additional stress in this figure can be obtained based on the elasticity, which is called the Boussinesq problem.

3.5.1 Additional Stress Due to Vertical Concentrated Load A vertical concentrated load F is applied on the surface of a homogeneous and isotropic elastic half-space ground soil, as shown in Fig. 3.8. The solutions of the stress and displacement at an arbitrary point M (x, y, z) in the the elastic body were first provided by Boussinesq as. 1. Three normal stress components at point M x 2 (2R + z) 3F x 2 z 1 − 2μ R 2 − Rz − z 2 − 3 σx = + 2π R 5 3 R 3 (R + z) R (R + z)2

Fig. 3.7 Gravity pressure and additional stress diagram of soil

(3.7a)

3.5 Additional Stress in Soils

45

Fig. 3.8 Stress state of soil due to concentrated load

σy =

y 2 (2R + z) 3F y 2 z 1 − 2μ R 2 − Rz − z 2 − + 2π R 5 3 R 3 (R + z) R 3 (R + z)2 σz =

3F z 3 . 2πR 5

(3.7b)

(3.7c)

2. Three-shear stress components at point M τx y = τ yx

3F x yz 1 − 2μ x y(2R + z) · 3 = − 2π R 5 3 R (R + z)2

(3.8a)

τ yz = τzy =

3F yz 2 · 2π R 5

(3.8b)

τzx = τx z =

3F x z 2 · 2π R 5

(3.8b)

3. Three displacement stress components at point M F(1 + μ) x z x − − 2μ) (1 2πE R3 R(R + z) y F(1 + μ) yz v= − − 2μ) (1 2πE R3 R(R + z) 2 1 F(1 + μ) z , w= + 2(1 − μ) 2πE R3 R

u=

(3.9a) (3.9b)

(3.9c)

46

3 Stresses in Soil Masses

where R is the distance from this point to the action point of ground force, r is the distance from the horizontal projection point to the action point of ground force, and E and μ are the elastic modulus and Poisson’s ratio of soil, respectively. The above equations are well-known Boussinesq solutions. The vertical stress σ z and vertical displacement w are the most commonly used in engineering. For simplicity and practical application, the normal stress can be rewritten as given in the following form: σz = where R =

√

3F z 3 3F z 3 F = 5/2 = α 2 , 2 2 2πR 5 z 2π r + z

x 2 + y 2 + z 2 and α =

3 5/2 2π [(r/z)2 +1]

(3.10)

is the coefficient of additional

stress under foundation underside due to vertical concentrated load. When several vertical concentrated loads are applied on the surface of an elastic half-space, the additional stress can be calculated by superposition method: n 1 σz = 2 αi Fi . z i=1

(3.11)

3.5.2 Additional Stress Due to Distributed Load If the shape of the foundation base and the distributed load are regular, the additional stress in the foundation soil can be obtained by applying the method of integration. Assume a distributed load p (x, y) acting on the surface of a semi-infinite soil, as shown in Fig. 3.9. ¨ σz =

dσz = A

3z 3 2π

¨ A

p(x, y)dεdη

5/2 (x − ε)2 + (y − η)2 + z 2

(3.12)

The above solution is related to three conditions as follows: (1) The distribution pattern of the distributed load p (x, y) and its magnitude; (2) The geometry and size of the distribution area A of the distributed load; (3) The values of the coordinates x, y, z of the point where the stress is calculated. The result of the integration is complex, but it is a function of l/b. z/b(z/r 0 ). In order to facilitate the engineering application, the dimensionless parameterization is carried out for the calculation formula, that is, using l/b. z/b(z/r 0 ) to draw up forms. When used, as long as to lookup table, the additional stress coefficient α f under distributed load can be obtained. Then σ z is given as

3.5 Additional Stress in Soils

47

Fig. 3.9 Stress state of soil due to distributed load

σ z = α f p0 .

(3.13)

3.5.3 Stress Analysis in Spatial Problems Common spatial problems include uniformly distributed rectangular and circular loads, triangularly distributed rectangular loads, etc. A schematic diagram of a rectangular area acting as a uniform load and a triangular load, respectively, is shown in Fig. 3.10. 1. Calculation of vertical additional stress in soil under the corners of a rectangular foundation underside due to a vertical uniform load: The length and width of a rectangular foundation are l and b, respectively, as shown in Fig. 3.10a. The vertical uniform load p0 is applied on the foundation as expressed by σz = αc p0 ,

(3.14)

where αc is the coefficient of additional stress under the corner O of the underside of a rectangular foundation due to vertical uniform load, namely

lbz l 2 + b2 + 2z 2 1 lb αc = + arctan √ √ 2π l 2 + z 2 b2 + z 2 l 2 + b2 + z 2 z l 2 + b2 + z 2

48

3 Stresses in Soil Masses

Fig. 3.10 Rectangular foundation underside objected to vertical load. a Vertical uniform load, b vertical triangular load

When the stress points are not in the corner points, the corner-points method will be used to calculate. 2. Calculation of vertical additional stress in soil under the corners of a rectangular foundation underside due to a vertical triangular load: If the maximum intensity of the triangular load on the rectangular foundation underside is p0 , as shown in Fig. 3.10b, when the load at corner 1 is zero, the coordinates of point M at depth z are (0, 0, z) and p (x, y) = x/bp0 . Therefore, the additional stress at an arbitrary depth z under the corner O induced by this point load can be calculated as given by ¨ σz =

dσz = A

3z 3 2π

¨ A

p(x, y)dεdη

5/2 , 2 (x − ε) + (y − η)2 + z 2

(3.15)

where αt1 is the coefficient of additional stress which is corresponding to the minimum value of the vertical triangular load 1 lz z2 αt1 = − . √ √ 2π b l 2 + z2 b2 + z 2 l 2 + b2 + z 2

(3.16)

Similarly, when the load at corner 2 is maximum, the additional stress at an arbitrary depth z under the corner O induced by this point load can be calculated as follows σz = (αc − αt1 ) p0 = αt2 p0 ,

(3.17)

3.6 Additional Stresses in Plane Strain State

49

where αt2 is the coefficient of additional stress which is corresponding to the maximum value of vertical triangular load, αt2 = αc − αt1 .

3.6 Additional Stresses in Plane Strain State If an infinitely strip distributed load is applied to the surface of the infinite elastic body, the distribution of load is arbitrary in the width direction, while in the length direction is the same. When calculating the stress at any point in a soil mass, it is only related to the coordinate plane but has nothing to do with the coordinate of length. This is a plane strain problem. In practice, the footing of the wall, subgrade, dam foundation, and earth-retaining wall foundation are considered as the plane strain problems.

3.6.1 Vertical Line Load Vertical line load is defined as the vertical uniform load applied on an infinitely long line, which is represented by p, as shown in Fig. 3.11a. When a vertical line load is acted on the Y –Y axis, the additional stress at an arbitrary point induced by the line load pdy on the infinitesimal width dy can be taken as a concentrated load, i.e., dF = pdy, as given by dσz =

3z 3 pdy . 2π R 5

(3.18)

By integral, there is

Fig. 3.11 Stress state due to line load and uniform strip load. a Line load, b uniform strip load

50

3 Stresses in Soil Masses

+∞ σz =

−∞

3z 3 pdy

2π x 2 + y 2 + z 2

2/ 5 =

2z 3 p

π x 2 + z2

2 .

(3.19)

Similarly, σ x and τ xz can be obtained as follows: σx =

2x 2 z p

π x 2 + z2

τx z = τzx =

2

2z 2 x p

π x 2 + z2

(3.20a)

2

(3.20b)

According to the generalized Hooke’s law and conditions ε y = 0, τ x y = τ yx = τ yz = τ zy = 0 σ y = μ(σx + σz ).

(3.21)

Equation (3.21) is called the Flamant solution in elasticity.

3.6.2 Vertical Uniform Strip Load In practical engineering, the strip load that is often encountered is shown in Fig. 3.11b, so the uniform strip load p0 on an infinitesimal width dx can be replaced by line load p, and the angle between the OM line and z axis is introduced, we obtained p = p0 dx =

p0 R 1 dβ. cos β

(3.22)

The additional stress at an arbitrary point M on the ground is expressed in polar coordinates as follows: β2 σz =

dσz = β1

p0 [sin β2 cos β2 − sin β1 cos β1 + (β2 − β1 )]. π

(3.23a)

Similarly available σx =

p0 [− sin(β2 − β1 ) cos(β2 + β1 ) + (β2 − β1 )] π p0 2 sin β2 − sin2 β1 . τzx = τx z = π

(3.23b) (3.23c)

3.6 Additional Stresses in Plane Strain State

51

Fig. 3.12 Principal stress of the additional stress caused by strip load and stress bulb

When M point is located between the two ends of the load distribution width, β 1 will take a negative value, otherwise takes a positive value. The expression of the major principal stress σ 1 and minor principal stress σ 2 in point M is σz + σ x σ1 = ± σ3 2

/

σz − σ x 2

2 2 = + τzx

p0 [(β2 − β1 ) ± sin(β2 − β1 )]. (3.24) π

The major and minor principal stresses of the additional stress caused by uniform strip load are shown in Fig. 3.12, which are parallel and perpendicular to angle bisector direction, respectively. In the figure, the direction and length of the axes of each stress ellipse represent the direction and magnitude of the major and minor principal stresses at different positions, respectively. The contour line of the principal stress is an arc that passes through two points on the edge of a strip foundation, and the circumference angles of each point in the arc are equal. In order to facilitate the calculation, the above three formulas can also be expressed as a rectangular coordinate form. Taking the midpoint of the strip load as the original point, the three stress components are given by 4m 4n 2 − 4m 2 − 1 1 + 2n p0 1 − 2n + arctan − σz = = αsz p0 arctan 2 π 2m 2m 4n 2 + 4m 2 − 1 + 16m 2 (3.25a)

2 4m 4n − 4m 2 − 1 1 + 2n p0 1 − 2n + arctan + σx = = αsx p0 arctan 2 π 2m 2m 4n 2 + 4m 2 − 1 + 16m 2 (3.25b) τx z = τzx =

32m 2 n p0 = αsx z p0 , π 4n 2 + 4m 2 − 1 2 + 16m 2

(3.25c)

52

3 Stresses in Soil Masses

Fig. 3.13 σ z contour maps under uniform strip load (left), σ x (at top right), and τ xz

where αsz , αsx , and αsx z are three additional stress coefficients, respectively. They are all a function of n = x/b and m = z/b. Using the above formulas, the contour map of σ z , σ x , and τ zx under the action of strip load can be plotted. This kind of curve is also known as the stress bubble, as shown in Fig. 3.13. In order to have a more comprehensive understanding of foundation additional stress distribution, some σ z contour maps (stress bubble) of the different foundation under typical loads are given. Figure 3.14a is the σ z contour map under the action of concentrated load Q, and Fig. 3.14b is the σ z contour map under the action of line load P. Figures 3.15 and 3.16 are the σ z contour maps under the action of rectangular uniform load for L = 1.5B and L = 2B, L = 3B and L = ∞ (L is long enough), respectively. According to these figures, it is possible to judge the influence sphere of the additional stress caused by the external load when the building has a different length–width ratio. As can be seen from Fig. 3.15b, when L = 2B and vertical additional stress is − 0.1p, the influence depth will increase to about 3B. Figure 3.16b shows that when L ≈ ∞ and vertical additional stress is − 0.1p, the influence depth will increase to about 6.3B.

3.6 Additional Stresses in Plane Strain State

53

Fig. 3.14 σ z Contour maps under the action of concentrated load line load a load Q, b load P.

Fig. 3.15 σ z contour maps (rectangular uniform load for a L = 1.5B and b L = 2B)

54

3 Stresses in Soil Masses

Fig. 3.16 σ z contour maps (rectangular uniform load for a L = 3B and b L = ∞)

3.7 Stresses Induced by Uniform Circular Load Assuming that the radius of the circular load area is r 0 , the vertical uniform load acting on the ground surface is p0 . In Fig. 3.17a, the calculation of additional stress is solved by polar coordinates. At this time, dA = r dr dθ , dF = p0 r dr dθ , the additional stress is obtained by the coordinate transformation: r0 2π σz = 0

0

3/2 3 p0 r z 3 dr dθ z2 = p0 1 − 2 = αr p0 , 2π(r 2 + z 2 )5/2 z + r2

(3.26)

where αr is the additional stress coefficient at the center point of uniform circular load. Similarly, the additional stress under around uniform circular load is obtained: σz = αt p0 ,

(3.27)

where αt is the additional stress coefficient around the uniform circular load. The stress distribution under uniform circular load is shown in Fig. 3.17b. As a comparison, Fig. 3.18a, b is the σ z contour maps under the action of circular and square uniform load, respectively. It can be seen from the above figures that when the load is a circular load (or square load L = B) and vertical additional stress is 0.1p, the influence of the vertical additional stress of − 0.1p is about 2D (2B).

3.7 Stresses Induced by Uniform Circular Load

55

Fig. 3.17 Additional stress and stress bubble under circular uniform load. a Additional stress, b stress bubble

Fig. 3.18 σ z contour maps under the action of concentrated load line load. a Square uniform load, b circular uniform load

56

3 Stresses in Soil Masses

3.8 A Brief Introduction of Soil Elasto-plastic Stress Analysis When the stress of the soil exceeds the elastic limit, the plastic deformation will be produced, and the stress distribution cloud map will be different with the different strength theories. Figure 3.19 shows the calculated results obtained from different Unified Strength Theory parameter b by Fan et al. (2017).

Fig. 3.19 Elasto-plastic stress distribution of soil in different b values

References

57

It can be seen that the range of stress distribution is maximum when the Unified Strength Theory parameter b = 0, and the result is the same as that of the Mohr– Coulomb strength theory; the range of stress distribution is minimum when b = 1. This indicates that the material has more parts involved in the limit state to play its own potential strength, so that the ultimate load of the structure is improved (Fan 2007).

References Boussinesq J (1897) Théorie de l’écoulement tourbillonnant et tumultueux des liquides dans les lits rectilignes a grande section. 1. Gauthier-Villars Chen WF et al (1994) Constitutive equations for engineering materials: plasticity and modeling, vol 2. Elsevier, Amsterdam De Boer R (1988) On plastic deformation of soils. Int J Plasticity 4:371–391 Fan W, Yu MH, Deng LS (2017) Strength theory of geotechnical structure. Beijing Science Press, Beijing Han L, Song XL, Tan S et al (2012) Numerical analysis for the additional stress distribution in the strip foundation base. Build Sci s1:37–40 (in Chinese) Lambe TW, Whitman RV (1979) Soil mechanics. Wiley Pearson C (1959) Theoretical elasticity. Harvard University Press Qian JH, Yin ZZ (1988) Soil mchanics. Hohai University Press, Nanjing (in Chinese)

Chapter 4

Strength Characteristics of Soil

4.1 Introduction The traditional soil mechanics mainly focus on the shear strength of soils. In this chapter, we will discuss some basic strength characteristics of soils in general stress states based on a large number of experimental results. These experimental results are very important for studying the strength of soils. Soils are composed of many components, such as solid aggregates, water, and gas, and have a wide variety of properties. The microscopic structure of the soil is discontinuous. However, it is worth remarking that the macroscopic properties of soils show great regularity. This regularity is shown in the results of uniaxial tests and multi-axial tests. A triaxial test was carried out on crushed stone by Yangtze River Scientific Research Institute. The triaxial testing setup and a three-dimensional (3D) stress state of a sample of crushed rock are shown in Fig. 4.1a, b, respectively. The relations between σ 1 − σ 3 and σ 1 + σ 3 , σ 1 and σ 3 , σ 1 -σ 3 and σ 3 are shown in Fig. 4.2 for broken rock under loads. It is shown that although the crushed rock particles are randomly distributed, the simple linear response of the sample can still be observed. Owing to (σ 1 − σ 3 ) = 2τ 13 and (σ 1 + σ 3 ) = 2σ 13 , the curve in Fig. 4.2a–c represents the relations between shear strength and normal stress for broken rock. Figure 4.3a shows the relations between shear stress and mean stress of Weald clay in the triaxial test (Parry 1956). Figure 4.3b illustrates the relations between shear stress and mean stress of London clay with 42 specimens in triaxial test. A good regularity can be found in these results. The mechanical properties of other soils with different structures, such as coarsegrained soil and various rockfill materials (Bai et al. 2002), coal gangue, granular light mixed soil, shear strength curve of soil rock mixture (40% rock content), and loose rock and soil in mine waste dump, have certain regularity. This regularity is the basis of soil mechanics.

© Springer Nature Singapore Pte Ltd. and Zhejiang University Press 2023 M.-H. Yu, Soil Mechanics, https://doi.org/10.1007/978-981-99-2781-4_4

59

60

4 Strength Characteristics of Soil

Fig. 4.1 Triaxial test apparatus and a 3D stress state of crushed stone. 1—specimen; 2—hydraulic steel pillow; 3—steel formwork; 4—steel plate; 5—pressure I-steel row; 6—table skeleton, 7—Isteel (length is 4 m); 8—test table; 9—tension transfer column (there are altogether eight pillars); 10—bolt wedge

Fig. 4.2 Strength variation of broken rock. a σ 1 − σ 3 ~ σ 1 + σ 3 , b σ 1 ~ σ 3 , c σ 1 − σ 3 ~ σ 3

Fig. 4.3 Relationship between τ and σ m for Weald clay and London clay. a Weald clay Parry (1956), b London clay

4.3 Shear Strength and Effect of Normal Stress

61

Fig. 4.4 Shear strength of municipal solid waste

Another example is solid waste soil. The composition and structure of refuse soils are very complex. Due to the rapid development of modern urbanization, the stability problems and landslides arising from urban refuse deposits have attracted attention from all over the world. A lot of research has been carried out on them and summarized by Kavaganjian et al. 1995 Figure 4.4 shows the relationship between shear strength and normal stress of solid waste soils. It can be seen that there is also a certain regularity between them. The basic strength characteristics of soil will be summarized as follows.

4.2 Strength Different Effect (SD Effect) The strength of soils is greater under compression than that under tension, that is, σ c /= σ t . The difference between the compressive and the tensile strengths is referred to as the SD effect. As their compressive strengths are much greater than their tensile strengths, the compressive strength is mainly utilized in engineering design and construction. The compression is usually taken as positive in the stress–strain curve. Therefore, the single-parameter failure criteria such as the Tresca criterion and Hubervon Mises criterion with α = σ t /σ c = 1 are not suitable for soils.

4.3 Shear Strength and Effect of Normal Stress It is worthy to note that the strength of soils usually depends on shear stresses. Hence, many efforts are devoted to research on the shear strengths of geomaterials and the relation between shear and normal stresses on the sliding surface of the material under the action of a load.

62

4 Strength Characteristics of Soil

There is a considerable amount of literature in this area. Figure 4.6 shows the results of sand, cohesive soil, and loess under different pressure conditions obtained by French scholar Coulomb. Although the strengths of different materials are different, the linear relation is observed. Similar results were obtained by other researchers. Chen (2005) summarized the different methods and the relations between shear stress and normal stress for glass balls, Toyoura sand-soil and crushed sand, as shown in Fig. 4.5. The strength parameters of three kinds of collapsible loess in Xi’an area are C 1 = 40 kPa, ϕ1 = 21.3◦ , C 2 = 55 kPa, ϕ2 = 24.5◦ , and C 3 = 65 kPa, ϕ3 = 26.7◦ . Coulomb proposed the shear strength expression of sand based on the test results, which is also known as Coulomb’s law. It indicates that a linear relationship between the shear strength and normal stress on sliding surface can be found, where C and ϕ are the shear strength index of soil. This relation can satisfy the precision requirements for most engineering practice. It is a basic law for studying the shear strength of soil and is supported by a large number of experimental results.

Fig. 4.5 Results obtained by using new shear apparatus and triaxial experiments a glass ball, b Toyoura sand-soil, and c crushed sand Chen (1982)

Fig. 4.6 Relations between shear strength and normal stress of sand, cohesive soil, and loess. a Sand, b cohesive soil, c Xi’an loess

4.5 Effect of Hydrostatic Stress

63

The above relationship between normal stress and shear stress is similar to the Coulomb’s law of friction in physics. It has been extended to the law of material failure, which has been a fundamental concept of strength of materials theory in common use for many years. It should be pointed out that: (1) There are only one surface and one shear stress in sliding friction, and there are three principal shear stresses in the interior of a material. (2) In dealing with the experiment results, one generally only takes the maximum and minimum principal stresses σ 1 and σ 3 into account on the failure of materials and plots the stress circle with the diameter of (σ 1 − σ 3 ). The effect of the intermediate principal stress σ 2 is not taken into account.

4.4 Normal Stress Effect of Twin-Shear Strength The effects of normal stress on shear strength of the material can be extended to the effects of other shear stresses. The shear stress and the normal stress on the surface are taken into account in the normal stress effect of soil shear strength. In fact, it is the extension of the dry friction theorem between objects to the internal object strength. Thanks to there are three principal shear stresses in three-dimensional stress state of soil, so the normal stress effect of soil shear strength can also be expressed as the relation between the three principal shear stresses and normal stress. However, considering that there are only two independent components among the three principal shear stresses, therefore, we consider two larger shear stresses to study the relationship between the twin-shear stress and the normal stress: (τ13 + τ12 ) = f (σ13 + σ12 ). According to the experimental data of Tang (1981), the linear relation between twin-shear stresses and normal stresses at the shear sections (τ13 + τ12 ) = β(σ13 + σ12 ) or (τ13 + τ12 ) = 2τ0 + β(σ13 + σ12 ) can be obtained, as shown in Fig. 4.7, where τtw = (τ13 + τ12 ), σtw = (σ13 + σ12 ). In fact, in the triaxial experiments, because there is a relation τ13 = τ12 , so the relation between single-shear stress and twin-shear stress is equivalent, namely τ13 = βσ13 is equivalent to (τ13 + τ12 ) = β(σ13 + σ12 ), or τ13 = 2τ0 + βσ13 is equivalent to (τ13 + τ12 ) = 2τ0 + β(σ13 + σ12 ). They all show a certain linear relationship. Therefore, the single-shear theory can be extended to the twin-shear theory.

4.5 Effect of Hydrostatic Stress Hydrostatic stress σm = (σ1 + σ2 + σ3 )/3 has a great influence on the strength of soils. The stress–strain curve of the material under triaxial test can be obtained by applying a certain confining pressure to the specimen, then keeping the confining pressure constant, and then gradually increasing the axial pressure. Similarly, the stress–strain curves of materials under different confining pressures can be obtained,

64

4 Strength Characteristics of Soil

Fig. 4.7 Relation between twin-shear strength and relevant normal stresses. a Medium sand, b compacted sand

as shown in Fig. 4.8. It can be seen that the strength limit of the soil increases as the envelope pressure increases, so it is also possible to derive the law that the ultimate stress circle varies with the change in envelope pressure. So the law of the limit stress circle changing with the change of confining pressure can be obtained. In the axisymmetric triaxial test, the axial stress σ 1 minus hydrostatic pressure σ 3 yields the maximum shear stress, that is, τ max = (σ 1 − σ 3 )/2. Therefore, the results are often expressed as the relationship between shear strength and confining pressure of soil. These curves are not only a reflection of the tension and compression effect, but also a comprehensive reflection of the SD effect and the hydrostatic stress effect. The advantage of the triaxial compression test device is that the structure is simple, and the material parameters, the internal friction angle ϕ, and cohesion C of the soil can be conveniently measured, as well as the drainage conditions can be strictly controlled and the change of pore water pressure in the test piece can be measured. Therefore, the triaxial compression test is widely used in engineering.

Fig. 4.8 Relations between limiting stress circle and hydrostatic pressure

4.6 Effect of Intermediate Principal Stress

65

4.6 Effect of Intermediate Principal Stress Some research topics on the strength theory of geotechnical materials are often caused by existing problems in the strength theory. The effect of intermediate principal stress on soil was the least problem, because the strength of soil under the action of threedimensional stress (σ 1 , σ 2 , σ 3 ) is related to these three actions. Research on the effect of intermediate principal stress is important in both theory and practical engineering. However, the earliest expression for the Tresca yield criterion (1864) f = σ1 − σ3 = σs , the Mohr–Coulomb strength theory (1773–1900) F = σ1 − ασ3 = σt , as well as many empirical criteria all predict that the intermediate principal stress has no effect on the strength of materials. There were no other theories at that time, so they were widely accepted and widely used in engineering. Although this problem has been proposed at the beginning, the study on the intermediate principal stress is difficult. This is because research on complex stress requires more sophisticated facilities, more advanced technology, and much more funds. Moreover, the effect of the intermediate principal stress usually exists in the test of hydrostatic stress, and it needs an explicit concept when testing it independently. Although many test results show that the effect of intermediate principal stress did exist, it is difficult to offer a new strength theory that can reflect the effect of intermediate principal stress with certain physical concepts and simple mathematical expressions. The limit lines on π-plane obtained by the experiment are all larger than that in the Mohr–Coulomb theory. Great efforts were dedicated to the development of true triaxial testing facilities, which then were used to test engineering materials. Some representative efforts were contributed by Tongji University, Hohai University, and Xi’an University of Technology. Many test results show that the effect of intermediate principal stress did exist. Shibata and Karube (1965) at Kyoto University in Japan who have published the results of clay in 1965 and concluded that the shape of stress– strain curve on clay is related to σ 2 . A series of experimental results obtained by Cambridge University, Imperial College London, and Glasgow University does not agree with the Mohr–Coulomb strength theory. Figure 4.9a shows the intermediate principal stress effect of sand summarized by Li GX. The effect of intermediate principal stress can be extended to the effect of the intermediate principal shear stress. The effect of the intermediate principal shear stress has not been discussed before. Some experiments, however, have shown this effect. The law is the same as that of the intermediate principal stress effect. The experimental results of the effect of intermediate principal shear stress on sandstone are obtained by Kwasniewski et al. as shown in Fig. 4.9b, where the horizontal ordinates are τ23 /τ13 and τ23 = (σ2 − σ3 )/2 (and also may be τ12 = (σ1 − σ2 )/2), respectively; The ordinates are friction angle and shear strength τ13 = (σ1 − σ3 )/2 of the material, respectively. The experimental results of other scholars can also be transformed into the results of the intermediate principal shear stress effect for soil.

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Fig 4.9 Effect of intermediate principal stress for sand and sandstone. a Sand Li (2004), b Sandstone (Kwasniewski Li)

4.7 Effect of Stress Angle It is interesting to note that the study of stress angle effect is the same as the intermediate principal stress effect and is also caused by the existing strength theory. This problem is more complex and difficult to find than the effect of intermediate principal stress in Tresca and Mohr–Coulomb criteria. Because their expressions reflect the three principal stresses (σ 1 , σ 2 , σ 3 ), those of two criteria are proposed by the world’s leading dynamicist. So this problem was not attached much attention to until it was again raised by the famous scientists Humpheson Nyalor and Zienkiewicz in 1975 and 1977, they point that: “Unfortunately, the Drucker-Prager criterion gives a very poor approximation to the real failure conditions.” In 1980s, Chen (1982) and Zienkiewicz (1977) pointed out that: “for geotechnical materials, the limit loci of the strength theories on deviatoric plane should not be a circle (that is the circle loci is independent of the stress angle), but it is related to the stress angle.” In 1992 and 1998, Yu MH analyzed the yield surfaces of four cones in detail, namely elongated cones, eclectic cones, compression cones, and inscribed cones, and pointed out the difference between them and geomaterial in which Fig. 4.10 shows the comparison of yield surfaces between four kinds of cone and Mohr–Coulomb strength theory, and Fig. 4.11 shows the comparison of yield surfaces between four kinds of cone and twin-shear strength theory. All circular yield loci must be convex. However, the circular loci cannot be matched with three tensile test points (triangle) as well as the three compression test points (square) of geotechnical materials (σ t /= σ c ), as shown in Fig. 4.12. At that time, however, only the inside hexagon was known as the Tresca yield criterion for non-SD materials and the Mohr–Coulomb failure criterion for geotechnical materials. Chen WF indicated that: “The advantages of the Drucker-Prager criterion are simple and smooth. The limitation of the Drucker-Prager criterion is circular deviatoric trace which contradicts experiments.”

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Fig 4.10 Comparison of yield surfaces between four kinds of cone and Mohr–Coulomb theory

Most experimental results show that the geotechnical material has the effect of stress angle, that is, the intersection line between the limit plane and deviatoric plane is not a circular, as shown in Fig. 4.12. From that point speaking, the Drucker-Prager yield criterion cannot satisfy the condition of stress angle effect. The intermediate stress is taken into account in the Drucker-Prager criterion. The deviation of the Drucker-Prager criterion, however, from the Mohr–Coulomb criterion is surprising, as indicated by Davis and Selvadurai (2002), as shown in Fig. 4.13. It is seen that the Drucker-Prager criterion cannot match the experimental points in uniaxial tension and uniaxial compression meanwhile. Ottosen and Ristinmaa (2005) indicated that: “The Drucker-Prager criterion should be used with caution. In practice, it can only be used with sufficient accuracy when α is small, i.e., when the influence of the hydrostatic stress It is moderate.” Neto et al. (2008) in the book

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Fig 4.11 Comparison of yield surfaces between four kinds of cone and twin-shear theory

“Computational Methods for Plasticity” also pointed out this problem. Tension and compression strength of the material is relatively small material close to the same tensile strength of the material, that is, at this time, the Drucker-Prager criterion is the Mises criterion. Yu (2010) shows clearly that: “The Drucker-Prager yield criterion has been used quite widely in geotechnical analysis. However, experimental research suggests that its circular shape on a deviatoric plane does not agree well with experimental data. For this reason, care is needed when the Drucker-Prager plasticity model is used in geotechnical analysis.” Yu (2018) also specifically pointed out in his monograph: “The criterion that the deviatoric plane is circular does not apply to geotechnical materials.”

