Limit Analysis Theory of the Soil Mass and Its Application 9811515719, 9789811515712

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Chuanzhi Huang

Limit Analysis Theory of the Soil Mass and Its Application

Limit Analysis Theory of the Soil Mass and Its Application

Chuanzhi Huang

Limit Analysis Theory of the Soil Mass and Its Application

123

Chuanzhi Huang Tianjin Port Engineering Institute Ltd. of CCCC First Harbor Engineering Company Ltd. Tianjin, China

ISBN 978-981-15-1571-2 ISBN 978-981-15-1572-9 https://doi.org/10.1007/978-981-15-1572-9

(eBook)

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, Hangzhou, China 2020 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, express 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

Foreword

In the geotechnical engineering, stability is a key factor of the engineering success. It is one of long-term objectives for scientists and engineers to strive for limit analysis theories and methods on and for stability of the slope, ground bearing capacity, and soil pressure. For the port construction, it is developing toward the large-size and deep water areas. As the natural environment conditions, including ground, are relatively poor, it puts forward higher requirements for analysis and calculation theory on geotechnical engineering. The author and his colleagues work together to, based on the former work, make some progress by original study methods. Hereby, the study achievements are collated into a book, which is of great significance. Main features of the book are described as follows: It provides the extremum condition of the yield function and regards it as one of fundamental equations so that one set of complete fundamental equations are available for the limit analysis, and it constitutes a complete limit equilibrium or variation issue. On this basis, a brand-new solution method—generalized limit equilibrium method—is provided, which can be mutually corroborated with limit equilibrium method, slip line method, and upper- and lower-bound plasticity analysis method. During the study on slope stability, it avoids such study model as solving the statically indeterminate problem by introducing assumptions, which is commonly applied in the classical methods; the proposed slope stability analysis method is more consummate, and the calculation results are more reliable. In the study on the ground bearing capacity, it solves not only the calculation problem on the ground bearing capacity of heterogeneous soil, but also the application scopes of calculation methods are more extensive. Therefore, all above are innovative achievements, and its brand-new limit equilibrium analysis theory is more scientific compared with the former work. Another feature of the book is the close combination of the study and engineering practices, together with the stress on practical problems in the engineering. For example, the calculation method for the ground bearing capacity is established in consideration of port engineering features; for tens of engineering examples, different calculation methods are applied for the calculation and analysis, and it has verified that the proposed method is of extensive application prospects. v

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Foreword

I believe that the publication of the book will play a very active role in enrichment and development of geotechnical limit analysis theories and methods as well as solving practical problems in the engineering. In addition, some new viewpoints and expression modes in the book will achieve development and get common understandings through the joint discussion and study of readers and the author. Tianjin, China

Huan Chen

Summary

The book summarizes learning, application, and study about the limit analysis theory of the soil mass. It comprises three parts as follows: Chap. 2 discusses failure mode and fundamental equation of soil mass or ground, which is the basis of the limit analysis theory of the soil mass; thereinto, the prominent feature of the book lies in that the extremum condition of the yield function is regarded as one of the fundamental equations and provides one set of complete fundamental equations for the limit analysis on soil mass. Chapters 3–7 discuss solution methods for limit analysis, including characteristic line method, stress field method, limit equilibrium method, virtual work equation-based generalized limit equilibrium method, and generalized limit equilibrium method for surface failure mode; thereinto, it mainly describes the generalized limit equilibrium method for surface failure mode which is easy to be generalized to the general conditions of heterogeneous soil. Chapters 8–12 discuss application of limit analysis theory of the soil mass. Chapters 8 and 9 are involved with the ground bearing capacity and mainly describe the calculation for the ground bearing capacity of the heterogeneous soil; Chaps. 10 and 11 are involved with the slope stability and mainly describe the composite slip surface method with an extensive application scope and the analysis method for the slope stability obtained without general assumptions or simplifications (except the slip surface); Chap. 12 describes soil pressure. The book may be referred to by geotechnical engineering researchers and technicians.

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Preface

Since the early 1980s, the author has been engaged in calculation and analysis of the geotechnical engineering. During the learning and application of soil mechanics theory for engineering analysis and calculation, he felt that the existing limit analysis theory of the soil mass still could not fully meet the demands so that he tried to conduct some study together with his colleagues. Based on the work, the book is the summary report on their learning, application, and study about the limit analysis theory of the soil mass over the years. However, due to limited energy and ability, there may be no relatively comprehensive reference or omission of some existing achievements; we hereby hope to get your understanding about the above all. Particular thanks must go to profs. Yan Shuwang and Zhang Xueyan from Tianjin University and Liu Yongxiu, professor and senior engineer of the CCCC First Harbour Consultants for their proofreading work and the precious modification suggestions. Considering the limits of resources and time, mistakes in the book might have been committed. I would be appreciated if readers help to correct the mistakes. Tianjin, China

Chuanzhi Huang

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Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Development of Limit Analysis Theory of the Soil Mass 1.3 Study Topics of Limit Analysis Theory of the Soil Mass 1.3.1 Failure Mode of the Soil Mass or Ground . . . . 1.3.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . 1.3.3 Basic Method for Limit Analysis . . . . . . . . . . . 1.4 Main Features of the Book . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Mathematical Modeling . . . . . . . . . . . . . . . . . . 1.4.2 Limit Analysis Methods . . . . . . . . . . . . . . . . . 1.4.3 Seeking for Specific Engineering Analysis and Calculation Methods . . . . . . . . . . . . . . . . . 1.5 Main Contents of the Book . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Failure Mode and the Fundamental Equation . . . . . . . . . . . 2.1 Equilibrium Equation and Yield Condition . . . . . . . . . . 2.1.1 Equilibrium Equation . . . . . . . . . . . . . . . . . . 2.1.2 Yield Function and Yield Condition . . . . . . . 2.2 Stress Relationship of a Point on a Curved Surface . . . . 2.3 The State and Failure Mode of the Soil Mass . . . . . . . . 2.3.1 The State of the Soil Mass . . . . . . . . . . . . . . 2.3.2 The Failure Mode of the Soil Mass . . . . . . . . 2.4 Stress Field in the Limit State . . . . . . . . . . . . . . . . . . . 2.4.1 Extremum Condition of Yield Function . . . . . 2.4.2 The Stress Field of the Soil Mass in the Limit State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Meaning of Stress Field in the Limit State . . . 2.5 Stress Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Generalized Flow Rule and Velocity Equation . . . . . . .

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2.6.1 Geometric Equation . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Generalized Flow Rule . . . . . . . . . . . . . . . . . . . . 2.6.3 Velocity Equation . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Associated Flow Rule and Velocity Equation . . . . . . . . . . . 2.8 A Fundamental Equation in the Surface Failure Mode . . . . . 2.8.1 The Stress Equation in the Surface Failure Mode . 2.8.2 Fundamental Equation for Slope Stability . . . . . . . 2.9 Limit Equilibrium and Variation . . . . . . . . . . . . . . . . . . . . 2.9.1 Load Boundary Condition and Kinematic Boundary Condition . . . . . . . . . . . . . . . . . . . . . . 2.9.2 Engineering Boundary Condition . . . . . . . . . . . . . 2.9.3 Limit Equilibrium of the Load Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.4 The Limit Equilibrium of the Kinematic Boundary Condition and the Mixed Boundary Condition . . . 2.9.5 Variation in the Surface Failure Mode . . . . . . . . . 2.10 The Simplest Velocity Field . . . . . . . . . . . . . . . . . . . . . . . 2.11 Solution Using a Limit Analysis . . . . . . . . . . . . . . . . . . . . 2.11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.2 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.3 Solutions in the Field Failure Mode . . . . . . . . . . . 2.11.4 Solutions in the Surface Failure Mode . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Characteristic Line Method for Limit Analysis . . . . . . . . . . . . . . 3.1 Characteristic Line Equation and Stress Equation . . . . . . . . 3.2 Calculation Process of Limit Load . . . . . . . . . . . . . . . . . . . 3.2.1 Differential Equation . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Boundary Condition . . . . . . . . . . . . . . . . . . . . . . 3.2.3 re and h at the Foundation Edge Point . . . . . . . . . 3.2.4 Two-Way Recursive Method for Calculation of Differential Equation . . . . . . . . . . . . . . . . . . . . 3.3 Calculated Results of Limit Load . . . . . . . . . . . . . . . . . . . . 3.4 Comparison with Existing Calculation Equations . . . . . . . . 3.4.1 Comparison with Total Limit Load Equation . . . . 3.4.2 Comparison of Limit Load Distribution . . . . . . . . 3.5 Discussion on Characteristic Line Method . . . . . . . . . . . . . 3.5.1 Treatment for Boundary Condition of Foundation Bottom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Calculation for Limit Load of Ground with Heterogeneous Soil . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Determination of Ground Bearing Capacity . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Stress Field Method for Limit Analysis . . . . . . . . . . . . . . . . . . . 4.1 The Simplest Stress Field . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Plane Slip Surface . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Purely Cohesive Material ðu ¼ 0Þ . . . . . . . . . . . 4.1.3 Approximation of Limit Stress Field by Stress Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stress Field of Plane Slip Surface . . . . . . . . . . . . . . . . . . 4.3 Stress Field of Common Helicoid Slip Surface . . . . . . . . . 4.4 Application Example: Critical Excavation Height . . . . . . . 4.4.1 Vertical Critical Excavation Height of Plane Slip Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Plane–Helicoid Slip Surface . . . . . . . . . . . . . . . 4.5 Limit Load in Field Failure Mode . . . . . . . . . . . . . . . . . . 4.5.1 Stress of Foundation Bottom . . . . . . . . . . . . . . . 4.5.2 Calculation Mode 1 . . . . . . . . . . . . . . . . . . . . . 4.5.3 Calculation Mode 2 . . . . . . . . . . . . . . . . . . . . . 4.5.4 Calculation Mode 3 . . . . . . . . . . . . . . . . . . . . . 4.5.5 Total Limit Load . . . . . . . . . . . . . . . . . . . . . . . 4.5.6 Limit Load in Surface Failure Mode . . . . . . . . . 4.6 Stress Field Along Slip Surface (Family) and Limit Load . 4.6.1 General Form of Stress Field . . . . . . . . . . . . . . . 4.6.2 General Forms of Limit Load . . . . . . . . . . . . . . 4.6.3 Example: Plane–Common Helicoid–Plane . . . . . 4.7 Limit Load in the Plane–General Helicoid–Plane Calculation Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Limit Load in Helicoid–Helicoid–Plane Calculation Mode . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limit Equilibrium Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Basic Idea of Limit Equilibrium Method . . . . . . . . . . . . . 5.2 Equilibrium Equations for Force and Moment on Slip Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Calculation of Limit Load . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Force Equilibrium-Based Limit Load Expression 5.3.2 Force and Moment Equilibrium-Based Limit Load Expression . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Example: Limit Load of Slip Surface in Plane–Helicoid–Plane Form . . . . . . . . . . . . . . 5.4 Slope Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Equivalence of the Extremum Condition of the Yield Function and the Extremum Condition of the Safety Factor . . . . . . . . . . . . . .

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5.4.2 Expression of Safety Factor . . . . 5.4.3 Examples . . . . . . . . . . . . . . . . . 5.5 Calculation of Soil Pressure . . . . . . . . . . . 5.5.1 Active Soil Pressure (Fig. 5.7a) . 5.5.2 Passive Soil Pressure (Fig. 5.7b) 5.6 Calculation Equation for Limit Load . . . . 5.6.1 Calculation Mode 1 . . . . . . . . . 5.6.2 Calculation Mode 2 . . . . . . . . . 5.6.3 Calculation Mode 3 . . . . . . . . . 5.6.4 Calculated Results . . . . . . . . . . 5.6.5 Corrected Calculation Equation . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

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Virtual Work Equation-Based Generalized Limit Equilibrium Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Approximate Solution of Limit Analysis . . . . . . . . . . . . . . 6.2 Virtual Work Equation Under Yield Criterion Condition . . 6.3 Upper and Lower Bound Theorems of Limit Load . . . . . . 6.3.1 Proving for Upper and Lower Bound Theorems of Limit Load . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Proving for Upper and Lower Bound Theorems of Limit Soil Pressure . . . . . . . . . . . . . . . . . . . . 6.4 Discontinuity of Velocity Field . . . . . . . . . . . . . . . . . . . . 6.5 Upper and Lower Bound Theorems Under Associated Flow Rule Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Required Condition for Theorem Proving . . . . . . 6.5.2 Difference in Actual Content of the Theorem . . . 6.5.3 Difference in the Corresponding Limit Solution . 6.5.4 Incompletion of the Existing Proving . . . . . . . . . 6.6 Generalized Limit Equilibrium Method for Limit Load . . . 6.7 Common Slip Surface Family and Velocity Field . . . . . . . 6.7.1 Velocity Field of Plane Slip Surface . . . . . . . . . 6.7.2 Velocity Field of Common Helicoid Slip Surface 6.7.3 Velocity Field of Arc Slip Surface . . . . . . . . . . . 6.7.4 Velocity Field of General Helicoid Slip Surface . 6.7.5 Combination Form of Slip Surface and Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Calculation Process of Limit Load . . . . . . . . . . . . . . . . . . 6.8.1 Example I: Slip Surface in Plane–Common Helicoid–Plane Form . . . . . . . . . . . . . . . . . . . . 6.8.2 Example II: Slip Surface in Plane–Arc Surface–Plane Form . . . . . . . . . . . . . . . . . . . . . 6.8.3 Example III: Slip Surface in Plane–General Helicoid–Plane Form . . . . . . . . . . . . . . . . . . . .