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Material with little difference between tension and compression strength actually is close to the material with the same tensile and compression strength, that is, σ t = σ c ; at this time, the Drucker-Prager criterion is the Huber-von Mises criterion. The Drucker-Prager criterion is a curve criterion, so it has not been applied in the analytical solution of soil mechanics problems and the textbook of soil mechanics. In recent years, many scholars have also put forward some theories about the outer boundary of constitutive model plus circle as failure criterion. Similarly, they are also incomplete in theory. However, since the Drucker-Prager criterion is written into many structural analysis commercial software, the application is very convenient, so it has been widely applied in numerical computation of soil mechanics and computational mechanics. After years of practice and research, the problems of the Drucker-Prager criterion have been gradually recognized. Owen, a member of the Royal Society, academicians of the Royal Academy of Engineering and US Academy of Engineering and Chinese Academy of Sciences, pointed out in their academic works that: “It should be emphasized here that any of the above Drucker-Prager criterion to the Mohr–Coulomb criterion can give a poor description of the material behavior for certain states of stress. Thus, according to the dominant stress state in a particular problem to be analyzed, other approximation may be more appropriate.” They also made a graph similar to Fig. 4.13 to illustrate this problem. Therefore, it can be considered that there are some problems in both theory and practice without considering the stress angle effect, which need to be very cautious and preferably not used. Recently, Yu MH summarized these problems theoretically in his monograph “The Unified Intensity Theory and Its Application” (2nd edition) in 2018. Five basic principles were put forward and compared with some typical failure criteria. Through the above analysis, we can also make an important conclusion: “For geotechnical materials, the circular criterion can not be the inner and outer boundaries of failure criteria.” Otherwise, it will be wrong not only in theory but also in concrete application. These have been warned by many famous scholars, Zienkiewicz, Chen, Yu, and Owen et al., and they wrote it into their works. Fig. 4.12 Circular loci cannot be matched with these all basic experimental points

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Fig. 4.13 Drucker-Prager criterion cannot matching the experimental points

An engineering example is the high-slope lock of the Three Gorges Hydropower Station. Researchers have used various failure criteria to calculate and compare. Finally, Drucker-Prager criterion was used as a design criterion. However, the result was that the bottom of the isolation pier was almost entirely calculated as “plastic zone.” In order to solve this problem, a large number of bolts have to be used for reinforcement, and high-strength cement slurry was poured in rock. This treatment not only increases the investment and prolongs the construction period, but also needs to be considered for a long time.

4.8 Convexity and Inter- and Outer Boundary of the Failure Limit Surface of Soils The strength of geotechnical materials under different stress is different. They also differ from each other in the eight quadrants of the stress space as shown in Fig. 4.14. How to use the simple mathematical formula to express the limit surface is a complex problem that needs to solve urgently. The convexity of yield surface provides a theoretical frame for the study of strength theory. A fundamental postulate was proposed by Drucker D.C. The Drucker postulate and its associated convexity of yield surface are regarded as the fundamental law of plasticity. In 1967, Palmer, Maier, and Drucker published new papers demonstrating that the Drucker facility could be extended to softening materials. In 1980s, Professors Li and Deng (University of Science and Technology of China) published a new argument, which also pointed out that Drucker’s postulate could be extended to softening materials and applied to dynamic problems. Due to Drucker’s postulate, the yield surface should be convex, which provides a theoretical frame for the study of strength theory. The yield curves can be composed of a single curve, various lines, or curves, and it can form a sharp point. The convexity and normality of Drucker’s postulate can be illustrated as in Fig. 4.15. However, the shape and size of the yield

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Fig. 4.14 Eight quadrants in principal stress space

surface are not arbitrary and need to be determined according to the experimental results and convexity. According to the Drucker postulate, the yield surface in the stress space must have in the stress a convex surface. If there are two different stress vectors σi j and space and if its following inequality holds: (4.1) where λ is real and 0 < λ < 1, the function of vector σ ij is convex, which is based on the coordinate origin or coordinate axis is taken as the reference. The geometric meaning of this definition is illustrated in Figs. 4.16 and 4.17 for the one-dimensional case. Fig. 4.15 Convexity of Drucker’s postulate

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Fig. 4.16 Convexity of yield surface

Fig. 4.17 Convexity and coordinate axis. a y'' < 0, b y'' > 0

As can be seen, the definition of convexity and the understanding of the word “non-concave” are the same. But it must be specified that convex or concave has different frame of reference. There are different meanings. For example, in Fig. 4.16 of the convex function, from the following axis looking up, it is convex; but from above looking down the curves, it is concave. For consistency, we use the coordinates of the origin or axis as a reference system, as shown in Fig. 4.17. A more general situation of the convex yield surface is shown in Figs. 4.18 and 4.19. In order to coordinate the origin as a reference, the figure of the yield surface is convex everywhere. The yield surface at any stress point should meet the yield condition f (σi J ) = 0. Drawing a tangent for the yield surface at any stress point σ ij , all possible stress point σ ij * within or on the yield surface must be on the side of the tangent A–A, namely the yield surface is convex. It can also be interpreted that a line linking any two points within the convex yield surface is still within this surface. According to convexity, the yield surface cannot be concave, as shown in Fig. 4.19, but it can be sub-smooth yield function together to form a yield surface and allow the formation of corner points, as shown in Fig. 4.20. In addition, because of the change of stress state within the yield surface, the material is elastic; therefore, yield locus is simply connected; it means that the stress vector from the starting coordinates of the origin cannot be intersected to the yield locus twice. It can also be expressed as follows: Connection line between any two points within the yield surface does not cross the yield surface. Otherwise, it is a non-convex shape, as shown in Fig. 4.19.

4.8 Convexity and Inter- and Outer Boundary of the Failure Limit Surface …

Fig. 4.18 Convex yield surfaces

Fig. 4.19 Non-convex yield surfaces

Fig. 4.20 Piecewise convex yield surface

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Fig. 4.21 Inner and outer boundaries and convexity of limit surface

The convexity of the limit surface can also be studied by the yield line on deviatoric plane. For geotechnical materials, the yield line on π-plane must pass through six points a1 , a2 , a3 and a’1 , a’2 , a’3 , as shown in Fig. 4.21a. By using the different curves to connect these six points, we can get different kinds of polygon yield lines. The inequilateral hexagon must be the smallest yield line rather than the inward curve a1 -n-a’3 . This inequilateral hexagon is the limit surface of Mohr–Coulomb strength theory. For SD materials, the limit loci of the Unified Strength Theory in deviatoric plane is shown in Fig. 4.21b. Furthermore, the convex curve on yield line connects these six points which should also be limited. In Fig. 4.21, for example, the convex curve a1 -m’-a’3 which connects a1 and a’3 is the yield line; according to axisymmetry, the inward cusp is formed at point a1 , which violated the convexity of limit surface. The convex yield surfaces have to pass through the tensile test point σ t and compression test point σ c , as well as situate between the two unequal sides hexagon (inside and outside), as shown in Fig. 4.22 (plane stress), where (a) and (b) were put forward by D.D. Ivlev (a professor at the Moscow State University) in 1958 and C.T. Candland (a researcher at the US Army Ballistic Research Laboratory) in 1975, respectively. Now, we all know hexagon inner boundary which is composed of straight lines connecting the experimental points (the star of Fig. 4.22), in the theory, is the Tresca yield criterion and the Mohr–Coulomb failure criterion. They are the lower bound limits of yield criterion for non-SD materials (σ t = σ c ) and SD materials (σ t /= σ c ), respectively. None of the other convex yield surfaces can be less than them. They can be also referred as the single-shear criterion owing to only single-shear stress is taken into account in the mathematical modeling equation.

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Fig. 4.22 Two bounds of yield loci in plane stress state. a non-SD materials (σ t = σ c ), b SD materials (σ t /= σ c )

The single-shear strength theory can be expressed as follows: ( ) f σi j = τ13 = C (Tresca criterion)

(4.2)

( ) f σi j = τ13 + βσ13 = C (Mohr − Coulomb criterion).

(4.3)

The hexagon outer bound which is composed of straight lines connecting the experimental points, in the theory, is the twin-shear strength theory. They are the upper bound limits of yield criterion for non-SD materials (σ t = σ c ) and SD materials (σ t /= σ c ), respectively, as shown in Fig. 4.23. None of the other convex yield surfaces can be larger than them. The twin-shear yield criterion can be expressed as follows: 1 σ1 + σ3 F = τ12 + τ13 = σ1 − (σ2 + σ3 ) = σs , when σ2 ≤ 2 2 F ' = τ23 + τ13 =

1 σ1 + σ3 . (σ1 + σ2 ) − σ3 = σs , when σ2 ≥ 2 2

(4.4a) (4.4b)

The mathematical model of twin-shear strength theory can be expressed as follows: F = τ12 + τ13 + β(σ12 + σ13 ) = C, when τ12 + βσ12 ≥ τ23 + βσ23

(4.5a)

F ' =τ23 +τ13 +β(σ23 + σ13 ) = C, when τ12 + βσ12 ≤ τ23 + βσ23

(4.5b)

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Fig. 4.23 Several possible convex limit surfaces. a Linear limit loci, b Nonlinear limit loci (curve-3, 4)

The principal stress of twin-shear strength theory can be expressed as follows: F = σ1 − F' =

α σ1 + σ3 (σ2 + σ3 ) = σs , when σ2 ≤ 2 2

1 σ1 + σ3 (σ1 + σ2 ) − ασ3 = σs , when σ2 ≥ 2 2

(4.6a) (4.6b)

The determination of limit surface is of great significance. Due to historical reasons, it is only necessary to know the inner bound of the yield surface for geotechnical materials. If the outer bound of the yield surface is also understood in teaching and research, it will be more perfect in theory and better understanding the experimental results (Song 2000). Several possible convex limit surfaces are shown in Fig. 4.23. It should be pointed out that the general criterion cannot reach the outer bound, most of them are situated within the half of inner and outer bounds regions to twothirds. Beyond this range, the limit surface is concave, such as the limit locus-5 in Fig. 4.23b is also a non-convex curve. Therefore, these curves cannot be extended to the full extent, if called it as the unified yield criterion, it is generally regarded as a kind of local unified yield criterion or “pseudo” unified yield criterion. The outer bound of convex yield surface is composed of two straight lines. Therefore, the outer bound cannot be expressed by a general curve, but a piecewise linear criterion. For example, two kinds of three-shear curved yield surface are shown in Fig. 4.24, and their convex yield surfaces can only be extended to a relatively small range. It will be a non-convex yield surface in a large area, which is contrary to the convexity. Therefore, it is impossible for various kinds of curves to cover all convex domains.

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Fig. 4.24 Two kinds of three-shear curved yield surface. a Only cover partial domains, b nonconvex yield surface

Many scholars have made a great deal of research on the limit surface of geotechnical materials under complex stress states. However, these experimental results generally did not agree with the Mohr–Coulomb strength theory. The limit surfaces are situated between the inner and outer boundaries of the Mohr–Coulomb theory and twin-shear strength theory. Some experimental results will be discussed in Chap. 5. For experimental results, four kinds of limit loci of the strength theories have been drawn: Drucker-Prager criterion, a circle as shown in Fig. 4.25; the Unified Strength Theory (b = 0), i.e., the Mohr–Coulomb strength theory; the Unified Strength Theory (b = 0.5), i.e., a new failure criterion; the Unified Strength Theory (b = 1), i.e., the twin-shear strength theory. It is obvious that the Drucker-Prager criterion or other circular criterion cannot match the experimental results of rock. In the following two sections, it also can be found that it does not agree with the results for concrete and soil. For the Unified Strength Theory with different values of b, the limit locus of b = 1 is the upper bound of the convex failure locus, while of b = 0 is the lower bound. The limit locus of the Unified Strength Theory with b = 0.5 agrees with the experimental results of volcanic rock obtained by Mogi. Some of these basic properties are related to each other. Study on the basic characteristics of geotechnical materials under complex stress not only has the significance for the proposed new yield criterion but also for judging, selection, and application of reasonable yield criterion and analysis of geotechnical structure. There are hundreds of yield and failure criteria that can be seen. Various yield criteria and failure criteria have been proposed in the past; however, all of them must be situated between the bounds if the convexity is considered. The lower bound is the single-shear strength theory (the Mohr–Coulomb strength theory) and the singleshear yield criterion (the Tresca yield criterion). The upper bound is the twin-shear strength theory and the twin-shear yield criterion. Other failure criteria, which can

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Fig. 4.25 Comparison of experimental results of two kinds of rock

be approximated by the Unified Strength Theory and the unified yield criterion, are situated between these two bounds. The limit loci of the Unified Strength Theory cover all the regions.

References Bai ST, Zhou XG, Chao HY (2002) Physico-mechanical properties of soft rock materials. J Hydroelectric Power 04:34–44 (in Chinese) Chen ZY, Wang XG (2005) Rock slope stability analysis: theory, method and programs. China Water Power Press, Beijing (in Chinese) Chen WF (1982) Plasticity in reinforced concrete. McGraw–Hill, New York Davis RO, Selvadurai APS (2002) Plasticity and geomechanics. Cambridge University Press, Cambridge, pp 74–75 Kavazanjian E, Matasovic N, Bonaparte R, Schmertmann GR (1995) Evaluation of MSW properties for seismic analysis. In: Proceedings of geo-environment, New Orleans, p 46 Neto EA, Peric D, Owen DRJ (2008) Computational methods for plasticity. Wiley, UK Ottersen NS, Ristinmaa M (2005) The mechanics of constitutive modeling. Elsevier, Amsterdam Parry RH (1956) Strength and deformation of clay. Ph.D. thesis, London University Shibata T, Karube D (1965) Influence of the variation of the intermediate principal stress on the mechanical properties of normally consolidated clays. Proc Sixth ICSMFE 1:359–363 Song GY (2000) Soil mechanics. Luo D, Yao YP translation. China Water Power Press Tang L (1981) The failure criterion of sand. Chin J Geotech Eng 3(2):1–7 (in Chinese) Yu HS (2010) Plasticity and geotechnics. Springer, New York, p 80 Yu MH (2018) Unified strength theory and its applications, 2nd edn. Springer and XJU Press Zienkiewicz OC, Pande GN (1977) Some useful forms of isotropic yield surfaces for soil and rock mechanics. Finite Elem Geomech Gudehus G ed. Wiley, London, pp 179–190

Chapter 5

Yu Unified Strength Theory

5.1 Introduction In the common soil mechanics, only the shear strength τ 0 of soils is discussed in most cases. With consideration of τ 0 = τ 13 0 = (σ 1 − σ 3 )/2, the strength of soil in common soil mechanics only related to the maximum principal stress σ 1 and minimum principal stress σ 3 . The influence of the intermediate principal stress σ 2 is ignored. Craig indicated “The Mohr–Coulomb failure criterion, because of its simplicity, is widely used in practice although it is by no means the only possible failure criterion for soils” (Craig 2004). The Mohr–Coulomb failure criterion can be expressed as f = τ + βσ = τ13 + βσ13 =

1 [( σ1 − σ3 ) + β(σ1 + σ3 )] 2

It is found that the above equation only takes the maximum principal stress σ 1 and the minimum principal stress σ 3 into account, and the influence of the intermediate principal stress σ 2 is neglected. Thus, the Mohr–Coulomb strength theory can also be referred to as the single-shear strength theory. The common soil mechanics based on the Mohr–Coulomb strength theory may also be referred to as “single-shear soil mechanics.” However, most of the soils in engineering are subjected to the threedimensional stresses including the three basic problems of soil mechanics (Drucker 1951; Naghdi 1960; Palmer 1967; Deng 1987; Li et al. 1988; Chen 1982).

© Springer Nature Singapore Pte Ltd. and Zhejiang University Press 2023 M.-H. Yu, Soil Mechanics, https://doi.org/10.1007/978-981-99-2781-4_5

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It is desired that there is a Unified Strength Theory which can be easily used. For isotropic soil materials, a Unified Strength Theory should have the following characteristics: (1) It should agree well with the experimental data. (2) It should be consistent with the Drucker postulate. (3) The yield loci of the Unified Strength Theory should be convex and cover all region from inner bound to outer bound. (4) It should have a clear physical meaning and a unified mechanical model. (5) The strength theory should be expressed by a simple and unified mathematical equation. (6) It should take into account the influence of all stress components σ 1 , σ 2 , and σ 3 on the material failure. (7) The material parameters should be as few as possible and easily determined by experiments. (8) It is hoped that the Unified Strength Theory has a simple linear form, which is convenient for application to analytical solutions, general manual calculations, engineering design and theoretical analysis. In addition, in order to meet various experimental results, it is possible to make the soil strength theory become very complex. And there are so many parameters that affect the deformability and strength, which will cause problems in practice. From the historical development of strength theory in the twentieth century, it is difficult to have a Unified Strength Theory that is easy to apply.

5.2 Voigt-Timoshenko Conundrum What is the relationship among various strength theories? Can we propose a new criterion which can be suitable to more kinds of materials? Considerable effort has been devoted to this important problem by many scientists, such as Bauschingerin (1833–1893) at Munich Technical University, Mohr (1835–1918) at Stuttgart University and Dresden University, A. Foppl (1854–1924) at Munich Technical University, Voigt (1850–1919), Prandtl (1875–1953) and von Karman at Gottingen University, and others. After the presentation of the Mohr strength theory by Mohr (1835–1918) at Dresden University in 1900, a lot of experiments of rock and other materials under complex stress state were done by Professor Voigt for checking the Mohr theory at Gottingen University, Germany. These experimental results, however, were not in agreement with Mohr’s theory. In 1901, Voigt came to the conclusion that the question of strength is too complicated, and it is impossible that a single strength criterion can be applied to all kinds of structural materials (Voigt 1901). Voigt (1850–1919) acquired an interest in theory of elasticity and prepared his doctor’s thesis on the elastic properties of rock salt in 1874. He was elected to a chair at Gottingen University, Germany, in 1883, where he instituted courses in theoretical physics. He also set up a laboratory in which worked on the theory of elasticity. He

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was especially interested in the elastic properties of materials and did outstanding work in that subject. Fifty years later, Voigt’s conclusion has not yet been settled. In 1953, Timoshenko wrote in History of Strength of Materials that “A number of tests were made with combined stresses with a view to checking Mohr’s theory. All these tests were made with brittle materials and the results obtained were not in agreement with theory. Voigt came to the conclusion that the question of strength is too complicated and that it is impossible to devise a single theory for successful application to all kinds of structural materials” (Timoshenko 1953). Voigt’s conclusion turns into the Voigt-Timoshenko Conundrum. At the same period, several new strength theories were proposed. The Hubervon Mises yield criterion was proposed by Huber in 1904 and by Mises in 1913. The Drucker-Prager criterion was proposed in 1952. The Huber-von Mises criterion was considered as the best one. In 1968, a monograph of plasticity was published by Mendelson. He stated that: “From an engineering viewpoint the accuracy of the von Mises criterion for yielding is amply sufficient, …the search for more accurate theories, particularly since they are bound to be more complex, seems to be a rather thankless task”. (Mendelson 1968). It means that the simpler and better theory than the Mises criterion is impossible. The same thought as the Voigt-Timoshenko Conundrum was expressed in Encyclopedia of China (1985) and in the paper of Yu et al. (1985). It was said that “it is impossible to establish a Unified Strength Theory for various materials.” This conclusion was cleared away in the second edition of Encyclopedia of China (2009). Can we propose a new criterion that is suitable to more kinds of materials? Fortunately, a theoretical frame relating the yield criterion was appeared in 1951. This is the Drucker postulate. The study of yield criterion may be developing on a more reliable theoretical basis.

5.3 Mechanical Model of the Yu Unified Strength Theory Mechanical and mathematical models are powerful means for establishing and understanding the development of a new theory. Mechanical modeling is an abstraction, a formation of an idea or ideas that may involve the subject with special configurations. Mathematical modeling may involve relationships between continuous functions of space, time and other variations. To express the general nature of the strength theory, the cubic element is often used in solid mechanics and engineering. It is clear that there are three principal stresses σ 1 , σ 2 , and σ 3 acting on the cubic element as shown in Fig. 5.1. The Mohr–Coulomb strength theory was widely used in mechanics and engineering. The mechanical model can be illustrated by a hexagonal prism element, as shown in Fig. 5.2. A series of single-shear stress theories, such as the Tresca criterion and the Mohr–Coulomb strength theory, can be obtained from this model.

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Fig. 5.1 Principal stresses model

Fig. 5.2 Single-shear model

It can be seen that the single-shear strength theory only takes the maximum principal shear stress τ 13 and the corresponding normal stress σ 13 into account on the failure of materials. The effect of the intermediate principal stress σ 2 is not taken into account in the Tresca and the Mohr–Coulomb strength theory. So they have obvious shortcomings in describing the realistic characteristics of materials, since even if the value of σ 2 is zero, the other principal shear stresses also reflect the effect of σ 2 . The principal stress state (σ 1 , σ 2 , σ 3 ) can be converted into the principal shear stress state (τ 13 , τ 12 , τ 23 ). There are only two independent principal shear stresses since τ 13 = τ 12 + τ 23 , which has been described in Chap. 2. The shear stress state can be converted into the twin-shear stress state (τ 13 , τ 12 ; σ 13 , σ 12 ) or (τ 13 , τ 23 ; σ 13 , σ 23 ). It can be illustrated by the new orthogonal octahedral element model as follows: The twin-shear stress model is different from the Isoclinic octahedral model. Orthogonal octahedron can also be called double shear octahedron. The orthogonal octahedral model consists of two groups of four sections that are perpendicular to each other and are acted on by the maximum shear stress τ 13 and the intermediate principal stress τ 12 or τ 23 . The orthogonal octahedral element can be divided into two cases, as shown in Fig. 5.3 and in the chromatic figure before Chap. 2. They are available to use for the mechanical model of strength theory. Based on the orthogonal octahedral element, the UST can be developed.

5.4 Mathematical Modeling of the Yu Unified Strength Theory

(a)

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(b)

Fig. 5.3 Twin-shear model. a Twin-shear element τ13 , τ12 , b Twin-shear element τ13 , τ23

5.4 Mathematical Modeling of the Yu Unified Strength Theory Based on this new model, and taking into account the all stress components acting on the twin-shear element and the different effects of various stresses on the failure of materials, the mathematical modeling of the UST was established by Yu in 1990 (Yu 1994, 2004) as follows: F = τ13 + bτ12 + β(σ13 + bσ12 ) = C, when τ12 + βσ12 ≥ τ23 + βσ23 (5.1a) F = τ13 + bτ23 + β(σ13 + bσ23 ) = C, when τ12 + βσ12 ≤ τ23 + βσ23 (5.1b) where b is a parameter that reflects the influence of the intermediate principal shear stress τ 12 or τ 23 on the failure of material; β is the coefficient that represents the effect of the normal stress on failure; C is a strength parameter of material; τ 13 , τ 12 , and τ 23 are principal shear stresses and σ 13 , σ 12 , and σ 23 are the corresponding normal stresses acting on the sections where τ 13 , τ 12 , and τ 23 act. They are defined as 1 1 1 (σ1 − σ3 ); τ12 = (σ1 − σ2 ); τ23 = (σ2 − σ3 ) 2 2 2 1 1 1 = (σ1 + σ3 ); σ12 = (σ1 + σ2 ); σ23 = (σ2 + σ3 ) 2 2 2

τ13 = σ13

(5.2)

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5.5 Experimental Determination of Material Parameters The magnitude of β and C can be determined by experimental results of uniaxial tension strength σt and uniaxial compression strength σc ; the experimental conditions are: σ1 = σt , σ2 = σ3 = 0 and σ1 = σ2 = 0, σ3 = −σc

(5.3)

So the material constants β and C can be determined: β=

σc − σt 1−α (1 + b)σc σt (1 + b) , C= σt = = σc + σt 1+α σc + σt 1+α

(5.4)

5.6 Mathematical Expression of the Yu Unified Strength Theory Substituting β and C into the Eqs. (5.1a) and (5.1b), the UST is now obtained. It can be expressed in terms of principal stresses as follows: F = σ1 − F =

α (bσ2 + σ3 ) = σt , when 1+b

σ2 ≤

σ1 + ασ3 1+α

1 σ1 + ασ3 (σ1 + bσ2 ) − ασ3 = σt , when σ2 ≥ 1+b 1+α

(5.5a) (5.5b)

The UST is the result of successive research by the author from 1961 to 1991. It is the extension of the twin-shear yield criterion for non-SD material, the twin-shear strength theory for SD material (Yu et al. 1985), the twin-shear ridge model, and the twin-shear multi-parameter criteria. The UST gives a series of failure criteria and establishes a relationship among various failure criteria. This UST has all of the desired characteristics mentioned above; it can be easily extended and developed in the future, for example, the establishment of the related unified elasto-plastic constitutive equation (Yu et al. 1994, 1997). Though the mathematical expression of the UST is very simple and linear, it has rich and varied contents. It gives good agreement with existing experimental data. The comparisons of the UST with the experimental results will be discussed in the next chapter.

5.7 Other Formulations of the Yu Unified Strength Theory

85

5.7 Other Formulations of the Yu Unified Strength Theory The UST is expressed in terms of the principal shear stresses and principal stresses. It can also be expressed in other terms as follows:

5.7.1 Yu Unified Strength Theory Cohesion C and Friction Angle Φ Expression In actual engineering, the material parameters of soil are often expressed in terms of cohesion C and friction angle ϕ; in this case, the Yu Unified Strength Theory can be expressed as: 1 1 F = σ1 − (bσ2 + σ3 ) + σ1 + (bσ2 + σ3 ) sin ϕ0 = 2C0 cos ϕ0 1+b 1+b

1 sin ϕ0 (5.6a) (σ1 + σ3 ) + (σ1 − σ2 ) 2 2 1 1 F (σ1 + bσ2 ) − σ3 + (σ1 + bσ2 ) − σ3 sin ϕ0 = 2C0 cos ϕ0 1+b 1+b σ2 ≤

σ2 ≥

1 sin ϕ0 (σ1 + σ3 ) + (σ1 − σ3 ) 2 2

(5.6b)

The relationship between C0 and ϕ0 and other material parameters is: α=

1 − sin ϕ0 2C0 cos ϕ0 σt = 1 + sin ϕ0 1 + sin ϕ0

(5.7)

5.7.2 Unified Theoretical Expression of Strength with Positive Compressive Stress Unified theoretical expression of strength with positive compressive stress 1 − sin ϕ0 bσ2 + σ3 2C0 cos ϕ0 σ1 − = 1 + sin ϕ0 1+b 1 + sin ϕ0 1 sin ϕ0 when σ2 ≤ (σ1 + σ3 ) − (σ1 − σ3 ) h 2 2

F=

(5.8a)

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5 Yu Unified Strength Theory

1 − sin ϕ0 (σ1 + bσ2 ) − σ3 = 2Cϕ0 cos0 (1 + b)(1 + sin0 ) 1 sin ϕ0 when σ2 ≥ (σ1 + σ3 ) − (σ1 − σ3 ) h 2 2

F =

(5.8b)

Starting from a unified mechanical model, the Yu Unified Strength Theory considers all the stress components of the stress state and their different effects on the yield and failure of materials. It establishes a brand-new Yu Unified Strength Theory and a series of new typical calculation criteria, which can be flexibly applied to different materials. Although the mathematical expression of the Yu Unified Strength Theory is very simple, in the subsequent elaboration, we can see that it has a very wide and rich connotation and is consistent with most of the true triaxial test results we have seen. In Sect. 5.9, we will compare the Yu Unified Strength Theory with the experimental results.

5.8 Relation Among the Parameters of the UST The relation among shear strength τ 0 , the uniaxial tensile strength σ t , and uniaxial compressive strength σ c can be determined as follows: b=

(1 + α)τ0 − σt 1+α− B σt 1+b+α σt = = ,α = , B = σt − τ0 B−1 τ0 τ0 1+b

(5.9)

The ratio of shear strength to tensile strength of materials can be introduced from the UST as follows: ατ =

τ0 1+b = σt 1+b+α

(5.10)

It is shown that: 1. The ratio of shear strength to tensile strength α τ = τ 0 /σ t of brittle materials (α τ < 1) is lower than that of ductile materials (α τ = 1). This agrees with the experimental data. 2. The limit surface may be non-convex when the ratio of shear strength to tensile strength α τ < 1/(1 + α) or α τ > 2/(2 + α). 3. The shear strength of the material is lower than the tensile strength of the same material. This is true for metallic materials. It needs, however, further study for other materials. The UST with the tension cutoff (similar to the Mohr–Coulomb theory with tension cutoff suggested by Paul in 1961) has to be supplemented in the state of three tensile stresses. 4. The UST can be expressed in another form as follows:

5.9 Special Cases of the UST for Different Parameter b

87

F = σ1 − (1 + α − B)σ2 − (B − 1)σ3 = σt , when

F =

1+α− B B−1 σ1 + σ2 − ασ3 = σt , α α

σ2 ≤

when

σ2 ≥

σ1 + ασ3 1+α (5.11a) σ1 + ασ3 1+α (5.11b)

5.9 Special Cases of the UST for Different Parameter b The Yu Unified Strength Theory (UST) contains two families of yield criteria and failure criteria. The first family is the convex failure criteria. A series of convex failure criteria can be deduced from the UST by giving a certain value to parameter b. The series of convex yield criteria (α = 1) is its special cases. Another family is the non-convex criteria, which can be obtained when b < 0 or b > 1. The convex failure criteria will be studied in this section, and the non-convex failure criteria will be discussed in the next section. The parameter b reflects the influence of the intermediate principal shear stress τ 12 or τ 23 on the failure of a material. It also reflects the influence of the intermediate principal stress σ 2 on the failure of a material. We can see below that b is also the parameter that determines the formulation of a failure criterion. A series of convex failure criteria can be obtained when the parameter varies in the range of 0 ≤ b ≤ 1. The five types of failure criteria with the values of b = 0, b = 1/4, b = 1/2, b = 3/4, and b = 1 are introduced from the UST in the following sections. 1. b = 0 The Mohr–Coulomb strength theory can be deduced from the UST with b = 0 as follows: F = F = σ1 − ασ3 = σt F = F =

1 σ1 − ασ3 = σc α

(5.12a) (5.12b)

2. b = 1/4 A new failure criterion is deduced from the UST with b = 1/4 as follows: α σ1 + ασ3 (σ2 + 4σ3 ) = σt , σ2 ≤ 5 1+α

(5.13a)

1 σ1 + ασ3 (4σ1 + σ2 ) − ασ3 = σt , σ2 ≥ 5 1+α

(5.13b)

F = σ1 − F =

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5 Yu Unified Strength Theory

3. b = 1/2 A new failure criterion is deduced from the UST with b = 1/2 as follows: F = σ1 − F =

α σ1 + ασ3 (σ2 + 2σ3 ) = σt , σ2 ≤ 3 1+α

1 σ1 + ασ3 (2σ1 + σ2 ) − ασ3 = σt , σ2 ≥ 3 1+α

(5.14a) (5.14b)

Since the Drucker-Prager criterion cannot match with the practice, this criterion is more reasonable and can be substituted for the Drucker-Prager criterion. 4. b = 3/4 A new failure criterion is deduced from the UST with b = 3/4 as follows F = σ1 − F =

α σ1 + ασ3 (3σ2 + 4σ3 ) = σt , σ2 ≤ 7 1+α

1 σ1 + ασ3 (4σ1 + 3σ2 ) − ασ3 = σt , σ2 ≥ 7 1+α

(5.15a) (5.15b)

5. b = 1 A new failure criterion is deduced from the UST with b = 1. The mathematical expression is F = σ1 − F =

α σ1 + ασ3 (σ2 + σ3 ) = σt , when σ2 ≤ 2 1+α

1 σ1 + ασ3 (σ1 + σ2 ) − ασ3 = σt , when σ2 ≥ 2 1+α

(5.16a) (5.16b)

This is the generalized twin-shear strength model proposed by Yu in 1983 (1985).