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Limit Load in Helicoid–Helicoid–Plane Calculation Mode . 6.9.1 Calculation Mode 1 . . . . . . . . . . . . . . . . . . . . . 6.9.2 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.3 Calculation Equation for Limit Load . . . . . . . . . 6.9.4 Calculation Mode 2 . . . . . . . . . . . . . . . . . . . . . 6.9.5 Calculated Results . . . . . . . . . . . . . . . . . . . . . . 6.10 Discussion on Variational Principle for Limit Load . . . . . . 6.10.1 Variation of Limit Load . . . . . . . . . . . . . . . . . . 6.10.2 Equivalence Between Variation and Limit Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Generalized Limit Equilibrium Method in Plane Failure Mode . 7.1 Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Generalized Limit Equilibrium Method for Slope Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 General Calculation Equation of Safety Factor . . . 7.2.2 Upper and Lower Bound Theorem of the Slope Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Several Discussions . . . . . . . . . . . . . . . . . . . . . . 7.3 Common Slip Surface Family and Velocity Field . . . . . . . . 7.3.1 Velocity Field of Plane Slip Surface . . . . . . . . . . 7.3.2 Velocity Field of Common Helicoid Slip Surface . 7.3.3 Velocity Field of Arc Slip Surface . . . . . . . . . . . . 7.3.4 Velocity Field of General Helicoid Slip Surface . . 7.4 Calculation Examples of Slope Stability and Excavation Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Arc Slip Surface . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Plane–Arc Surface . . . . . . . . . . . . . . . . . . . . . . . 7.5 Discussion on Surface Failure Mode and Field Failure Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Moment Equilibrium Equation-Based Generalized Limit Equilibrium Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Limit Load in Plane–Helicoid–Plane Calculation Mode . . . . 7.8 Limit Load in Helicoid–Helicoid–Plane Calculation Mode . . 7.9 Limit Load in Plane–General Helicoid–Plane Calculation Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Discussion on Limit Load and Critical Load . . . . . . . . . . . . 7.11 Calculation Example of Slope Stability . . . . . . . . . . . . . . . . 7.11.1 Expression of the Safety Factor . . . . . . . . . . . . . . 7.11.2 Example I: For Plane–Common Helicoid Without Load on Slope Surface . . . . . . . . . . . . . . . . . . . . 7.11.3 Example II: For Arc Slip Surface Without Load on Slope Surface . . . . . . . . . . . . . . . . . . . . . . . .

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Variational Principle of Generalized Limit Equilibrium Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12.1 Variational Principle of Limit Load . . . . . . . 7.12.2 Variational Principle of Slope Stability . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Limit Load on Ground with Heterogeneous Soil . . . . . . . . . . . . . 8.1 Basic Consideration on Calculation of Limit Load . . . . . . . 8.1.1 Determination of Calculation Condition . . . . . . . . 8.1.2 Study Method for Limit Load . . . . . . . . . . . . . . . 8.1.3 Determination of Slip Surface (Family) . . . . . . . . 8.1.4 Basic Consideration on Calculation of Limit Load on Ground with Heterogeneous Soil . . . . . . 8.2 Helicoid Calculation Mode in Surface Failure Mode . . . . . . 8.2.1 Common Helicoid Calculation Mode . . . . . . . . . . 8.2.2 Homogeneous Soil . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Limit Load on Heterogeneous Soil . . . . . . . . . . . 8.2.4 Layered Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Helicoid Calculation Mode in Field Failure Mode . . . . . . . . 8.3.1 Calculation Equation for Limit Load . . . . . . . . . . 8.3.2 Calculated Result of Ground with Homogeneous Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Calculated Results of Two Layers of Soil Grounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Distribution of Limit Load on Ground . . . . . . . . . 8.4 Analysis and Comparison of Surface Failure Mode and Field Failure Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Analysis of Surface Failure Mode . . . . . . . . . . . . 8.4.2 Analysis on Field Failure Mode . . . . . . . . . . . . . . 8.4.3 Comparisons of Two Failure Modes . . . . . . . . . . 8.5 General Calculation Mode by Generalized Limit Equilibrium Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Helicoid–Helicoid–Plane Calculation Mode . . . . . . . . . . . . 8.6.1 Slip Surface Family . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Calculated Result of the Ground with Homogeneous Soil . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Distribution of Limit Load . . . . . . . . . . . . . . . . . 8.6.4 Calculated Result of Two Layers of Soil Grounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Calculation of Limit Load According to Load Inclination . . 8.7.1 Treatment for Boundary Condition of Foundation Bottom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 Comparison of Calculation Results Under Two Types of Boundary Conditions . . . . . . . . . .

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Calculation of Limit Load with Undrained Shear Strength Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1 Sand Cushion Under Foundation Bottom . . . . . . 8.8.2 u0 ¼ 0 for Soil on Lower Foundation Bottom . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9

Ground Bearing Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Design Load and Ground Failure Mode . . . . . . . . . . . . . . 9.1.1 Design Load of Actual Engineering . . . . . . . . . . 9.1.2 Failure Mode of Ground Bearing Capacity . . . . . 9.2 Ground Bearing Capacity and Allowable Bearing Capacity 9.2.1 Limit Load and Ground Bearing Capacity . . . . . 9.2.2 Allowable Load on Ground and Allowable Bearing Capacity . . . . . . . . . . . . . . . . . . . . . . . 9.3 Measurement for Stability of Ground Bearing Capacity . . . 9.3.1 Measurement of Ground Stability with Ultimate Bearing Capacity . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Evaluation of Ground Stability in Accordance with Allowable Bearing Capacity . . . . . . . . . . . . 9.4 Determination of Ground Bearing Capacity with Limit Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Determination of Ground Bearing Capacity Without Prejudice to Yield Criterion . . . . . . . . . 9.4.2 Determination of Ground Bearing Capacity with Requirements of Basic Effective Width . . . . 9.4.3 Partial Bearing Capacity of Partial Failure . . . . . 9.4.4 Several Discussions . . . . . . . . . . . . . . . . . . . . . 9.5 Ground Bearing Capacity of Gravity Wharf . . . . . . . . . . . 9.5.1 Calculation Method . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Calculated Results of Bearing Capacity . . . . . . . 9.5.3 Analysis of Calculated Results . . . . . . . . . . . . . . 9.5.4 Calculation for Allowable Bearing Capacity . . . . 9.5.5 Reliability of Ground Bearing Capacity . . . . . . . 9.5.6 Calculation for Bearing Capacity Stability of Ground by Overall Stability Analysis Method 9.6 Ground Bearing Capacity of Breakwater . . . . . . . . . . . . . . 9.6.1 Possible Failure Mode of Breakwater Ground . . . 9.6.2 Calculation Mode for Ground Bearing Capacity of Breakwater . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Calculated Result of Bearing Capacity . . . . . . . . 9.6.4 Analysis of Calculated Result . . . . . . . . . . . . . .

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9.7

Ground Bearing Capacity by Generalized Limit Equilibrium Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

10 Slope 10.1 10.2 10.3

10.4

10.5

10.6

10.7

10.8 10.9

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis Method for Homogeneous Soil Slope . . . . . . . . . . Analysis Method for Helicoid Slip Surface . . . . . . . . . . . . . 10.3.1 Basic Solving Idea . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Analysis Method for Simple Slope with Two Types of Soils . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Analysis Method for General Slope . . . . . . . . . . . Several Common Analysis Methods . . . . . . . . . . . . . . . . . . 10.4.1 Simple Slice Method . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Simplified Bishop Method . . . . . . . . . . . . . . . . . . 10.4.3 Unbalanced Thrust Transmission Method . . . . . . . 10.4.4 Morgenstern–Price Method . . . . . . . . . . . . . . . . . 10.4.5 Method of Chen Zuyu . . . . . . . . . . . . . . . . . . . . 10.4.6 Janbu Method . . . . . . . . . . . . . . . . . . . . . . . . . . . Limit Equilibrium Method . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Static Indetermination . . . . . . . . . . . . . . . . . . . . . 10.5.2 Slip Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 Engineering Case . . . . . . . . . . . . . . . . . . . . . . . . 10.5.4 Application for Extremum Condition of the Yield Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis Method for Composite Slip Surface . . . . . . . . . . . 10.6.1 General Form of Analysis Method . . . . . . . . . . . . 10.6.2 Calculation Equation for Arc Slip Surface . . . . . . 10.6.3 Calculation Equation for Slip Surface of Plane–Arc Surface–Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.4 Calculation Equation for Plane–Helicoid–Plane Slip Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculated Results and Their Discussion . . . . . . . . . . . . . . . 10.7.1 Treatment in Particular Case . . . . . . . . . . . . . . . . 10.7.2 Calculated Results . . . . . . . . . . . . . . . . . . . . . . . 10.7.3 Comparison of Calculation Methods . . . . . . . . . . 10.7.4 Comparison of Different Slip Surfaces . . . . . . . . . 10.7.5 Reliability Analysis for Slope Stability . . . . . . . . . Analysis Method for Slope with Weak Interlayer . . . . . . . . Generalized Limit Equilibrium Method-Based Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10.9.1 Basic Solving Idea . . . . . . . . . . . . . . 10.9.2 Calculation Equation of Safety Factor 10.9.3 Calculated Results and Comparison . . 10.10 Slope Stability Analysis Method . . . . . . . . . . . 10.10.1 Soil Mass Cannot Withstand Tension . 10.10.2 Slip Surface . . . . . . . . . . . . . . . . . . . 10.10.3 Analysis Method . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Stability During Construction and Pore Water Pressure . Slope Stability During Construction . . . . . . . . . . . . . . . . . Effective Stress Method . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Slice Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composite Slip Surface Method . . . . . . . . . . . . . . . . . . . . Calculated Results and the Comparison . . . . . . . . . . . . . . Pore Water Pressure in Terzaghi’s Consolidation Theory . . 11.6.1 Terzaghi’s Consolidation Equation . . . . . . . . . . . 11.6.2 General Equation for the Pore Water Pressure . . 11.6.3 Pore Water Pressure Under Class I Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.4 Pore Water Pressure Under Class II Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.5 Approximate Calculation of Pore Water Pressure 11.7 Non-slip Role of Geosynthetic Reinforced Cushion . . . . . . 11.7.1 Basic Consideration for the Analysis Method . . . 11.7.2 Non-slip Mechanism of Geosynthetic Reinforced Cushion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.3 Simple Slice Method . . . . . . . . . . . . . . . . . . . . . 11.7.4 Engineering Practice of Non-slip Mechanism and Model Test Verification . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Slope 11.1 11.2 11.3 11.4 11.5 11.6

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12 Soil Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Stress Field Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Active Soil Pressure with Plane–Helicoid Fracture Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2 Passive Soil Pressure of Plane–Helicoid Fracture Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Stress Equation-Based Stress Field Method . . . . . . . . . . . . . 12.2.1 Soil Pressure of Plane–Helicoid Fracture Surface . 12.2.2 Soil Pressure of Plane–Arc Fracture Surface . . . . . 12.3 Comparison of Existing Methods . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Preamble

1.1 Introduction For the solution of specific engineering problems, soil mechanics is mainly involved in the study on deformation and stability, which are related to each other; oversize deformation of the soil mass frequently causes instability; on the contrary, before the failure of the soil mass or ground, it is often accompanied by relatively large deformation. As the engineering properties of soil have not been fully recognized and mastered, it is still difficult to solve the engineering problems through the study in combination with the deformation and stability at present; as a result, they are often separated for the study. For the deformation, it studies the stress and deformation of the soil mass or ground under normal use conditions of the engineering; for the stability, it studies the stress and safety degree of the soil mass or ground in the limit state, which is also referred to as the limit analysis on the soil mass, including all the stability study theories and methods. In the actual engineering, safety construction and use of the engineering are one of the most concerned issues by engineers. In the process of the engineering construction and use, under loading or unloading (excavation) conditions, the soil mass, as the ground, may still be in a stable state or may be seen with a failure (failure state) or just in the critical state (limit state) from the stable state to the failure state. In the above states, it is the easiest to accurately measure the limit state and apply it in the engineering. And, the book discusses the limit analysis theory of the soil mass or ground and its application. Of course, the limit state is just an ideal state of the soil mass. The practical soil mass is generally impossible to be just in the limit state and the soil mass in the engineering ground is also not allowed to be in the limit state, instead required to be in the stable state. In order to make the soil mass in the limit state, considering from the theoretical study, it may be assumed to achieve it by the following methods:

© Springer Nature Singapore Pte Ltd. and Zhejiang University Press, Hangzhou, China 2020 C. Huang, Limit Analysis Theory of the Soil Mass and Its Application, https://doi.org/10.1007/978-981-15-1572-9_1

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2

1 Preamble

(1) increasing the load applied to the soil mass; (2) reducing soil strength. And the two classical stability issues in the soil mechanics are ground bearing capacity and slope stability; another one in the soil mechanics comes to the soil pressure. The limit load is just the load to make the soil mass just in the limit state. The ground bearing capacity and the soil pressure are just the limit loads on a certain section of different boundary surfaces. The limit load shall also include active limit load and passive limit load. The former is just the commonly referred active earth pressure; namely, the load on the partial boundary is known, and the limit load generated on another partial boundary is solved when the soil mass is in the limit state. The latter is just the commonly referred passive earth pressure; namely, the load on the partial boundary is known, and the limit load that another partial boundary may bear is solved when the soil mass is in the limit state. In the engineering, safety factors or partial safety factors are generally used to evaluate its stability or reliability. In the deterministic method, the ratio of the ground bearing capacity to the design load is generally referred to as the safety factor of the ground bearing capacity (bearing capacity safety factor). When the shape of the boundary surface and the load applied are known, the soil mass state is deterministic. So does the slope stability study, namely, when the load on the boundary surface is known, it may be assumed that the soil strength is reduced (increased) to make the soil mass in the limit state; the reduced (increased) ratio is referred to as safety factor (stability safety factor). In the probabilistic method, limit state expression is regarded as the fundamental equation for the calculation of the reliable probability. The reliability analysis-based partial safety factor method has been widely used in the engineering. Such limit state design method, with the limit state of the soil mass or ground as the principle and the limit analysis theory of the soil mass as its theoretical basis, has been the basic method for engineering stability analysis for the intermediate and long-term engineering. Limit analysis on the soil mass is a practical science; by far, the limit analysis theories and methods based on the force and moment equilibrium, the principle of work and energy and Coulomb yield criterion have been used to solve the practical engineering problems. The discussion in the book will be made within the above range.