5.10 Special Cases of the UST by Varying Parameter a 1. α = 1, The Unified Yield Criterion When the tensile strength and the compressive strength are identical, the tension– compressive strength ratio α = σ t /σ c equals 1, or the friction angle ϕ = 0. The unified yield criterion can be obtained as a special case of the UST. The mathematical

5.11 Limit Loci of the UST by Varying Parameter b in the π-Plane

89

expression of the unified yield criterion is expressed in Chap. 3. It also contains a series of yield criteria that were described already. 2. α = 1/2 A new series of failure criteria can be obtained from the UST with α = 1/2. The mathematical expressions of this series are σ1 + ασ3 1 (bσ2 + σ3 ) = σt , when σ2 ≤ 2(1 + b) 1+α

(5.17a)

1 σ1 + ασ3 1 (σ1 + bσ2 ) − σ3 = σt , when σ2 ≥ 1+b 2 1+α

(5.17b)

F = σ1 − F =

Figure 5.4 shows the relationship between the UST and some existing strength theories as well as some new failure criteria. A great number of new failure criteria are given.

5.11 Limit Loci of the UST by Varying Parameter b in the π-Plane The mathematical expression of the UST in terms of principal stresses is as follows: F = σ1 − F =

α σ1 + ασ3 (bσ2 + σ3 ) = σt , when σ2 ≤ 1+b 1+α

σ1 + ασ3 1 (σ1 + bσ2 ) − ασ3 = σt , when σ2 ≥ 1+b 1+α

(5.18a) (5.18b)

The relationships between the coordinates of the deviatoric plane and hydrostatic stress axis z with the principal stresses are: 1 1 1 x = √ (σ3 − σ2 )y = √ (2σ1 − σ2 − σ3 )z = √ (σ1 + σ2 + σ3 ) 2 6 3 √ √ √ 1 √ 1 √ ( 6y + 3z), σ2 = (2 3z − 6y − 3 2x), 3 6 √ √ 1 √ σ3 = (3 2x − 6y + 2 3z) 6

(5.19)

σ1 =

(5.20)

By substituting Eqs. (5.19) and (5.20) into Eqs. (5.5a) and (5.5b), the equations of the UST in the deviatoric plane can be obtained: √

2(1 − b) F =− αx + 2(1 + b)

√

6(2 + α) y+ 6

√ 3(1 − α) z = σt 3

(5.21a)

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5 Yu Unified Strength Theory

Fig. 5.4 UST and its special cases

√ √ √ 2−b 2 6 3(1 − α) b +α x+ +α y+ z = σt F =− 1+b 2 1+b 6 3

(5.21b)

A great number of new failure criteria can be generated from the UST by changing α and b. The general shape of the limit loci of the UST on the deviatoric plane is shown in Fig. 5.5. Material parameters α and σ t are the tension–compression strength ratio and the uniaxial tensile strength, respectively, and b is a material parameter that reflects the influence of intermediate principal shear stress. A series of limit surfaces can be

5.11 Limit Loci of the UST by Varying Parameter b in the π-Plane

91

Fig. 5.5 Serial yield loci of the Unified Strength Theory (Yu 1994)

obtained by varying b. Five special cases will be discussed with values of b from b = 0, b = 1/4, b = 1/2, b = 3/4 and b = 1. 1. b = 0 Substituting b = 0 into Eqs. (5.21a) and (5.21b), we have √ F = F = −

√ √ 2 6 3 αx + (2 + α)y + (1 − α)z = σt 2 6 3

(5.22)

This is the Mohr–Coulomb strength theory. The limit locus of the Mohr–Coulomb strength theory is the lower bound of the convex limit loci, as shown in Fig. 5.5. 2. b = 1/4 Substituting b = 1/4 into Eqs. (5.21a) and (5.21b), we have √ √ √ 6 3 3 2 αx + (2 + α)y + (1 − α)z = σt F =− 10 6 3 √ √ √ 7 2 6 3 1 +α x+ +α y+ (1 − α)z = σt F =− 5 2 5 6 3

(5.23a)

(5.23b)

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5 Yu Unified Strength Theory

This is the limit surface of a new failure criterion. The limit locus is close to the broken line shown in Fig. 5.5. 3. b = 1/2 Substituting b = 1/2 into Eqs. (5.21a) and (5.21b), we have √

√ √ 2 6 3 αx + (2 + α)y + (1 − α)z = σt F =− 6 6 3 √ √ √ 1 2 6 3 F =− +α x + (1 + α) y+ (1 − α)z = σt 3 2 6 3

(5.24a)

(5.24b)

This is a new failure criterion. It is intermediate between the Mohr–Coulomb strength theory and the twin-shear strength theory. The limit locus of the new criterion on the deviatoric plane is also shown in Fig. 5.5. 4. b = 3/4 Substituting b = 3/4 into Eqs. (5.21a) and (5.21b), we have √

√ √ 2 6 3 αx + (2 + α)y + (1 − α)z = σt F =− 14 6 3 √ √ √ 3 5 2 6 3 F =− +α x+ +α y+ (1 − α)z = σt 7 2 7 6 3

(5.25a)

(5.25b)

This is the limit surface of a new failure criterion. The limit locus is close to the limit locus of the twin-shear strength theory, as shown in Fig. 5.5. 5. b = 1 Substituting b = 1 into Eqs. (5.21a) and (5.21b), we have √

√ 6 3 (2 + α)y + (1 − α)z = σt F= 6 3 √ √ √ 1 2 6 3 1 +α x+ +α y+ (1 − α)z = σt F = − 2 2 2 6 3

(5.26a)

(5.26b)

This is the twin-shear strength theory proposed by Yu M-H in 1985. The limit locus of the twin-shear strength theory is the upper bound of the convex limit loci, as shown in Fig. 5.5. A great number of failure criteria can be deduced from the UST. As discussed above, the UST gives a series of new yield and failure criteria, establishes a relationship among various failure criteria, and encompasses previous yield criteria and failure criteria as its special cases or linear approximations, as shown in Fig. 5.6. In particular, the UST with b = 1/2 and b = 3/4 can serve as a new criterion, which can conveniently replace the smooth-ridge models. The UST has

5.12 Limit Loci of the Yu Unified Strength Theory in Plane Stress State

93

Fig. 5.6 System of the UST (Yield loci of the unified yield criterion on deviatoric plane)

clear physical meaning and a unified mechanical model. It embraces all the criteria from the lower bound to the upper bound. The UST is very simple but can be used widely.

5.12 Limit Loci of the Yu Unified Strength Theory in Plane Stress State The yield loci of the Yu Unified Strength Theory in the plane stress state are the intersection line of the yield surface in principal stress space and the plane σ 1 − σ 2 . Its shape and size depend on the values of b and α. It will be transformed into hexagon when b = 0 or b = 1 and into dodecagon when 0 < b < 1.

94

5 Yu Unified Strength Theory

(a)

(b)

Fig. 5.7 Variation of the limit loci of the UST in plane stress state, a α = 1/2, b α = 2/3

The equations of the 12 yield loci of the Yu Unified Strength Theory in the plane stress state can be given as follows. αb σ2 = σt σ1 − 1+b αb σ2 − 1+b σ1 = σt α σ1 − 1+b σ2 = σt α σ2 − 1+b σ1 = σt α + σ2 ) = −σt (bσ 1 1+b α + σ1 ) = −σt 1+b (bσ2

1 + bσ2 ) = σt 1+b (σ1 1 + σ2 ) = σt (bσ 1 1+b 1 σ − ασ2 = σt 1+b 1 1 σ − ασ1 = σt 1+b 2 b σ − ασ2 = σt 1+b 1 b σ − ασ1 = σt 1+b 2

(5.27)

A series of new failure criteria and new yield loci in the plane stress state can be obtained from the Yu Unified Strength Theory. The yield loci of the Yu Unified Strength Theory (UST) in the plane stress state with different values of b are shown in Fig. 5.7a for α = 1/2 material, Fig. 5.7b for α = 2/3 materials) and Fig. 5.8 (for α = 1/4 material). The relations between some traditional strength theory and various yield loci of the Yu Unified Strength Theory in the plane stress state are shown in Fig. 5.9.

5.13 Limit Surface of the Yu Unified Strength Theory The Yu Unified Strength Theory is not a single criterion; it is a series of failure criteria and a system of strength theory. The yield surfaces in stress space of the Yu Unified Strength Theory are usually a semi-infinite hexagonal cone with unequal sides and a dodecahedron cone with unequal sides. The shape and size of the yield hexagonal cone depend on the parameter b and on the tension–compression strength ratio α.

5.14 Limit Surfaces of the Yu Unified Strength Theory Drawed …

95

Fig. 5.8 Yield loci of the UST in the plane stress state (α = 1/4)

The Mohr–Coulomb criterion and the twin-shear strength criterion are classical and commonly used empirical theoretical formulas, and they are all special cases of the Yu Unified Strength Theory. As shown in Eq. (5.5), a series of convex failure criteria can be deduced from the Yu Unified Strength Theory by giving a certain value to parameter b (0 ≤ b ≤ 1). Equation (5.5) is the Mohr–Coulomb criterion when b = 0, as shown in Fig. 5.10; Eq. (5.5) is the generalized Tresca criterion when b = 0 and ϕ = 0; Eq. (5.5) is the new yield criterion when b = 0.5, as shown in Fig. 5.11. Equation (5.5) is the twin-shear strength criterion when b = l.0, as shown in Fig. 5.12. Limit surfaces of the Yu Unified Strength Theory are shown in Fig. 5.13. When the tensile and compressive strength of the material is equal, the Yu Unified Strength Theory is degenerate into the unified yield criterion. Its spatial yield surface is shown in Fig. 5.14.

5.14 Limit Surfaces of the Yu Unified Strength Theory Drawed by Kolupave-Altenbach The Yu Unified Strength Theory has been widely studied and applied by domestic and foreign scholars since it was proposed by Yu in 1991. Recently, German scholars have studied the visualization of the Yu Unified Strength Theory (Kolupaev et al. 2013). The Yu Unified Strength Theory provides the fundamentals for the systematic study of various strength hypotheses and yields criteria for isotropic materials. Kolupaev and Altenbach summarized the variation of limit surfaces of the Yu Unified Strength Theory on deviatoric plane with α and k, as shown in Fig. 5.15.

96

5 Yu Unified Strength Theory

Fig. 5.9 Variation of the UST and the relationships among the criteria (plane stress state)

5.14 Limit Surfaces of the Yu Unified Strength Theory Drawed …

97

UST, b=0 (single shear strength theory) Fig. 5.10 Yield surface and the yield loci of the single-shear strength theory (Mohr–Coulomb theory) (b = 0)

UST, b=0.5

Fig. 5.11 Yield surface and the yield loci of a typical case of the UST (b = 0.5)

The abscissa is the ratio of tension to compression of the material, and the ordinate is the comprehensive variable k related to the tensile strength, compressive strength, √ and the Yu Unified Strength Theory parameter b, and k = 3(1 + b) (1 + b + α). . The lower bound is provided by the single-shear strength theory (the Mohr–Coulomb strength theory, or the Yu Unified Strength Theory with b = 0). The upper bound is given by the generalized twin-shear strength theory or the Yu Unified Strength Theory with b = 1. The median is a new series of yield criteria deduced from UST with b = 1/2. Other series of new yield criteria can also be deduced from Yu Unified Strength Theory with b = 1/4 or b = 3/4, as shown in Fig. 5.15. All the convex yield surfaces are situated between two bounds of the twin-shear theory and the single-shear theory.

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5 Yu Unified Strength Theory

UST, b=1.0 (twin shear strength theory)

Fig. 5.12 Yield surface and the yield loci of the twin-shear strength theory (b = 1) Fig. 5.13 Limit surfaces of the Yu unified strength theory

Fig. 5.14 Limit surfaces of the unified yield criterion

5.15 Comparison Between the UST and Experimental Results

99

Fig. 5.15 Variation of the shape of yield loci of the UST with parameters b and α

Yu Unified Strength Theory has also been written into academic books by German scholars in 2014. Details can be found in the literature (Altenbach and Öchsner 2014).

5.15 Comparison Between the UST and Experimental Results In addition, it needs to be pointed out that although some complex stress tests can produce especial complex stresses σ 1 , σ 2 , and σ 3 , they are all situated in a special plan. For example, the triaxial compression test (axisymmetric triaxial test) is widely used in soil mechanics. A special triaxial stress state can be produced, i.e., the stress state of σ 1 ≥ σ 2 = σ 3 or σ 1 = σ 2 ≥ σ 3 for axisymmetric triaxial experiments can be obtained, as shown in Fig. 5.16. In this experiment, the soil is subjected to the action of σ 1 , σ 2 , σ 3 , so this kind of test is usually called the triaxial test, although it involves only very special combinations of triaxial stress. However, no matter what the isotropic consolidation, K 0 consolidation, triaxial compression shear, or triaxial elongation shear, two stresses in them are always equal, i.e., σ 2 = σ 3 or σ 1 = σ 2. In fact, this kind of complex stress in the three-dimensional stress space is in a special plane, as shown in Fig. 5.17. But it should be noted that the axisymmetric triaxial compression test is only to obtain the material parameters of soil, while cannot be compared and verified on various strength theory.

100

5 Yu Unified Strength Theory

(a)

(b)

(c)

(d)

Fig. 5.16 Stress and strain conditions of various tests in triaxial instrument Fig. 5.17 Corresponding plane in stress space is produced by axisymmetric triaxial test

Over the past three decades, many experimental results for limit surface of soil materials under complex stress states have been published, and they do not agree with the Mohr–Coulomb strength theory. The relative independence of study on strength theory and experimental verification is available. These experiments were carried out by many researchers. Figure 5.18a, b shows the limit loci of Ottawa fine sand given by Dakoulas and Sun. The inner bound is the Mohr–Coulomb strength theory, the outer bound is the twin-shear strength theory, and the middle bound is the Yu Unified Strength Theory with b = 1/2. The Mohr–Coulomb strength theory, twinshear stress theory, many other new criteria, and the concave strength theory can

5.16 Significance of the Yu Unified Strength Theory

(a)

101

(b)

Fig. 5.18 Limit loci for Ottawa fine sand (Dakoulas and Sun 1992), a loose sand, b dense sand

also be obtained from the UST. At present, most of the deviatoric plane experimental results are within the range of the UST.

5.16 Significance of the Yu Unified Strength Theory From simple shear theory (Tresca1884–Moh1900) to three-shear strength theory (Drucker Prager 1952 Argyris Gudehus, Matsuoka Nakai 1973), to double shear theory (Yu, 1961–1983), and finally to Unified Strength Theory (Yu 1994). Its three development can be described by the changes in the shape of the limit loci in the πplane, as shown in Fig. 5.19. The inner bound is the Mohr–Coulomb strength theory, and the outer bound is the twin-shear strength theory. The limit loci of the UST covers all areas from the inner bound to the outer bound, which has a more fundamental theoretical significance. In 2008, the Chairman of Chinese Society of Rock Mechanics and Engineering, Qian QH, pointed out that: “The further development of single-shear theory is the twin-shear theory, and the further development of the twin-shear theory is the Yu Unified Strength Theory. Single-shear theory and twinshear theory and the other failure criteria between the single-shear and twin-shear theories are special cases of Yu Unified Strength Theory or linear approximations. It can be said that the UST is making an outstanding contribution to the development history of strength theory”. The UST is the generalization and materialization of the convexity of the Drucker postulate, which is a significant progress following the convexity of the Drucker postulate. It is not a single yield criterion suitable only for one kind of material, but a completely new system. All the yield loci and the failure loci conform the convexity of the Drucker postulate. The UST embraces many well-established criteria as its

102

5 Yu Unified Strength Theory

Fig. 5.19 Development of strength theory: from single shear to three shear to twin shear to uniform

special cases or linear approximations. It also gives a series of new failure criteria and can be developed to a more widely used theory. In addition, it can extended to other broader theoretical and computational criteria; this will be further developed in the following chapters. The relationship between the Yu Unified Strength Theory and the existing strength theory is shown in Fig. 5.4.

5.17 Summary

103

The significance of the UST is summarized as follows: (1) It is suitable for various kinds of materials. (2) It is the generalization and materialization of the convexity of the Drucker postulate, which establishes a complete relationship between the Drucker postulate and strength theory. (3) It covers all region between the inner bound to the outer bound; there are no other rules in the world that can covers all areas

5.17 Summary In this chapter, based on a unified physical model that takes into account the effects of all the shear stress components and their normal stress components on the failure of materials, a new strength theory was proposed by Yu. It has a unified mathematical expression and contains a series of yield criteria. The mathematical expression of the UST can be expressed into various forms. More than ten kinds of expressions are discussed in this chapter. The two material parameters of UST are σ t and α = σ t /σ c, which are the same as the parameters used in the Mohr–Coulomb strength theory, Drucker-Prager criterion, the twin-shear strength theory, and other two-parameter criteria. The UST parameter b is a coefficient that reflects the influence of the intermediate principal shear stress and corresponding normal stress on the yield of materials. The Yu Unified Strength Theory is a series of basic theoretical innovation achievements made by Yu from 1961 to 1991. A new twin-shear element model, new mathematical modeling method, that is, the two equations, and additional discriminant and the skillful coordination of multiple parameters are the key innovative achievements of Yu Unified Strength Theory. The mathematical modeling of the UST is given as Eq. (5.1); the principal stress expression is given as Eq. (5.5). The yield surface of the UST in stress space and yield loci on plane stress, deviatoric plane, and meridian plane are illustrated in this chapter. The UST embraces many well-established yield surfaces and yield loci as its special or asymptotic cases, such as yield surfaces of the Tresca yield criterion, the Huber-von Mises yield criterion, and the Mohr–Coulomb strength theory, as well as the twin-shear yield criterion (Yu 1961), the twin-shear strength theory, and the unified yield criterion. The UST forms an entire spectrum of convex criteria, which can be used to describe many kinds of engineering materials. A paper entitled “Considerations on the Unified Strength Theory due to Mao-Hong Yu” is made by Altenbach and Kolupaev, see literature (Kolupaex 2010). Reviews of “Unified Strength Theory and Its Applications” are made by Teodorescu (2006). The Unified Strength Theory can also be called Yu Maohong strength theory or Yu strength theory (Kolupaev and Altenbach 2010).

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5 Yu Unified Strength Theory

References Altenbach H, Öchsner A (2014) Plasticity of pressure-sensitive materials. Springer, Berlin Chen WF (1982) Plasticity in reinforced concrete. McGraw-Hill Book Company Craig RF (2004) Craig’s soil mechanics, 7th edn. CRC Press Deng YK (1987) On the generalized use of Drucker postulate. J Chang’an Uni (Nat Sci Edn) 5(1):75–83 (in Chinese) Drucker DC (1951) A more foundational approach to stress-strain relations, In: Proceedings of the 1st U.S. National Congress of applied mechanics, ASME, pp 487–491 Kolupaev VA, Altenbach H (2010) Einige Überlegungen zur Unified Strength Theory von MaoHong Yu (Considerations on the Unified Strength Theory due to Mao-Hong Yu). Forsch Ingenieurwes 74:135–166 (in German) Kolupaev VA, Yu MH, Altenbach H (2013) Visualization of the unified strength theory. Arch Appl Mech 83(7):1061–1085 Li YC, Tang ZJ, Hu JZ (1988) Further study on Drucker postulate and plastic constitutive relations. J Univ Sci Technol China 18(3):339–345 (in Chinese) Mendelson A (1968) Plasticity: theory and application. MaCmillan, New York Naghdi PM (1960) Stress-strain relations in plasticity and thermoplasticity. In: Palmer AC, Maier G, Drucker DC (1967) Plasticity: normality relations and convexity of yield surfaces for unstable materials or structural elements. J Appl Mech 34(2):464–470 Teodorescu PP (2006) Review to unified strength theory and its applications. Springer, Berlin, 2004 Zentralblatt MATH Cited in Zbl. Reviews 1059.74002 (02115115) The Editor Committee of Encyclopedia of China (1985) Encyclopedia of China. The Encyclopedia of China Press, Beijing Timoshenko SP (1953) History of strength of materials. McGraw-Hill Publishing Co Voigt W (1901) Zur Festigkeitslehre. Annalen Der Physik 309(3):567–591 Yu MH (1994) Unified strength theory for geomaterials and its application. Chin J Geotech Eng 16(2):1–10 (in Chinese) Yu MH (2004) Unified strength theory and its applications, 2nd edn. Springer, Berlin (Springer and XJU Press) Yu MH, He LN, Song LY (1985) Twin shear stress theory and its generalization. Sci China A 28(12):1113−1121 (in Chinese) (English: 1985, 28(11):1175–1183) Yu MH, Yang SY, Fan SC et al (1997) Twin shear unified elasto-plastic constitutive model and its applications. Chinese J Geotech Engrg 21(6):9–18. (in Chinese, English abstract) Yu MH (1961) General behaviour of isotropic yield function. Res report of Xi’an Jiaotong University. Xi’an, China (in Chinese)

Chapter 6

Compression and Settl

6.1 Introduction The study of the compression and settlement of soils is of theoretical and practical engineering importance. Figure 6.1 is the Holsten Gate at Lübeck, Germany, which is built between 1464 and 1478. The settlement of the soil foundation of the Holsten Gate over time is shown in Fig. 6.2. It can be seen that the settlement takes about two years to reach stability. This characteristic is closely related to the properties of the soils (Dimitri 1941). Soil is composed of solids, liquids, and gases and is highly compressible due to its properties. The compression of soil is usually divided into three parts: (1) the compression of the solid particles; (2) the compression of the pore water and pore air; (3) the pore water and pore air are expelled from the soil skeleton. Experimental studies have shown that the compression of soil particles and water under normal pressure (100–600 kPa) is small compared to the total compression of the soil and can therefore be completely neglected. The volume of pore water that drains from the soil skeleton is a measure of the volume change of the water saturated soil subjected to the applied loads. The compression of high permeable saturated soil subjected to a load needs a short time to complete; however, for the saturated soil with low permeability, the water can only be expelled slowly, and it takes a length of time to complete compression. Such a time related compression process is called consolidation. It is an important issue for saturated cohesive soil.

© Springer Nature Singapore Pte Ltd. and Zhejiang University Press 2023 M.-H. Yu, Soil Mechanics, https://doi.org/10.1007/978-981-99-2781-4_6

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Fig. 6.1 Schematic diagram of Holstentor

Fig. 6.2 Foundation settlement of Holstentor

In this chapter, the compression and consolidation of soils and the settlement of foundations will be discussed. Soil consolidation and settlement are of great importance to practical engineering problems, both for new and old buildings, and their analysis is closely related to the theory chosen. Figure 6.3a, b shows the deformation results of different sections of Xi’an city wall using the double shear strength theory. Figure 6.3b shows the case of a hole in the wall. The circumference of the ancient city wall of Xi’an is 14 km, but the air-raid shelters are crossed vertically and horizontally for a total of 41 km. From the analysis in Fig. 6.3b, it is clear that the air-raid holes are the cause of deformation and collapse of the wall.

6.1 Introduction

107

Fig. 6.3 Deformation of wall under concentrated force. a P = 400 kg, b P = 800 kg

Particular attention must also be paid to the buildings with uneven settlement. The Leaning Tower of Pisa is a well-known example. Figure 6.4 shows a diagram of settlement arising from two adjacent silos in Canada. Settlement of buildings often takes place over a long period of time. Figure 6.5 shows the settlement curve of a post office building in Bregenz, Germany, built in 1894, during its first 18 years. The record of settlement is of great importance for the use and maintenance of buildings.

Fig. 6.4 Tilting of structures caused by stress overlap in Canada (Bozozuk 1976)

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Fig. 6.5 Settlement of a postoffice building in Bregenz, Germany. a Section, b plan of building

6.2 Compression Test and Compression Index 6.2.1 Compression Test and Compression Curve In a compression test, the soil sample is generally enclosed in a stiff metal ring and placed between two permeable stones. The upper permeable stone transmits the applied load to the soil sample. As no lateral deformation occurs, as shown in Fig. 6.6, this method is also known as the oedometer test. Suppose the initial height of a soil sample is H 0 and the height after compression be H. Then, H = H 0 − s and s is the deformation of the soil sample subjected to the external pressure p. Based on the definition of soil void ratio, if we assume that the volume V s is a constant, the pore volume of soil sample is e0 × V s before compression and is e × V s after compression, as shown in Fig. 6.7. In order to obtain the void ratio e0 of soil sample after compression, sing the two conditions of constant volume of soil particles before and after compression and constant cross-sectional area of the soil sample, it follows that H0 − s H0 H = = 1 + e0 1+e 1+e e = e0 −

s (1 + e0 ) H0

(6.1) (6.2)

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109

Fig. 6.6 Schematic diagram of oedometer test

Fig. 6.7 Schematic diagram of vertical compression deformation

where e0 = d s (1 + w0 )γ w /γ 0 − 1 is the initial void ratio. w0 , γ 0 , and γ w are the initial moisture content, soil particle density, and initial density of soil samples, respectively, and they can be determined by laboratory experiments. So as long as the compression s of soil sample loads at different levels of pressure is determined, the corresponding void ratio e can be calculated and then the compression curve can be drawn. The compression curve is plotted in two ways, as shown in Figs. 6.8 and 6.9. One is the e-p curve; in conventional experiment, the load generally can be added five levels as follows p = 50, 100, 200, 300, 400 kPa; the other is e-lgp curve, where the experiment starts with a little pressure, takes small increments of multi-stage loading, and adds until a larger load (e.g., 1000 kPa) is applied.

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Fig. 6.8 Calculation of a in curve e-p

Fig. 6.9 Calculation of C c in curve e-lgp

6.2.2 Compressibility Index (1) Compression coefficient a The slope of the tangent line a at any point on the e-p curve indicates the compressibility of the soil subjected to the pressure p: a=−

de dp

(6.3)

The negative sign in Eq. (6.3) indicates that e decreases as the pressure p increases. In engineering practice, the compressibility characteristics of the soil from the original stress p1 to the external load p2 at a point is generally studied. Generally, the tangent slope is used to represent the compressibility of soil, suppose α is the angle

6.2 Compression Test and Compression Index

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Fig. 6.10 Compression curve for soil. a Curve of e-p for soil, b Curve of e-lgp for soil

between tangent slope and horizontal ordinate and we find a ≈ tan α =

e1 − e2 Δe = Δp p2 − p1

(6.4)

Figure 6.10 is the compression curve of soft clay and dense sand for soil. (2) Compression index C c The compression index C c (dimensionless) is the slope of the e-lgp compression line and can be calculated as follows Cc =

e1 − e2 p2 = (e1 − e2 )/ lg lg p2 − lg p1 p1

(6.5)

The higher the compression index C c , the more compressible the soil is. Different from a, C c does not change with the pressure in the straight line segment. (3) Compression modulus Es The ratio of the vertical additional stress increment in the soil at zero lateral strains to the corresponding strain increment is referred to as the compression modulus E s , which can be expressed as follows: Es =

1 + e1 a

(6.6)

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The modulus of compression of soil E s is an indicator of the compressibility of soil, expressed in kPa or MPa. It can be found from Eq. (6.6) that E s is inversely proportional to a. The greater the E s , the smaller the a and less compressible the soil.

6.2.3 Rebound Curve and Recompression Curve In the laboratory compression test, when the soil pressure is added to a certain value pi (corresponding to the b point on an e-p curve), and the pressure is removed step by step, the rebound of the soil sample is observed. If the void ratio is measured, the corresponding void ratio versus pressure curve can be plotted (as shown in the in Fig. 6.11) and called the rebound curve. Figure 6.11 shows during unloading or expansion, the specimen rebounds along the rebounding curve bc, rather than the virgin compression curve ab, which indicates the deformation cannot be fully restored. This shows that the soil deformation is made up of reversible part and irreversible part, in which the former is called elastic deformation and the latter is the main one and known as the residual deformation. As seen in Fig. 6.11a, the rebound curve after decompression does not follow the compression curve ab back but is much flatter, which indicates that the deformation cannot be fully recovered; it is composed of two parts, of which the recoverable part is called elastic deformation and the irrecoverable part is called residual deformation. The same phenomenon can be seen in the e-lgp curve, as shown in Fig. 6.11b.