1.2 Development of Limit Analysis Theory of the Soil Mass From the theory and its application, the limit analysis study on the soil mass may be traced back to 1773 when Coulomb put forward the famous Coulomb yield criterion and laid a foundation for the failure theory of soil mass. In the 1920s, Fellenius et al. established a study method for the limit analysis, limit equilibrium method. Although the current theory had not been completed, the proposed stability analysis method for the side slope arc has still been applied in the engineering so far. In the 1940s, Sokolovskii et al. searched out another method for limit analysis, characteristic line

1.2 Development of Limit Analysis Theory of the Soil Mass

3

method; the calculation formula for ground bearing capacity obtained according to the characteristic line method was also extensively applied in engineering. It is generally considered that since 1950, the limit analysis theory of the soil mass has been greatly developed. Above all, the extremum theorem for the limit analysis of the soil mass, upper and lower bound theorem was proposed, which symbolized that the limit analysis theory had already entered its mature stage; another extremum theorem-based method, upper and lower bound solution (or referred to as the limit analysis method) received extensive attention. Such work was introduced in the works of Chen [1]. In the meanwhile, Chinese scholars also made many contributions, for example, contributions of Shen [2], Zheng and Gong [3], Zhuang and Yan [4], Chen [5], and Luan [6, 7]. It must be admitted that stability analysis and calculation methods (including ground bearing capacity, slope stability, and soil pressure) provided in the limit analysis theory of the soil mass play a large role in various engineering designs and constructions. Meanwhile, with the continuous development of the theoretical study, the discovered and proposed problems are increasing. More extensive application of the theoretical study achieved in practice and much higher theory requirements were proposed in the practice, which will further promote the development and application of the theoretical study. Huang indicates that [8] the soil mechanics and foundation engineering is a technical science and depends on the development of the engineering practices. Nowadays, the development in the discipline still goes far behind that in the engineering practices. A lot of experience and knowledge obtained in practices can often be seen in the regular books after they have been applied for 10–15 years. Therefore, the success of the geotechnical engineering study, separated from the engineering practices, is unthinkable. Jiang indicates that [2]: the soil mechanics theory is derived from the summary on practices and it has developed very slowly; up to today, it has not formed a complete theory system yet. Just because the theory is still not very mature now, it seems impossible to solve complex geotechnical engineering problems. From the practical applications, although the soil mechanics has put forward many analysis methods or calculation formulas for the engineering, engineers still feel that they can not be applied to solving many practical problems in the specific engineering. For example, after a lot of surveys and analyses are conducted on the reason of partial failure in the northern jetty at Phase II Yangtze River Estuary, it is considered that [9], after the wave load is transferred to the ground, it softens the ground soil and reduces its bearing capacity, which is the main reason for ground failure and severe caisson subsidence; under the current technical level, a quantitative judgment on whether the ground soil will be softened still can not be achieved through a theoretical analysis. This common view of many experts, in fact, shows their helplessness on the existing ground analysis and calculation theory. Moreover, for the calculation of the ground bearing capacity, when the heterogeneous soil has greatly different strength indexes for soil layers and ground surfaces are irregular, calculation methods satisfactory to the engineering circle are still unavailable at present [10].

4

1 Preamble

From the theoretical study, it shall be admitted that some theoretically mature methods can not be rapidly approved and applied by the engineering circle due to various reasons; such a puzzle to the theoretical circle is indeed existing [11]. However, to be blunt, the exiting limit analysis theory study still can not fully meet the engineering demands and there are still many issues to be further studied and improved. The fundamental equation, the basis for the limit analysis theory of the soil mass, is still not uniform or complete yet. It is not uniform because different equations are used for different solution methods. For example, the limit equilibrium method is adopted for the Coulomb yield condition; the characteristic line method is adopted for the Mohr–Coulomb yield condition; the upper bound solution method based on the upper bound theorem still needs to regard the associated flow rule as the fundamental equations. For different problems, it is correct that corresponding equations are selected as the fundamental equation and that appropriate conversion is made for the fundamental equation as the equation for the specific solution method. However, a set of complete and uniform fundamental equations shall be provided for the limit analysis on the soil mass and such a set of fundamental equations shall be the complete description in mathematical language for the limit state of the soil mass and shall not vary with different solution methods. Based on such one set of complete and uniform fundamental equations, various solution methods may be applied to obtain corresponding solutions; moreover, advantages and disadvantages of various methods may be discussed under the same uniform standard. Otherwise, people will misunderstand that different limit analysis theories or limit analysis problems have different solutions (for example, limit load), which causes a relatively confused limit analysis theory. It is not complete because quantity of the established fundamental equations is always less than that of the unknown quantities. For example, in the study on the slope stability, it is just a statically indeterminate [12]; the associated flow rule is regarded as the fundamental equation, but the geotechnical materials do not comply with the associated flow rule [13]. In fact, it is one of the main tasks for the limit analysis on the soil mass to obtain the most dangerous slip surface or the possible failure area enclosed by the slip surface and the soil mass surface; however, the slip surface has not to be defined yet by far; namely, there is no general equation to determine the slip surface. Just have a think, for an equation (virtual work equation or moment equilibrium equation) divided along the slipping area, the slip surface is not known what it is so that it will be very difficult to have a clear discussion. The slip surface is just one of the limit analysis theory difficulties; if only specific analysis methods are established but it is not discussed how the slip surface is selected, it is undoubtedly that the problem is left to the engineering circle that applies the analysis methods. From the combination of the theoretical study and practical application, the existing limit analysis theory is usually established under the “ideal” conditions. For example, the soil yield condition is regarded as the basic theory for the study on the soil mass failure; however, the yield condition applied at present is just the “shear failure in the ideal soil [14].” In the classical failure theory of soil mass, a rigid

1.2 Development of Limit Analysis Theory of the Soil Mass

5

plasticity assumption is adopted and the actual soil mass failure process is far different from the theory [2]. The limit analysis theory of the soil mass is about the stress field under the limit state conditions of the soil mass; for the general heterogeneous soil conditions in the engineering, it is just an “ideal” failure mode whether the soil mass is in the limit state (surface failure mode) along the slip surface or in the limit state (field failure mode) in the slip mass. The stress field in the limit state under such “ideal” conditions is not that under the actual load conditions of the soil mass or ground. Therefore, it is very important to combine the theoretical study and practical application experiences and it still needs to be combined with the practical experiences accumulated for years to apply the theory in practical problems. For example, during the solution of the limit load, some may be combined with the actual loads, for example, the load inclination under the joint action of the horizontal and vertical loads; but some not, for example, the distribution mode of the vertical load. Generally, the calculation of the limit load is discussed under uniform load on the foundation surface; however, the limit load is impossible to be a uniform load, except in few special cases (such as the internal friction angle of the soil ϕ = 0 or the soil mass weight γ = 0). Certainly, the calculation of the ground bearing capacity may be discussed under known load on the foundation surface. However, the ground bearing capacity of the engineering shows the capacity of the ground to bear the actual load; unless the distribution of the actual load is just consistent with that of the limit load, the limit load of the ground and the ground bearing capacity will not be the same thing. The calculation formula for the limit load obtained according to the limit analysis theory can not be regarded as that for the ground bearing capacity. Establishment and development of the plastic mechanics of the rocksoil create further development space for the limit analysis theory of the soil mass; the upper bound solution method is just an application in the limit analysis on the plastic mechanics of the rocksoil. However, there is also the problem to combine its theory with the practical experiences accumulated for years. For example, “in the early stage when the finite element method introduces geotechnical engineering calculation, someone once has calculated the safety factor for each unit and then the integral safety factor is obtained on average in some way. Such a method is not advisable because it confuses the design stress state and the limit stress state and does not meet the requirements of the basic principles [2] for the limit design.” Actually, constitutive model-based finite element analysis on slope stability is corresponding to the stress field [15, 16] under the normal use of the engineering, which is an analysis method different from the limit state design method. In conclusion, it is indisputable that study on the limit analysis theory of the soil mass and its application fall behind the engineering practices and can not fully meet the engineering demands; therefore, there is still a very long way for the study to go.

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1 Preamble

1.3 Study Topics of Limit Analysis Theory of the Soil Mass As aforesaid, the book discusses the stress and safety degree of the limit state of the soil mass or ground with the main study work covering failure mode of the soil mass or ground, mathematical model as well as solution methods for limit analysis and their application, etc.

1.3.1 Failure Mode of the Soil Mass or Ground Before the discussion about the stability of the soil mass or ground, possible failure modes of the soil mass shall be discussed and corresponding mathematical model and calculation mode shall be established on the basis of the failure mode. It is a starting point for the discussion about the limit analysis on the soil mass to make a quantitative description of the stability, failure, and limit states. The soil mass state may be described in the yield function; in the current study, Coulomb yield function is mostly adopted; namely, the difference between the total shear stress at any point in the soil mass and the shear strength is used to define the yield function, which may be used to give an accurate quantitative description on the limit state of the soil mass. If the soil mass or ground is stable, any point in the ground soil shall be stable. If the soil mass or ground is destroyed, the ground soil would be destroyed along certain curved surface or in certain area. If the ground soil along certain curved surface is in the limit state and the soil mass on both sides of the curved surface is both in the stable state, the curved surface will be just the slip surface; such failure mode is referred to as the surface failure mode. If the ground soil is in the limit state in certain area and the soil mass beyond this area is in the stable state, such area will be just the slip mass, and the failure mode will be referred to as the field failure mode. Surface failure mode and field failure mode are two basic failure modes of the soil mass and both of them shall be considered in the limit analysis theory. If only either of them is taken into consideration, it would just form part of the theory but not complete. Under known load on the boundary surface, the failure mode generally can only be the surface failure mode. Therefore, the slope stability can only be studied according to the surface failure mode. For the limit load, the failure mode may be considered according to the surface failure mode or field failure mode. The surface failure mode can only be used to calculate the total load on certain section of the boundary surface while the field failure mode may calculate the distribution of the limit load. In the actual engineering, for the slope stability, integral failure is considered; for the limit soil pressure, partial soil mass in the limit state is considered; for the ground bearing capacity, both the integral failure and partial failure shall be considered. The possible engineering hazards from the partial failure of ground soil can not be

1.3 Study Topics of Limit Analysis Theory of the Soil Mass

7

ignored [9]; for example, partial abnormal deformation on the gravity wharf causes inclination of the wharf structures and even building instability due to the chain reaction. In addition, since edge loads on both sides of the foundation are different from each other, there are also failure modes such as one-way failure and two-way failure. If the gravity wharf shall be a one-way failure, the vertical breakwater shall be a twoway failure [9]. Certainly, the aforesaid two-way failure indicates that the ground soil, instead of the engineering structure, is possibly destroyed toward both sides of the foundation. During the discussion about the calculation mode of the soil stabilization, such failure modes shall be taken into consideration.

1.3.2 Mathematical Model A mathematical model for the limit analysis on the soil mass is to establish a fundamental equation for the soil mass in the limit state, which is the basis for the limit analysis on the soil mass and to obtain the limit load or safety factor required for the engineering and the most dangerous slip surface or possible soil mass failure area. Out of doubt, for the limit analysis on the soil mass, it is most difficult to build a mathematical model. In-depth study for the soil yield criterion or yield function and equations to exactly and completely describe conformity of the soil mass with the yield criterion are the key to building a mathematical model. Mathematical model finally shall come down to a definite solution and the limit equilibrium is just one of the specific definite solutions mathematically; here, it is a quantitative description that the soil mass is in the limit state in mathematical language and shall include relevant fundamental equations and corresponding boundary conditions. Therefore, the limit analysis on the field failure mode may come down to the limit equilibrium. For the surface failure mode, the soil mass is required to be in the limit state along the slip surface while the soil mass on both sides of the slip surface is in the stable state; however, an extremum is required to be taken for the limit load or safety factor. Therefore, it generally may come down to the variation (conditional extremum).

1.3.3 Basic Method for Limit Analysis For the limit analysis theory, it is one of the main tasks to seek solution methods for limit equilibrium or conditional extremum and obtain solutions with practical value. As the fundamental equation for the limit analysis contains nonlinear partial differential equation, even for the ground with homogeneous soil, it is very difficult to obtain an exact analytical expression for the limit load (or safety factor). By far, exact solutions have been obtained in only a few special cases, for example, the Rankine’s soil pressure formula without regard to friction between the wall and the soil and

8

1 Preamble

Plandtl’s limit load formula under weightless soil conditions [17]. At least for now, it is difficult to obtain exact solutions of stress field, slip surface (family), velocity field, and limit load (or safety factor) under satisfactory equation and boundary conditions. Therefore, the study has to turn to seek approximate solutions or numerical solutions. Two kinds of limit analysis methods are available; one is to directly solve the limit equilibrium problem and obtain the stress field and the limit load on this basis mainly by the characteristic line method for the numerical solution and stress field method for the approximate solution; the other is to avoid the difficulty to obtain the stress field and instead the limit load or safety factor. Such methods shall be carried out under the assumption of the slip surface and an approximate solution will be obtained mainly by slip surface force and moment equilibrium-based limit equilibrium method, extremum theorem-based upper bound solution method, and virtual work equation-based or stress equation-based (along slip surface) generalized limit equilibrium method. The characteristic line method is to convert the static equilibrium equation and yield criterion to a stress equation established on two groups of characteristic lines, and thus, the mathematically mature characteristic line method may be applied to calculate stress component at the intersection of two groups of characteristic lines and then to obtain the stress field and limit load. This method may be called ingenious because it may be proofed that one group of characteristic lines is just the slip line [namely slip surface (family)]. The characteristic line method requires the soil mass inside and outside the slip mass to be in the limit state (in accordance with the yield criterion) and stable state (without prejudice to the yield criterion), respectively; by this method, numerical solution in the field failure mode will be obtained. If the characteristic line network is a relatively dense and higher-order solution is calculated, the calculated result will be close to the exact solution. By the characteristic line method, the limit load may be obtained; based on the calculation of the high-accuracy limit load, the obtained ground ultimate bearing capacity method is applied in the engineering [18, 19]. For the solution of the limit equilibrium, the best way is to obtain the analytical expression for the stress field, even though the approximate stress field. With a stress field, it will be easy to obtain the approximate value of the limit load and judge the state of the soil mass. However, it is relatively difficult to determine the stress field; at present, the stress field method is still applied less. Actually, the lower bound solution method also belongs to the stress field method, just the stress field of the structure is required to be in a stable state. Traditional limit equilibrium methods consider that the soil mass failure happens on the slip surface and the soil mass along the slip surface meets the yield condition. It is assumed that the type of the slip surface is known, such as plane, arc surface, logarithmic helicoid, or other irregular surfaces; the limit load or safety factor is determined by considering the static equilibrium and moment equilibrium of each isolator in the slip mass when the soil mass along the slip surface is in the limit state. The limit equilibrium method is mainly involved with the solution methods for the surface failure mode; by the method, only the approximate solution of the surface

1.3 Study Topics of Limit Analysis Theory of the Soil Mass

9

failure mode can be obtained generally. Even though many slip surfaces are selected for the calculation and the minimum limit load or safety factor is taken as the calculated result. As the calculation for each slip surface is mutually independent, it can not make the soil mass enclosed by the slip surface and the surface in the limit state simultaneously. Certainly, the limit equilibrium method may also be used to find a solution for the ground bearing capacity and the soil pressure and obtain the calculated result of the field failure mode; however, it only requires making solutions when every point on the foundation bottom (the bearing capacity) is taken for the slip surface family. The upper and lower bound solution method is based on the extremum theorem of the limit analysis [1–4], namely the upper bound theorem and lower bound theorem. The upper bound theorem may be expressed as follows: all the loads on the boundary surface corresponding to the velocity fields (or displacement rate) in accordance with the flow rule and kinematic boundary condition are not less than the real limit load; or it may be expressed as follows with the stress field [20]: All the loads on the boundary surface corresponding to the stress field that makes the soil mass in the failure state are not less than the real limit load. It is crucial to determine a velocity field (moving velocity field) including assumed slip surface for the upper bound solution. The lower bound theorem may be expressed as follows: All the loads on the boundary surface corresponding to the stress field that makes the soil mass in the stable state are not greater than the real limit load. It is critical to construct such a stress field (statically admissible stress field), including assumed slip surface, for the lower bound solution. Generally, it is relatively easy to construct a velocity field in accordance with the flow rule and kinematic boundary condition, but not to construct a stress field that makes the soil mass in the stable state; therefore, the upper bound solution is easier to obtain than the lower bound solution. If a stress field (approximate) can be constructed, the required limit load will be able to be directly obtained with the stress field. At present, for the application of the upper bound solution method, generally, the curved surface through one point (foundation rear toe with regard to the bearing capacity and bottom surface of the wall with regard to the soil pressure) may be regarded as the potential slip surface and an approximate solution of the surface failure mode will be obtained. The generalized limit equilibrium method, a novel limit analysis method, was proposed by the author on the basis of the fact that the extremum condition of the yield function is regarded as one of the fundamental equations for the limit analysis, from which it can be concluded that limit analysis solution can be obtained only after we know the slip surface. For the virtual work equation and velocity equation-based generalized limit equilibrium method, the velocity field is solved through the velocity equation, and then the limit analysis solution is obtained through the virtual work equation. For the moment equation-based and stress equation-based (along the slip surface) generalized limit equilibrium methods, the normal stress on the slip surface is solved through the stress equation along the slip surface and then the limit analysis solution

10

1 Preamble

may be obtained through the moment equilibrium equation. The key to making the obtained solution be close to the exact solution lies in the selection of the slip surface. According to the calculation example, as long as the selected slip surface is close to the real slip surface, the obtained approximate solution will be close to the exact solution.

1.4 Main Features of the Book 1.4.1 Mathematical Modeling It is known that the yield function is established on the potential slip surface (family) and the yield criterion has two layers of exact meanings: (1) the soil mass is in the limit state along the slip surface; (2) the soil mass is in the stable state along other curved surfaces. With the tangent slope of the curved surface (namely, potential slip surface), the relationship between the nominal/tangential stress along the curved surface and the normal stress component may be obtained [21]. Such relation has been introduced in general statics textbooks [22]; the only difference is that it is expressed in the normal direction angle of the curved surface. Thus, the yield function expressed in the normal stress and tangential stress may be expressed in the positive stress component and slip surface (family) and the two layers of yield criterion meanings may be expressed in a mathematical equation, which is the yield condition and extremum condition of the yield function [23]. In fact, the extremum condition of the yield function lies in the limit Mohr’s circle; however, it is still not regarded as an independent equation. One feature of the book is that the relationship between the nominal/tangential stress and the positive stress component is expressed in the tangent slope of the curved surface and that the extremum condition of the yield function is regarded as one of the fundamental equations for the limit analysis. For the field failure mode, under the plane strain conditions, quantity of the fundamental equations of limit analysis including the slip surface family is the same as that of the unknown quantities; for the boundary condition of the load, it has constituted a complete limit equilibrium and renders an explicit definition for the slip surface family; namely, the relationship between the stress field in the limit state explicitly expressed in the mathematical equation and the slip surface family provides a basis for the determination of the slip surface. Velocity equation for the relationship between the velocity field and the slip surface family is established according to the direction of the maximum shear stress and energy dissipation condition, which is another feature of the book. Thus, for the kinematic boundary condition, not only the quantity of the fundamental equation is the same as that of the unknown quantities, but also it constitutes a complete limit equilibrium problem.

1.4 Main Features of the Book

11

For the surface failure mode, with static equilibrium equation, yield condition and extremum condition of the yield function, the stress equation along slip surface may be obtained, which is an ordinary differential equation expressed in the slip surface and normal stress; if the slip surface is determined, the stress field may be obtained so that the functional, whose independent variable of the limit load and slope stability is a slip surface, may be established with the moment equilibrium equation and stress equation along the slip surface and it constitutes a variational problem.

1.4.2 Limit Analysis Methods Two new limit analysis methods are proposed: stress field method and generalized limit equilibrium method. With the stress equation along the slip surface, if the slip surface is determined, the stress field may be obtained. Moreover, it will be easy to obtain the calculation formulas for the limit load and limit soil pressure. For the limit load and slope stability, the upper and lower bound theorems can be strictly proofed with the virtual work equation, and further, the generalized limit equilibrium method for the limit analysis may be developed. Variational principle of the limit load problem and slope stability may be proven with the moment equation and the stress equation along the slip surface; for the limit load, the extremum condition of the yield function is equivalent to that of the limit load; for the slope stability, the extremum condition of the yield function is equivalent to that of the safety factor. Further, the moment equilibrium equation-based generalized limit equilibrium method may be developed and calculation formulas for the safety factor without any other assumed or simplified limit loads, soil pressure, and slope stability may be obtained. If the selected slip surface is close to the real slip surface, safety factor with enough accurate limit load, soil pressure, and slope stability may be obtained.

1.4.3 Seeking for Specific Engineering Analysis and Calculation Methods For general conditions of the ground with heterogeneous soil, calculation formulas for the limit loads corresponding to different slip surfaces may be obtained by a generalized limit equilibrium method. On this basis and with the surface boundary load conditions considered, calculation methods for the ground bearing capacity may be obtained. Identically, for the general side slope with heterogeneous soil, analysis method for the slope stability without any force assumption and applicable to any slip surface may be obtained by generalized limit equilibrium method.

12

1 Preamble

According to the calculated results of tens of engineering examples, the obtained methods are of extensive application value.

1.5 Main Contents of the Book In Chap. 2, it discusses the failure mode and fundamental equation of the soil mass or ground, which are the most basic core contents of the limit analysis theory of the soil mass. Thereinto, after the establishment of the extremum condition of the yield function as one of the fundamental equations, one set of complete and uniform fundamental equations are available for the limit analysis theory, which is distinguished from the existing limit analysis theory or it may be called as a development of the existing limit analysis theory. In Chaps. 3, 4, 5, 6, and 7, the solution methods for limit analysis are studied based on the complete and uniform fundamental equations of limit analysis, with the details as follows. In Chap. 3, it proposes a two-way recursive and iterative calculation process for solution by the slip lines one by one with regard to the characteristic line method according to the characteristics of the boundary condition; so, the calculation process is more convenient and much easier to calculate the numerical solution of the limit load at a high accuracy. In Chap. 4, after a slip surface is assumed, the equilibrium equation and yield condition are both linear equations and it is easy to obtain a stress field. With the stress equation along the slip surface (family), the corresponding stress field may be obtained for slip surfaces in any functional form. If the selected slip surface is close to the real slip surface, the stress field will be close to the stress field in the limit state. With the obtained stress field, it is easy to obtain the approximate analytical expression for the limit load. If the slip surface is properly selected, the calculated limit load and the characteristic line method will be very close. In Chap. 5, the force on slip surface and moment equation are established with the fundamental equation and it indicates that the horizontal force equilibrium equation and moment equilibrium equation are not mutually independent, which is not easy to be discovered by traditional limit equilibrium method with the division of several isolators for the force analysis as its basic idea and attention shall be paid to. For the slip surface in some form, the limit equilibrium method and the upper bound solution method are identical, which is a fact that can be proved. In Chap. 6, upper and lower bound theorems of the limit load are strictly proved. As the velocity equation (equation system) is only related to the slip surface family, it is independent of the stress field. Therefore, as long as the slip surface is known, the velocity field may be solved with the velocity equation. After the virtual work equation is substituted, the limit load may be calculated with the virtual work equation. As long as the selected slip surface is close to the real slip surface, the limit load close to the genuine solution may be obtained.

1.5 Main Contents of the Book

13

As the upper and lower bound theorem is only a narrow theorem, if the selected slip surface is not a real slip surface, the calculated limit load can not be proved to be an upper bound solution; thus, the above solution method is a generalized limit equilibrium method. In Chap. 7, it discusses the solution method for the surface failure mode; with the stress equation along the slip surface, the unknown stress in the moment equilibrium equation may be eliminated and then solution methods for the limit load and slope stability may be established without any assumptions or simplifications, which are referred to as the generalized limit equilibrium method for the surface failure mode. The variational principles of the limit load and slope stability laid a theoretical foundation for the generalized limit equilibrium method for the surface failure mode. It is well known that the study on the ground limit load has a long history in the soil mechanics community; however, better calculation methods for the ground with heterogeneous soil are still not available up to the present, which needs to be solved in the field of engineering and will be discussed in Chap. 8. By virtue of the generalized limit equilibrium method for the surface failure mode, it is easy to establish the limit load calculation mode of the ground with heterogeneous soil; with logarithmic helicoid and helicoids–helicoid–plane calculation modes, many cases are calculated so as to have a better understanding about the limit load calculation method for the ground with heterogeneous soil and provide effective methods for the calculation of the ground bearing capacity. In Chap. 9, it discusses the calculation of the ground bearing capacity, which is generally carried out in two steps: Step 1, the limit load is calculated; Step 2, the ground ultimate bearing capacity is determined according to the design load and limit load. Moreover, it considers that it is a better method to determine the ground bearing capacity according to the soil mass without prejudice to the yield criterion. Limit load-based calculation mode gives rise to one set of calculation methods for the ground bearing capacity with extensive application scope, and they are free from the restriction of ground and load conditions as it were, such as ground with heterogeneous soil including partial replacement, and eccentric and inclined load. Chapter 10 discusses the analysis methods for the slope stability. Based on the limit equilibrium method, slope stability analysis method and composite slip surface method with more reasonable theory and reliable calculated results may be obtained by applying the extremum condition of the yield function. By the generalized limit equilibrium method for the surface failure mode, slope stability analysis method without any assumption or simplification and applicable to any slip surface may be established. For general side slope conditions, slip surfaces such as arc surface, helicoid, plane–helicoid–plane and plane–arc surface–plane may be selected for the calculation. In Chap. 11, for the purpose of the analysis on the slope stability during construction period, it discusses the calculation of the pore water pressure by twodimensional consolidation theory. In addition, it discusses stability analysis methods for the non-slip role of geotextile reinforced cushion.

14

1 Preamble

Chapter 12 discusses the calculation of the limit soil pressure. By stress field method, it may be very convenient to give the calculation formula for the soil pressure of fracture plane with plane–helicoid and plane–arc surface.

References 1. Chen HF (1995) Limit analysis and soil plasticity (trans: Zhan S, Proofread by Han D). China Communications Press, Beijing 2. Shen ZJ (2000) Theoretical soil mechanics. China Water Power Press, Beijing 3. Zheng YR, Gong XN (1989) Plastic theory of geotechnics. China Architecture and Building Press, Beijing 4. Zhang XY, Yan SW (2006) Fundamentals of geotechnics plasticity, 2 edn. Tianjin University Press, Tianjin 5. Chen ZY (2003) Soil slope stability analysis–theory, methods and programs. China Water Power Press, Beijing 6. Luan MT, Lin G, and Guo Y et al (1995) Generalized sliding—wedge method and its application to stability analysis in soil mechanics 17(4):1–9 7. Luan MT, Jin CQ, Lin G (1988) Ultimate bearing capacity of shallow footings on nonhomogeneous soil foundations. Chin J Geotech Eng 10(4):14–27 8. Huang WX (1986) Shear strength of soil and introduction to academic discussion on constitutive relation. Chin J Geotech Eng 8(1):1–5 9. Fan QJ, Li NY (2004) Reasons and countermeasures for North bank’s part failure in the second phase regulation project of Yangtze Estuary. China Harbour Eng (2):1–8 10. Luan MT, Nian TK, Zhao SF (2003) Summary of research process of earth structure and side slope. In: 9th conference proceedings on soil mechanics and geotechnical engineering. Tsinghua University Press, Beijing, pp 37–55 11. Shen ZJ, Lu PY (1997) Comments on the conservative tendency of rock and soil engineering practice in the current. Chin J Geotech Eng 19(4):115–118 12. Qian JH, Yin ZZ (1996) Principle and calculation of geotechnical. China Water Power Press, Beijing 13. Zheng YR (2003) New development of geotechnical plastic mechanics—generalized plastic mechanics. Chin J Geotech Eng 25(1):1–10 14. Terzaghi (1960) Theoretical soil mechanics (trans: Xu Z). Geological Publishing House, Beijing 15. Yin ZZ, Lv QF (2005) Finite element analysis of soil slope based on circular slip surface assumption. Rock Soil Mech 26(10):1525–1529 16. Zhang LY, Shi WM, Zheng YR (1999) Slope stability analysis by FEM under the plane strain condition. Chin J Geotech Eng 24(4):487–490 17. Tianjin University (1980) Soil mechanics and foundation. China Communications Press, Beijing 18. Xiao DP, Zhu WY, Chen H (1998) Progress in slip lines method to solve the bearing capacity problem. Chin J Geotech Eng 20(4):25–29 19. Xiao DP, Chen H, Yang JL (1995) Ultimate bearing capacity formula for solution of exact value based on slip line method. Master Dissertation of Tianjin University 20. Huang CC (2006) Verification of theorem of limit analysis of soil mass and generalized flow rule. Chin J Geotech Eng 28(6):700–704

References

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21. Huan CZ, Zhang J, Sun WH (2002) Stress field and ultimate load for soil mass in limit state. Chin J Geotech Eng 24(3):389–391 22. Timoshenko SP, Goodier JN (1964) Theory of elasticity (trans: Xu Z, Wu Y). Higher Education Press, Beijing 23. Huang CZ (2003) Extremum condition of the yield function—one of fundamental equations for the limit analysis on the soil mass. In: 9th conference proceedings on soil mechanics and geotechnical engineering. Tsinghua University Press, Beijing, pp 37–55

Chapter 2

Failure Mode and the Fundamental Equation

Under the action of load, is the soil mass in a stable state, failure state, or just in the limit state? In order to provide a concrete answer to this question, a certain judgment criterion will have to be established. Using this criterion, we analyzed the possible failure mode of the soil mass or ground. On the basis of the failure mode, a corresponding mathematical model is constructed, which is one of the main tasks in performing a limit analysis of the soil mass. This mathematical model is a quantitative description of the conformity rule when the soil mass is in the limit state; namely, it should be in accordance with the fundamental equation and its definite conditions. Therefore, a complete and uniform fundamental equation is the basis for a limit analysis of the soil mass.