Fig. 6.11 Rebound curve and compression curve of soil. a e-p curve, b e-lgp curve

6.3 Influence of Stress History on Settlement

113

6.3 Influence of Stress History on Settlement 6.3.1 Stress History of Natural Soil Layers The change of stress in the geological age of soil formation is defined as the stress history. Cohesive soil is subjected to different geological and stress changes in the process of formation and existence, which leads to different densification process and consolidation state. The maximum consolidation pressure of natural soil layers in the history is called the pre-consolidation pressure pc . The over-consolidation ratio, OCR, is defined as the ratio of the pre-consolidation pressure pc to existing self-weight pressure p1 of the overburden soil. According to the over-consolidation ratio, natural soil layers can be divided into three consolidation states as following: Over-consolidation state. The pre-consolidation pressure pc of natural soil layers in the geological history is greater than the current overburden pressure p1 , that is OCR > 1. A part of the ground may be eroded by the rise of the ground or the erosion of the river, or the soil beneath an ancient glacier has been compressed by the ice load, and then, due to global warming and the melting of ice and snow, the overburden pressure began to decrease. Normal consolidation state. Soil is compressed and stabilized under the action of maximum consolidation pressure, then the thickness of the soil layers is not changed after silence, and no subsequent action of other loads. Namely, pc = p1 = γ z, OCR = 1. Unconsolidation state. Soil is stabilized under the action of pre-consolidation pressure, and then for some reason, the soil layer continue to compress. Finally, the pre-consolidation pressure pc is less than the overburden pressure p1 , that is, OCR < 1. The consolidation state has not yet been completed due to the time is not long, so this state is called unconsolidated state.

6.3.2 Determination of Pre-consolidation Pressure There are many methods to determine the pre-consolidation pressure, and the most widely used method is the empirical drawing method proposed by A. Casagrande in 1936. Specific drawing steps are as follows (see Fig. 6.12): (1) First estimate the point of smallest curvature from the e-lgp curve, A, then draw a horizontal line through A (A1) and the tangent to the curve at A (A2). (2) Bisect the angle 1A2 to give the line A3, and locate the straight part of the compression curve, then project the straight part of the curve upwards to cut A3 in B. (3) The point B then gives the value of the pre-consolidation pressure pc . This method is only applicable for soil with obvious curvature variation in the e-lgp curve; otherwise, the radius r min will be difficult to determined.

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Fig. 6.12 Determination of pc

In addition, the curvature of the e-lgp curve varies with the change of the e axis coordinate scale, and there is not a uniform coordinate proportion at present, and human factors exert a tremendous influence, so the value of pre-consolidation pressure pc is not reliable. Therefore, in general, it is necessary to combine the historical survey data of site topography and geomorphology to determine the pre-consolidation pc .

6.3.3 Final Settlement of Foundation Considering the Effect of Stress History In the layer-wise summation method, the influence of stress history on the settlement of the foundation can be considered as long as the compressibility index is determined from the original compression curve (e-lgp curve). 1. Normally consolidated soil After the compression index C c is determined by the original compression curve, the final settlement can be calculated by Eq. (6.7): s=

n n Δei hi p1i + Δpi Cci lg hi = 1 + e0i 1 + e0i p1i i=1 i=1

(6.7)

where Δei is the variation of void ratio in the ith layer determined by the original compression curve; Δpi is the average value of additional stress in the ith layer soil of soil. The original compression curve is shown in Fig. 6.13; the drawing steps are as follows:

6.3 Influence of Stress History on Settlement

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Fig. 6.13 Void ratio of normally consolidated soil

(1) Make e-lgp curve and determine the pre-consolidation pressure pc . (2) Draw a line e0 , and intersect pc in point b. (3) Draw e = 0.42e0 to obtain point c, then connect bc that is the original compression curve. (4) From the slope of straight line bc, the compression index C c can be obtained. 2. Over-consolidated soil The compression index C c and rebound index C e of soil are determined by the original compression curve and the original recompression curve, respectively, as shown in Fig. 6.14a. The drawing steps are as follows: (1) Make e-lgp curve and determine the pre-consolidation pressure pc . (2) Make rebound recompression curve (unload from pi to p1 ). (3) Draw a line e0 , and intersect pc in point b. Fig. 6.14 Void ratio of over-consolidated and unconsolidated soils. a Over-consolidated soil, b unconsolidated soil

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(4) Draw parallel line b1 b||fg, the rebound index C e is obtained the slope of straight line fg. (5) Draw e = 0.42e0 to obtain point c. (6) Then connect bc that is the original compression curve. Treat these two situations differently in the calculations (1) If Δp > (pc − p1 ) (see Fig. 6.14a), the porosity ratio of the layered soil will decrease Δe' along the curve segment b1 b of the original recompression curve and then decrease Δe'' along the curve segment bc of the original compression curve, that is, the void ratio Δe corresponding to Δp should be equal to the sum of the two parts, where Δe' and Δe'' are:

pc Δe = Ce lg p1 p1 + Δp Δe'' = Cc lg pc '

The variation of total void ratio Δe is p1 + Δp pc Δe = Δe' Δe'' = Ce lg + Cc lg p1 pc

(6.8) (6.9)

(6.10)

Therefore, for Δp > (pc − p1 ), the total settlement sn of layered soil is sn =

n i=1

hi ptextci p1i + Δp Cei lg + Cci lg 1 + e0i p1i pci

(6.11)

(2) If the effective stress increment of layered soil is Δp ≤ (pc − p1 ), the void ratio Δe will only changes along the recompression curve b1 b, which is: p1 + Δp Δe = Cc lg pc

(6.12)

Therefore, for Δp ≤ (pc − p1 ), the total settlement sm of layered soil is sm =

m i=1

hi p1i + Δpi Cci lg 1 + e0i p1i

(6.13)

The total settlement is the sum of the two parts mentioned above: s = sm + s n

(6.14)

6.4 Relationship Between Deformation and Time

117

3. Unconsolidated soil The variation of void ratio of unconsolidated soil can be determined by the same method as that of normal consolidated soil, as shown in Fig. 6.14b. The consolidation settlement consists of two parts: ➀ Settlement due to additional stress of foundation; ➁ Settlement due to self-weight stress of soil. The formula is as follows: p1i + Δpi Δei = Cci lg pci

(6.15)

The total settlement is s=

n i=1

hi p1i + Δpi Cci lg 1 + e0i pci

(6.16)

It can be seen that the results obtained from the calculation of settlement of the consolidated soil under normally consolidated soil may be much less than the actual settlement.

6.4 Relationship Between Deformation and Time In the construction practice of soft soil foundation, it is often necessary to deal with the relationship between settlement and time. For example, to determine the settlement during the construction period or a certain moment after completion, in order to control the construction rate or the use restrictions and security measures of designated buildings. The relationship between the deformation of foundation and time should also be considered when dealing with the foundation. Because of the good permeability of gravel soil and sandy soil, the deformation time is short, and it can be considered that the deformation is stable when the external load is applied. However, for cohesive soil, it takes a few years or even decades to complete the consolidation, so the relationship between deformation and time of saturated soil is only discussed in here.

6.4.1 Seepage Deformation of Saturated Soils The seepage consolidation is the process of saturated clay under pressure and the pore water gradually discharged with time, and the pore volume decreases. The time required is related to the permeability and thickness of soil, the smaller the permeability, the thicker the soil, and the longer the time of consolidation. The spring-piston model shown in Fig. 6.15 helps to give an understanding of the seepage consolidation process of saturated soil. In the analogy, the spring represents

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Fig. 6.15 Seepage consolidation model of saturated soil. a t = 0, u = σz , σ ' = 0, b 0 < t < +∞, u + σ ' = σz , σ ' > 0, c t = ∞, u = 0, σ ' = σz

the compressible soil skeleton, and the water represents the water in the voids of soil; the size of weep hole is analogous to the permeability of soil. When load is applied to the piston, it will be carried initially by the water pressure created, but due to the weep hole, there will be a slow bleeding of water from the cylinder accompanied by a progressive settlement of the piston until the spring is compressed to its corresponding load. Let the pressure of spring be the effective stress σ ' , the pressure borne of water be pore pressure u, there is σz = σ ' + u It is obvious that the physical meaning of above formula is the effect of pore water pressure of soil and effective stress on the external force, which is related to time. (1) At time t = 0, that is, the piston top is suddenly subjected to a additional stress σ z of the moment, the water cannot be discharged and the spring has no deformation and stress, and the entire additional stress will be carried by water at all depths, namely u = σz , σ ' = 0. (2) At time t > 0, after the application of the additional stress, to the clay layer, the water in the void spaces will start to be squeezed out and the piston begins moving down. By this process, the excess pore water pressure u at any depth in the clay layer will gradually decrease, and the stress σ ' carried by the soil solids will increase. In a word, u + σ ' = σz , u < σz , σ ' > 0. (3) At time t → ∞, the entire excess pore water pressure would be dissipated by drainage from all points of the clay layer, and the entire additional stress σ z will be carried by the spring. Seepage consolidation of saturated soil is completed. That is, u = 0, σz = σ ' . This gradual process of drainage under an external load application and the associated transfer of pore water pressure to additional stress causes the time-dependent settlement in the clay soil layer.

6.4 Relationship Between Deformation and Time

119

Fig. 6.16 Distribution of pore water pressure in compressible soil

6.4.2 Terzaghi’s One-Dimensional Consolidation Theory 1. Basic hypothesis In order to obtain the deformation of saturated soil in consolidation process at any time, we usually using the one-dimensional consolidation theory to calculate, which is proposed by Terzaghi in 1925. The condition is that the load area is much larger than the thickness of compressed soil, and the pore water in foundation mainly flows along the vertical seepage. As shown in Fig. 6.16a, it is one of the cases of one-dimensional consolidation, in which the top surface of the saturated cohesive soil with a thickness of H is permeable, and bottom is impermeable. It is assumed that the consolidation of soil under self-weight has been completed, but only caused by uniform load p0 . The assumptions made in the theory of one-dimensional consolidation are: (1) The soil is homogeneous, isotropic and fully saturated. (2) The distribution of additional stress in soil on the wall slab with horizontal plane is infinite, so the compression and the seepage flow are one-dimensional. (3) In comparison with pore of soil, the soil particles and water are incompressible. (4) The seepage flows of water in soil obeys Darcy’s law. (5) In the process of seepage consolidation, permeability coefficient k and compressibility coefficient a remain constant throughout the process. (6) The external load is assumed to be applied instantaneously.

6.4.3 One-Dimensional Consolidation Differential Equation Consider an element of fully saturated soil having dimensions dx, dy and dz in the x, y, and z directions, respectively, within a clay layer of thickness H, as shown in

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Fig. 6.16b. An increment of total vertical stress p0 is applied to the element with flow taking place in the z direction only. Because the consolidation seepage can only be from bottom to top, after applying external load, the volume of water entering and leaving the element per unit time are ⎫ ∂h ⎪ dxdy ⎪ q = ki A = k − ⎬ ∂z ⎪ ∂h ∂ 2h ⎭ q '' = k − − 2 dz dxdy ⎪ ∂z ∂z '

(6.17)

Therefore, the difference between the volume of water entering and leaving the element per unit time is q ' − q '' = k

∂ 2h dxdydz ∂z 2

(6.18)

The change rate of pore volume V v in element is known that ∂ Vv ∂ = ∂t ∂t

e dxdydz 1+e

(6.19)

According to the continuity condition of consolidation seepage, water changes of element at a time should be equal to pore volume change rate at the same time and also because the element of soil particle volume dxdydz/(1 + e) is a constant, so k

1 ∂e ∂ 2h = ∂z 2 1 + e ∂t

(6.20)

And according to stress–strain relationship of the lateral limit conditions, have de = −adp = −adσ ' or ∂e/∂t = −a∂σ ' /∂t. then k(1 + e) ∂ 2 h ∂σ ' = a ∂ z2 ∂t

(6.21)

According to the principle of effective stress ∂σ ' ∂u =− ∂t ∂t

(6.22)

∂ 2h 1 ∂ 2u = 2 ∂z γw ∂z 2

(6.23)

and

We can obtain

6.4 Relationship Between Deformation and Time

121

k(1 + e) ∂ 2 u ∂u = γw a ∂z 2 ∂t

(6.24)

Owing to k(1 + e)/aγ w = C v , Eq. (6.24) can be expressed as Cv

∂ 2u ∂u = ∂ z2 ∂t

(6.25)

This is the one-dimensional consolidation differential equation of saturated soil, in which C v being defined as the coefficient of vertical consolidation, suitable unit being m2 /year. The initial conditions and boundary conditions are expressed as follows: when t = 0 and 0 ≤ z ≤ H, u = σz ; when 0 < t < ∞, u = 0; when 0 < t < ∞ and z = H, ∂u/∂z = 0; when t = ∞ and z = H, u = 0; According to the above conditions, u z,t can be calculated by the decomposition variable method: u z,t

m=∞ A 1 mπ z m2π 2 = σz sin exp − Tv π m=1 m 2H 4

(6.26)

where m are odd valves, T v is the coefficient of vertical consolidation time, and T v = tC v /H 2 . When the soil under single drainage condition, H is equals to the thickness of soil layer; when double drainage, H is equals to half the thickness of soil layer.

6.4.4 Calculation of Consolidation Degree With the solution of pore water pressure varying with time and depth, the consolidation settlement of foundation at any time can be obtained using the basic computational formulae of soil compression. The degree of consolidation can be often used in engineering, which is defined as follows: U=

sct sc

(6.27)

where sct is the consolidation settlement of foundation in moment t; sc is the final consolidation settlement of foundation. For one-dimensional consolidation, because the consolidation settlement of soil layer is proportional to the area of effective stress. Therefore, the ratio of the effective stress area at any time to the ultimate effective stress area is called the average

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consolidation degree of the unidirectional consolidation of soil U z (see Fig. 6.16): H u z,t dz area abce − area ade area abcd = = 1 − 0 H Uz = area abce area abce σz2 dz 0

Substituting Eq. (6.26) into the above equation, as given by Uz = 1 −

m=∞ 8 1 m2π 2 T exp − v π 2 m=1,3 m 2 4

(6.28)

It can be seen from the above equation that the average degree of consolidation is a single valued function of the time factor T V . The series in above bracket converges very quickly, and you can only take the first item of Eq. (6.28) to calculate when U z > 30%, namely Uz = 1 −

8 m2π 2 T exp − v π2 4

6.4.5 In Late-Stage Settlement Calculated by Settlement Observation Data For most practical engineering problems, the secondary consolidation settlement is much less significant than for primary consolidation settlement. Therefore, the final settlement of the foundation is mainly composed of the instantaneous settlement and the consolidation settlement, that is to say s = sd + sc , correspondingly, the settlement after the construction period T (t > T ) is st = sd + sct

(6.29a)

st = sd + U z sc

(6.29b)

or

Settlement amount in above formula is calculated using one-dimensional consolidation theory; the results are not consistent with the measured ones. Because the settlement of foundation belongs to the three-dimensional problem, and the actual situation is very complex, it is of great practical significance to calculate the later settlement (including the final settlement) by using the settlement observation data.

6.4 Relationship Between Deformation and Time

123

Here are two common methods: logarithmic curve method (three point method) and hyperbolic method (two-point method). 1. Logarithmic curve method A formula for calculating the degree of consolidation U z under different conditions can be summarized by a general expression: Uz = 1 − A exp(−Bt)

(6.30)

where A and B are two parameters, compared with the one-dimensional consolidation theory; it can be seen that the parameter A in above formula is a constant value 8/π 2 in theory, and the parameter B is related to the coefficient of consolidation and the drainage distance in time factor T v . The values of A and B are undetermined if they as the parameter in measured settlement and time relation curves. Substituting Eq. (6.30) into Eq. (6.29b), as given by st − sd = 1 − A exp(−Bt) sc

(6.31)

Substituting s = sd + sc into the above equation, and use final consolidation settlement s∞ in place of s, as given by

st = s∞ 1 − A exp(−Bt) + sd A exp(−Bt)

(6.32)

If s∞ and sd are also unknown, together with A and B, the above formula contains four unknowns. Three time points t 1 , t 2 , and t 3 are selected from the early s-t curves (see Fig. 6.17) after the load is applied, where t 3 should be as close as possible to the end of the curve. Time differences (t 2 − t 1 ) and (t 3 − t 2 ) must be equal and as large as possible. Substituting the time into Eq. (6.32), respectively, as given by

⎫ st1 = s∞ 1 − A exp(−Bt1 ) + sd A exp(−Bt1 )⎪ ⎬

st2 = s∞ 1 − A exp(−Bt2 ) + sd A exp(−Bt2 ) ⎪

⎭ st3 = s∞ 1 − A exp(−Bt3 ) + sd A exp(−Bt3 )

(6.33)

And the additional condition is exp[B(t2 − t1 )] = exp[B(t3 − t2 )]

(6.34)

Combining Eqs. (6.33) and (6.34), there are B=

1 st2 − st1 ln t2 − t1 st3 − st2

(6.35)

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Fig. 6.17 Settlement-time curves

s∞ =

st3 (st2 − st1 ) − st2 (st3 − st2 ) (st2 − st1 ) − (st3 − st2 )

(6.36)

Substituting Eqs. (6.35) and (6.36) into Eq. (6.33), sd can be expressed as follows:

st1 − s∞ 1 − A exp(−Bt1 ) sd = A exp(−Bt1 )

(6.37)

where the parameter A is an approximate value 8/π 2 , and then the after-settlement st at any time can be calculated using Eq. (6.34). All of the above time t should be revised after the zero point O' , when a load to the construction period increases at a constant speed, the point O' is at the midpoint of the loading period (see Fig. 6.17). 2. Hyperbolic method The settlement observation data of buildings show that the relation curve between settlement and time, s-t is close to hyperbolic curve (except for construction period). Empirical formula of hyperbolic is as follows: st1 = s∞ t1 /(at + t1 )

(6.38a)

st2 = s∞ t2 /(at + t2 )

(6.38b)

where s∞ is the final settlement, the theoretical time is t = ∞; st1, st2 are the settlement after time t 1 and t 2 , the time should start from half of the construction period (assuming that load is a first order constant loading); at is a curve constant. In Eqs. (6.38a, 6.38b), the two groups st1 , t 1 and st2 , t 2 are measured values, s∞ and at can be computed as follows: s ∞ = t2 − t1 /

t2 t1 − st2 st1

(6.39)

6.5 Settlement Analysis for the City Wall of East Gate in Xi’an

125

Fig. 6.18 Settlement calculated by hyperbolic method

a t = s∞ ·

t1 t2 − t1 = s∞ · − t2 st1 st2

(6.40)

In order to eliminate the possible error of observation data, including the system error of instrument and equipment, human error and random error be negligent, the observation point sti and t i of rear section must be used, and then the values t i /sti can be calculated. In rectangular chart, the rear section should be a straight line (points with larger errors are eliminated) as shown in Fig. 6.18. Two representative points (t 1 , st1 ) and (t 2 , st2 ), which are selected from the straight line, can be used to determine the final settlement s∞ and at using Eqs. (6.39) and (6.40); and substituting these two values into Eqs. (6.38a, 6.38b) to determine the settlement at any time.

6.5 Settlement Analysis for the City Wall of East Gate in Xi’an In 1990s, scientists from all over the world began to show their interest in structural mechanics of ancient architectures. This interest was triggered by the requirement for the preservation for cultural heritage, which was aroused by the worldwide peaceful environment and development of economics. In the repair project of East Gate of Xi’an, Yu cooperated with Xi’an Cultural Relics Bureau again to study the structural mechanics characteristics and the bearing capacity of the East Gate City. Participants include Wang Y, Zhao JH and Yang SY. Wang Y and Yang SY took part in the revision, supplement, and debugging of the elasto-plastic finite element program of Yu Unified Strength Theory. In addition, Wang Y, Zhao JH, and Yu MH did the

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Fig. 6.19 Finite element model of the East Gate stylobate

static and dynamic finite element analysis on the City Wall over the East Gate. Finite element model of the East Gate stylobate is shown in Fig. 6.19. Under the condition that three piles of columns are existed, taking the typical plane, the columns can be simplified to three centralized force N = 137.5 kN on the wall (see Fig. 6.19). Materials used are the Ming Dynasty soil: E = 6.9 × 104 kPa, Poisson’s ratio ν = 0.347, cohesion C = 36.3 kPa, friction angle ϕ = 25.65°; Ming Dynasty’s brick: E = 2.23 × 106 kPa, Poisson’s ratio ν = 0.1, σ c = 3225 kPa, σ t = 289 kPa. The main material of the tower stylobate is compacted loess. The differences between the limit loci of various yield criteria in the deviatoric plane are shown in Fig. 6.20, in which the limit locus 1 is from the Mohr–Coulomb strength theory proposed in 1900; it can be thought of as the lower bound of all yield criteria; locus 2 from the twin-shear strength theory proposed by Yu et al. in 1985, and it can be thought of as the upper bound of all yield criteria; the other loci 2, 3, 4, 5 are situated between those two bounds. The three types of yield criterion can be obtained from the Yu Unified Strength Theory when b = 0, b = 1/2 and b = 1, as shown in Fig. 6.21. They represent three typical yield criteria for the upper, middle, and lower yield criteria. The five typical yield criteria are shown in Fig. 6.21. Results of static finite element analysis for the City Wall of East Gate in Xi’an are as follows: The relationship between the displacement and load coefficient (the load coefficient is f = F/N, which is the transient load of progressive loading, structure load P = ∑N = 137.5 × 3 = 412.5 kN) at each point in all directions was achieved by Zhao JH through the analysis and calculation. The relationship between the displacement and load coefficient of the 1037th point and 1053rd point is shown in Fig. 6.22, respectively. It is suggested that the choice of the strength theory would have significant influence on the calculation result. The Yu Unified Strength Theory provides the theoretical basis for the study on this influence (see Chap. 5).

6.5 Settlement Analysis for the City Wall of East Gate in Xi’an

127

Fig. 6.20 Different yield criteria

Fig. 6.21 Several basic yield criteria for UST

Figure 6.23 shows the principal stress trace of the East Gate City Wall when b = 1, which shows the direction and relative magnitudes of principal stresses. The relationship between the displacement and load coefficient of the 1061th point is shown in Fig. 6.24. The deformation of the East Gate City Wall when b = 1 is shown in Fig. 6.25. This figure also reflects the relative magnitudes of deformation of each node. From the above analysis, we can see that: (1) Different yield criteria have different load coefficients; the quantitative description of deformation can be obtained from the load–displacement curve. It can be seen from the figure that the influence of different parameters b values in the Yu Unified Strength Theory. According to Fig. 6.24, there is a big difference between the load coefficient obtained by different strength theory, and the load coefficient under the three yield criteria are: f 1 = 0.9 (b = 0), f 2 = 1.1 (b = 1/ 2), f 3 = 1.2 (b = 1), the maximum discrepancy is by 33.3%. It is shown that

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Fig. 6.22 Displacement–load coefficient relation of 1037th point and 1053rd point. a 1037th point, b 1053rd point

Fig. 6.23 Principal stress trace of the East Gate City Wall

the ultimate load of the structure will be improved after taking the effect of intermediate principal stress into account. This conclusion is consistent with the actual results of some geotechnical engineering. Therefore, it is suggested that the choice of the strength theory would have significant influence on the calculation result. Yu Unified Strength Theory provides the theoretical basis for the study on this influence (2) The closer the distance to the loading point, the larger the deformation is, the greater impact on the efficiency. Under the vertical load, the vertical displacement increases with the increase of the load, and the deformation of the horizontal direction is smaller.

6.5 Settlement Analysis for the City Wall of East Gate in Xi’an

129

Fig. 6.24 δ-f relation of 1061th point

Fig. 6.25 Deformation of the East Gate Wall

(3) The deformation produced by point loading is much larger than that of far away from loading action point. Under the vertical load, the points that are farther from loading point have a tendency to tilt. The deformation of the wall under building load is small. (4) Considering the force from the angle of gate hole wall has a certain effect on the strength of the platform, which needs further study. From the point of load exerted on structure, the hole in gate wall has a certain effect on the platform strength. Further research is needed in this area.

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6.6 Analysis of Unidirectional Compression Consolidation in Saturated Soft-Soil Unidirectional compression consolidation analysis of saturated soft soil Academician Shen ZJ was the first to implemented twin-shear strength theory into his some special finite element programs and made a detailed numerical analysis for the foundation. He compares five kinds of yield functions by analyzing the results of the three numerical examples (Shen 1993). The first example is the uniaxial compression test. Assuming that the initial stresses at the surface of the specimen are σ z0 = 10 kPa, σ x0 = σ y0 = 5 kPa, the order of vertical load are 20, 40, 60, 100, 150, 200, 250, 300, 400, 600, 800, 1000, 1300, 1600 kPa, and then unloading in reverse order to 100 kPa. The results are shown in Fig. 6.26, in which symbol M represents the Mohr– Coulomb strength theory (single-shear theory); symbol D represents the twin-shear strength theory; symbol T represents the three-shear strength theory; symbol S represents the defect strength theory; 5 symbol O represents the Mises strength theory. The left in figure is a common εv -logσ 0 curve (note that the void ratio e has been replaced by bulk strain εv ), and the right is σ x -σ z curve. This figure shows that the calculation results of three models of O, S and D are exactly the same, and the lateral pressure σ x is strictly equal to σ y . However, the numerical of the M model is instability, when σ z reaches 150 kPa (arrow in figure), σ x and σ y are not equal each other, and the amount of compression becomes smaller.

Fig. 6.26 Shear parameters of uniaxial compression

6.8 Consolidation Analysis of in Saturated Soft-Soil Foundation Under …

131

Fig. 6.27 Compression parameters of single-shear test

6.7 Analysis of Shear Consolidation in Saturated Soft-Soil The second example is the single-shear test. The soil sample is assumed to be in a state of uniform pressure, σ x0 = σ y0 = σ z0 = 100 kPa, after the vertical pressure remains unchanged, the order of increase in shear load are 10, 18, 24, 30, 34, 35, 41, 44, 46, 48, 50, 52, 54, 56, 58, 60 kPa. The results are shown in Fig. 6.27, in which the left is the compression test parameter, and the right is the shear test parameter. The stress–strain curve on the left side does not tend to be horizontal, which is obviously unreasonable. It is obvious that the parameters obtained from the compression test cannot be used to calculate the shear curve. The shape of the shear curve on the right is reasonable in comparison. As can be seen from the figure, the results of O model and S model are too large, and the results of other three models of M, D and T are reasonable.

6.8 Consolidation Analysis of in Saturated Soft-Soil Foundation Under Uniform Load The third example is a saturated soft soil with a thickness of 10 m, which is subjected to a uniform load with width 10 m. Assuming that the ground surface within the load surface is impervious to water and is impermeable and outside is permeable. Taking half of foundation as the computational domain, as shown in Fig. 6.28. According to the Biot consolidation theory, the initial stresses in the foundation are set up using the buoyant unit weight 10 kPa/m3 , and the lateral pressure coefficient is taken as 0.53. The load is applied in ten stages, each grade for three days. The order of increase in load is 30, 50, 70, 85, 100, 110, 120, 130, 140, and 150 kPa. The calculated settlement and pore water pressure processes are shown in Fig. 6.29. Figure 6.30 shows the surface subsidence and uplift when the load are

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Fig. 6.28 Element mesh of saturated soft soil foundation

120 kPa and 60 kPa, respectively. Figure 6.31 shows the horizontal displacement on the vertical edge of the load and the distribution of pore water pressure on the centerline. These calculations show that: It is feasible to implemented twin-shear strength theory into finite element programs and calculations for solving geotechnical problems; The calculated results of these three geotechnical problems are obtained from the twin-shear strength theory are reasonable; The results of twin-shear model can be compared with the other two models, that is, the single-shear model and three-shear model. They have the same law, but the numbers are different.

Fig. 6.29 Pore water pressure and settlement of load center

6.9 Settlement Analysis of the Foundation of Big Goose Pagoda

133

Fig. 6.30 Surface subsidence and uplift. a Compression parameters, b shear parameters

Fig. 6.31 Horizontal displacement and pore water pressure. a Compression parameters, b shear parameters

6.9 Settlement Analysis of the Foundation of Big Goose Pagoda The Big Goose Pagoda was constructed during the Tang Dynasty in A.D. 653 and was repaired in A.D. 930. The Pagoda is 64.1 m high from the spire to the ground. Until now, there is no obvious subsidence and destruction of the Big Goose Pagoda, only slight inclination caused by the uneven settlement of the ground under the building load. The Big Goose Pagoda is the national key protection cultural relics and a worldrenowned Buddhist holy land. It prohibited from any destructive test. The shape of foundation is unknown. Therefore, based on the several documents and historical

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records of other ancient pagodas in Xi’an which was built at the similar period to that of the Big Goose Pagoda, Yu proposed and divided the foundation structure of the Big Goose Pagoda into four main forms: rectangular foundation structure; stepped foundation structure; inverted stepped foundation structure; inverted stepped foundation structure with underground chamber, as shown in Fig. 6.32. The structure is simplified to a spatial axial symmetric structure. The parameters are as follows: p = 1.38 MPa, σ t = 0.0056 MPa, σ 0t = 0.56 MPa, γ = 0.07. The bearing capacity of foundation of the Big Goose Pagoda has been studied. Plastic zones obeying the single-shear theory (b = 0) and twin-shear theory (b = 1) for four foundations are shown in Figs. 6.33 and 6.34, respectively. According to the results of analysis, the size of plastic area decreases with parameter b increases. Under the same load and the same foundation structure, the area of plastic zone is largest when b = 0; the area of plastic zone is smallest when b = 1. It could be concluded that considering the effect of intermediate principal stress, the limiting load of the structure will increase. This conclusion is consistent with the actual results of some geotechnical engineering and also proves the correctness of Yu Unified Strength

Fig. 6.32 Possible structure of foundation of the Big Goose Pagoda. a Rectangular foundation, b stepped foundation, c inverted stepped foundation, d inverted stepped foundation with chamber

6.9 Settlement Analysis of the Foundation of Big Goose Pagoda

135

Fig. 6.33 Plastic zones for four foundations using the UST with b = 0. a Rectangular foundation, b stepped foundation, c inverted stepped foundation, d inverted stepped foundation with chamber

Theory. Therefore, it is important to choose a proper value of the UST parameter b in order to make efficient use of material. The size of plastic area increases when cap model is used to calculate. These results prove that it is feasible to apply the theory of twin-shear cap model to soil mechanics. The analysis results also show that the influence of different forms of foundation on bearing capacity has its own advantages and disadvantages. However, the existence of the cavern chamber (that is underground palace) in foundation reduces the weight pressure of foundation on basement, which is beneficial to the foundation stability of the ancient Pagoda. This seems to have been known to the ancients. From the view of present information, whether it is the Famen Temple, Big Goose Pagoda, Small Goose Pagoda, and Leifeng Temple in Hangzhou, they have been defined the existence of underground palace, which has preserved a large number of cultural relics. Therefore, the protection of ancient Pagoda should also include the protection of underground palace. The underground palace of Famen Temple and Big Goose Pagoda is larger and Small Goose Pagoda is smaller. The author has entered the underground palace of Small Goose Pagoda, a cavern chamber about 1.8 m2 , but the interior has been empty. This structure can also be seen in tall monuments, such as the Washington Monument, which also has an underground palace.

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Fig. 6.34 Plastic zones for four foundations using the UST with b = 1. a Rectangular foundation, b stepped foundation, c inverted stepped foundation, d inverted stepped foundation with chamber

The settlement of foundation is the process of elasto-plastic deformation of soil mass under the load of foundation and superstructure. Its theoretical analysis results are related to the selected strength theory. Figure 6.35 shows the different results of the foundation deformation analysis with different parameters of the Unified Strength Theory (Fan et al. 2017).

6.9 Settlement Analysis of the Foundation of Big Goose Pagoda

137

Fig. 6.35 Influence of different failure criteria on the calculation results of foundation displacement. a UST, b = 0.0, b UST, b = 0.25, c UST, b = 0.5, d UST, b = 0.75, e UST, b = 1.0, f Variation curve of maximum Z-direction displacement of foundation under different b values

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References Dimitri PK (1941) Soil mechanics, 1st edn. McGRAW-HILL BOOK COMPANY, Inc., New York, London Fan W, Yu MH, Deng L (2017) Strength theory of geotechnical structural. Beijing Science Press, Beijing (in Chinese) Shen ZJ (1993) Comparison of several yield function. Rock Soil Mech 14(1):41–50 (in Chinese)

Chapter 7

Earth Pressure Theory

7.1 Introduction Retaining structure such as retaining walls, basement walls, and bulkheads are commonly encountered in foundation engineering as they support slopes of earth masses. Its purpose is to retaining earth lateral movement and guaranteeing the stability of earth structure or the earth. Earth pressure is the main load on retaining structure, so the calculation of earth pressure on retaining structure is a key aspect in structure design. However, it is still difficult to calculate the exact solution by theory. The calculation of earth mechanics is a very complicated problem, which involves the joint action between the packing, the wall body, and the foundation. In the design and construction of basic engineering and slope engineering, it is important to determine the earth pressure on the supporting structure whether the conventional design method or the elastic foundation beam method. Especially in the excavation of large-scale underground engineering, it is very important to estimate earth pressure correctly to ensure the safety construction of engineering (Rankine 1857). In general, the magnitude of earth pressure and its distribution pattern are related to the direction and size of lateral displacement of the retaining structure, the nature of the soil, and the height of the retaining structure. The earth pressure can be classified into three types according to the direction and size of the lateral displacement of the retaining structure: (1) At-rest earth pressure (Fig. 7.1a) If a rigid retaining wall remains stationary in its original position, the earth pressure acting on the wall is called the static earth pressure. In this case, the earth pressure is referred to as the at-rest earth pressure. The resultant force of at-rest earth pressure

© Springer Nature Singapore Pte Ltd. and Zhejiang University Press 2023 M.-H. Yu, Soil Mechanics, https://doi.org/10.1007/978-981-99-2781-4_7

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7 Earth Pressure Theory

(a)

(b)

(c)

Fig. 7.1 Three types of earth pressure. a Earth pressure at rest, b active earth pressure, c passive earth pressure

on the unit length retaining wall is expressed by E 0 (kN/m), which can be obtained from the non-lateral deformation theory. The strength is expressed by P0 (kPa). (2) Active earth pressure (Fig. 7.1b) If the retaining wall moves away from the filling direction under the action of the backfill pressure of the wall, the earth pressure acting on the wall will gradually decrease from the resting earth pressure. When the soil behind the wall reaches the limit equilibrium state and a continuous sliding surface appears, the earth pressure is reduced to the minimum, which is referred to as the active earth pressure. The resultant force and strength of active earth pressure are expressed by E a (kN/m) and Pa (kPa), respectively. (3) Passive earth pressure (Fig. 7.1c) If the retaining wall moves toward the filling direction under the action of external force, the earth pressure acting on wall will gradually increase from the static earth pressure until the soil reaches the limit equilibrium and a continuous sliding surface appears. Then, the earth after the wall will extrude upward and the earth pressure increases to the maximum, which is referred to as the passive earth pressure. The resultant force and strength of passive earth pressure are expressed by E p (kN/m) and Pp (kPa), respectively. When the height of the retaining wall is the same with the filling conditions, the relationship among above three types of earth pressures is shown in Fig. 7.2, that is Ea < E0 < Ep. In the current design of foundation pit, regardless of whether it is a cantilevered support structure or a supporting bracing structure, the calculation of earth pressure in geotechnical engineering is usually performed using the Rankine’s earth pressure theory. Based on the Mohr–Coulomb strength theory and in the limit equilibrium state, Rankin studied the stress conditions of a semi-unlimited soil under the action of self-weight in 1857 and derived the calculation formula of soil pressure, which is the well-known Rankine’s soil pressure theory. However, this theory also considers the earth pressure as a plane problem and only considering the effect of σ 1 and

7.1 Introduction

141

Fig. 7.2 Relationship between soil pressure and displacement of retaining wall

σ 3 , without considering the intermediate principal stress σ 2 . The calculation of soil pressure is a space problem, so the effect of σ 2 should be considered. Since the Yu Unified Strength Theory was proposed, some scholars introduced it into the calculation of earth pressure and extended the original single solution to a series of unified solution with taking the effect of intermediate principal stress into account (Xie et al. 2003; Yuan 2011; Ying et al. 2004; Zhang et al. 2010a, b; Dong et al. 2012; Zheng 2013). Based on the twin-shear Unified Strength Theory and energy theory, Gao et al. (2006) derived the formulas of active and passive earth pressure by calculating the volume of the arch area and the slip volume. Based on Unified Strength Theory, Fan et al. (2004, 2005a, b), considered multi-triangle failure mechanism in the framework of upper-bound method of the plastic limit analysis theory and deduced some basic formulae, the program for calculating earth pressure was compiled. Assuming that the intermediate principal stress is σ 2 = Kσ 1 , σ 2 = σ 1 σ 3 , or σ 2 = m(σ 1 + σ 3 )/2, Xie et al. (2003) propose a new calculation method for soil pressure by using the Unified Strength Theory and the Rankine’s earth pressure analysis principle. The key to this method is to perform stress analysis on the earth elements of the earth-filled soil behind the retaining wall, determine two of the three principal stresses, and then combine the Unified Strength Theory with the principal stress type expression to solve another principal stress. However, this method is often difficult to determine the appropriate intermediate principal stress during stress analysis. The key to this method is to perform stress analysis on the earth elements of the earth-filled soil behind the retaining wall, determine two of the three principal stresses, and then combine the Unified Strength Theory to calculate another principal stress. However, it is difficult to determine the proper intermediate principal stress during stress analysis. Based on the Unified Strength Theory, Zhai et al. (2004) also deduced the calculation formulas of active and passive earth pressure and discussed the influence of parameter b on the earth pressure. It overcomes the problem that the Mohr–Coulomb criterion does not take into account the influence

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of the intermediate principal stress, so that the calculation results are more practical. These documents are all directed against the Rankine or Coulomb earth pressure formulas, and which are derived based on the principle of limit equilibrium. The active and passive earth pressure can also be studied by finite element analysis and experiment. Figure 7.3 is the deformation of earth pressure obtained by Borja and Regueiro (2001) in Stanford University. Figure 7.4 is the experiment result of earth pressure obtained by Matsuoka (2001) of Nagoya industrial university in Japan. The two results are very similar in shape. The earth pressure problem studied in this chapter is the same as the traditional earth pressure, but the Yu Unified Strength Theory is adopted instead of the traditional Mohr–Coulomb strength theory. Its results also developed from a solution to a series of uniform solutions. Therefore, it can be applied to more materials and structures.

(a)

(b) Fig. 7.3 Deformation diagram of active and passive earth pressure of soil (Borja and Regueiro 2001). a Active earth pressure, b passive earth pressure

7.2 Theoretical Solution of Rankine’s Earth Pressure

143

(a)

(b) Fig. 7.4 Results of active and passive earth pressure of soil (Matsuoka 2001). a Active earth pressure, b passive earth pressure

7.2 Theoretical Solution of Rankine’s Earth Pressure Yu Unified Strength Theory is a new strength theory that takes twin-shear element as the mechanics model, and considers all stress components acting on the element, which fully considers the effect of the intermediate principal stress σ 2 on the yield or failure of the material under different stress conditions. If the earth pressure acting on retaining wall is regarded as a plane strain problem, the intermediate principal stress σ 2 = ν(σ 1 + σ 3 ) can be determined by the generalized Hooke’s law, and another principal stress can be determined according to Rankine’s earth pressure analysis principle, On this basis, the formula of Rankine earth pressure is derived when the expression of principal stress in Yu Unified Strength Theory is used. The weighted coefficient b considering effects of intermediate principal stress and the Poisson’s ratio v are introduced to the formula. It discusses the calculation method of Rankine’s earth pressure theory from another way. The Yu Unified Strength Theory can be expressed as in term of classical cohesion C 0 and internal friction ϕ 0 as follows:

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7 Earth Pressure Theory

ϕ0 (1 + b)σ1 − bσ2 σ3 = tan2 45◦ − 2 σ1 − σ3 ϕ0 σ1 + σ3 ◦ C0 σ2 ≤ − sin ϕ0 − 2(1 + b) tan 45 − 2 2 2 ϕ0 bσ2 + σ1 σ3 = tan2 45◦ − 2 1+b σ1 − σ3 ϕ σ1 + σ3 0 ◦ − 2 tan 45 − C0 σ2 ≥ − sin ϕ0 , 2 2 2

(7.1a)

(7.1b)

where b is the coefficient of intermediate principal shear stress, because the limit surface of the geotechnical is usually convex, so the value of b is 0 ≤ b ≤ 1.

7.2.1 Theoretical Analysis Model Rankine’s earth pressure theory (maximum-normal stress theory), developed in 1857 by Rankine, is a stress field solution that predicts active and passive earth pressure. It assumes that the soil is cohesionless, the wall is frictionless, the soil-wall interface is vertical, the failure surface on which the soil moves is planar, and the resultant force is angled parallel to the backfill surface, as shown in Fig. 7.5. Now the state of the soil element at any depth z near the retaining wall will be analyzed. When the soil mass is in a static state, the stress of the soil micro-element is σ z = γ z, in which γ is the unit weight and z is the depth of the calculation point. When the retaining wall is allowed to move forward, the soil is able to expand, the horizontal stress σ decreases, while the vertical stress σ z remains unchanged, the stress is known as the active earth pressure strength Pa when the soil unit reaches the active limit equilibrium state. Similarly, when the retaining wall is allowed to move pushed, the soil will tend to be compressed, and there will be an increase in the value of the horizontal stress σ . The vertical stress σ z remains unchanged, the Fig. 7.5 Stress state of a point in soil

7.2 Theoretical Solution of Rankine’s Earth Pressure

145

stress is known as the passive earth pressure strength PP when the soil unit reaches the passive limit equilibrium state. Plane strain state is widely existent in geotechnical engineering such as slope, strip foundation, and retaining wall. Intermediate principal stress can be determined by using the generalized Hook’s law when analyzing the strength and deformation condition of rock and soil body with the nonlinear method. Suppose that the cross section of retaining wall as xz plane, thus the direction vertical to the cross section is y-direction. With the elastic solution of plane strain problem, we know that ε y = 0.

(7.2)

According to the generalized Hooke’s law, εy =

1 σ y − ν(σz + σx ) , E

(7.3)

where ν is the Poisson ratio of the filling material, 0 < ν < 0.5. Combining Eq. (7.2) with (7.3), we get σ y = ν(σz + σx ).

(7.4)

In xz plane, σ z + σ x = σ 1 + σ 3 , the intermediate principal stress is tentatively defined as σ2 = ν(σ1 + σ3 ).

(7.5)

Substituting Eq. (7.5) into Eq. (7.1a, 7.1b), there are ϕ0 − νb σ3 (1 + νb) = σ1 (1 + b) tan2 45◦ − 2 σ1 − σ3 ϕ0 σ1 + σ3 ◦ σ2 ≤ − sin ϕ0 − 2C0 (1 + b) tan 45 − 2 2 2 (7.6a) ϕ ϕ 0 0 = σ1 (1 + νb) tan2 45◦ − σ3 1 + b − vb tan2 45◦ − 2 2 ϕ0 σ1 + σ3 σ2 ≥ − 2C0 (1 + b) tan 45◦ − 2 2 σ1 − σ3 (7.6b) sin ϕ0 − 2 Based on the UST and plane strain assumption, the active earth pressure can be derived in two different cases as follows.

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7 Earth Pressure Theory

7.2.2 Formula Derivation When the Intermediate Principal Stress Is Larger In this section, we will discuss the situation when intermediate principal stress is larger, that is, the expression of the earth pressure in Eq. (7.6b). 1. Formula derivation of Rankine’s active earth pressure When the soil mass enters the active limit equilibrium state, it can be known that σ1 = σz = γ z

(7.7)

σ2 = ν(σ1 + σ3 ).

(7.8)

Substituting Eqs. (7.7) and (7.8) into Eq. (7.6a, 7.6b), the Rankine’s active earth pressure strength can be obtained

tan 45◦ − ϕ20

Pa = σ3 = 1 + b − νb tan2 45◦ −

ϕ0 ◦ tan 45 + νb)γ z − 2C − . + b) (1 (1 0 ϕ0 2 2 (7.9)

Let K a = tan2 45◦ − ϕ20 , that is the Rankine’s active earth pressure coefficient, while Eq. (7.9) can be written as Pa = σ3 = γ z K a

1 + νb

1 + b − νb tan2 45◦ −

ϕ0 2

− 2C0 K a

1+b

. 1 + b − νb tan2 45◦ − ϕ20 (7.10)

√ When b = 0, there is Pa = γ z K a − 2C0 K a . By comparing Eq. (7.10) with the classic formula of Rankine’s active earth pressure, we can see that the former has add two coefficients on two sub-items, that is 1 + νb

1 + b − νb tan2 45◦ −

ϕ0 2

and

1+b

1 + b − νb tan2 45◦ −

ϕ0 2

.

The formula shows that the active earth pressure consists of two parts. The existence of cohesion reduces the earth pressure on the wall and forms a negative pressure zone (tensile stress zone) on the upper part of the wall. The soil will be disconnected if there is a small tensile stress between the back of the wall and filling. In practice, however, this tension cannot be relied upon to act on the wall, since cracks are likely to develop within the tension zone and the part of the pressure distribution diagram above should be neglected. When C is greater than zero, the value of Pa is zero at a particular depth z0 , there is

7.2 Theoretical Solution of Rankine’s Earth Pressure

z0 =

147

(1 + b)2C0 √ (Pa = 0). (1 + νb) K a

(7.11)

In the design of a retaining wall, the first step is to judge whether the height H of retaining wall is greater than the depth z0 by using a discriminant. And if H > z0 , Rankine’s active earth pressure should be calculated; If H ≤ z0 , no calculation is needed and the retaining wall is only designed according to the structure requirement. The action direction of Pa is perpendicular to the back of the wall and distributes in a triangle along the wall height. If the wall height is H, then the Rankine active earth pressure of unit wall length is 1 + νb 1 1

Pa (H − z 0 ) = γ H 2 K a 2 2 1 + b − νb tan2 45◦ − √ 2C0 H K a (1 + b)

− 1 + b − νb tan2 45◦ − ϕ20

Ea =

+

2C02 (1 + b)2

γ 1 + b − νb tan2 45◦ −

ϕ0 2

ϕ0 2

. (1 + νb)

(7.12)

The direction of E a is perpendicular to the back of the wall, which is act at distances of (H − z 0 )/3, above the bottom of the wall surface. If b = 0, there is Ea =

1 2C02 γ H 2 K a − 2C0 H K a + . 2 γ

(7.13)

Equation (7.13) is the classical Rankine active earth pressure formula. So Eq. (7.12) can be written as ⎧ ⎨

2 2(1 + b)C0

Ea = H− ⎩ γ (1 + νb) tan 45◦ − ϕ20

4(1 + b)2 C02 1 − tan2 45◦ − ϕ20 γ (1 + νb)2 tan2 45◦ − ϕ20

.

+ 2 1 + b − νb − tan2 45◦ − ϕ20 γ 2 (1 + νb)2 tan2 45◦ − ϕ20 (7.14) When H =

2(1+b)C0 , ϕ γ (1+νb) tan(45◦ − 20 )

E a takes the minimum value.

2. Formula derivation of Rankine passive earth pressure When the soil mass enters the passive limit equilibrium state, there are σ3 = γ z

(7.15)

σ2 = ν(σ1 + σ3 ).

(7.16)

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7 Earth Pressure Theory

Substituting Eqs. (7.15) and (7.16) into Eq. (7.6b), the Rankine passive earth pressure strength can be obtained: Pp = σ1 =

◦ ϕ0 2 + 2C 0 tan 45 (1 + νb) tan2 45◦ − ϕ20

ϕ ϕ0 1 + b − νb tan2 45◦ − 20

σ3 1 + b − νb tan2 45◦ −

= γ z tan2 45◦ +

+ 2C0

1 + νb

2

−

ϕ0 2 (1 + b)

ϕ0 1+b tan 45◦ + . 1 + νb 2 (7.17)

Let K p = tan2 45◦ + ϕ20 , that is the Rankine passive earth pressure coefficient. When b = 0, Pp = γ z K p + 2C0 K p . The direction of Pp is perpendicular to the back of the wall and distributes in a triangle along the wall height. If the wall height is H, then the Rankine passive earth pressure of unit wall length is

1 + b − vb tan2 45◦ − 1 2 Ep = γ H K p · 2 1 + vb

ϕ0 2

+ 2C0 H K p

1+b . 1 + vb

(7.18)

The Pp is act at distances of H/3, above the bottom of the wall surface. If b = 0, there is Ep =

1 ϕ0 ϕ0 + 2C0 H tan 45◦ + . γ H 2 tan2 45◦ + 2 2 2

This is the calculation formula of classical Rankine passive earth pressure.

7.2.3 Formula Derivation When the Intermediate Principal Stress Is Smaller In this section, we will discuss the situation when intermediate principal stress is smaller, that is, the expression of the earth pressure in Eq. (7.6a). 1. Formula derivation of Rankine’s active earth pressure Substituting Eqs. (7.7) and (7.8) into Eq. (7.6a), the Rankine active earth pressure strength can be obtained

ϕ0 1 + b − vb tan2 45◦ + ◦ · Pa = σ3 = γ z tan 45 − 2 1 + vb 1+b ϕ 0 − 2C0 tan 45◦ − . 2 1 + vb 2

ϕ0 2

(7.19)

7.2 Theoretical Solution of Rankine’s Earth Pressure

149

Let K a = tan2 45◦ − ϕ20 , that is the Rankine active earth pressure coefficient, while Eq. (7.19) can be written as

1 + b − vb tan2 45◦ + ϕ20 1 + vb Pa = γ z K a · − 2C0 K a . (7.20) 1 + vb 1+b √ When b = 0, Pa = σ3 = γ z K a −2C0 K a , that is the strength formula of classical Rankine active earth pressure. The height z0 of the tensile zone can be calculated by Eq. (7.10) z0 =

2C0 (1 + b)

√ (Pa = 0). 1 + vb − tan2 45◦ + ϕ20 γ K a

(7.21)

So the Rankine active earth pressure can be obtained

1 + b − vb tan2 45◦ + ϕ20 1 1 2 . E a = Pa (H − z 0 ) = γ H K a 2 2 1 + vb √ 2C0 H K a (1 + b) 2C02 (1 + b)2

− (7.22) + 1 + vb γ 1 + b − vb tan2 45◦ + ϕ20 (1 + vb) The action direction of E a is perpendicular to the back of the wall, which is act at distances of (H − z 0 )/3, above the bottom of the wall surface. When b = 0, there is Ea =

1 2C02 γ H 2 K a − 2C0 H K a + . 2 γ

That is the calculation formula of the classical Rankine active earth pressure. In the same way, 0 When H = γ 1+vb−νb−tan2(1+b)C , E takes the minimum value. [ (45◦ + ϕ20 )] tan(45◦ − ϕ20 ) a 2. Formula derivation of Rankine’s passive earth pressure Substituting Eqs. (7.15) and (7.16) into Eq. (7.6a), the Rankine’s passive earth pressure strength can be obtained 1 + vb ϕ0

· Pp = σ1 = γ z tan2 45◦ + 2 1 + b − vb tan2 45◦ + ϕ20 1+b ϕ0

. + 2C0 tan 45◦ + · 2 1 + b − vb tan2 45◦ + ϕ20

(7.23)

Let K p = tan2 45◦ + ϕ20 , that is the Rankine passive earth pressure coefficient. When b = 0, pp = γ z K p + 2C0 K p , this is the calculation formula of classical Rankine passive earth pressure.

150

7 Earth Pressure Theory

The direction of Pp is perpendicular to the back of the wall and distributes in a triangle along the wall height. If the wall height is H, then the Rankine passive earth pressure of unit wall length is Ep =

1 + vb 1

γ H 2 Kp 2 1 + b − vb tan2 45◦ +

ϕ0 2

+ 2C0 H K p

1+b

1 + b − vb tan2 45◦ +

ϕ0 . 2

(7.24) The Pp is act at distances of H/3, above the bottom of the wall surface. When b = 0, there is Ep =

1 ϕ0 ϕ0 + 2C0 H tan 45◦ + . γ H 2 tan2 45◦ + 2 2 2

That is the calculation formula of the classical Rankine passive earth pressure.

7.3 Expression of UST with Shear Strength The stress state at a point (element) is determined by the combination of the three principal stresses (σ 1 , σ 2 , σ 3 ). Based on the characteristics of the stress state and by introducing a certain parameter, it can be divided into several types. According to the definition of Lode parameters and twin-shear stress state parameters, 2σ2 − σ1 − σ3 σ1 − σ3

(7.25)

μτ =

τ12 σ1 − σ2 s1 − s2 = = τ13 σ1 − σ3 s1 − s3

(7.26)

μ'τ =

τ23 σ2 − σ3 s2 − s3 = = τ13 σ1 − σ3 s1 − s3

(7.27)

μτ =

1 − μσ = 1 − μ'τ 2

(7.28)

μ'τ =

1 + μσ = 1 − μτ . 2

(7.29)

μσ (σ1 − σ3 ) σ1 + σ3 + . 2 2

(7.30)

μσ =

Transform Eq. (7.25), there is σ2 =

Substituting Eq. (7.30) into Eq. (7.1a, 7.1b), we can obtain When μσ ≤ − sin ϕ0 ,

7.3 Expression of UST with Shear Strength

σ1 =

151

(1 + sin ϕ0 )(2 + b − bμσ )σ3 + 4(1 + b)C0 cos ϕ0 . 2(1 + b)(1 − sin ϕ0 ) − b(1 + μσ )(1 + sin ϕ0 )

(7.31a)

When μσ > − sin ϕ0 , σ1 =

2(1 + b)(1 + sin ϕ0 ) − b(1 − μσ )(1 − sin ϕ0 ) 4(1 + b)C0 cos ϕ0 σ3 + . (2 + b + bμσ )(1 − sin ϕ0 ) (2 + b + bμσ )(1 − sin ϕ0 ) (7.31b)

Let σ1 =

1 + sin ϕUST 2CUST cos ϕUST σ3 + . 1 − sin ϕUST 1 − sin ϕUST

(7.32)

There are When μσ ≤ − sin ϕ0 , 2(1 + b) sin ϕ0 2 + b(1 − μσ ) − b(1 + μσ ) sin ϕ0

2(1 + b)C0 cos ϕ0 cot 45◦ + ϕUST 2 = . 2 + b(1 − μσ ) − (2 + 3b + bμσ ) sin ϕ0

sin ϕUST = CUST

(7.33a)

When μσ > − sin ϕ0 , 2(1 + b) sin ϕ0 2 + b(1 + μσ ) + b(1 − μσ ) sin ϕ0 2(1 + b)C0 cos ϕ0

= (2 + b + bμσ )(1 − sin ϕ0 ) tan 45◦ +

sin ϕUST = CUST

ϕUST 2

.

(7.33b)

Introducing into the twin-shear stress state parameters μτ (or μ'τ ), Eqs. (7.33a) and (7.33b) can be written as ϕ0 When μ'τ ≤ 1−sin , 2 2(1 + b) sin ϕ0 2 + 2b(1 − μ'τ ) − 2bμ'τ sin ϕ0

2(1 + b)C0 cos ϕ0 cot 45◦ + ϕUST 2 = . 2 + b(1 − μ'τ ) − 2(1 + b + bμ'τ ) sin ϕ0

sin ϕUST = CUST When μ'τ >

(7.34a)

1−sin ϕ0 , 2

2(1 + b) sin ϕ0 2 + 2bμ'τ + 2b(1 − μ'τ ) sin ϕ0 2(1 + b)C0 cos ϕ0

= ' 2(1 + bμτ )(1 − sin ϕ0 ) tan 45◦ +

sin ϕUST = CUST

ϕUST 2

.

(7.34b)

152

7 Earth Pressure Theory

Equation (7.33) can be written as σ1 + σ3 σ1 − σ3 = sin ϕUST + CUST cos ϕUST . 2 2

(7.35)

According to the Mohr circle of a point stress state, the stress on the surface which at an α angle to the maximum principal stress is τ=

σ1 + σ3 σ1 − σ3 σ1 − σ3 sin 2α, σ = + cos 2α. 2 2 2

Substituting it into Eq. (7.35), there is τ=

sin ϕUST sin 2α CUST cos ϕUST sin 2α σ+ . 1 + cos 2α sin ϕUST 1 + cos 2α sin ϕUST

(7.36)

In order to obtain the maximum shear stress on a certain plane, according to the extreme value method and ∂τ/∂α = 0, there is cos 2α = − sin ϕUST ,

(7.37)

so α = 45◦ +

ϕUST . 2

(7.38)

Therefore, the shear failure occurs along a plane at an angle of 45◦ + ϕUST /2 to the maximum principle stress. Substituting Eq. (7.38) into Eq. (7.36), τ = σ tan ϕUST + CUST .

(7.39)

The above formula can choose the values of C UST and ϕ UST by according to the discriminant.

7.4 Unified Solution of Theory on Sliding Wedge for Earth Pressure Based on the Unified Strength Theory, the unified solution of the earth pressure theory can be derived. The basic assumption for calculation is as follows (Zhai et al. 2004): (1) The soil after the wall is homogeneous and isotropic and cohesionless soil mass. (2) It belongs to the problem of plane strain. (3) The soil surface is a plane, which at an angle β with the horizontal plane.

7.4 Unified Solution of Theory on Sliding Wedge for Earth Pressure

153

(4) Active state: If the retaining wall moves away from the filling direction under the action of the backfill pressure of the wall, so that the soil reaches the active limit equilibrium state and forms the sliding surface BC (see Fig. 7.6); Passive state: If the retaining wall moves toward the filling direction under the action of external force, the soil reaches the passive limit equilibrium and a continuous sliding surface BC appears. (5) The force on the sliding surface satisfies the limit equilibrium relationship T = N tan ϕUST ; the force on the back of the wall satisfies the limit equilibrium relationship T = N ' tan δ, where ϕ UST is the unified friction angle and δ is the friction angle between the soil and the wall back. According to the balance of the sliding wedge, it can be obtained

Ea = Ep =

sin(θ −ϕUST ) W sin(α+θ −ϕUST −δ) sin(θ +ϕUST ) W sin(α+θ +ϕUST +δ)

,

(7.40)

where W is self-weight of sliding wedge and it can be obtained by formula: W = 1 γ AB · AC · sin(α + β). 2 The active earth pressure will inevitably occur on the wedge surface that E a is maximum, while the passive earth pressure must be generated on the wedge surface that E p is minimum. Therefore, the most dangerous slip surface can be obtained when E a and E p are directed to the θ, respectively, and then the active and passive earth pressure can be obtained

E a = 21 γ h 2 K a , E p = 21 γ h 2 K p

(7.41)

where γ is the unit weight of soil; h is the height of retaining wall; K a and K p are the active and passive earth pressure coefficient, respectively, which can be expressed as follows: ⎧ sin2 (α+ϕUST ) ⎪ 2 / Ka = ⎪ ⎪ sin(ϕUST −β ) sin(ϕUST +δ ) ⎨ sin2 α·sin(α−δ) 1+ sin(α+β) sin(α−δ) . (7.42) sin2 (α−ϕUST ) ⎪ / 2 Kp = ⎪ ⎪ sin ϕ +β sin ϕ +δ ( UST ) ( UST ) 2 ⎩ sin α·sin(α−δ) 1−

sin(α+β) sin(α+δ)

The direction of earth pressure is at an angle δ with the back of the wall, but the direction is opposite (see Fig. 7.6). When the action point of earth pressure is not overloaded, the Pp is act at distances of H/3, above the bottom of the wall surface. When the distributed load p acts on the top of the wall surface, the weight of the sliding wedge should be added an overload item (see Fig. 7.7), that is W =

1 2q sin α cos β . γ AB · AC · sin(α + β) 1 + 2 γ h sin(α + β)

(7.43)

154

7 Earth Pressure Theory

(a)

(b)

Fig. 7.6 Calculation diagram of earth pressure. a Active state, b active state

Let K q = 1 +

2q sin α cos β , γ h sin(α+β)

and then Eq. (7.43) can be written as

W =

1 γ K q AB · AC · sin(α + β). 2

(7.44)

In the same way, the active and passive earth pressure can be obtained

E a = 21 γ h 2 K a K q . E p = 21 γ h 2 K p K q

(7.45)

The direction of the earth pressure is still in the δ angle of the back of the wall. The earth pressure is located in the shape of trapezoid shape, and the heel height of the wall is

Fig. 7.7 Distribution of uniform loads on the soil surface

7.5 Examples for Unified Solution of Rankine’s Earth Pressure

ZE =

155

h 2 pa + pb h γ h + 3q = · · , 3 p a + pb 3 γ h + 2q

(7.46)

where pα , pb pα , pb are the distribution earth pressure at the top of the wall and the heel of the wall.