2.1 Equilibrium Equation and Yield Condition In the equations for the limit analysis of the soil mass, one part is the same as the fundamental equation of statics and may be applied directly and another part is specific to the limit analysis and needs to be established on the basis of the failure mode.

2.1.1 Equilibrium Equation A two-dimensional limit analysis is generally considered according to the plane strain and the stress components σx , σz , τx z satisfy the following equilibrium equations: ∂σx ∂x ∂σz ∂z

+ ∂τ∂zx z = 0 + ∂τ∂ xx z = γ



© Springer Nature Singapore Pte Ltd. and Zhejiang University Press, Hangzhou, China 2020 C. Huang, Limit Analysis Theory of the Soil Mass and Its Application, https://doi.org/10.1007/978-981-15-1572-9_2

(2.1)

17

18

2 Failure Mode and the Fundamental Equation

where γ is the weight of the soil mass. The equilibrium equation is essential for statics. In the limit analysis of the soil mass, any method is indispensable.

2.1.2 Yield Function and Yield Condition The Coulomb yield function f is defined as follows: f (σ, τ ) = τ − (σ tan ϕ + c)

(2.2)

where c, ϕ are the cohesion and internal friction angle of the soil mass, respectively, σ is the normal stress on the curved surface, and τ is the tangential shear stress on the curved surface. The yield function is the difference between the total tangential shear stress and the total strength at any point on the curved surface. The definition of the yield function f is related to the curved surface (family) without loss of generality, and dz = h  (x, z), which is the curved surface (family) may be defined by the equation dx  an ordinary differential equation. If h (x, z) is determined, only with any entry point on the curved surface (family), it will be able to determine a curved surface through the point. When a certain section of the whole surface boundary (for example, the foundation width) is taken for (x0 , z 0 ), a curved surface family will be formed. The yield criteria (Coulomb yield criteria) for the soil mass are as follows: f = 0 with regards to all aspects along a certain curved surface (family) and f ≤ 0 along other curved surfaces. If the soil mass satisfies the yield criterion, max{ f } = 0. If the soil mass does not violate the yield criterion, max{ f } ≤ 0. f = 0 is referred to as the Coulomb yield condition. If f = 0 is true for the dz = h  (x, z) and f ≤ 0 for the other curved curved surface defined in the equation dx surfaces, the curved surface will be referred to as a slip surface (or yield surface/slip dz = h  (x, z) that satisfies the yield line). The curved surface (family) defined in dx criterion is referred to as a slip surface (family). In the limit analysis on the soil mass, the slip surface (family) h  (x, z) should be regarded as an unknown function to be considered. It is important to note that the Mohr–Coulomb yield criterion derived according to the Mohr’s stress circle method for the limiting stress prevails in some problems.  fm =

   1 1 (σz − σx )2 + τx2z − (σz + σx ) sin ϕ + c cos ϕ = 0 4 2

(2.3)

Later, we will see that the Mohr–Coulomb yield criterion is just a special form (namely max{ f } = 0) of Eq. (2.2) when the soil mass is in the limit state along the slip surface and does not violate the yield condition ( f ≤ 0) along other curved surfaces.

2.1 Equilibrium Equation and Yield Condition

19

If we know the load on the surface of the soil mass, it is generally possible to determine the soil mass state. In other words, perhaps there is no such slip surface and stress field to make the yield function (2.2) meet the yield condition. At this time, it may be defined as: f =τ−

1 (σ tan ϕ + c) Fs

(2.4)

After the total strength is divided by Fs , the yield function is the difference between the total tangential shear stress at any point on the curved surface and the total strength divided by Fs . Thus, Fs is referred to as the safety factor (strength safety factor). It should be noted that, from the perspective of a probabilistic method (reliability), the variability in the soil mass strength is one of the main factors that affect the stability of the ground bearing capacity. In the partial safety factor method based on reliability theory, the partial safety factor of the strength and that of the load (action) are jointly used to measure the stability of the ground bearing capacity, which has been applied in engineering, and the yield function in Eq. (2.4) is applicable.

2.2 Stress Relationship of a Point on a Curved Surface dz The yield function is defined on the curved surface family  : dx = h  (x, z); in order to serve the yield condition and the equilibrium equation jointly for a solution, the relationship between the normal stress/tangential shear stress and the positive stress/shear stress shall also be established [1]. For any point on the curved surface, its differential triangle (Fig. 2.1) is drawn. From the differential triangle, the following equations are satisfied:

X ds = σx dz − τx z dx Z ds = σz dx − τx z dz



x

σz

τ xz α

σX

x

z

α z = h (x )

(a) Differential Triangle

Fig. 2.1 Schematics of the differential triangle and forces

σ

z

τ

τ xz

(b) Forces acting on the differential triangle

20

2 Failure Mode and the Fundamental Equation

where dx, dz, and ds are the side lengths of the differential triangle and X , Z are the stress components on the hypotenuse. Thus, we have  X = σ sin α − τ cos α Z = σ cos α + τ sin α  dz dz dx sin α = , cos α = , ds = (dx)2 + (dz)2 and h  = ds ds dx Therefore, X= Z=

√ 1 (σx h  − τx z ) 1+h 2 √ 1 (σz − τx z h  ) 1+h 2

X= Z=

√ 1 (σ h  − τ ) 1+h 2 √ 1 (σ + τ h  ) 1+h 2

 (2.5)

 (2.6)

The stress relation expressed in Eq. (2.5) is applicable for any curved surface, including the boundary surface. However, Eq. (2.6) shows the relationship between the total vertical force/total horizontal force and the normal stress/tangential shear stress of any point on the curved surface. By comparing Eqs. (2.5) and (2.6), the relationship between the normal stress/tangential shear stress and the positive stress/shear stress is obtained: σx h  − τx z = σ h  − τ σz − τ x z h  = σ + τ h 

 (2.7)

According to Eq. (2.7), we get the solutions for σ, τ as follows: 1    σ = 1+h 2 [σz − τ x z h + h (σ x h − τ x z )] 1   τ = 1+h 2 [h (σz − τx z h ) − (σx h  − τx z )]

 (2.8)

It is actually the same as the stress relationship given in statics [2]; only that it is expressed in the tangent slope of the curved surface, instead of the sin α, cos α of angle α between the horizontal direction and the tangential direction of the curved surface (it is also the same when α is taken as the angle between the horizontal direction and the normal direction of the slip surface).

2.3 The State and Failure Mode of the Soil Mass The primary difficulty of limit analysis lies in the treatment of the yield criterion, so it is necessary to discuss the basic features of the yield function and its quantitative description of the state of the soil mass. The yield function f is the difference between

2.3 The State and Failure Mode of the Soil Mass

21

the total shear stress and the shear strength of any point on the curved surface and may be regarded as one standard of judgment for the quantitative description of the state of the soil mass. Therefore, whether the soil mass is in a stable state or in a state of failure may be judged theoretically based on the yield function.

2.3.1 The State of the Soil Mass (1) Point state: the state of any point (x, z) in the soil mass: For any point (x, z) in the soil mass, regardless of the shape of the curved surface through the point and the tangent slope h  (x, z) of the curved surface, if f < 0 is always satisfied, the soil mass at the point will be in the stable state. If a specific value of a certain h ∗ (x, z) causes f > 0, the soil mass at the point will be in the failure state. If f = 0 is true for a certain h ∗ (x, z) and f ≤ 0 for any other h  (x, z), the soil mass at the point will be in the limit state. It is noted that this is only an ideal description of the state of the soil mass, and f > 0 is impossible according to the requirements of yield criterion conformity. However, when an approximate solution of the ground limit load is found, f > 0 is possible (for example, an upper bound solution). It should be noted that the above is only an ideal description of the state of the soil mass and f > 0 is impossible if the requirements of the yield criterion are satisfied. However, when the approximate solution of the ground limit load is solved, f > 0 is possible (for example, the upper bound solution). (2) Plane state: the state of any curved surface in the soil mass: If all of the points on the curved surface (x, z(x)) are in a stable state or some of the points (one or several sections on the curved surface) are in a stable state and some of them are in a limit state, the soil mass on the curved surface will be in a stable state; if all of the points on the curved surfaces (x, z(x)) are in a limit state, the soil mass on the curved surface will be in a limit state; if all of the points on the curved surface (x, z(x)) are in a state of failure or some of the points are in a state of failure and some are in a limit state, the soil mass on the curved surface will be in a state of failure. If some of the points are in a state of failure and some are in a limit state while the other points are in a stable state, the soil mass on the curved surface will be in a state of partial failure. The stress state on the curved surface is completely different from that along the curved surface. The former refers to the stress state of every point on the curved surface, including many other curved surfaces through every point on the curved surface; however, the latter refers to the stress state along a specific curved surface. (3) Field state: the state of all the points in the soil mass: If all of the points in the soil mass (x, z) are in the stable state or some of the points (one or several areas) are in the stable state and some are in the limit state, the soil

22

2 Failure Mode and the Fundamental Equation

mass in the entire area will be in the stable state. If all of the points in a specific area (x, z) are in the limit state, the soil mass in the entire area will be in the limit state. If all of the points in a certain specific area are in the failure state or some of the points are in the failure state and some are in the limit state, the soil mass in this area will be in the failure state. If some of the points in a specific area are in the failure state, some are in the limit state, and some are in the stable state, the soil mass will be in the partial failure state. The state of the stress field is consistent with that of the soil mass; namely, the stress field that causes the soil mass to be in the stable state is referred to as the stable stress field. Similarly, the stress fields that cause it to be in the failure and limit states are referred to as the failure stress field and limit stress field, respectively. For the state of the soil mass from an engineering viewpoint, if the soil mass is stable, it is in the stable state in the entire area; namely, for all the points in the soil mass (x, z), f ≤ 0 will be always true, regardless of the value of h  (x, z). If the soil mass is destroyed, it will only need to be in the failure state for a certain curved surface. If the soil mass is in the failure state along a certain curved surface, it will only need to be in the failure state at some of the points on the curved surface and in the limit state at other points. Namely, f ≥ 0 is true for any point (x, z) on the curved surface, and the equals sign can be true only at some of the points on the curved surface. Certainly, when the soil mass is in the failure state for several curved surfaces or a certain specific area, the soil mass will also be subject to failure.

2.3.2 The Failure Mode of the Soil Mass The limit state is a critical state between the stable state and the state of failure and it may be in the following modes: Plane failure mode: For all points (x, z) in the soil mass, regardless of the value of h  (x, z), f ≤ 0 will be always true, and the equals sign will be true only for certain (several) curved surfaces. Namely, the soil mass is in the limit state for certain (several) curved surfaces and in the stable or limit state at other points. In other words, in the surface failure mode, the soil mass is required to be in the limit state on the slip surface and in the stable or limit state on both sides of the slip surface. Field failure mode: For any point (x, z) in the soil mass, regardless of the value of h  (x, z), f ≤ 0 will be always true, and the equals sign will be true for a family of curved surfaces, which are throughout a certain specific area. Namely, the soil mass is in the limit state for a certain specific area and in the stable state for other areas. Regarding the surface failure mode, the slip surface in a limit state is the most dangerous; the area enclosed by the slip surface and the boundary surface of the soil mass is referred to as the slip mass and the soil mass in the slip mass should be in a stable state or some of the soil mass should be in a stable state and some in a limit state. For the field failure mode, the area enclosed by the envelope surface (the largest slip surface) of the slip surface family and the boundary surface of the soil mass is

2.3 The State and Failure Mode of the Soil Mass

23

referred to as the slip mass, and the soil mass in the slip mass should be in a limit state. Beyond the slip mass, in any failure mode, the soil mass should be in a stable state. In all practical problems, on the basis of specific conditions, it shall be determined whether the surface failure mode or the field failure mode has to be adopted. Generally, under a known load of the soil mass surface, the state of the soil mass is determinate; In this case, the surface failure mode should be adopted for study, e.g., the slope stability. For an unknown limit load (ultimate bearing capacity and limit soil pressure) on a certain section of the boundary surface, the field failure mode or surface failure mode may be adopted for study. The plane failure mode and field failure mode are the most fundamental for the limit analysis; if only one of them is considered, the limit analysis theory will be incomplete. Compared with the ground soil conditions and load conditions in engineering, the above failure mode can only be considered as an “ideal” state of the soil mass. If the field failure mode requires the soil mass in the slip mass to always be in a limit state, the distribution form of the corresponding limit load should be unique and that of the actual load applied to the ground would generally not be exactly consistent. In other words, regarding the ground under the action of the actual load, the soil mass in the slip mass cannot always be put in a limit state. However, if the limit load under such “ideal” conditions can be obtained, it will be possible to determine whether the ground soil is in a stable state under the action of the actual load, which is the attainable objective for studying the ground stability by applying such an “ideal” failure mode. The above two failure modes are considered from the viewpoint of the possible failure of the ground soil. From the viewpoint of the surface load (including the edge load) of the ground, namely the surface shape, there are complete failure, partial failure, one-way failure, and two-way failure modes, which will be discussed regarding a specific issue (ground bearing capacity).