7.5 Examples for Unified Solution of Rankine’s Earth Pressure 7.5.1 Example 1 (Zhang J, Hu RL, et al.) Based on the Unified Strength Theory and considering the effect of the intermediate principal stress, Zhang J and Hu RL et al. analyzed the following examples: The fill parameters of retaining wall are the unit weight is 18 kN/m3 ; the internal friction angle is ϕ = 15°, cohesion is C = 10 kPa, and Poisson’s ratio is ν = 1/3. According to the Unified Strength Theory (Zhang et al. 2010a, b), the Rankine earth pressure Pa , Pp and E a , E p in different b and H can be obtained. The calculation results are shown in Tables 7.1, 7.2, 7.3, Figs. 7.8, and 7.9. From Tables 7.1, 7.2, 7.3 and Figs. 7.8, 7.9 we can know that the values of E a , E p increase with the increase of H value. For the active earth pressure, they decreases Table 7.1 z0 values under different parameter b b value

0.25

0.50

0.75

1.00

z0 value

2.22

2.48

2.70

2.89

Table 7.2 Pa and E a values under different parameters b and H b value

Active earth pressure

b=0

pa /kPa E a /kN

b = 0.25 b = 0.5 b = 0.75

4.0

4.5

5

5.5

6.0

27.04639

32.34551

37.64462

42.94374

48.24286

34.51081

49.35879

66.85632

87.00341

109.8001

pa /kPa

22.26808

27.04829

31.82849

36.60869

41.3889

E a /kN m−1

25.93339

38.26248

52.98168

70.09097

89.59037

pa /kPa

18.85955

23.2696

27.67964

32.08969

36.49974

E a /kN m−1

20.16322

30.6955

43.43282

58.37515

75.52251

pa /kPa

16.30564

20.43834

24.57103

28.70373

32.83643

E a /kN b=1

m−1

The height of retaining wall H/m

m−1

16.08356

25.26956

36.5219

49.84059

65.22563

pa /kPa

14.32073

18.23787

22.15501

26.07215

29.98929

E a /kN m−1

13.08885

21.2285

31.32672

43.38351

57.39887

156

7 Earth Pressure Theory

Table 7.3 Pp and E p values under different parameters b and H b value b=0 b = 0.25

Passive earth pressure

4

4.5

5

5.5

6

pp /kPa

148.349

163.6346

178.9202

194.2057

209.4913

E p /kN m−1

348.8271

426.823

512.4617

605.7432

706.6675

pp /kPa

165.6335

182.5784

199.5233

216.4682

233.4131

E p /kN b = 0.5 b = 0.75 b=1

The height of retaining wall H/m

m−1

391.4159

478.4689

573.9943

677.9922

790.4625

pp /kPa

180.4488

198.8159

217.1831

235.5502

253.9174

E p /kN m−1

427.9206

522.7367

626.7365

739.9198

862.2867

pp /kPa

193.2887

212.8885

232.4883

252.088

271.6878

E p /kN m−1

459.558

561.1022

672.4464

793.5905

924.5345

pp /kPa

204.5236

225.2019

245.8803

266.5586

287.237

487.2407

594.672

712.4426

840.5523

979.0012

E p /kN

m−1

Fig. 7.8 Active earth pressure in different b

Fig. 7.9 Passive earth pressure in different b

7.5 Examples for Unified Solution of Rankine’s Earth Pressure

157

with the increase of parameter b, while for the passive earth pressure, they increases with the increase of parameter b. Under a same value H, the maximum effect of parameter b on E a is about 50%; under a same value H, the maximum effect of parameter b on E p is about 40%. The above results show that the active earth pressures computed by Rankin’s active earth pressure theory are obviously higher than the actual value, and the maximum is about 50%, which have a certain safety reserve. The passive earth pressures are obviously smaller than the actual value, and the maximum is about 40% (Zhang et al. 2010a, b).

7.5.2 Example 2 (Zhang J, Hu RL, Yu WL, et al.) Based on the formula for the intermediate principal stress, by Zhang J, Hu RL and Yu WL et al. derived the formula for shear strength parameters in the Unified Strength Theory. So the shear strength parameters under the Mohr–Coulomb criterion can be transformed into the shear strength parameters under the Unified Strength Theory. Finally, an example is given to carry out sensitivity analysis of six factors in terms of the design theory of orthogonal tests (Zhang et al. 2010a, b). The height of the retaining wall is H = 5 m, and the six influential factors are considered including the cohesion C 0 , internal friction angle ϕ 0 , intermediate principal shear stress parameter b, Poisson’s ratio ν, angle of inclined surface β, and unit weight γ . The analysis results are given in Tables 7.4 and 7.5. They come to a conclusion that among the influential factors of Rankine’s active earth pressure, the unit weight is the most important one, followed by the intermediate principal shear stress parameter, the cohesion, the internal friction angle, the angle of inclined surface, and the Poisson’s ratio; among those of Rankine’s passive earth pressure, the internal friction angle is the most important one, followed by the intermediate principal shear stress parameter, the Poisson’s ratio, the cohesion, the angle of inclined surface, and the unit weight. So the effect of the intermediate Table 7.4 Calculation results of active earth pressure Difference

C 0 /kPa

ϕ 0 /°

b

ν

β/kN m

γ /kN m−3

K 1j

572.8

545.4

547

491.5

481.5

411.0

K 2j

528.9

507.7

529.7

435.6

541.5

452.6

K 3j

462.5

452.2

484.5

507.6

479.7

502.0

K 4j

453.7

472.3

490.6

503.9

454.3

524.4

K 5j

435.7

475.6

401.4

514.4

496.2

563.3

Range

137.5

93.2

145.6

71.8

87.2

152.3

Result

γ >b>C>ϕ>β>v

158

7 Earth Pressure Theory

Table 7.5 Calculation results of passive earth pressure Difference

C 0 /kPa

ϕ 0 /°

b

ν

β/kN m

γ /kN m−3

K 1j

2050.2

1662.9

1889.0

2794.8

2658.7

2396.8

K 2j

2361.0

2074.3

2111.6

2765.3

2323.5

2280.0

K 3j

2581.8

2489.5

2589.5

2358.5

2599.8

2513.9

K 4j

2577.6

2886.9

2452.0

2204.6

2610.7

2385.5

K 5j

2665.8

3122.8

3104.3

2113.2

2043.7

2660.2

615.6

1459.9

1215.3

681.6

615.0

380.2

Range Result

ϕ>b>v>C>β>γ

principal stress on the calculated results of the Rankine’s earth pressure is a critical factor (Zhang et al. 2010a, b).

7.5.3 Example 3 (Fan W, Shen ZJ, et al.) Based on Unified Strength Theory, multi-triangle failure mechanism was considered in the framework of upper-bound method of the plastic limit analysis theory. Then some basic formulae were deduced, the program for calculating earth pressure was compiled, and the earth pressure formula that considered the effect of intermediate principal stress was obtained. A series of results could be presented by means of changing the parameter of Unified Strength Theory. These results were compared with other results in literature, and the calculating method could be used to study the effect of the strength theory (Wei 1995). Table 7.6 is the results of soil pressure calculation at γ 1 = 18.5 kN/m3 , C cu = 10 kPa, ϕ cu = 19°, depth z = 6 m. They give the calculation results of five cases when b = 0, b = 0.25, b = 0.5, b = 0.75, and b = 1. It can be seen that: A series of criteria for the Unified Strength Theory provide an effective theoretical basis for the study of the structural strength theory. It gives a series of continuous results from the lower bound to the upper bound. Table 7.6 Earth pressure calculation results Formula number

Earth pressure (kPa)

Unified strength theory parameter b b=0

b = 0.25

b = 0.50

b = 0.75

b = 1.0

Undrained shear strength index of consolidation

Pa

42.2

38.5

35.8

33.8

32.2

Pp

246.2

261.2

273.2

283.1

291.3

Undrained shear strength index

Pa

55.9

49.8

44.9

40.8

37.5

Pp

166.1

172.2

177.1

181.2

184.5

7.5 Examples for Unified Solution of Rankine’s Earth Pressure

159

It can be seen from an example that when b = 0, the results in this paper are basically consistent with that of the literature (Yuan 2011). However, when the parameter b increases, the active earth pressure becomes smaller and the passive pressure becomes larger. In other words, when the intermediate principal stress is considered in different degrees, it has a great influence on the earth pressure, which is in line with the law of earth pressure. When soil pressure is calculated by effective stress method, the soil and water separation is considered, so the concept is clearer. However, it is sometimes impossible to consider the effect of the excess pore pressure on the soil under undrained shear condition. Therefore, the total pressure method is sometimes used to calculate the earth pressure, and there are two ways to calculate, that is, using the consolidated undrained and undrained shear strength indexes to calculate. These two indexes will have different degrees of influence on earth pressure. In the actual calculation, the appropriate parameter b and stress state can be selected according to the soil properties, and the size of the soil pressure can be determined reasonably.

7.5.4 Example 4 (Yuan JL) The parameters of retaining wall and fill are unit weight 18 kN/m3 ; internal friction angle ϕ = 30°, cohesion C = 10 kPa, and Poisson’s ratio v = l/3. According to the Rankine’s earth pressure theory and the formula derived in this paper, the calculation of active and passive earth pressure strength Pa and Pp are carried out. The results obtained by Yuan (2011) are shown in Figs. 7.10 and 7.11. As can be seen that the calculation results by using Rankine’s earth pressure theory are only one case of the Unified Strength Theory with parameter b = 0. The active earth pressures computed by Rankin’s active earth pressure theory are obviously higher than the actual value, and the maximum is about 50%, which have a certain safety reserve. The passive earth pressures are obviously smaller than the actual value, and the maximum is about 40%. The values of Pa and Pp increase with the Fig. 7.10 Pa at different values of H and b

160

7 Earth Pressure Theory

Fig. 7.11 PP at different values of H and b

increases of the depth and parameter b, the curve becomes steeper less gradually. The greater the parameter b is, the greater the role of Intermediate principal stress is.

7.5.5 Example 5 Retaining wall material known data: high retaining wall for H = 5 m, metope and the angle between the horizontal α = 80°, ϕ = 30°, C = 6 kPa, γ = 18 kN/m3 , δ = 10°, β = 15°, θ = 30°. The active earth pressure Pa , active earth pressure coefficient K a and passive earth pressure Pp and passive earth pressure coefficient K b values are calculated. The calculation results are listed in Tables 7.7 and 7.8. The parameters of retaining wall and fill are the height of the retaining wall is H = 5 m, the angle between wall and horizontal line is α = 80°, unit weight 18 kN/ m3 ; friction angle ϕ = 30°, cohesion C = 10 kPa, δ = 10°, β = 15°, θ = 30°. The Table 7.7 Calculation results of active earth pressure of retaining wall with different b UST parameter b

b=0

b = 0.2

b = 0.4

b = 0.6

b = 0.8

b = 1.0

ϕ UST (°)

18.00

19.14

20.05

20.80

21.41

21.94

C UST (kPa)

6.00

6.41

6.74

7.01

7.24

7.46

Ka

0.253

0.148

0.063

− 0.007

− 0.064

− 0.116

Pa (kN)

53.70

31.30

13.40

− 1.38

− 13.62

− 24.60

Table 7.8 Calculation results of passive earth pressure of retaining wall at different b UST parameter b

b=0

b = 0.2

b = 0.4

b = 0.6

b = 0.8

b = 1.0

ϕ UST (°)

18.00

19.14

20.05

20.80

21.41

21.94

C UST (kPa)

6.00

6.41

6.74

7.01

7.24

7.46

Kb

4.926

5.163

5.359

5.526

5.667

5.795

Pb (kN)

1046.9

1097.1

1138.8

1174.3

1204.3

1231.5

7.5 Examples for Unified Solution of Rankine’s Earth Pressure

161

calculation of K a , Pa , K p and Pp are calculated. The calculation results are listed in Tables 7.6 and 7.7. It can be seen from Table 7.7 that the active earth pressure and its coefficient decrease with the increase of parameter b. When b = 0, the result of Unified Strength Theory is the same as than that obtained by Mohr–Coulomb strength theory. So the Mohr–Coulomb theory is only a special case of the Unified Strength Theory. When b > 0.6, the soil has sufficient shear strength and self-stability, and the earth pressure acting on the retaining wall is zero, when b > 0.6, K a and Pb are negative, which only indicate the degree of filling self-stability can depend on its own shear strength. The filling has no pressure on the retaining wall. According to Table 7.8, the passive earth pressure and its coefficient increase with the increase of parameter b. Similarly, when b = 0, the result of Unified Strength Theory is the same as than that obtained by Mohr–Coulomb strength theory.

7.5.6 Example 6 (Ying J, Liao HJ, et al.) In 2004, based on the Unified Strength Theory, the ultimate equilibrium stress equation under plane strain state is deduced by Ying et al. (2004) introduced into the Rankine earth pressure theory. Then, the coefficients of active and passive earth pressure and the calculation formulas of earth pressure based on the Unified Strength Theory are proposed. It is shown that the formula of Rankine earth pressure based on the Mohr–Coulomb theory is a particular case of the presented formulas. The earth pressure along the depth in a practical project is calculated with different weight coefficients which reflect the influence of intermediate principal stress, and the results prove that calculation results of earth pressure considering intermediate principal stress are closer to the practical situation than those without considering intermediate principal stress. The depth of foundation pit is 10.8 m, and the area is about 3.69 m2 . The supporting structure adopts the diaphragm wall, the thickness of wall is 0.8 m, and the wall depth into the soil is 21.8 m. The earth pressure was calculated by using the Mohr–Coulomb theory and Unified Strength Theory, respectively. Five results using the UST with b = 0, b = 0.25, b = 0.5, b = 0.75, b = 1 were shown in Fig. 7.12. It can be seen that the earth pressure decreases with the increase of parameter b. When b = 0, the result of Unified Strength Theory is the same as than that obtained by Mohr– Coulomb strength theory. So the Mohr–Coulomb theory is a special case of the UST. When b = 1, that is the twin-shear strength theory, from Fig. 7.12, it can be seen that the calculation result is closer to the measured value than the Mohr–Coulomb theory. So it is more economical and practical. Numerical comparison showed that the earth pressure formula based on the twin-shear theory smaller than Rankine’s earth pressure about 20–25%, which proves that the Intermediate principal stress σ 2 cannot be ignored on the calculation of earth pressure.

162

7 Earth Pressure Theory

Fig. 7.12 Distribution curve of earth pressure

They concluded conclusion as follows: (1) Based on the Yu Unified Strength Theory, the ultimate equilibrium stress equation under plane strain state is deduced and introduced into the Rankine earth pressure theory. It is shown that the formula of Rankine earth pressure based on the Mohr–Coulomb theory is a particular case of presented formulas. (2) The derivation formula takes into account the influence of the intermediate principal stress σ 2 , and different earth pressure values can be obtained for different UST parameter b. When b = 1, the Yu Unified Strength Theory is reduced to twin-shear strength theory, and the results prove that calculation results are more in line with the actual situation. It can be applied in practical engineering to achieve good economic benefits. (3) The Unified Strength Theory covers or approximates the strength theories in existence, which can be suitable for all kinds of materials. For geotechnical materials, a more reasonable result can be obtained by selecting appropriate parameter b according to the indexes and the properties of soil.

7.6 Study on J. Karstedt Space Earth Pressure of Reinforced Retaining Wall Based on the UST Based on the Unified Strength Theory, studied on J. Karstedt space earth pressure of reinforced retaining wall based on the UST. The reinforced soil was considered as a kind of composite material with characteristic of aeolotropy, the formulae of J. Karstedt space earth pressure of reinforced retaining wall were derived based on the UST. The solutions given are appropriate to reinforced soil of different reinforcements and different fillings. The strength parameters of reinforced soil can be

7.7 Unified Solution of Space Earth Pressure Computing Theory

163

Table 7.9 Results of different b for J. Karstedt space earth pressure UST parameter b

ϕ UST (°)

P (kN)

λ

P (kN/m)

0.0

37.00

806.83

0.19

216.68

0.2

38.50

754.79

0.17

192.45

0.4

39.65

712.92

0.15

175.32

0.6

40.57

678.93

0.14

162.57

0.8

41.32

650.96

0.13

152.71

1.0

41.94

627.61

0.12

144.85

calculated by using the UST when parameter b is of different values, which contain the contributions of the principal stresses σ 1 , σ 2 , and σ 3 . The unified solution can give full play to the strength potential of the reinforced soil and help obtain a certain degree of economic benefits in practical engineering. The parameters of reinforced retaining wall are H = 7.5 m, ϕ = 37°, B = 8 m, γ = 19 kN/m3 , the calculating results of different b for J. Karstedt space earth pressure are given in Table 7.9. λ is the earth pressure coefficient. They come to the conclusions that: Based on the Unified Strength Theory, this paper gives the calculation formula of the J. Karstedt space earth pressure of reinforced retaining wall. The calculation results show that the space earth pressure decreases with the increase of the UST parameter b. The unified solution can be applied flexibly to the calculation of earth pressure on the retaining wall. The earth pressure coefficient decreases with the increase of the UST parameter b.

7.7 Unified Solution of Space Earth Pressure Computing Theory Using the traditional Rankine theory, Coulomb theory, and limit equilibrium theory to calculate the earth pressure, the retaining wall is studied as a plane problem, that is, the wall is regarded to be a unit length in an infinite wall in length. However, in fact, the length of all retaining walls is limited, but their relative length is different. The earth pressure acting on the retaining wall varies with the height and length of wall. Along the length of the wall, the earth pressure acting on the middle fracture section is obviously different from that on the section at both ends, indicating that the earth pressure acting on the retaining wall is a space problem rather than a plane problem. As early as 1930s, in Terzaghi’s works, the spatial characteristics of earth pressure have been pointed out, but the actual research on this issue began only in 1950s, especially in the past 20 years. Many scholars have carried out experimental research on this issue, which has made it possible to calculate the spatial pressure. In 1977,

164

7 Earth Pressure Theory

Klein proposed that the slippery soil is a semi-cylindrical truncated cylinder and derived the calculation method of the earth pressure. However, the Mohr–Coulomb criterion is still adopted, only the effect of principal stresses σ 1 and σ 3 on the yield of soil are considered, this is obviously not in line with reality. With the development of urban construction and the extensive exploitation and utilization of underground space, deep foundation pit engineering has been paid more and more attention. The design and construction of deep foundation pit has gradually become a research hotspot in engineering and academia. At present, the theoretical calculations for the relevant problems of deep foundation pits are generally based on two-dimensional plane situation. In fact, the foundation pit is a three-dimensional spatial structure with finite length, width, and depth. A large number of engineering practices and experimental studies have shown that the earth pressure and displacement values in the middle area of the foundation pit are larger than that of those in a certain range at both ends of foundation pit. There is obvious spatial effect on the slope of deep foundation pit and supporting structures. The study of the spatial effect of foundation pits is to study the variation law of the stress and deformation in the middle and both ends of the foundation pit slope, so as to guide the design and construction effectively. The study of earth pressure is generally based on the failure mechanism of the foundation pit slope. Based on the Unified Strength Theory, the formula of the space earth pressure computing theory was derived by Gao JP, et al. The solutions have been given used to calculate the space earth pressure with kinds of different special property materials. The strength’s parameter contains the contributions of the principal stresses σ 1 , σ 2 , and σ 3 to material strength. The unified solutions can be fully applied in the backfill and help obtain outstanding economical benefit. The results of the example analysis are given in Table 7.10. The results show that (1) With the increase of coefficient of intermediate principal stress value of b, the space active earth pressure decreases, and the active earth pressure coefficient increases slightly. The vertical distance of action point from the top of wall can be slightly reduced. Table 7.10 Calculation results of space earth pressure of retaining wall at different b UST parameter b

b=0 b= 0.2

b= 0.4

b = 0.6 b = 0.8

b = 1.0

ϕ UST (°)

30

31.45 32.58 33.49

34.23 34.85

K

2.264 2.285 2.297 2.304

2.308 2.310

Active earth pressure Pa (kN)

353.8 337.9 325.7 316.0

308.0 301.5

The earth pressure per unit length of wall e (kN/m)

29.49 28.16 27.14 26.33

25.67 25.12

Vertical distance of action point from the top 2.624 2.624 2.624 2.6239 2.624 2.6238 of wall y (m)

7.7 Unified Solution of Space Earth Pressure Computing Theory

165

(2) Use the active earth pressure of various materials in space solutions given by the application of the unified strength parameter formula can calculate different b values of the soil strength parameters, the value contains both the principal stresses σ 1 , σ 2 , and σ 3 on material strength contribution and can be with the plane corresponding to the conventional geotechnical tests measured the strength indexes of C 0 , ϕ 0 for the space problem required strength indexes C UST and ϕ UST . Unified solutions can exert the potential strength of retaining wall filler. As for the spatial effect of deep foundation pits, Zheng (2013) conducted a theoretical analysis based on the Unified Strength Theory. By analyzing the threedimensional sliding wedge model of the cohesive-free soil slope of foundation pit, the distribution law of earth pressure under the Unified Strength Theory is obtained, and compared it with the results of Mohr–Coulomb strength theory, the former is closer to actual measurement values. An example is The depth of a foundation pit is 10 m and a rectangle with a plane of 20 × 32 m; the physical and mechanical properties of soil layer are γ = 18.4 kg/ m3 , ϕ = 25.8°, α = 0, δ = 16.4°. When b takes different values, the distribution of soil pressure along the long direction of foundation pit is shown in Fig. 7.13. The results show that the shear strength parameter ϕ UST is increased with the increase of b value. That is, the value calculated by using the Unified Strength Theory considering the intermediate principal stress is increased relative to Mohr–Coulomb strength theory; the height H of the balance arch decreases with the increase of the b value, namely both the space earth pressure and the volume of the sliding wedge are decreased. She came to the conclusion as follows (1) For the calculation of earth pressure in three-dimensional space, it is more reasonable to use the Unified Strength Theory considering intermediate principal stress. Fig. 7.13 Distribution of soil pressure along the long direction of foundation pit

166

7 Earth Pressure Theory

(2) The value of shear strength calculated by using twin-shear strength theory is larger than that obtained by the Mohr–Coulomb theory. It shows that the existence of intermediate principal stress increases the soil shear strength. (3) The results obtained by the Mohr–Coulomb theory are conservative. (4) Compared with the Mohr–Coulomb strength theory, there has an increasing trend in spatial effect zone of the foundation pit when computed by the twinshear strength theory. (5) The measured results of earth pressure and displacement of the foundation pit slope are generally less than the theoretical values. The effect of the intermediate principal stress is one of the factors. The result of twin-shear strength theory is closer to actual measurement values.

7.8 Summary The calculation of earth pressure is an important part of soil mechanics, which is mostly calculated by the Rankine earth pressure theory. Based on the Mohr–Coulomb theory, the formulas of the active and passive earth pressure for clay and non-cohesive soil are derived by this theory. However, the Mohr–Coulomb criterion is still adopted; only the effect of principal stresses σ 1 and σ 3 on the yield of soil are considered; this obviously deviates from the measured value; and the distribution of Rankine earth pressure differs from the practical one. In this chapter, we introduce the Yu Unified Strength Theory into the analysis of earth pressure and thus come up with a series of new results that can be better adapted to different materials and structures. (1) The Yu Unified Strength Theory covers or approximates the strength theories in existence, which can be suitable for metal and geotechnical materials. For geotechnical materials, a more reasonable result can be obtained by selecting appropriate parameter b according to the indexes and the properties of soil. (2) The earth pressure acting on retaining wall is regarded as a problem of plane strain, and the Rankine active earth pressure and passive earth pressure are deduced by combined with the Rankine earth pressure theory and the Yu Unified Strength Theory. The formula introduces the intermediate principal stress coefficient b, which can be applied to various rock and soil materials with different characteristics. So the classical Rankine’s earth pressure theory is a special case of the UST. (3) In practice, the active earth pressure obtained by the Rankine’s earth pressure formula is usually larger than the measured values. The reason is that the influence of intermediate principal stress is not taken into account. It can be seen that the intermediate principal stress has a certain influence on Rankine’s earth pressure. The Yu Unified Strength Theory can exert the strength potential of the material better than the Mohr–Coulomb strength theory. An example analysis shows that it can better exert the material’s strength potential by 20–50%, which can produce certain economic benefits.

References

167

(4) When b = 0, it corresponds to the Mohr–Coulomb criterion, and the intermediate principal stress has no effect on the strength; When b = 1, it corresponds to the twin-shear strength theory. The influence of intermediate principal stress on strength is equivalent to the minimum principal stress. Parameter b can be determined by using the true triaxial test. (5) The calculation formula of the sliding wedge pressure is based on the plane sliding surface, and it has certain difference for the actual surface slip surface. According to the research, it is known that the curvature of the active sliding surface is smaller, so the deviation is not large when the plane slip surface is used instead to calculate. However, in the passive state, there will be a large error.

References Borja RI, Regueiro RA (2001) Strain localization in frictional materials exhibiting displacement jumps. Comput Methods Appl Mech Eng 190(20–21):2555–2580 Dong Q, Mi J, Jing HJ et al (2012) Application of retaining wall in the Rankine active earth pressure based on twin shear UST. Highway 8:32–35 (in Chinese) Fan W, Liu C, Yu MH (2004) Formula for earth pressure based on UST. J Chang’an Univ (Natural Science Edition) 24(6):43–46 (in Chinese) Fan W, Bai XY, Yu MH et al (2005a) Unified solution of formulae for earth pressure. Coal Geol Explor 33(2):52–55 Fan W, Sheng ZJ, Yu MH (2005b) Upper-bound limit analysis of earth pressure based on UST. Chin J Geotech Eng 27(10):1147–1153 (in Chinese) Gao JP, Liu YL, Yu MH (2006) UST with applications to earth pressure. J Xi’an Jiaotong Univ 40(3):357–359 (in Chinese) Matsuoka (2001) Soil mechanics (trans: Luo D, Yao YP). China Water Power Press Rankine WJM (1857) On the stability of loose earth. Philos Trans R Soc Lond 19–27 Wei RL (1995) Some problems of calculating earth pressure by total stress method. Chin J Geotech Eng 17(6):120–125 (in Chinese) Xie QD, Liu J, He J (2003) Unified twin shear strength theory for calculation of earth pressure. Chin J Geotech Eng 25(3):343–345 (in Chinese) Ying J, Liao HJ, Pu WC (2004) Earth pressure theory based on UST under plane strain state. Chin J Rock Mech Eng 23(s.1):4315–4318 (in Chinese) Yuan JL (2011) Sensitivity analysis of parameters of Rankine earth pressure based on UST. Coal Geol Explor 39(1):47–51 (in Chinese) Zhai Y, Lin YL, Fan W, Yu MH (2004) Unified solution of the theory on sliding wedge for earth pressure. J Earth Sci Enivron 26(1):24–28 (in Chinese) Zhang J, Hu RL, Liu HB et al (2010a) Calculation study of Rankine earth pressure based on UST. Chin J Geotech Eng 29(A01):3169–3176 (in Chinese) Zhang J, Hu RL, Yu WL et al (2010b) Sensitivity analysis of parameters of Rankine’s earth pressure with inclined surface considering intermediate principal stress. J Geotech Eng 32(10):1566–1572 Zheng HH (2013) Analysis of soil pressure distribution of foundation pit slope based on twin shear strength theory. J China Foreign Highw 33(4):58–61 (in Chinese)

Chapter 8

Ultimate Bearing Capacity of Strip Footings

8.1 Introduction In geotechnical engineering, the ultimate bearing capacity of soils under strip footing is an important problem. The ultimate bearing capacity is defined as the least pressure that will cause complete shear failure of the soil in the vicinity of the foundation. Loads from a structure are transferred to the underlying soil through a foundation such as a footing. The foundation and soils must not collapse or become unstable under any conceivable loading. This chapter mainly studies the bearing capacity of strip foundation, and the construction of a shallow foundation is shown in Fig. 8.1 (Budhu 2007). Generally speaking, there are two main methods for calculating the ultimate load: One is with the static equilibrium and the limit equilibrium condition, the differential equation is established, and then the stress solution of each point is obtained according to the boundary condition. The results obtained from this method are more accurate, but it is difficult to solve the problem in a little complicated condition. The other is with the assumption of sliding surface, and the ultimate bearing capacity can be calculated based on the static equilibrium condition of the soil. This method is widely used because of its simplicity in calculation. For strip foundation, the present theory is basically based on the assumption of sliding surface, which has some limitations. Prandtl firstly suggested the bearing capacity equation for a weightless and semiinfinite space under a vertical strip load using characteristic method in 1920. The corresponding theoretical solution is obtained according to the limit equilibrium. With the plane strain slip line field theory, the limit analysis of strip foundation was studied by Hencky (1923), Geiringer (1930), Hill (1950), Prager (1949), Johnson and Mellor (1962). CokolovckiN et al. further popularized the slip line field theory. The slip field theory is supported by other observations. Similar results are obtained for numerical analysis of strip foundations (Yu and Li 2012). For the actual foundation condition, Reissner modified the Prandtl’s equation in 1924, which taking into account the embedded depth d of foundation and obtained the calculation equation of earth pressure on foundation. Prandtl and Reissner’s equations without taking © Springer Nature Singapore Pte Ltd. and Zhejiang University Press 2023 M.-H. Yu, Soil Mechanics, https://doi.org/10.1007/978-981-99-2781-4_8

169

170

8 Ultimate Bearing Capacity of Strip Footings

Fig. 8.1 Construction of a shallow foundation (Budhu 2007)

into account the effect of shear strength, gravity, and intermediate principal stress of soil. Therefore, the calculation is not very accurate and is limited by the variety of properties. Terzaghi et al. (1996) also proposed the formulas of bearing capacity coefficients based on rigid and rough strip footing in 1943. In 1961–1973, based on the Prandtl theory, the famous scholar Hansen and Vesic have put forward the formula for calculating the ultimate bearing capacity of foundation with different central shapes and different embedded depths under the central inclined load, and the effects of shape, eccentricity and inclination, shear strength of soil on both sides of foundation, the tilt of foundation, and the compressibility of soil are also studied. Terzaghi applied Prandtl’s theory to a strip footing with the assumption that the soil is a semi-infinite, homogeneous, isotropic, weightless rigid-plastic material. The Rankine, Meyerhof, Terzaghi’s formula are all derived based on the Tresca criterion or Mohr–Coulomb criterion, which does not consider the influence of intermediate principal stress. The Yu Unified Strength Theory takes into account the influence of intermediate principal stress and has a simple linear expression, which provides a new theoretical basis for the study of ultimate bearing capacity of foundation. In 1997, Yu extended the Yu Unified Strength Theory to the problem of plane strain, and put forward the plane strain unified slip line field theory (Yu et al. 1997). Then, he obtained the unified solution of the ultimate bearing capacity of foundation by using the unified slip field theory. After twentieth century, some scholars have introduced the Yu Unified Strength Theory into the study of ultimate bearing capacity of foundation, which taking into account the σ 2 on the calculation results. So the unified solution obtained in their work is a series of results, rather than one specific result. Zhou et al. (2002a, b), Wang et al. (2006), Yang et al. (2005), Fan et al. (2003, 2005), Lu et al.