2.4 Stress Field in the Limit State 2.4.1 Extremum Condition of Yield Function As the above description of the limit state of the soil mass is still not convenient for application, an accurate description of the limit state of the soil mass determined by the yield function will now be discussed mathematically. Substituting Eq. (2.8) into Eq. (2.2), we can rewrite the yield function as follows: Equation (2.8) is substituted into Eq. (2.2) and the yield function will be rewritten as below:

24

2 Failure Mode and the Fundamental Equation

f (σ, τ ) = f (σx , σz , τx z , h  ) 1 = [(σz − h  τx z )(h  − tan ϕ) − (σx h  − τx z )(1 + h  tan ϕ) 1 + h 2 − c(1 + h 2 )] (2.9) Note that for any point (x, z) in the soil mass, the stress component is only a function of (x, z) unrelated to the shape (tangent slope h  of the curved surface) of the curved surface through the point. Moreover, there may be many curved surfaces through the point (x, z), namely h  at the point is a variable; therefore, f is the function of both (x, z) and h  . Among these curved surfaces, the one that can always meet the yield condition f ≤ 0 and become a real slip surface will satisfy max{ f }h  = 0. In other words, if max{ f }h  = 0 (max{ f }h  = 0 and max{ f } = 0 will be proven to be equivalent later), the above-mentioned conditions of the limit state will always satisfy the yield condition ( f ≤ 0) and equality must be held on the real slip surface (family). According to the necessary condition [3] of the extremum, it shall be as follows:  1 ∂f 2  2 2 2 = − σ )(1 + 2h tan ϕ − h ) − τ (2h − tan ϕ + h tan ϕ) =0 (σ z x x z ∂h  (1 + h 2 )2 2

(2.10) Equation (2.10) is referred to as the extremum condition of the yield function, which is a fundamental equation for the relationship between the slip surface family and the stress field. Hence, the following may be obtained: 

h =

1 (σ 2 z

− σx ) tan ϕ − τx z ± 1 (σ 2 z



1 (σ 4 z

− σx )2 + τx2z (1 + tan2 ϕ)

− σx ) + τx z tan ϕ

Substituted it back into Eq. (2.9) to obtain:  f =±

  1 1 (σz − σx2 ) + τx2z (1 + tan2 ϕ) − (σz + σx ) tan ϕ + c 4 2

which is maximized when the plus sign is chosen and minimized when the minus sign is chosen. The maximum is of a precise meaning. If max{ f }h  = 0, it will be as follows: 

h =

1 (σ 2 z

− σx ) tan ϕ − τx z + 1 (σ 2 z

 1 4

(σz − σx )2 + τx2z (1 + tan2 ϕ)

− σx ) + τx z tan ϕ

(2.11)

2.4 Stress Field in the Limit State

 max{ f }h  =

25

  1 1 (σz − σx )2 + τx2z (1 + tan2 ϕ) − (σz + σx tan ϕ + c) = 0 4 2 (2.12)

 It is learned that max{ f }h  / 1 + tan2 ϕ = 0 is the Mohr–Coulomb yield criterion of Eq. (2.3). In other words, max{ f }h  = 0 is the Mohr–Coulomb yield criterion. The extremum condition of the yield function may also be derived by the limit in Mohr’s stress circle method [4]. Actually, Eq. (2.7) is rewritten as below: (σx − σm )h  − τx z = (σ − σm )h  − τ σz − σm − τx z h  = σ − σm + τ h 

 (2.13)

where σm =

1 (σz + σx ) 2

(2.14)

After the parts on both sides of each equation in Eq. (2.13) are squared respectively and then added together, Mohr’s stress circle will be obtained as below: τ 2 + (σ − σm )2 = pt2

(2.15)

where

pt =

1 (σz − σx )2 + τx2z 4

(2.16)

pt is the maximum shear stress. After introduction of the parameter θ , the polar form of Mohr’s stress circle is as below:  τ = pt cos θ (2.17) σm − σ = pt sin θ Substituting Eq. (2.17) into the Coulomb yield function, f = pt (cos θ + tan ϕ sin θ ) − (σm tan ϕ + c)

(2.18)

It is easy to prove the following. When tan θ = tan ϕ, the maximum of the Coulomb yield function is attained: max{ f } =

1 [ pt − (σm sin ϕ + c cos ϕ)] cos ϕ

When max{ f } = 0, the Mohr–Coulomb yield criterion is derived.

(2.19)

26

2 Failure Mode and the Fundamental Equation

Fig. 2.2 Limit of Mohr’s circle

τ ϕ

τf

c

σz

τm 2θ

τ xz

σx

σ

It also proves that the Mohr–Coulomb yield criterion is the extreme form (max{ f } = 0) of the Coulomb yield condition f = 0 and it will be as follows: τ = pt cos ϕ σ = σm − pt sin ϕ

 (2.20)

According to Eqs. (2.8) and (2.20), σ , τ , and pt are eliminated to get: 1 (σz − σx )(1 + 2h  tan ϕ − h 2 ) − τx z (2h  − tan ϕ + h 2 tan ϕ) = 0 2 This is the same as the necessary condition (2.10) for the extremum of the yield function. ¯ is supposed; thereinto, θ is the commonly In addition, h  = tan(θ − μ)  referred direction angle of the major principal stress μ¯ = π4 − ϕ2 . Together with (1 + 2h  tan ϕ − h 2 )/(1 + h 2 ) = sin(2θ) and (2h  − tan ϕ + h 2 tan ϕ)/(1 + cos ϕ

h 2 ) = − cos(2θ) , the extremum condition of the yield function may be rewritten cos ϕ 1 as 2 (σz − σx ) sin(2θ ) + τx z cos(2θ ) = 0. The relation has been given in the limit of Mohr’s stress circle [5] (Fig. 2.2). In other words, as the extremum form (max{ f } = 0) of the Coulomb yield condition f = 0, the Mohr–Coulomb yield criterion includes the Coulomb yield condition and the extremum condition of the yield function. However, when the Mohr–Coulomb yield criterion is applied, the extremum condition of the yield function is still not regarded as an independent equation. Here, it is connected with the slip surface and regarded as one of the fundamental equations for the limit analysis of the soil mass. For convenience, λ = tan ϕ in the following discussion.

2.4.2 The Stress Field of the Soil Mass in the Limit State On the one hand, according to the yield condition, Eq. (2.7) may be rewritten as follows:        c c h (h + − τ = σ + − λ) σ x x z λ λ     (2.21) σz + λc − τx z h  = σ + λc (1 + λh  )

2.4 Stress Field in the Limit State

27

On the other hand, according to Eqs. (2.11) and (2.12), it will be as follows: 

1 (σz − σx ) + τx z λ h  = σz λ − τx z + c 2

(2.22)

According to Eqs. (2.21) and (2.22), the following equations may be solved: ⎫   2 +(h  −λ)2 ⎪ σ + λc σx + λc = 1+λ 1+h 2  ⎬ (1+λ2 )h 2 +(1+λh  )2 c c σ+λ σz + λ = 1+h 2 ⎪  2 ⎭ −(h  −λ)2 c σ + τx z = λ 1+λ 1+h 2 λ

(2.23)

In Eq. (2.23), the stress component is expressed as a function of the slip surface (h  ) and the normal stress σ on the slip surface, and it is referred to as the stress field of the soil mass in the limit state. Further, the following relationship between the stress components satisfying the yield condition and extremum condition of the yield function may be obtained: σx + σz +

1+λ2 +(h  −λ)2 σ (1+λ2 )(1+h 2 ) e 2 2 )h +(1+λh  )2 = (1+λ σe (1+λ2 )(1+h 2 ) 1+λ2 −(h  −λ)2 λ (1+λ2 )(1+h 2 ) σe

c λ c λ

τx z =

=

⎫ ⎪ ⎬ ⎪ ⎭

(2.24)

where σe =

 c 1 c (σx + σz ) + = σ + (1 + λ2 ) 2 λ λ

(2.25)

According to h  = tan(θ − μ), it is easy to get Eq. (2.24) as below: ⎫ σz = σe (1 − sin ϕ cos 2θ ) − c/λ ⎬ σx = σe (1 + sin ϕ cos 2θ ) − c/λ ⎭ τx z = σe sin ϕ sin 2θ

(2.26)

This is the same as the relation [5] derived by direct application of the Mohr– Coulomb yield condition and the angle θ in the major principal stress direction.

2.4.3 Meaning of Stress Field in the Limit State First, considering the extremum feature of the yield function, one equation may be increased: the extremum condition of the yield function. The increased equation to describe the relationship between the slip surface (family) and the stress component is of great importance and contributes to an accurate definition of the real slip surface.

28

2 Failure Mode and the Fundamental Equation

According to Eq. (2.20), pt = 41 (σz − σx )2 + τx2z is the maximum shear stress and it is observed that the action direction of τ is the tangential direction of the slip surface; therefore, the action direction of the maximum shear stress will be the direction that constitutes angle ϕ with a tangential direction of the slip surface. Second, the stress field of Eq. (2.23) in the limit state has five unknown quantities [three stress components, slip surface (h  ) and normal stress σ on the slip surface] and three equations. In other words, Eq. (2.23) and equilibrium equations (2) together constitute one complete set of fundamental equations or the theoretical foundation for the limit analysis theory of the soil mass. For the field failure mode, with proper boundary conditions, the complete set of fundamental equations may be directly applied for the solution so as to obtain a stress field in the limit state, including the slip surface (family) and the limit load required for engineering. Certainly, if the Mohr–Coulomb yield condition and the equilibrium equation are directly applied, equations and unknown functions will have the same quantity. However, not just any equation will be available to give an accurate definition of the slip surface. Without any knowledge about the real slip surface, many issues will not be convenient to be discussed or cannot be discussed at all. For example, it may be confirmed that, when partial areas are in a stable state and partial ones are in a failure state, the limit load which is the same as the exact solution of the corresponding limit load under the limit state conditions of the soil mass in the limit state may also be derived. The latter can only be proven not to be an exact solution by an analysis of the slip surface or stress field. In addition, the mechanical meanings of τ and σ in Eq. (2.20) lie in that, in the framework of the Coulomb yield function, the shear stress τ is the relative maximum while the normal stress σ is the relative minimum. Therefore, the stress state shown in Eq. (2.23), among all the possible stress states, is the most likely to cause a shear failure on the soil mass. Thereinto, “possible stress states” refer to the stress fields in accordance with the basic rules of soil mechanics (such as equilibrium equation and boundary condition). In other words, it is of definite engineering meaning to apply Eq. (2.23) and the equilibrium equation for limit analysis of the soil mass, and the stress state is the most likely to cause a shear failure of the soil mass.

2.5 Stress Equation Stress components are expressed as Eq. (2.23) with h  , σ . As long as h  , σ can be obtained, the stress filed in the limit state will be obtained. A partial derivative is calculated according to Eq. (2.23) and then the result is substituted in the equilibrium equation and rearranged as below:

2.5 Stress Equation

29

⎫ ∂  ∂  c  c ⎪ ⎪ λ(1 + λ2 − (h  − λ)2 ) σ+ + (1 + λ2 )h 2 + (1 + λh  )2 σ+ ⎪ ⎪ ∂x λ ∂z λ ⎪ ⎪  ⎪     ⎪ ∂h ∂h c 2λ  2  2 2 2 ⎪ ⎪ ⎪ − λ + λh ) λ − h ) ) σ + − (2h − (1 + 2h = γ (1 + h ⎬ 2 1+h λ ∂x ∂z     ∂ ∂ c c ⎪ ⎪ ⎪ (1 + λ2 + (h  − λ)2 ) σ+ + λ(1 + λ2 − (h  − λ)2 ) σ+ ⎪ ⎪ ∂x λ ∂z λ ⎪ ⎪  ⎪     ⎪ c ∂h ∂h 2λ ⎪  2 2  2 ⎪ ⎭ σ+ (1 + 2h λ − h ) + (2h − λ + λh ) =0 − 2 1+h λ ∂x ∂z

The above equations can also be simplified as follows: 

∂ ∂x σ  ∂ ∂x σ

+ +









 ∂h 

c c 2λ c  ∂ λ + h ∂z σ + λ − 1+h 2 σ + λ  1+λh  ∂    c c 2λ λ + λ−h  ∂z σ + λ + 1+h 2 σ +

⎫ ⎬



 ∂h  = γ h  −λ ∂ x + h ∂z 1+λ  2  c ∂h  1+λh  ∂h  = +  λ ∂x λ−h ∂z

γ



1+λh  λ−h 

+λ



1 1+λ2

⎭

(2.27) These two equations in Eq. (2.27) are referred to as the stress equations. Theoretically, with proper boundary conditions, σ, h  may be derived according to Eq. (2.27) and then the stress field is obtained according to Eq. (2.23). A similar stress equation in accordance with the equilibrium equation can also be derived according to Eq. (2.24): ∂σe ∂x ∂σe ∂x

+





  2λ = γ (h  − λ) σ ∂h + h  ∂h 1+h 2 e ∂ x ∂z     1+λh  ∂σe 2λ ∂h  ∂h  = γ 1+λh + 1+h + 1+λh 2 σe λ−h  ∂z ∂x λ−h  ∂z λ−h 

e + h  ∂σ − ∂z

⎫ ⎬

 +λ ⎭

(2.28)

Further, the following may also be derived with Eq. (2.23): z x − h  (1 + h  λ) ∂σ + (1 + 2h  λ − h 2 ) ∂τ∂ xx z = 0 (h  − λ) ∂σ ∂x ∂x ∂σz ∂σx    (h − λ) ∂z − h (1 + h λ) ∂z + (1 + 2h  λ − h 2 ) ∂τ∂zx z = 0



Thus, ∂∂ xf = 0, ∂∂zf = 0. In other words, for the stress field and slip surface family in accordance with the equilibrium equation, the yield condition, and the extremum condition of the yield function, the yield function also meets the condition of ∂∂ xf = 0, ∂∂zf = 0, which actually proves that max{ f }h  = 0 and max{ f } = 0 are equivalent.

30

2 Failure Mode and the Fundamental Equation

2.6 Generalized Flow Rule and Velocity Equation 2.6.1 Geometric Equation x εx = − ∂v ∂x ∂vz εz = − ∂z x γx z = − ∂v + ∂z

⎫ ⎪ ⎬ ∂vz ∂x

⎪ ⎭

(2.29)

where vx , vz are the horizontal velocity and vertical velocity, respectively.