8.2 Bearing Capacity of Strip Footings

171

(2007), Liu and Zhao (2005), Yu et al. (1994, 2006), Yu and Li (2012), Ma et al. (2013, 2014a, b), Sui and Wang (2011), Zhu et al. (2015) are all obtained new results in their studies. The ultimate bearing capacity of strip foundation studied in this chapter is the same as the traditional one, only the Yu Unified Strength Theory will be used instead of the traditional Mohr–Coulomb strength theory. The results are also developed from a solution to a series of ordered solutions. Therefore, it can be applied to more materials and structures and provides more information, results, references, and reasonable choices for engineering applications. These new results may also achieve some economic benefits.

8.2 Bearing Capacity of Strip Footings Experimental results show that due to the insufficient bearing capacity, the shear failure will be taken place in foundation under the action of loading. And the failure modes can be divided into three types, namely the overall shear failure, the local shear failure, and the punching shear failure, which can be judged by field load test. The concept of overall shear failure was first proposed by Prandtl in 1920. Its destructive characteristics are the foundation produces an approximately linear elastic deformation under the action of loading, such as the first section of the p-s curve in Fig. 8.2a is linear, and P0 is known as the ultimate load. Continuous complete that destroyed the foundation of the sliding surface above the ground, based on both sides of the ground has a hump phenomenon. When the foundation occurs the destruction of this type, buildings will suddenly dumped. The damage type of p-s curve has obvious turning point. This is a typical soil strength damage, and damage has certain suddenness. For smaller soil compression, such as hard close-grained sand or clay, when the pressure is large enough, usually this type of damage occurs. The concept of the local shear failure was first proposed by Terzaghi in 1943. Its destructive characteristics are the damage type of p-s inflection point of curve are difficult to determine. Destruction of the sliding surface of the foundation is not complete, also does not extend to the ground. There may be slight bulge on the ground, but the foundation is not significantly tilt. The foundation settlement is large compared with the foundation overall shear failure, because of the large settlement, the foundation loses to continue carrying capacity. The p-s curve as shown in Fig. 8.2b, for loose sand and soft clay, the base with the relatively larger depth has such a destructive type. The concept of the punching shear failure was first proposed by de Beer and Vesic in 1959. Such the destruction types of p-s curve inflection point cannot be determined. Ground foundation damage is due to the soft soil deformation below the foundation and vertical shear along the base perimeter, so that making the foundation sink. Destroyed the foundation settlement is very big, also called pierce shear failure. The p-s curve as shown in Fig. 8.2c, for loose sand and soft soil which have large

172

8 Ultimate Bearing Capacity of Strip Footings

Fig. 8.2 Failure forms of foundations. a Bulk shear failure, b local shear failure, c punch shear failure

compression, or the foundation with the large relative depth situation will appear this kind of the damage types.

8.2.1 Formula of Ultimate Bearing Capacity The UST can be expressed as follows: α σ1 + ασ3 σ2 ≤ F = σ1 − (bσ2 + σ3 ) = σt 1+b 1+α 1 σ1 + ασ3 σ2 ≥ , F= (σ1 + bσ2 ) − ασ3 = σt 1+b 1+α

(8.1)

8.2 Bearing Capacity of Strip Footings

173

where α is the ratio of tension to compression of material, parameter b is a coefficient that reflects the influence of the intermediate principal shear stress and corresponding normal stress on the yield of materials. 1. Basic assumption (1) The substrate with completely rough When the foundation is to destroy and occur continuous slip surface, part of the soil under basal will move down with the base together and in the elastic equilibrium state. The part of the soil is called elastic wedge. The soil in this area is less prone to shear displacement, but in a compacted state. Boundary ab and the horizontal angle depends on the substrate roughness, as shown in Fig. 8.3. Except for the elastic wedge, all soils in the sliding range are in a plastic state. The sliding zone is composed of radial shear zone II and Rankine passive zone X, and radial shear zone boundary bc is expressed by logarithmic spiral curve: r = r0 eθtgϕUST ,

(8.2)

in which ϕUST = arcsin

b(1 − m) + (2 + bm + b) sin ϕ0 , 2 + b + b sin ϕ0

(8.3)

where m = 2σ 2 /(σ 1 + σ 3 ), ϕ 0 is the internal friction angle of geotechnical materials r 0 is the initial radius, θ is the angle between the arbitrary angle and initial radius, the boundary cd of Rankine passive zone is a line, and the angle between the boundary and the horizontal surface is 45° + ϕ UST /2, as shown in Fig. 8.3. The both sides of foundation above the base are not taken into account the effect of shear strength, while the corresponding uniformly distributed load is expressed by q = rD. 2. Determination of ultimate bearing capacity of foundation

Fig. 8.3 Basement with the rough substrate

174

8 Ultimate Bearing Capacity of Strip Footings

According to the above basic hypothesis, the ultimate load at the overall shear failure can be obtained by the equilibrium condition of the elastic wedge aba1 in Fig. 8.4: 1 Q u = 2Pp cos(ψ − ϕUST ) + CUST B tan ψ − γ B 2 tan ψ, 4

(8.4)

where CUST =

2(1 + b)C0 cos ϕ0 1 · . 2 + b + b sin ϕ0 cos ϕUST

(8.5)

C 0 is the cohesive force of geotechnical materials, B is the base width, γ is the bulk density of foundation soil, and Pp is the resultant force of passive earth pressure acting on the boundary ab of an elastic wedge, that is Pp = Ppc + Ppq + Ppγ B 1 γ B tan ϕ C k + qk + k UST pc pq UST pγ 2 cos2 ϕUST 4 3π ϕUST ( 2 +ϕUST −2ψ ) tan φUST (1 + sin ϕUST ) − 1 kpc = cos e cot ϕ UST cos ψ where .

2 3π kpq = coscosϕψUST e( 2 +ϕUST −2ψ ) tan ϕUST tan π4 + ϕUST 2 Pp =

(8.6) (8.7)

(8.8)

in which k pγ is the passive earth pressure coefficient of the γ term, which must be determined by tentative calculation. Substituting Eqs. (8.6) and (8.7) into Eq. (8.4), there is Fig. 8.4 Mechanical model of elastic wedge

8.2 Bearing Capacity of Strip Footings

qu =

Qu 1 = CUST Nc + q Nq + λB Nγ , B 2

175

(8.9)

where q is the equivalent surcharge of the footing embedment, N c and N q are bearing capacity coefficients of cohesion and surcharge. The expression for N c N γ and N q can be written as cos(ψ − ϕUST ) ( 3π +ϕUST −2ψ ) tan ϕUST e 2 (1 + sin ϕUST ) − 1 cos ψ sin ϕUST π ϕUST cos(ψ − ϕUST ) ( 3π +ϕUST −2ψ ) tan ϕUST e 2 + tan Nq = cos ψ 4 2 kpr cos(ψ − ϕUST ) 1 Nγ = tan ψ −1 . (8.10) 2 cos ψ cos ϕUST Nc = tan ψ +

Equation (8.9) is obtained under the condition of rough substrate, wherein the angle ψ between the boundary ab of elastic wedge and the horizontal plane is an undetermined value. For that reason, this section assumes that. Assuming that the substrate with completely rough (Fig. 8.5), it can be assumed that the angle between boundary ab of elastic wedge and the horizontal plane is equal to ψ = ϕ UST , Eq. (8.10) can be written as follows:

Nc = Nq − 1 cot ϕUST 3π e( 2 −ϕUST ) tan ϕUST

Nq = 2 cos2 π4 + ϕUST 2 kpr 1 Nγ = tan ϕUST −1 . 2 cos2 ϕUST

(8.11)

The coefficient of bearing capacity is related to the friction angle, and the coefficient k pγ of passive earth pressure must be determined by tentative calculation. In order to facilitate the calculation, combined with Terzaghi formula, there is

Nγ = 1.8 Nq − 1 tan ϕUST

Fig. 8.5 Substrate is completely rough

(8.12)

176

8 Ultimate Bearing Capacity of Strip Footings

Fig. 8.6 Substrate is completely smooth

Assuming that the foundation with the completely smooth substrate (Fig. 8.6), the elastic wedge has become Rankine active zone, and the entire area of the slide has evolved into exactly the same than that of Prandtl active zone. The angle between boundary of Rankine active zone and the horizontal plane is ψ ψ=

1 π + ϕUST . 4 2

(8.13)

Substituting Eq. (8.12) into Eq. (8.9), the bearing capacity coefficients N c , N q and N r of the foundation with smooth substrate can be obtained.

8.2.2 The Influence of Various Parameters on Ultimate Bearing Capacity There is a strip foundation, width is 4 m, the embedded depth is 3 m, and the foundation is homogeneous cohesive soil. The bulk density of foundation soil is 19.5 kN/m3 . The influence of various parameters on ultimate bearing capacity will be discussed in below. 1. For the rough substrate (m = 1) Assuming that the substrate is rough, Eqs. (8.9)–(8.12) are used to calculate, the results are shown in Figs. 8.7, 8.8, 8.9, 8.10, 8.11, 8.12, 8.13, 8.14, 8.15, 8.16, 8.17, 8.18, 8.19, 8.20, 8.21, 8.22, 8.23 and 8.24. Figures 8.7, 8.8, 8.9, 8.10, 8.11, 8.12, 8.13, 8.14 and 8.15 are the rough substrate results when m = 1; the relation between ϕ 0 and qu is shown in Figs. 8.7, 8.8 and 8.9; the relation between C 0 and qu is shown in Figs. 8.10, 8.11 and 8.12; the relation between UST parameter b and qu is shown in Figs. 8.13, 8.14 and 8.15. 2. For the rough substrate (m = 0.8) Figures 8.16, 8.17, 8.18, 8.19, 8.20, 8.21, 8.22, 8.23 and 8.24 are the rough substrate results when m = 0.8; the relation between ϕ 0 and qu is shown in Figs. 8.16, 8.17

8.2 Bearing Capacity of Strip Footings

177

Fig. 8.7 Relation between qu and ϕ 0 (b = 1)

Fig. 8.8 Relation between qu and ϕ 0 (b = 0.5)

Fig. 8.9 Relation between qu and ϕ 0 (b = 0.3)

and 8.18; the relation between C 0 and qu is shown in Figs. 8.19, 8.20 and 8.21; the relation between UST parameter b and qu is shown in Figs. 8.22, 8.23 and 8.24. 3. For the smooth substrate (m = 1) Assuming that the substrate is smooth, Eqs. (8.8), (8.9) and (8.13) are used to calculate, the results when m = 1 are shown in Figs. 8.25, 8.26, 8.27, 8.28, 8.29, 8.30,

178 Fig. 8.10 Relation between qu and C 0 (b = 1)

Fig. 8.11 Relation between qu and C 0 (b = 0.5)

Fig. 8.12 Relation between qu and C 0 (b = 0.3)

Fig. 8.13 Relation between qu and b (ϕ 0 = 20°)

8 Ultimate Bearing Capacity of Strip Footings

8.2 Bearing Capacity of Strip Footings Fig. 8.14 Relation between qu and b (ϕ 0 = 30°)

Fig. 8.15 Relation between qu and b (ϕ 0 = 40°)

Fig. 8.16 Relation between qu and ϕ 0 (b = 1)

Fig. 8.17 Relation between qu and ϕ 0 (b = 0.5)

179

180 Fig. 8.18 Relation between qu and ϕ 0 (b = 0.3)

Fig. 8.19 Relation between qu and C 0 (b = 1)

Fig. 8.20 Relation between qu and C 0 (b = 0.5)

Fig. 8.21 Relation between qu and C 0 (b = 0.3)

8 Ultimate Bearing Capacity of Strip Footings

8.2 Bearing Capacity of Strip Footings

181

Fig. 8.22 Relation between qu and b (ϕ 0 = 20°)

Fig. 8.23 Relation between qu and b (ϕ 0 = 30°)

Fig. 8.24 Relation between ultimate load qu and UST parameter b (m = 0.8, ϕ 0 = 40°)

8.31, 8.32 and 8.33. Figures 8.25, 8.26, 8.27, 8.28, 8.29, 8.30, 8.31, 8.32 and 8.33 are the rough substrate results when m = 1; the relation between ϕ 0 and qu is shown in Figs. 8.25, 8.26 and 8.27; the relation between C 0 and qu is shown in Figs. 8.28, 8.29 and 8.30; the relation between UST parameter b and qu is shown in Figs. 8.31, 8.32 and 8.33. 4. For the smooth substrate (m = 0.8) Figures 8.34, 8.35, 8.36, 8.37, 8.38, 8.39, 8.40, 8.41 and 8.42 are the rough substrate results when m = 0.8; the relation between ϕ 0 and qu is shown in Figs. 8.34, 8.35 and 8.36; the relation between C 0 and qu is shown in Figs. 8.37, 8.38 and 8.39; the relation between UST parameter b and qu is shown in Figs. 8.40, 8.41 and 8.42.

182

8 Ultimate Bearing Capacity of Strip Footings

Fig. 8.25 Relation between ϕ 0 and qu (b = 1)

Fig. 8.26 Relation between ϕ 0 and qu (b = 0.5)

Fig. 8.27 Relation between ϕ 0 and qu (b = 0.3)

8.3 Ultimate Bearing Capacity of Footings Caused by Cohesion and Overloading This section mainly studies the unified formula for the ultimate bearing capacity of foundation caused by the cohesion and the overloading of soil on both sides of foundation.

8.3 Ultimate Bearing Capacity of Footings Caused by Cohesion …

183

Fig. 8.28 Relation between qu and C 0 (b = 1)

Fig. 8.29 Relation between qu and C 0 (b = 0.5)

Fig. 8.30 Relation between qu and C 0 (b = 0.3)

8.3.1 Formula Deduction of Ultimate Bearing Capacity of Foundation 1. Basic assumptions (1) As shown in Fig. 8.43, when the foundation is subjected to an overall shear failure, its sliding surface extends continuously to ground and intersects at the

184 Fig. 8.31 Relation between qu and b (ϕ 0 = 20°)

Fig. 8.32 Relation between qu and b (ϕ 0 = 30°)

Fig. 8.33 Relation between qu and b (ϕ 0 = 40°)

Fig. 8.34 Relation between qu and ϕ 0 (b = 1)

8 Ultimate Bearing Capacity of Strip Footings

8.3 Ultimate Bearing Capacity of Footings Caused by Cohesion … Fig. 8.35 Relation between qu and ϕ 0 (b = 0.5)

Fig. 8.36 Relation between qu and ϕ 0 (b = 0.3)

Fig. 8.37 Relation between qu and C 0 (b = 1)

Fig. 8.38 Relation between qu and C 0 (b = 0.5)

185

186 Fig. 8.39 Relation between qu and C 0 (b = 0.3)

Fig. 8.40 Relation between qu and b (ϕ 0 = 20°)

Fig. 8.41 Relation between qu and b (ϕ 0 = 30°)

Fig. 8.42 Relation between ultimate load qu and UST parameter b (m = 0.8, ϕ 0 = 40°)

8 Ultimate Bearing Capacity of Strip Footings

8.3 Ultimate Bearing Capacity of Footings Caused by Cohesion …

187

E point, while the sliding surface consists of three parts: line AC, spiral curve CH, and line HE, in which the angle between line AC and the horizontal plane is 45° + ϕ UST /2. The interaction between the side BF of foundation and the soil or the influence of soil weight on both sides BEF of foundation can be replaced by the equivalent stresses σ 0 and τ 0 on plane BE, respectively. Therefore, the soil BEF can be removed and replaced by the “equal free surface BE” when the soil balance is taken into account. It is assumed that the angle between the BE and the horizontal plane is β, which increases with the embedded depth of the foundation. If the friction angle between the side of the foundation and the soil is δ, the average normal stress σ α and tangential stress τ α acting on the side of the foundation are σα =

1 k0 γ D 2

τα = σα tan δ =

1 k0 γ D tan δ, 2

(8.14) (8.15)

where k 0 is the static lateral pressure coefficient of soil and D is the embedded depth of foundation. 2. Cohesion and bearing capacity caused by overloading of soil on both sides of foundation (1) Calculation of normal and tangential stresses (σ 0 , τ0 ) on an equal free surface BE. The normal stress on the BE surface can be obtained from the equilibrium condition of all the forces in the normal direction of the BE plane

Fig. 8.43 Ultimate bearing capacity model of the foundation

188

8 Ultimate Bearing Capacity of Strip Footings

σ0 =

1 1 γ D k0 sin2 β + k0 tan δ sin 2β + cos2 β . 2 2

(8.16)

Similarly, the tangential stress on the BE plane is 1 1 2 τ0 = γ D (1 − k0 ) sin 2β + k0 tan δ sin β . 2 2

(8.17)

By the nature of the logarithmic spiral curve and the geometric relation of BHE in Fig. 8.42, there is

2D sin π4 − ϕUST cos(η + ϕUST ) 2 sin β = , B1 cos ϕUST eθ tan ϕUST

(8.18)

where θ is the central angle of the logarithmic spiral curve (θ = 135◦ +β−η− 12 ϕUST ). Equations (8.16) and (8.17) show that the normal stress and tangential stresses on the BE surface are functions of β, the solution must be determined by tentative calculation, that is, the value β is assumed to be calculated values of σ 0 and τ0 by Eqs. (8.16) and (8.17), then to calculate the value β by the limit stress diagram on Fig. 8.44, until the assumed value is consistent with the inverse value. (2) Calculation of normal and tangential stresses (σ b , τb ) on surface H As seen in Fig. 8.44, ∠dce = 2η. By geometrical relationship, there is

Fig. 8.44 Mohr’s circle

(8.19)

8.3 Ultimate Bearing Capacity of Footings Caused by Cohesion …

σb = σ0 + ce sin(2η + φUST ) − cd sin φUST .

189

(8.20)

Owing to the BH surface is in the limit equilibrium state, the relationship between the stresses τb and σ b is τb = CUST + σb tan ϕUST .

(8.21)

Substituting Eq. (8.21) into Eq. (8.20), there is σb =

cos2 ϕUST σ0 + CUST cos ϕUST [sin(2η + ϕUST ) − sin ϕUST ] . cos2 ϕUST − sin ϕUST [sin(2η + ϕUST ) − sin ϕUST ]

(8.22)

(3) Calculation of normal and tangential stresses (σ b , τb ) on plane BC. As shown in Fig. 8.45, the normal stress on the BC surface can be obtained when the sum of the moments of all forces on the BCH surface is zero at B σc = [(CUST + σb )e2θ tan ϕUST − CUST ] cot ϕUST .

(8.23)

Owing to the BC surface is in the limit state, so the relation between the tangential stress and the normal stress of this plane is τc = CUST + σc tan ϕUST = (CUST + σc tan ϕUST )e2θ tan ϕUST .

(8.24)

Considering the triangular wedge ABC as an object, as shown in Fig. 8.46, the balance equation of vertical force can be obtained ϕUST . qu1 = σc + τc cot 45◦ + 2

(8.25)

Substituting Eqs. (8.22)–(8.24) into Eq. (8.25), the ultimate bearing capacity can be obtained qu1 = CUST Nc + σ0 Nq , Fig. 8.45 Forces on the BCH surface

(8.26)

190

8 Ultimate Bearing Capacity of Strip Footings

Fig. 8.46 Mechanical model

where Nc = Nq − 1 cot ϕUST , Nq =

(1+sin ϕUST )e2θ tan ϕUST . 1−sin ϕUST sin(ϕUST +2η)

3. Ultimate bearing capacity caused by soil self-weight At this point, it is assumed that the cohesion of soil and the overloading on both sides of foundation are all equal to zero, that is, C 0 = 0, σ 0 = τ 0 = 0, and the center of the logarithmic spiral curve moves to the O point and needs to be determined by trial and error method. Taking ACHG in Fig. 8.43 (left) as an object, the passive earth pressure on the AC surface can be obtained by the resultant moment of force through the O point is equal to zero: p=

p1 L 1 + W L 2 , L3

(8.27)

where p is the passive earth pressure on plane AC; W is the soil self-weight; and p1 is the passive earth pressure on plane GH. As shown in Fig. 8.47, the bearing capacity produced by soil self-weight can be obtained by the equilibrium conditions in the vertical direction of the force acting on the plane ABC 1 B1 γ Nγ , (8.28) 2

where Nγ = λ42 Bp 1 sin 45◦ + 21 ϕUST − 21 tan 45◦ + 21 ϕUST , γ is the bulk density of foundation soil below the base; B1 is the foundation width; and N γ is the coefficient of bearing capacity. qu11 =

8.3 Ultimate Bearing Capacity of Footings Caused by Cohesion …

191

Fig. 8.47 Mechanical model

Finally, the ultimate bearing capacity of homogeneous strip foundation under the central load can be obtained by the superimposing Eqs. (8.26) and (8.28) qu = qu1 + qu11 = CUST Nc + σ0 Nq +

1 B1 γ Nγ . 2

(8.29)

The bearing capacity coefficients N c , N q , N γ in above equation are related to ϕ 0 , β, η, while η is controlled by the dynamic coefficient of the shear strength on the “equal free surface,” as well as the relation between η and n can be obtained through the geometric relation in Fig. 8.44: cos(2η + ϕUST ) =

n(CUST + σ0 tan ϕUST ) cos ϕUST τ0 cos ϕUST = , τb CUST + σb tan ϕUST

(8.30)

where n is the dynamic coefficient of shear strength.

8.3.2 The Influence of Parameters on the Ultimate Bearing Capacity There is a strip foundation, width is 4 m, the embedded depth is 3 m, and the foundation is homogeneous cohesive soil. The bulk density of foundation soil is 19.5 kN/ m3 , supposing the resting lateral pressure coefficient is k 0 = 0.45, the friction angle between foundation and soil is δ = 12°, m = 1. The influence of various parameters on ultimate bearing capacity will be discussed below, see Figs. 8.48, 8.49, 8.50, 8.51, 8.52, 8.53, 8.54, 8.55 and 8.56.

192 Fig. 8.48 Relation between qu and ϕ 0 (b = 1)

Fig. 8.49 Relation between qu and ϕ 0 (b = 0.5)

Fig. 8.50 Relation between qu and ϕ 0 (b = 0.3)

Fig. 8.51 Relation between qu and C 0 (b = 1)

8 Ultimate Bearing Capacity of Strip Footings

8.3 Ultimate Bearing Capacity of Footings Caused by Cohesion … Fig. 8.52 Relation between qu and C 0 (b = 0.5)

Fig. 8.53 Relation between qu and C 0 (b = 0.3)

Fig. 8.54 Relation between qu and b (ϕ 0 = 20°)

Fig. 8.55 Relation between qu and b (ϕ 0 = 30°)

193

194

8 Ultimate Bearing Capacity of Strip Footings

Fig. 8.56 Relation between qu and b (ϕ 0 = 40°)

8.4 Examples 8.4.1 Compared with the Calculated Results of Terzaghi Formula The known conditions are the same with Sect. 8.2, in which the consolidated undrained shear strength index is C 0 = 20 kPa, ϕ 0 = 30°. The following is the comparison results of formula in Sect. 8.3 and the Terzaghi formula. 1. Terzaghi formula Assuming that the substrate is completely rough: When b = 0 and m = 1: N c = 38.8, Nγ = 23.26, N q = 23.4. qu =

Qu 1 = C Nc + q Nq + γ B Nγ = 3052 kPa. B 2

2. Unified Strength Theory formula (Sect. 8.2) Assuming that the substrate is completely rough: When b = 0.2 and m = 1, N c = 41.92, Nγ = 28.226, N q = 26.64. qu =

Qu 1 = CUST Nc + q Nq + γ B Nγ = 3547.43 kPa. B 2

When b = 0.5, m = 1, N c = 48.47, Nγ = 37.08, N q = 32.60, qu = 4409 kPa. When b = 0.8, m = 1, N c = 53.30, Nγ = 44.38, N q = 37.24, qu = 5121 kPa. When b = 1.0, m = 1, N c = 57.05, Nγ = 49.79, N q = 40.73, qu = 5651 kPa. Assuming that the substrate is completely smooth: When b = 0.0, m = 1, N c = 30.10, Nγ = 18.05, N q = 18.37, qu = 2379 kPa. When b = 0.2, m = 1, N c = 33.86, Nγ = 22.79, N q = 21.70, qu = 2853 kPa. When b = 0.5, m = 1, N c = 38.92, Nγ = 29.77, N q = 26.37, qu = 3552 kPa. When b = 0.8, m = 1, N c = 43.00, Nγ = 35.81, N q = 30.24, qu = 4143 kPa. When b = 1.0, m = 1, N c = 45.50, Nγ = 39.60, N q = 32.60, qu = 4509 kPa.

8.5 Slip Line Unified Solution of Ultimate Bearing Capacity for Strip Footings

195

From the above example, it can be known that the ultimate bearing capacity of foundation increases significantly with the increase of the intermediate principal stress coefficient b, which shows that the intermediate principal stress has an obvious influence on the ultimate bearing capacity of foundation.

8.4.2 Comparison Between Unified Strength Formula and Meyerhof Formula There is a strip foundation, width is 4 m, the embedded depth is 3 m, and the foundation is homogeneous cohesive soil. The bulk density of foundation soil is 19.5 kN/ m3 , supposing the resting lateral pressure coefficient is k 0 = 0.45, the friction angle between base and soil is δ = 12°. The consolidated undrained shear strength index is C 0 = 20 kPa, ϕ 0 = 22°. The following is the comparison results of formula in Sect. 8.3 and the Meyerhof formula. 1. For Meyerhof formula (Craig 2004), there are σ0 = 28.87 kPa, τ0 = 4.21 kPa, η = 30◦ , θ = 1.9 rad, β = 15◦ σb = 57.57 kPa, n = 0.2, Nc = 28, Nq = 12, Nγ = 9.5. The ultimate bearing capacity of Meyerhof is obtained: 1 qu = C Nc + σ0 Nq + γ B Nγ = 1276.94 kPa. 2 2. For Unified Strength Theory, there are When b = 1.0, m = 1, qu = CUST Nc + σ0 Nq + 21 γ B Nγ = 1655.5 kPa. When b = 0.5, m = 1, qu = CUST Nc + σ0 Nq + 21 γ B Nγ = 1362 kPa. When b = 0.3, m = 1, qu = CUST Nc + σ0 Nq + 21 γ B Nγ = 1272 kPa. When b = 1.0, m = 0.9, qu = CUST Nc + σ0 Nq + 21 γ B Nγ = 1889 kPa.

8.5 Slip Line Unified Solution of Ultimate Bearing Capacity for Strip Footings Yu Unified Strength Theory is not only a systematic strength theory of material, but also a series of ordered results can be obtained in the analysis of structural strength problems. In 1997, combined the Yu Unified Strength Theory and the plane strain slip line field theory, Yu et al. obtained the unified solution of the ultimate bearing capacity of the strip foundation. The results are shown in Fig. 8.57 and Eq. (8.31) (see Yu et al. 1997).

196

8 Ultimate Bearing Capacity of Strip Footings

w P

q

¦Ð -¦Õ 4

D

¢ ó ¢ ò

2

¦Á

¦Â

x

¢ ñ

y Fig. 8.57 Slip line field of strip foundation soil

qUST = CUST · cot ϕUST

1 + sin ϕUST exp(π · tan ϕUST ) − 1 , 1 − sin ϕUST

(8.31)

where ϕ UST and C UST are unified material parameters obtained from the unified slip line field theory (Yu et al. 1997, 2006). These two parameters can be expressed as follows: sin ϕUST = CUST =

2(b + 1) sin ϕ0 2 + b(1 + sin ϕ0 )

2(b + 1) cos ϕ0 C0 . · 2 + b(1 + sin ϕ0 ) cos ϕUST

It is interesting to note that this result is same as traditional soil mechanics in form; however, the parameters in formula are replaced by unified cohesion C UST and friction angle ϕ UST of the Unified Strength Theory (Yu 2012), so the result is not a single, but a series of results, as shown in Fig. 8.58. Applying Unified Strength Theory to solve the bearing capacity of strip foundation soil, a series of results can be obtained. In fact, the Mohr–Coulomb criterion (b = 0) is a special case of the UST. Therefore, the UST can be provided more reference, results, comparison, and selection for the engineering application.