2.6.2 Generalized Flow Rule Only the yield condition cannot be used to accurately describe the conformity of the soil mass with the yield criterion, so the yield condition and extremum condition of the yield function should be adopted to describe the conformity of the soil mass with the yield criterion. If the following equation is always true for any function μ(x, z) and η(x, z): f n = μf + η f h  = 0

(2.30)

The yield condition and extremum condition of the yield function will necessarily be true and f n is referred to as the generalized yield function. However,  ⎫  ∂ fh ∂ fn ∂f ⎪ = − μ + η εx = − ∂σ ⎪ ⎪ ∂σx  x ⎬  ∂σx ∂ fh ∂ fn ∂f εz = − ∂σz = − μ ∂σz + η ∂σz ⎪  ⎪ ∂ fh ⎪ ∂ fn ∂f ⎭ = − μ + η γx z = − ∂τ ∂τx z ∂τx z xz

(2.31)

The above is referred to as the generalized flow rule. If η = 0, Eq. (2.31) is exactly the same as the commonly associated flow rule at present. Thus, the generalized flow rue may be considered as the generalization of the associated flow rule and it will be as follows: ⎫ ∂ fh ∂f ∂vx = μ ∂σ + η ⎪ ⎬ ∂x ∂σ x x ∂ fh ∂vz ∂f = μ + η ∂z ∂σz ∂σz ∂ fh ⎪ ⎭ ∂vx z + ∂v = μ ∂τ∂ fx z + η ∂τ ∂z ∂x xz Namely,

2.6 Generalized Flow Rule and Velocity Equation ∂vx ∂x ∂vz ∂z ∂vx ∂z

⎫    λ) λ−h 2 = −μ h (1+h − η 1+2h ⎪ ⎬ 1+h 2 (1+h 2 )2 h  −λ 1+2h  λ−h 2 = μ 1+h 2 + η (1+h 2 )2 ⎪   λ−h 2 −λ+h 2 λ ⎭ z + ∂v = μ 1+2h − 2η 2h(1+h 2 )2 ∂x 1+h 2

31

(2.32)

2.6.3 Velocity Equation After μ and η are eliminated according to Eq. (2.32), it will be as follows:    2  ∂vz   2  ∂vx + 1 + λ2 + h  − λ 1 + λ2 h 2 + 1 + h  λ ∂z ∂x   ∂v    ∂v 2 x z + λ 1 + λ2 − h  − λ + =0 ∂z ∂x

(2.33)

  Both sides of Eq. (2.33) are multiplied by σ + λc /(1 + h 2 ) and the following equation may be derived according to Eq. (2.23): 

σx +

    ∂vx ∂vz c  ∂vx c ∂vz + σz + + τx z + =0 λ ∂x λ ∂z ∂z ∂x

(2.34)

This is actually the commonly referred to energy dissipation condition; thus Eq. (2.33) and the energy dissipation condition are equivalent. Moreover, the velocity component in the equation is only related to the slip surface family and the stress component is not contained any longer so that it is more convenient for the application. In addition, Eq. (2.33) is derived from the generalized flow rule with μ and η eliminated so that it is equivalent to the generalized flow rule; in other words, the generalized flow rule and the energy dissipation condition are equivalent. Therefore, Eq. (2.33) is one of the fundamental equations for the kinematic velocity to meet. Another equation for the velocity of the movement to meet is derived below. On the one hand, it is assumed that the angle between the direction of the movement of the soil mass on the slip surface and the horizontal direction is α, the velocity of the movement along the direction will be the highest, and it will be as follows: v(α) =

vx2 + vz2 = vx cos α + vz sin α, or vx = v cos α, vz = v sin α

Hence,    π π π = vx cos α + + vz sin α + = vz cos α − vx sin α = 0 (2.35) v α+ 2 2 2 It shows that the soil mass on the slip surface has zero velocity of movement in the direction with the horizontal angle of α + π/2.

32

2 Failure Mode and the Fundamental Equation

On the other hand, from the above, it is known that the direction of the maximum shear stress is the direction that constitutes angle ϕ with the tangential direction of the slip surface. From the perspective of strength, h  = tan β is supposed and, according to Eq. (2.21), the yield condition and extremum condition of the yield function may be rewritten as:  Z λ + c cos β = (σ λ + c) cos(β − ϕ)/ cos ϕ (2.36) X λ + c sin β = (σ λ + c) sin(β − ϕ)/ cos ϕ It can be observed that (Z λ + c cos β) = τx is the strength of the slip surface per unit of arc length in the horizontal direction of the soil mass while (X λ+c sin β) = τz is the strength in the vertical direction. Thus, the strength of the soil mass on the slip surface in the direction that constitutes angle β − ϕ with the horizontal direction is:

τm =

(Z λ + c cos β)2 + (X λ + c sin β)2 = (σ λ + c)/ cos ϕ

(2.37)

It is consistent with the maximum shear stress and it is the radius of the limit of Mohr’s stress circle (Fig. 2.3). From the above discussion, the soil mass in the limit state along the slip surface has the maximum strength in the direction that constitutes angle β −ϕ with the horizontal direction. Because, (Z λ + c cos β) sin(β − ϕ) − (X λ + c sin β) cos(β − ϕ) = 0, namely,   π π + (X λ + c sin β) sin β − ϕ + =0 (Z λ + c cos β) cos β − ϕ + 2 2 This shows that the soil mass on the slip mass has zero strength in the direction that constitutes angle β − ϕ + π/2 with the horizontal direction.

x

τm τf

ϕ Movement direction Slip surface

z

τm

c ϕ

Fig. 2.3 Limit of Mohr’s circle for any point on the slip surface

τx

τz

2.6 Generalized Flow Rule and Velocity Equation

33

The direction of the maximum shear stress on the slip surface will be consistent with the direction of the movement (or the direction of the maximum soil mass strength is consistent with the direction of the movement), namely α = β − ϕ. Thus, the following may be derived using Eq. (2.29): vx sin(β − ϕ) − vz cos(β − ϕ) = 0. Therefore, the relationship between the velocity field and the slip surface is obtained: vx (h  − λ) − vz (1 + h  λ) = 0

(2.38)

In other words, angle ϕ is constituted between the direction of the movement and the tangential direction of the slip surface. For the plane failure mode, this equation is true on the slip surface. This is identical to the existing theory of limit analysis [vn + vt λ = 0, see Eq. (2.63)] [4, 5]. For the field failure mode, according to the characteristic line method, we know that if the slip mass is always in the limit state, a slip surface family (slip line family) shall be provided for in the slip mass; for any slip surface in the slip surface family, Eq. (2.38) is always true. Moreover, the slip surface family spreads over the slip mass, so Eq. (2.38) will also be true in the slip mass, which has the same meaning as the yield condition being true on the slip surface and then true in the slip mass. Equations (2.33) and (2.38) are velocity equations and one of the fundamental equations for limit analysis. In the velocity equation, there are two equations and three unknown functions (velocity components vz and vx as well as the slip surface family determined accorddz = h  (x, z)). Thus, the velocity field shall be obtained simultaneously ing to dx with the solution of the limit state. In addition, the velocity equation establishes the relationship between the velocity field and the slip surface. If the slip surface is determined, the velocity may be determined according to the velocity equation, which obviously provides convenience for the method of solution for the approximate solution of the limit analysis under the assumed slip surface. Generally, the geometric equation and flow rule are regarded as one of the fundamental equations for limit analysis according to the requirements for treatment of the kinematic boundary condition. Actually, it is not necessary to regard the flow rule as one fundamental equation because it is not the essential and unique universal rule for limit analysis. Moreover, the ground bearing capacity, slope stability, and the limit of the soil pressure are almost load boundary conditions so that it is not required to regard the geometric equation and the flow rule as one of the fundamental equations for limit analysis.

2.7 Associated Flow Rule and Velocity Equation Speaking of the flow rule, we must never forget to mention the currently associated flow rule. The geometric equation and the currently associated flow rule are applied below to obtain the velocity equation [5, 6].

34

2 Failure Mode and the Fundamental Equation

εx = −γm ∂∂σfmx εz = −γm ∂∂σfmz ∂ fm γx z = −γm ∂τ xz

⎫ ⎪ ⎬ ⎪ ⎭

(2.39)

Substituting the expression for f m in Eq. (2.3), ⎫  2 ⎪ − σ εx = − 2σγme λ σe λ2 + 1+λ (σ ) z x ⎪ 2  ⎬ 2 γm 1+λ εz = − 2σe λ σe λ2 − 2 (σz − σx ) ⎪ ⎪   ⎭ γx z = σγemλ 1 + λ2 τx z

(2.40)

If the yield function in Eq. (2.9) is instead substituted, we have ⎫   λ) ⎪ εx = −γm h (1+h 2 ⎬ 1+h h  −λ εz = γm 1+h 2  ⎭ λ−h 2 ⎪ γx z = γm 1+2h 1+h 2

(2.41)

Equation (2.11) is substituted and it is observed that f = 0, so Eq. (2.40) can be obtained as well. In other words, Eqs. (2.41) and (2.40) are equivalents. However, Eq. (2.41) is more convenient to apply because Eq. (2.41) replaces three stress components in Eq. (2.40) with one unknown quantity h  . Hence, Eq. (2.9) is equivalent to the following energy dissipation condition: γm f = σ z ε z + σ x ε x + τ x z γ x z − γ m c = 0

(2.42)

whereas the extremum condition of the yield function is equivalent to the following equation: γm (1 + h 2 )

∂f = (σz − σx )γx z − 2τx z (εz − εx ) = 0 ∂h 

(2.43)

The yield condition is equivalent to the energy dissipation condition because the stress fields in the limit state, stable state, and failure state are supposed as σz , σx , τx z , f , σz# , σx# , τx#z , f # , and σz∗ , σx∗ , τx∗z , f ∗ . Because γm ≥ 0 and f = 0, f ∗ ≥ 0 is equivalent to the following equation: γm ( f ∗ − f ) = (σz∗ − σz )εz + (σx∗ − σx )εx + (τx∗z − τx z )γx z ≥ 0

(2.44)

whereas f # ≤ 0 is equivalent to the following equation: γm ( f # − f ) = (σz# − σz )εz + (σx# − σx )εx + (τx#z − τx z )γx z ≤ 0

(2.45)

After γm is eliminated according to Eqs. (2.30) and (2.33), the following equations are obtained:

2.7 Associated Flow Rule and Velocity Equation ∂vx ∂x ∂vx ∂x



 h − λ +

35

   +  λh = 0   z x h 1 + hλ = 0 + ∂v + ∂v ∂x ∂z

∂vz   h 1 ∂z  2

 1 + 2h  λ − h

(2.46)

Equation (2.46) may be rewritten as: ∂vx ∂x ∂vx ∂x

x + h  ∂v + h ∂z

+

1+h  λ ∂vx λ−h  ∂z



+



∂vz z =0 + h  ∂v ∂x ∂z     ∂v ∂v 1+h λ λ z z + 1+h λ−h  ∂x λ−h  ∂z

⎫ ⎬ = 0⎭

(2.47)

Equation (2.47) is just the velocity equation corresponding to the associated flow rule. Similarly, the velocity field will be obtained simultaneously with the solution of the slip surface family in the limit state. Additionally, in general, the Mohr–Coulomb yield function is directly adopted when the velocity equation is derived, which causes much inconvenience for the calculation of an approximate solution for limit analysis. During the calculation of the approximate solution for limit analysis, the solution usually needs to be carried out on condition that the slip surface is assumed (such as through the limit equilibrium method and the upper bound solution method). However, Eq. (2.40) is related to the stress field, and the stress fields in the failure state and stable state are not unique, so the velocity field still cannot be determined according to Eq. (2.40). It should be clear that Eq. (2.47) corresponding to the associated flow rule is different from that corresponding to the generalized flow rule. The former has two equations while the latter only have one (the other equation is unrelated to the flow rule), and the latter is the combination of the former. Because Eq. (2.33) may be written as:    ∂vx 2  ∂vx  ∂vz  ∂vz +h +h +h (1 + λ ) ∂x ∂z ∂x ∂z     ∂v 1 + h 1 + h  λ ∂vz 1 + h  λ ∂vz λ ∂v x x  2 + + + =0 + (h − λ) ∂x λ − h  ∂z λ − h ∂ x λ − h  ∂z (2.48) It should be noted that both Eqs. (2.47) and (2.38) cannot be simultaneously true in the slip mass. Otherwise, the velocity field and slip surface family would be determined by the three equations. However, is the slip surface family the real slip surface family? This issue will be discussed later on. According to Eq. (2.28), ∂σe ∂x ∂σe ∂z

where

= A1 σe + B1 = A2 σe + B2

 (2.49)

36

2 Failure Mode and the Fundamental Equation

A1 = A2 =



⎫    2λ ∂h  ⎬ + 1+h + 1 + h  λ ∂h 2 ∂ x ∂z       2λ ∂h  ⎭ − h  ∂h + 1+h + 1 + h  λ ∂h 2 ∂z ∂x ∂z ⎫    λ ⎬ B1 = γ λ − 2λ 1+h 2 1+h    B2 = γ 1 + 2λ λ−h2 ⎭

4λh  λ (1+h 2 )2   −4λ λ (1+h 2 )2

− h

 ∂h  ∂x

(2.50)

(2.51)

1+h

According to

∂ 2 σe ∂ x∂z

=

∂ 2 σe , ∂z∂ x

Eq. (2.50) is substituted to get:

 ∂ A1 ∂ B2 ∂ A2 ∂ B1 σe = (2.52) − − + A2 B1 − A1 B2 ∂z ∂x ∂x ∂z       ∂h   ⎫ 2 ⎬ A2 B1 − A1 B2 = −γ 2λ2 2 1 + 2h  λ − h 2 ∂h + 2h − λ + h λ ∂x ∂z (1+h )        ∂ B2 ⎭ − ∂∂zB1 = −γ 2λ2 2 1 + 2h  λ − h 2 ∂h + 2h  − λ + h 2 λ ∂h ∂x ∂x ∂z (1+h ) (2.53) 

Hence, If Eq. (2.38) showing the relationship between the velocity field and the slip surface and the associated flow rule are simultaneously true, Eq. (2.38) may be written as follows:  vx = (1 + h  λ)g (2.54) vz = (h  − λ)g where g is the function to be determined; it is substituted in Eq. (2.47) to get: 

∂g  ∂x ∂g ∂x

   ∂h   (h + h  ∂g − λ)(1 + h λ) + g λ(h  − λ) + ∂z  ∂x   (1 + h  λ)2 + g ∂h + h  ∂g (h  + λ + 2h  λ2 ) + ∂z ∂x



⎫

∂h   h (1 + h  λ) = 0 ⎬ ∂z  ∂h  λh  (1 + h  λ) = 0 ⎭ ∂z

(2.55) Hence, 

∂h  ∂h    (λ − h ) + (1 + h λ) = 0 gh ∂x ∂z 

h  = 0 and g = 0, Therefore, (λ − h  )

∂h  ∂h  + (1 + h  λ) =0 ∂x ∂z

(2.56)

2.7 Associated Flow Rule and Velocity Equation

37

Substituting this into Eq. (2.53) and assuming that γ = 0 and ϕ = 0 are true, we obtain (1 + λh  ) 

∂h  ∂h  + (h  − λ) =0 ∂x ∂z

(2.57)



Thus, ∂h = 0, ∂h = 0, which means that the slip surface can only be a plane. ∂x ∂z In other words, as long as γ = 0 and ϕ = 0 are true, if Eq. (2.38) and the associated flow rule are true in the slip mass, the slip surface can only be a plane. In addition, for general issues, the plane cannot be the real slip surface. It is universally known that the associated flow rule is an assumption in classical plastic mechanics; it perhaps shows the stress–stress relationship of the rigid (plastic) body and may be applied under the rigid (plastic) body assumption. However, the soil mass is neither a rigid (plastic) body nor complies with the rule [7]. Therefore, like the generalized flow rule, the associated flow rule cannot be regarded as a fundamental equation for limit analysis of the soil mass.