8.6 Summary For many years, the research of geomaterials destruction results conducted by scholars show that the intermediate principal stress has certain effects on the yield strength of geomaterials under complex stress states. Because Mohr–Coulomb strength theory does not consider the influence of intermediate principal stress, it constitutes the lower limit of yield surface, and the results are conservative and the potential strength of soil mass cannot be utilized sufficiently. Therefore, the strength of foundation has potential to be excavated. Of course, in the use of the formula, the accuracy of determining the parameter value of C and ϕ is higher. Because the

8.6 Summary

197

Fig. 8.58 Series unified solutions of ultimate load on strip foundation

determination of strength parameters is influenced by the complexity of geomaterials and the limitations of people’s understanding level, which can also cause error. With the development of the science and technology, by accurately determining, the strength parameters of geomaterials and selecting a reasonable model will help to better match the actual situation and save engineering investment. Since twentyfirst century, researchers from Chongqing University, Shanghai Jiao Tong University, Xi’an Jiao Tong University, Tongji University, Hunan University, Chang’an University, Xi’an University of technology, and others have studied the analytical and numerical solutions of bearing capacity of strip foundation by using Yu Unified Strength Theory. They have achieved a series of results, and the further applied them to practical problems. The results show that. Owing to the Mohr–Coulomb strength theory does not take the influence of intermediate principal stress into account, so the value of ultimate bearing capacity of foundation is the least, while the ultimate bearing capacity of foundation based on the twin-shear strength theory is the biggest. The internal friction angle ϕ 0 has a great influence on the ultimate bearing capacity of foundation, and with the increase of the internal friction angle, the ultimate bearing capacity of foundation increases significantly. According to examples, it is known that the larger the UST parameter b, the larger the ultimate bearing capacity of foundation. In this chapter, the unified formula of the ultimate bearing capacity of foundation is established by the Yu Unified Strength Theory. The corresponding solutions of different materials can be reasonably obtained by using this formula, which is of great significance to practical engineering. The application of Yu Unified Strength Theory can make better use of the strength potentiality of materials and achieve much economic benefit.

198

8 Ultimate Bearing Capacity of Strip Footings

References Budhu M (2007) Soil mechanics and foundations, 2nd edn. Wiley, Hoboken Craig RF (2004) Craig’s soil mechanics, 7th edn. CRC Press, Boca Raton Fan W, Lin Y, Qin Y (2003) Formula for critical load of foundation based on the UST. J Changan Univ Earth Sci Ed 25(3):48–51 (in Chinese) Fan W, Bai XY, Yu MH (2005) Formula of ultimate bearing capacity of shallow foundation based on UST. Rock Soil Mech 26(10):1617–1623 (in Chinese) Geiringer H (1930) Beit zum vollstandigen ebenen plastizitats-problem. In: Proceedings of the 3rd international congress on applied mechanics, vol 2, pp 185–190 Hencky H (1923) Ueber einige statisch bestimmte faelle des gleichgewichts in plastischen koerpern. Z Angew Math Mech 3:245–251 Hill R (1950) The mathematical theory of plasticity. Clarendon, Oxford Johnson W, Mellor PB (1962) Plasticity for mechanical engineers. Van Nostrand, London and New York Liu J, Zhao MH (2005) Researches on behaviour of composite foundation with single granular column based on unified twin shear strength theory. Chin J Geotech Eng 27(6):707–711 (in Chinese) Lu YE, Zheng JJ, Chen BG (2007) Determination of end bearing capacity of pile with double-shear slip-line theory. Chin J Rock Mech Eng 26(supp.2):4084–4089 (in Chinese) Ma ZY, Dang NF, Liao HJ (2013) Numerical solution for bearing capacity of strip footing considering influence of intermediate principal stress. Chin J Geotech Eng 35(2):253–258 (in Chinese) Ma ZY, Liao HJ, Dang FN (2014a) Effect of intermediate principal stress on flat-ended punch problems. Arch Appl Mech 84(2):277–289 Ma ZY, Liao HJ, Dang FN (2014b) Influence of intermediate principal stress on the bearing capacity of strip and circular footings. J Eng Mech ASCE 140(7):04014041 Prager W (1949) Recent developments in the mathematical theory of plasticity. J Appl Phys 20:235– 241 Sui FT, Wang SJ (2011) Research on application of UST to determining of bearing capacity of foundations. Rock Soil Mech 32(10):3038–3042 (in Chinese) Terzaghi K, Peck RB, Mesri G (1996) Soil mechanics in engineering practice, 3rd edn. Wiley, New York Wang XQ, Yang LD, Gao WH (2006) Calculation of bearing capacity about the strip foundation based on the twin shear UST. Chin Civil Eng J 39(1):79–82 (in Chinese) Yang XL, Li L, Du SC et al (2005) Double shear unified solution of Terzaghi ultimate bearing capacity of foundation. Chin J Rock Mech Eng 24(15): 2736–2740 (in Chinese) Yu MH, Li JC (2012) Computational plasticity: with emphasis on the application of the UST. Springer and ZJU Press Yu MH, Liu JY, Liu CY (1994) Orthogonal and non-orthogonal slip line field theory on the basis of twin shear strength theory. J Xi’an Jiao Tong Univ 28(2):122–126 (in Chinese) Yu MH, Yang SY, Liu CY (1997) Unified plane-strain slip line theory. China Civil Eng J 30(2):14–26 (in Chinese) Yu MH, Ma GW et al (2006) Generalized plasticity. Springer, Berlin Zhou XP, Huang YB, Ding ZC (2002a) Influence of intermediate principal stress on formula of Terzaghi ultimate bearing capacity of foundations. Chin J Rock Mech Eng 21(10):1554–1556 (in Chinese) Zhou XP, Wang JH Zhang YX (2002b) Calculation of the ultimate bearing capacity of foundation under triaxial compressive loading. J Chongqing Jianzhu Univ 24(3):28–32 (in Chinese) Zhu F, Nie L, Gao ZF et al (2015) Improved calculation method of critical filling height of embankment on soft ground. J Jilin Univ (Engineering) 45(2):389–393 (in Chinese)

Chapter 9

Slope Stability

9.1 Introduction Slope stability is one of the three classic engineering problems of soil mechanics. Slope stability and landslide treatment are important problems encountered in civil engineering, water conservancy, highway construction, and railway construction. Figure 9.1 is the slip field of a homogeneous slope under the action of loading. Figure 9.2 is a possible landslide diagram of soil slopes, in which the dashed part is the shape after landslide (Braja 2002). Their slope stability is of great significance. The slope in engineering practice, including natural slope and artificial slope, is a geological body with open-air lateral airport face on the earth surface. Natural slope refers to the natural formation of slopes, rivers, lakes, and other bank slopes. Artificial slope refers to the artificial excavation of the river, foundation pit, trench or embankment of earth dam, etc. Landslides happen because of the slope surface is tilted and in its self-weight and other external loads. If the sliding force of soil on an interior certain surface over soil anti-sliding ability in which a part of the soil slides over the other causing this phenomena. Landslides are often to the industrial and agricultural production and national life industry great loss, and some even put out the disaster. Figure 9.3 shows the collapse of a small high-rise in the lotus pond area in Shanghai. The accident was due to the excavation of an underground garage between residential buildings, causing the formation of man-made slopes. Fortunately, the distance between two rows of buildings is larger, so there is no harm to the opposite side of the building. If the dense buildings like Sha Po District of Xi’an Xingqing road, a building collapsed, it will inevitably lead to cascading collapse like the domino effect. Dense buildings cause insufficient sun on many floors, and there is no safe zone for people to escape during the earthquake. Dense buildings are not in conformity with the state policies and regulations, such as their appearance is often the result of the combination of individual rights departments and developers. Corruption is a social and long-term damage. Figure 9.4 shows a mountain in Hong Kong has caused landslides because of artificial construction and caused serious damage. © Springer Nature Singapore Pte Ltd. and Zhejiang University Press 2023 M.-H. Yu, Soil Mechanics, https://doi.org/10.1007/978-981-99-2781-4_9

199

200

Fig. 9.1 Slip line of soil slope

Fig. 9.2 A possible landslide diagram of soil slopes

Fig. 9.3 Small high-rise in Shanghai

9 Slope Stability

9.1 Introduction

201

Fig. 9.4 Landslide in Hong Kong

Movement occurred following heavy rain on a slope that had recently been cleared of forest cover (P183, Skinner and Porter, The dynamic earth and introduction to physical geology, Wiley, 1989). Debris flows are a conspicuous form of mass-wasting. They involve of the downslope movement. Slope stability analysis is often needed in engineering construction, and the results obtained under different failure criteria are often different. The conventional theoretical solutions are obtained on the basis of Mohr–Coulomb criterion. Due to the Mohr–Coulomb criterion does not take into account the influence of intermediate principal stress, there is a deficiency in theory and practice. A large number of studies, such as literatures have shown that soil materials, have obvious intermediate principal stress effect. Literature Yu (1992) shows that the yield criterion has a great influence on the calculation of soil slope stability and safety. The Yu Unified Strength Theory has a simple linear expression, which provides a theoretical basis for slope stability and the ultimate bearing capacity of foundation. At present, the Yu Unified Strength Theory has been widely used in engineering practice and has achieved good results. For geotechnical materials, using the Yu Unified Strength Theory to analysis can give full play to the strength potential of materials. Therefore, many scholars have carried out a great deal of research on the application of Unified Strength Theory in geotechnical materials and have given a

202

9 Slope Stability

positive evaluation, such as Yu et al. (1997), Zhang et al. (1998, 2000), Zhang and Shi (2010), Xu and Wang (2010), Liu et al. (2011). These research results have good engineering guiding significance.

9.2 Theoretical Derivation of Bearing Capacity There are many ways to analyze slope stability under the plane strain condition. The materials are subject to the variation law of Yu Unified Strength Theory. In this section, the unified stability theory and the common slice method are used to study the stability of slope. The expression of the stability safety factor is obtained, which will be of guiding significance to the design and construction of the slope. Generally, the stability of slope is considered in terms of plane strain problems. The plane strain of Unified Strength Theory is (Yu 1992) σ1 −

1 − sin ϕUST 2CUST cos ϕUST , (bσ2 + σ3 ) = (1 + b)(1 + sin ϕUST ) 1 + sin ϕUST

(9.1)

where b is the parameter that reflects the influence of intermediate principal stress on material failure. CUST , ϕUST are the cohesion and internal friction angle of Yu Unified Strength Theory, respectively. If let σ2 = m2 (σ1 + σ3 ), Eq. (9.1) can be written as (2 + bm)(1 − sin ϕUST )σ3 2 + 2b − bm + (2 + 2b + bm) sin ϕUST 4(1 + b)CUST cos ϕUST + . 2 + 2b − bm + (2 + 2b + bm) sin ϕUST

σ1 =

(9.2)

Generally, when the rock-soil mass is in an elastic state, m < 1; when the rock-soil mass is in yield state, m → 1. For the unit length of soil slope shown by Fig. 9.5 (left), supposing the possible sliding surface is an arc AD, with the center of the circle is O and radius is R. The sliding soil ABCDA is divided into many vertical soil bars, and the acting force on any soil bars is shown in Fig. 9.5. The size, action spot, and direction of self-weight of soil bar are known. Supposing the unknown normal force N i and tangential force T i is acting on the midpoint of the sliding surface ef. The normal forces on both sides of the soil bar are E i and E i+1 , and the vertical shear forces are X i and X i+1 . Among them, E i and X i can be obtained by the equilibrium condition of the former soil bar, while the size of E i+1 and X i+1 is unknown, and the location of the E i+1 action point is unknown. Without considering the forces acting on both sides of the soil bar, there is a stability safety factor for the slope corresponding to the sliding surface AD:

9.2 Theoretical Derivation of Bearing Capacity

203

Fig. 9.5 Slope stability is calculated by slices method a the unit length of soil slope b the acting force

∑i=n K =

i=1

[2(1 + b)C0i li cos ϕ0i − si Wi cos αi ]/ pi qi ∑n , i=1 Wi sin αi

(9.3)

/ where, si = b −bm +(2 +b +bm) sin ϕ0i , pi = 2 +b +b sin ϕ0i , qi = 1 − si2 / pi2 . The simple slices method does not take into account the interaction force between the soil bars, so the stability safety factor is smaller. This section will use the simplified method proposed by Bishop to discuss, and the basic assumption is that ➀ The force acting on both sides of the soil bar is not taken into account; ➁ neglecting the effect of vertical shear force between soil bars; ➂ the magnitude of the tangential force T i on slip surface is given and determined by formula (9.5) (Morgenstern and Price 1965). According to the vertical equilibrium condition of the soil bar, Wi − X i + X i+1 − Ti sin αi − Ni cos αi = 0.

(9.4)

If the stability safety factor of soil slope is K, the shear strength on the slip surface is only develop a part of it though, and Bishop assuming that it is in equilibrium with the tangential force on the sliding surface, that is Ti =

1 (Ni tan ϕUST + CUSTli ). K

(9.5)

According to Eq. (9.5), there is Ni =

Wi + (X i+1 − X i ) + cos αi +

1 K

CUST li K

sin αi

tan ϕUST sin αi

.

(9.6)

204

9 Slope Stability

Substituting Eq. (9.6) into Eq. (9.4), we get ∑i=n

1 i=1 ki

K =

(Wi + X i+1 − X i ) tan ϕUST + CUSTli cos αi , ∑i=n i=1 Wi sin αi

(9.7)

si sin αi 1 . pi qi K

(9.8)

where ki = cos αi +

According to the Bishop’s hypothesis that the vertical shear force is neglected, i.e., X i+1 − X i = 0, formula (9.7) becomes ∑i=n K =

i=1

[(Wi tan ϕUST + CUSTli cos αi )/ki ] . ∑i=n i=1 Wi sin αi

(9.9)

Owing to k i in formula (9.9) contains factor K, so it must be solved by the iterative method. Assume that a value K = K 1 (K 1 can take the value at 1.0–1.5), and substituting it into formula (9.8), then the obtained value k i will be substituted into formula (9.9), K = K 2 can be obtained. If the difference between K 1 and K 2 is small, then the K value is used as the stability safety factor for slope, otherwise, the re-select values were repeatedly checking computations.

9.3 Examples A homogeneous soil slope is shown in Fig. 9.6. The height of slope is H = 6 m, slope angle β = 55°, the volume density γ = 18.6 kN/m2 , the location of the center of the most dangerous slip surface O and the division of the soil bar are see in literature (Chen 2003). Fig. 9.6 Schematic diagram

9.3 Examples

205

The influence of parameter b on the safety factor of slope stability is discussed when cohesion and internal friction angle change. For the convenience, we use the same soil strip division and the same dangerous sliding surface. Figure 9.7 shows the relation between the safety stability factor K and ϕ when the cohesion is constant. Figure 9.8 shows the relation between K and C when the internal friction angle is constant. Figure 9.9 reflects the variation of safety factor K with the UST parameter b takes different values. When the internal friction angle is ϕ 0 = 12° and cohesion is C = 16.7 kPa, and Table 9.1 shows the difference in the stability safety factor K under different methods (Bishop and Unified Strength Theory method). From Table 9.1, we can see that the value of safety factor K increases with the increase of value b, which is consistent with the law of Fig. 9.9. Fig. 9.7 Relation between K and ϕ when C = 16.7 kPa

Fig. 9.8 Relation between K and b when ϕ = 12°

Fig. 9.9 Relation between K and C

206

9 Slope Stability

Table 9.1 Calculation results of K when b takes different values ∑

Numbers

1

2

3

4

5

6

7

αi

9.5

16.5

23.8

31.6

40.1

49.8

63

li

1.01

1.05

1.09

1.18

1.31

1.56

2.68

Wi

11.16 33.48 53.01 69.75 76.26 56.73 27.9

Wi sin αi

1.84

9.51

21.39 36.55 49.13 43.33 24.86 186.61

b=0 Wi tan ϕ m=1 Cli cos αi (M-C criterion) ki Ri

2.37

7.12

11.27 13.83 16.21 12.06 5.93

b = 0.25 m=1

Wi tan ϕ Cli cos αi

18.14 18.33 18.16 18.3

ki

1.016 1.01

Ri

20.4

25.84 30.84 36.45 40.8

Wi tan ϕ

2.75

8.26

Cli cos αi

19.31 19.51 19.33 19.48 19.42 19.51 23.57

ki

1.016 1.01

0.987 0.946 0.88

Ri

21.71 27.5

32.82 38.8

13.71 18.04 19.72 13.67 7.22

b = 0.5 m=1

b = 0.75 m=1

b=1 m=1

K

1.185

16.64 16.81 16.65 16.78 16.73 16.81 20.31 1.016 1.01

0.987 0.946 0.88

18.71 23.7

28.28 33.43 37.42 36.9

2.59

12.29 16.17 17.68 13.15 6.47

7.76

0.782 0.614 42.76 221.2

0.987 0.946 0.88

0.782 0.614 40.24 46.63 241.2

13.08 17.21 18.82 14

6.88

0.782 0.614

Wi tan ϕ

2.89

20.24 20.46 20.27 20.42 20.36 20.46 24.72

ki

1.016 1.01

0.987 0.946 0.88

Ri

22.77 28.84 34.41 40.67 45.53 44.9 3

52.04 269.16

Cli cos αi

21.01 21.23 21.03 21.19 21.13 21.23 25.65

ki

1.016 1.01

Ri

23.63 29.93 35.72 42.21 47.26 46.61 54.01 279.37

13.23 18.73 20.47 15.23 7.49 0.987 0.946 0.88

1.442

0.782 0.614

Wi tan ϕ

8.99

1.376

43.43 42.83 49.63 256.72

Cli cos αi

8.66

1.292

18.25 18.33 22.15

1.497

0.782 0.614

From the above calculation results, it is easy to see the conclusions that the internal friction angle and cohesion of soil have a great influence on the stability safety factor of slope. The stability safety factor of soil slope increases with the increase of soil friction angle and cohesion. At the same time, the stability safety factor of soil slope increases with the increase of parameter b, which indicated the intermediate principal stress has great influence on the stability safety factor, when the strength criterion parameter b = 0 is the calculation results of Bishop method under the M-C criterion.

9.4 Unified Solution of Bearing Capacity of a Trapezoid Structure When It …

207

9.4 Unified Solution of Bearing Capacity of a Trapezoid Structure When It Is Subjected to Uniform Load As shown in Fig. 9.10, when a trapezoidal structure or soil slope is partially submerged, according to the idea of the strip method, the weight of the underwater soil bar should be calculated by the saturated density, and the pore water pressure (hydrostatic pressure) on the sliding surface and the water pressure on the slope surface should also be taken into account. Taking the pore water in the sliding mass below the water surface EF as the detached body, besides the static pore water stress P1 on the sliding surface and the water pressure P2 on the slope surface, there are also an opposite reaction of the soil particles and the weight of the pore water on the center of gravity (Its size is equal to the seeper weight of the sliding soil under surface EF, which is expressed in GW1 ). These three forces form a balance force system. Therefore, the influence of water pressure on the sliding soil under the hydrostatic condition can be replaced by the buoyancy of the sliding soil below the static water surface. In fact, the weight of the underwater soil strip is calculated according to the floating weight. So the safety factor of the partially submerged slope is the same as that of the layered soil slope, as long as the gravity of the soil below the slope outside the water level is replaced by the heavy weight of the soil, that is ∑i=n ' ' ' i=1 C UST li + γi h 1i + γi h 2i + q bi cos αi tan ϕUST K = , ∑i=n ' i=1 γi h 1i + γi h 2i + q bi sin αi

(9.10)

where γ i is the natural gravity of soil, γ i ' is the floating weight of soil, h1i and h2i are the height of ith soil bar above the water level and below the water level, and q is the uniform load on the embankment or soil slope. When the water level of the reservoir descends, or the wharf rock slope is at low tide level and the groundwater level is relatively high, it will produce seepage and

Fig. 9.10 Calculation diagram under the uniformly distributed load

208

9 Slope Stability

endure the effect of seepage force. When considering the effect of seepage force, we must consider its influence when analyzing the stability of soil slope. If the combination of soil effective gravity and seepage force is used to taking into account the influence of seepage on the stability of slope, it is necessary to draw the flow network in the seepage area. At the same time, the seepage force generated by the seepage theory and its slip torque as well as the shear stress caused by the seepage force on the sliding surface are added to the calculation formula of the safety factor, so the safety factor of the trapezoidal soil slope under seepage can be obtained. The flow network is used to calculate the seepage force, as long as the flow network is drawn correctly enough, the accuracy of the flow network can be guaranteed. However, the calculation is tedious and it is difficult to draw the flow network at the same time. Therefore, the “substitution law” is the one most used at present. Instead of the seepage force, the sliding moment of the center of the circle is replaced by the same volume of water surrounded by the water level below the saturation line. As shown in Fig. 9.11, if the pore water under the immersion line is used as a detached mass above the sliding surface, the forces acting on it are (1) The pore water pressure on the sliding surface is Pw , and the direction points to the center of the circle. (2) The water pressure on the slope nC is P2 . (3) The combined force of the pore water weight in the nCl' range and the reaction of soil particles buoyancy is Gw1 , the direction is vertically downward. (4) The combined force of the pore water weight in the lmnl ' range and the reaction of soil particles buoyancy is Gw2 , the direction is vertical downward, and the center arm of force is d w . (5) The resistance of the soil to the percolation is T j , and the center arm of force is dj. Under stable percolation conditions, the above forces are composed of a balance force system. Through the equilibrium analysis of force, the safety factor of the

Fig. 9.11 Calculation diagram with percolation a the unit length of soil slope b the acting force

9.5 Slip Line Unified Solution of Trapezoidal Structures

209

trapezoid embankment or slope under the effect of steady seepage can be obtained as follows: ∑i=n ' ' ' ' i=1 C UST li + γi h 1i + γi h 2i + γi h 3i + q bi cos αi tan ϕUST K = , (9.11) ∑i=n ' i=1 γi h 1i + γsati h 2i + γi h 3i + q bi sin αi where γ sati is the saturated gravity of soil. h1i , h2i , h3i are the height of the ith soil bar above the phreatic line, between in phreatic line and the water level outside the slope, and the below water level outside the slope, respectively, as shown in Fig. 9.11b.

9.5 Slip Line Unified Solution of Trapezoidal Structures Generally, the stability of slope is considered in terms of plane strain problems. The Yu Unified Strength Theory for plane strain is shown in Eqs. (9.1) and (9.2). A symmetrical trapezoidal structure with a top angle 2ξ and the slip line field are also shown in Fig. 9.12. The uniformly distributed load is subjected to the top of slope, when 2ξ are respectively equal to 120°, 80°, and 60°, and the ultimate bearing capacity of structural can be calculated by using the unified plane strain slip line field theory. The unified solutions of the ultimate bearing capacity are shown in Eq. (9.12) and Fig. 9.13. qUST = CUST · cot ϕUST

1 + sin ϕUST exp(2ξ · tan ϕUST ) − 1 , 1 − sin ϕUST

(9.12)

where ϕUST and CUST are the Yu Unified Strength Theory parameters and they are equal to sin ϕUST = CUST =

Fig. 9.12 Slip field of a trapezoid structure

2(b + 1) sin ϕ0 2 + b(1 + sin ϕ0 )

2(b + 1) cos ϕ0 C0 . · 2 + b(1 + sin ϕ0 ) cos ϕUST

(9.13) (9.14)

210

9 Slope Stability

Fig. 9.13 Series results of unified solutions

It is interesting to note that this result is same as the traditional soil mechanics in form; however, the parameters in formula are replaced by unified cohesion C UST and unified friction angle ϕ UST of the Unified Strength Theory (Zhang et al. 2000), so the results are a series of results, as shown in Fig. 9.13. Traditional solution (b = 0) is just one of special cases of the Yu Unified Strength Theory. The series of unified solution can provide more information, reference, comparison, and selection for engineering applications. An experimental comparison of a trapezoidal structure with a slope angle 2ξ = 120° is also given. The experimental result is at the abscissa of b = 0.75, and the result obtained by the traditional Mohr–Coulomb strength theory is at the abscissa of b = 0. It can be seen that the results of the Yu Unified Strength Theory not only provide more data, reference, and selection for different materials, but also conform better to the experimental result (b = 0.75). Compared with the Mohr–Coulomb strength theory (b = 0), the bearing capacity of structure can be increased by 31% and remarkable economic benefits can be obtained. Figure 9.14 shows the relation curve of the slip angle 2μ with the parameter b. When the slope angle is equal to 2ξ = 180°, the embankment structure can be seen as the stress state of strip foundation, Eq. (9.12) is simplified as the unified formula of ultimate bearing capacity of strip foundation qUST = CUST · cot ϕUST

1 + sin ϕUST exp(π · tan ϕUST ) − 1 . 1 − sin ϕUST

(9.15)

9.6 Unified Solution of Slope Bearing Capacity When It Is Subjected …

211

Fig. 9.14 Slip angles at different b values

9.6 Unified Solution of Slope Bearing Capacity When It Is Subjected to Uniformly Distributed Load If the uniform load q is subjected to the top of slope or dome, as shown in Fig. 9.15, as long as the overload part are added to the weight of the soil stripe, respectively. The safety factor of soil slope is ∑i=n K =

1 i=1 ki

[(Wi + qbi ) tan ϕUST + CUSTli cos αi ] . ∑i=n i=1 (Wi + qbi ) sin αi

(9.16)

For an example in Sect. 9.3, if the uniform load is q = 10 kPa and the other conditions are the same, we can use Eq. (9.16) to calculate the stability coefficient of slope. Table 9.2 shows the change of slope stability factor K under different parameters b. It is easy to see the conclusions that when the intermediate principal stress (i.e., b > 0) is considered, the stability coefficient K of soil slope will increase, and it Fig. 9.15 Calculation diagram under uniformly distributed load

212

9 Slope Stability

Table 9.2 Relation between slope stability factor K and b under uniformly distributed load b

0

0.25

0.5

0.75

1

K

1.058

1.154

1.228

1.288

1.336

will increase with the increase of the intermediate principal stress. The result shows that the slope with uniform load will be more stable when taking the intermediate principal stress into account.

9.7 Slip Line Unified Solution of Slope The Mohr–Coulomb model or the Drucker-Prager criterion is always used for the numerical simulation of strip footing and circular foundation. Only one result is given by using of the Mohr–Coulomb model or the Drucker-Prager criterion, which is adapted only for one kind of material. In engineering practice, due to the Mohr–Coulomb strength theory does not take into account the influence of intermediate principal stress, so the result is biased against some materials. It only applies to materials with τ 0 = σ t σ c /(σ t + σ c ). The twin-shear strength theory can only be applied to materials with τ 0 = 2σ t σ c /(σ t + 2σ c ). The Yu Unified Strength Theory and the unified slip line field theory are used to analyze the ultimate load of the slope, and the unified solution of the structure can be obtained. It has many unique advantages. The known slope is in plane strain state and the material parameters are slope angle ∠BAE = γ > π /2, tension–compression strength ratio α = σ t /σ c , and the AB plane is subjected to the uniformly vertical load p, as shown in Fig. 9.16. According to the unified plane strain slip line theory and the slip line field in Fig. 9.16, the unified solution pUST of the ultimate bearing capacity acting on the AB surface can be obtained as follows: Fig. 9.16 Slip field of obtuse wedge

9.7 Slip Line Unified Solution of Slope

213

Fig. 9.17 Relationship between the ultimate bearing capacity and UST parameter b

pUST = (CUST · cot ϕUST ) tan2

π 4

+

ϕUST · exp (2γ − π ) · tan ϕUST 2

− CUST cot ϕUST .

(9.17)

It is precisely when the parameters ϕ UST and C UST are introduced to instead of ϕ 0 and C 0 , the intermediate principal stress effect is reflected in the ultimate bearing capacity expressed in Eq. (9.17). The two types of material parameters can be converted to each other, i.e., α = (1 − sinϕ)/(1 + sinϕ), σ t = 2Ccosϕ/(1 + sinϕ). If the internal friction angle of the material is ϕ 0 = 0, the material parameters of the Yu Unified Strength Theory are simplified as follows ϕ UST = 0, C UST = 2C 0 (b + 1)/(b + 2). Calculation example: If the slope angle is equal to γ = 0.8π, three kinds of tensile-compressive ratios be taken as α = 0.3, α = 0.5 and α = 0.8. The relationship between the ultimate bearing capacity pUST of slope and UST parameter b is shown in Fig. 9.17. When b = 0, ϕ UST = ϕ 0 , C UST = C 0 , the unified solution is reduced to a typical Mohr–Coulomb solution, that is

π ϕ0 + · exp (2γ − π ) · tan ϕ0 P0 = (C0 · cot ϕ0 ) tan2 4 2 − C0 cot ϕ0 . (9.18) When b = 0, α = 1, the ultimate load for the wedge based on the Tresca yield criterion is P0 = 2C0 (1 + γ − 2π ).

(9.19)

Scholars all over the world have conducted a lot of research on the stability of the slope and obtained rich research results. All kinds of theories and methods are

214

9 Slope Stability

colorful. According to the above analysis, the unified solution of slope stability analysis is not a solution, but a series of results. The results of this serialization can be better suited to different materials and structures. The unified solution can provide more comparison, information, reference, and reasonable selection for engineering application. There have been a number of different studies.

References Braja MD (2002) Principles of geotechnical engineering, 5th edn. Brooks/Cole-Thomson Learning, California Chen ZY (2003) Soil slope stability analysis: theory, method and programs. China Water Power Press, Beijing (in Chinese) Liu JJ, Li YM, Che AL (2011) Computation of dynamic safety factor of rock slope stability based on UST. Rock Soil Mech 32(Supp. 2):662–6722 (in Chinese) Morgenstern NP, Price VE (1965) The analysis of the stability of general slip surfaces. Géotechnique 15(1):79–93 Xu WL, Wang YJ (2010) Plastic analysis and supporting design and construction of Fenghuang mountain slope in Wuhan. Soil Eng Found 24(5):27–29 (in Chinese) Yu MH (1992) A new system of strength theory. Xian Jiaotong University Press, Xian (in Chinese) Yu MH, Yang SY, Liu CY (1997) Unified plane–strain slip line theory. China Civ Eng J 30(2):14–26 (in Chinese, English abstract) Zhang BH, Shi DG (2010) Study on the soil slope stability based on the UST. Chin J Undergr Space Eng 6(6):1174–1177 (in Chinese) Zhang YQ, Song L, Fan W (1998) Unified slip line solution of the wedge’ s ultimate load and its application in geotechnical engineering. J Xi’an Jiaotong Univ 32(12):59–62 (in Chinese) Zhang YQ, Fan W, Yu MH (2000) Unified slip line solution of ultimate load of slope. J Rock Mech Eng 19:994–996 (in Chinese)