2.8 A Fundamental Equation in the Surface Failure Mode 2.8.1 The Stress Equation in the Surface Failure Mode The surface failure mode must meet the yield condition f = 0 on the slip surface and the soil mass on both sides of the slip mass must meet the yield criterion f ≤ 0. Thus, the surface failure mode will be in accordance with the equilibrium equation, the yield condition on the slip surface, and the extremum condition of the yield function. Then Eq. (2.23) is only true on the slip surface, but wrong for the solution of the partial derivative. However, the slip surface is supposed as z = h, because f = 0 is true on the slip surface and the soil mass on both sides of the slip surface meets the requirement of f ≤ 0, the maximum will necessarily  be taken for the yield function = 0 are true along the slip on the slip surface, namely that max{ f } = 0 and ∂∂zf z=h

mass. According to the equilibrium equation, it will necessarily be as follows along the slip surface: (1 + h  λ)

dσx dτx z + (h  − λ) = γ (h  − λ) dx dx

whereas according to the stress equation along the slip surface obtained according to Eq. (2.23): d c 2λ  h − λ c  dh  σ+ − =γ σ+ 2 dx λ 1+h λ dx 1 + λ2

(2.58)

38

2 Failure Mode and the Fundamental Equation

This is the same as the first equation in Eq. (2.28). However, Eq. (2.58) is only true along the slipsurface.  = 0 will be true; however, because f = 0, ddxf = Additionally, ∂∂ xf z=h     ∂f ∂ f dh  ∂f  ∂f + h + = 0 will necessarily be true; thus, among = 0,  ∂x ∂z ∂h d x z=h ∂ x z=h     ∂f ∂f = 0, and ∂h = 0, only two are independent. With any two of them,  ∂z z=h

z=h

the above equations will be obtained.

2.8.2 Fundamental Equation for Slope Stability As mentioned above, when the load on the surface of the soil mass is determined, the state of the soil mass will be determined. The reduction of soil strength only with a safety factor can only make the soil mass on the slip surface be in the limit state; however, the soil mass on both sides of the slip surface is in the stable state. In other words, for the surface failure mode on the slip surface, the soil mass must meet the yield condition; for the soil mass on both sides of the slip surface, it is only required to abide by the yield criterion, which requires other equations except the equilibrium equation to be true only on the slip surface. If the function f 0 (x, z) is introduced and the yield function of the slope stability is rewritten as: f =τ−

1 (σ λ + c + f 0 ) Fs

(2.59)

where f 0 = 0 is true along the slip surface while f 0 ≤ 0 is true on both sides of the slip surface. Thus, the defined f 0 will be consistent with the yield criterion in the surface failure mode and the definition of the yield function will be extended from the slip surface to the whole area. Because c + f 0 is only equivalent to the reduction of the strength index c, after the introduction of the function f 0 (x, z), the soil mass on both sides of the slip surface is also made to be in the limit state by properly reducing the strength of the soil mass on both sides of the slip surface. Thus, it will be possible to study the surface failure mode in the same way as the field failure mode. When it meets the yield condition, it will be as follows: Fs =

1 (σ λ + c + f 0 ) τ

(2.60)

In order to make the soil mass abide by the yield criterion, max{ f }h  ≤ 0 must be true; correspondingly, h  is the curved surface to make the maximum yield function and the slip surface meets the max{ f }h  = 0. According to the necessary condition of the extremum, it should be [8]:

2.8 A Fundamental Equation in the Surface Failure Mode ∂f ∂h  ∂ Fs ∂h 

= =

39

∂ Fs ∂τ ∂σ 1 − Fλs ∂h  + F 2 (σ λ + c + f 0 ) ∂h  = 0 ∂h  s     2 −F ∂τ 1 ∂σ ∂τ λ ∂σ λ ∂h = σ λ+c+s f0 ∂h  − Fs ∂h   − F ∂h  τ s

=0

∂f ∂τ λ ∂σ Apparently, as long as ∂h  − F ∂h  = 0 is true, it will be necessary to make ∂h  = 0 s and ∂∂hFs = 0. Namely, the curved surface that attains an extremum in the yield function is the same as that in the safety factor. However, when the yield function is maximized, the safety factor will be minimized. Hence, the yield condition of the slope stability may be written as:

     1  σz − h  τ x z h  − λ F − σ x h  − τ x z 1 + h  λ F 2 1+h   −(c F + f 1 ) 1 + h 2 = 0

f =

(2.61)

The necessary condition for taking the extremum for the safety factor is as follows:  1 2  2  2 (σz − σx )(1 + 2h λ F − h ) − τx z (2h − λ F + h λ F ) = 0 fh = (1 + h 2 )2 2 (2.62) where λ F = λ/Fs , c F = c/Fs and f 1 = f 0 /Fs . dz = h  (x, z) in the slip mass is the curved Regarding the surface failure mode, dx surface family needed to take the maximum for the yield function; thereinto, the maximum : z = h is the slip surface. According to the yield condition and the extremum condition of the yield function, 



1+λ2F +(h  −λ F ) 1 F + σz ) + f1λ+c F (1+λ2F )(1+h 2 ) 2 (σx 2 1+λ2F )h 2 +(1+λ F h  ) 1 F F σz + f1λ+c = ( 1+λ + σz ) + f1λ+c 2 2 2 (σx 1+h F F ( F )( )   2 1+λ2 − h  −λ F τx z = λ F 1+λF 2 ( 1+hF2) 21 (σx + σz ) + f1λ+c F ( F )( )

σx +

f 1 +c F λF

=

2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(2.63)

After substituting the equilibrium equation and rearranging it, we obtain the stress equation along the slip surface    ∂σ f ∂σ f ∂h 2λ F  ∂h + h − + h σ f ∂x ∂z 1 + h 2 ∂x ∂z      1    ∂ f0 ∂ f 0 1 + hλF + h − λF = γ h − λF + λ ∂x ∂z where σf =

1 f0 + cF (σx + σz ) + 2 λF

(2.64)

40

2 Failure Mode and the Fundamental Equation

Because f 0 = 0 is true along the slip surface, ddxf0 = ∂∂fx0 + h  ∂∂zf0 = 0 will necessarily be true; moreover, f 0 ≤ 0 is true on both  of the slip surface. So  sides ∂ f0 = 0 will definitely be if the maximum is taken for f 0 on the slip surface, ∂z z=h

true. Thus, Eq. (2.64) along the slip surface will be as follows:     2λ F c F dh  cF d h − λF − σ + σ+ = γ dx λF 1 + h 2 λ F dx 1 + λ2F

(2.65)

Actually, the slip surface without a thickness is only a theoretically curved surface; however, in practice, when the soil mass has any shear failure, the actual slip surface has a thickness, namely it is a shear zone [4] (or narrow transition layer [5]). In the shear zone in the limit state, f 0 = 0 will be bound to be true; thus, Eq. (2.65) will be undoubtedly true along the slip surface (in the shear zone). Additionally, even though f 0 (x, z) is not introduced, Eq. (2.65) can still be obtained according to the  derivation process of Eq. (2.57). As a matter of fact,   ∂ f0 ∂f = 0 and ∂z = 0 are equivalent; in other words, for the surface ∂z z=h z=h   failure mode, the yield function has two extremum conditions: ∂∂zf = 0 and z=h   ∂f = 0. ∂h  z=h

Compared with the field failure mode, there is one more extremum condition for the field failure mode, but of a different meaning. The extremum condition of the field failure mode is met in the slip mass, including the slip surface, and the soil mass is required to always meet f = 0, ∂∂ xf = 0, ∂∂zf = 0 will certainly be true while the extremum condition of the plane failure mode is only met on the slip surface. Further, the following velocity equation is obtained: vz (1 + h  λ F ) − vx (h  − λ F ) = 0  ∂vz ∂vx (1 + λ2F )h 2 + (1 + h  λ F )2 + [1 + λ2F + (h  − λ F )2 ] ∂z ∂x   ∂v ∂v x z + =0 + λ F [1 + λ2F − (h  − λ F )2 ] ∂z ∂x

(2.66)

(2.67)

Because the unknown quantity f 0 is added in the yield condition and it cannot constitute a complete limit equilibrium problem, the problem of the surface failure mode cannot be directly solved through the above equations. Practically, for the surface failure mode, generally, the fundamental equation is established on the potential slip surface and the safety factor or the limit load is expressed through the moment equilibrium equation or the virtual work equation as the functional with the independent variable of the potential slip surface. If the extremum condition of the yield function (limit equilibrium method) is not considered or it (Eq. (2.65) of the generalized limit equilibrium method) is not completely considered when a corresponding function is established, it will be necessary to

2.8 A Fundamental Equation in the Surface Failure Mode

41

discuss the safety factor or the extremum of the limit load. Therefore, it is necessary to obtain the corresponding functional. Different functionals may be derived by different solution methods, e.g., by the force and moment equilibrium equation-based limit equilibrium method and virtual work equation-based upper bound solution method for the limit load. However, the functionals obtained for the limit load will be different. In addition, the corresponding function will be discussed during the discussion of solution methods.

2.9 Limit Equilibrium and Variation The limit equilibrium refers to the definite solution of the differential equation made up under certain boundary conditions by the differential equation (equations) in the limit analysis. Limit analysis equations have been discussed above, so corresponding boundary conditions will be discussed here.

2.9.1 Load Boundary Condition and Kinematic Boundary Condition The aforementioned equilibrium equation and yield condition, the extremum condition of the yield function as well as the velocity equation, and the corresponding boundary condition together constitute the limit equilibrium or variation available for the solution. The general boundary condition includes [5] the load boundary condition (load distribution is given at the surface boundary of the soil mass), the kinematic boundary condition (velocity distribution is given at the surface boundary of the soil mass), and the mixed boundary condition (the surface boundary of the soil mass shows the load boundary condition and another part of the boundary shows the kinematic boundary condition). For the load boundary condition: It is supposed that the normal stress and tangential stress of the external load given by the ground surface 0 : z = h s (x) are σs and τs , the following equation shall be true according to Eq. (2.8) as well as the vertical force pz (x) and horizontal force px (x) of the ground surface: √1

⎫ (τs − σs h s ) = px (x) ⎬

√1

(σs + τs h s ) = pz (x) ⎭

1+h 2 s 1+h 2 s

or,

(2.68)

42

2 Failure Mode and the Fundamental Equation

√1

(τx z − σx h s ) = 1+h 2 s √ 1 2 (σz − τx z h s ) = 1+h s

⎫ px (x) ⎬ (2.69)

pz (x) ⎭

On the contrary, if the boundary surface gives the vertical force pz (x) and the horizontal force px (x), the normal stress σs and the tangential stress τs may also be converted according to Eq. (2.68). Particularly, the soil mass at the surface boundary will meet the yield condition and the extremum condition of the yield function the same as the soil mass in the slip mass will; or the yield condition and the extremum condition of the yield function may also be regarded as one of load boundary conditions, which is useful to deal with the unknown limit load boundary and makes it convenient to solve the problems of limit load according to the load boundary condition.

2.9.2 Engineering Boundary Condition (1) Ground limit load (Fig. 2.4): Under a known load on the surface boundary b of the soil mass, when the soil mass in D is in the limit state and that beyond D is in the stable state, the limit loads pz , px that the surface boundary a of the foundation can bear are calculated. The ground limit load may be regarded as the load boundary condition for treatment: If the ratio of the total horizontal force to the total vertical force is given at the surface boundary a of the foundation and the vertical force is regarded as the load to be calculated, it may also be regarded as the boundary condition for treatment: The velocities vx0 , vz0 are given at the boundary a . However, it shall be noted that the velocity boundary is equivalent to the load boundary. In addition, the ground bearing capacity in engineering shows the load that the foundation can bear under the known distribution of the vertical force on the foundation surface, which is actually a conditional limit load. (2) Active soil pressure (Fig. 2.5): Under a known load on the boundary a , when the soil mass in D is in the limit state, the limit loads σb , τb produced on the boundary b are calculated. Fig. 2.4 Schematic diagram for ground limit load

px

pz

Γb

Γa

D

Γ

2.9 Limit Equilibrium and Variation

43

Fig. 2.5 Schematic diagram for active soil pressure

Γa

τb

D

Γ Γb

σb

(3) Passive soil pressure (Fig. 2.6): For a known load on the boundary b , when the soil mass in D is in the limit state, the limit loads σb , τb produced on a can be calculated. As the soil pressure is similar to the limit load, it may be regarded as the load boundary condition or the mixed boundary condition for the solution. In some cases, the kinematic boundary condition must be considered, including the soil pressure of the bearing wall and its rotation around the support point. (4) Slope stability (Fig. 2.7): Under a known load on the surface boundary a , when the soil mass on both sides of the slip surface is in a stable state, the soil mass on the slip surface is in a limit state after its strength is multiplied by 1.0/Fs . For practical problems, the minimum safety factor and the corresponding slip surface among all the potential slip surfaces will be calculated. During the solution of the slope stability problem with the safety factor of Fs , it is generally regarded as the load boundary condition for treatment. For the limit load, the unknown load on the boundary surface is regarded as one of the load boundary conditions for treatment and is a feature of the limit analysis of the soil mass. Practically, for the limit analysis on the soil mass, how to determine Γb

Fig. 2.6 Schematic diagram for passive soil pressure Γa

D

σa

Γ

τb

Fig. 2.7 Schematic diagram for slope stability

Γa

Γ

D

44

2 Failure Mode and the Fundamental Equation

the boundary condition of the foundation bottom (limit load) and the retaining wall surface (limit soil pressure) is also an issue to be discussed and it is tolerable in consideration of the kinematic boundary condition. However, it must still be considered according to the load boundary condition of the unknown load to be calculated, for example, the limit load; in addition, the kinematic boundary   condition must be coordinated with the load boundary conditions σz a = pz , τx z a = 0. In addition, some methods of solution can be followed under the assumed slip surface (family), when the selected the slip surface (family) is coordinated with the corresponding load boundary condition or kinematic boundary condition. Generally, for the load boundary condition, the boundary condition of the corresponding slip surface (family) may be determined according to the limit stress field in Eq. (2.23). For example: If the surface boundary is a horizontal plane, when the load boundary conditions σz = σz0 , τx z = τx z0 are given, according to Eq. (2.23), the slip surface on the boundary surface will meet the following condition. τx z0 [(1 + λ2 )h 2 + (1 + h  λ)2 ] = (σz0 λ + c)[1 + λ2 − (h  − λ)2 ] √   Especially, if τx z0 = 0 is true, h  = λ ± 1 + λ2 = tan ϕ ± π4 will be true. Correspondingly, the kinematical boundary will meet vx0 (h  −λ)−vz0 (1+h  λ) = 0. If the surface of the soil mass is a vertical surface, when the load boundary conditions σx = σx0 , τx z = τx z0 are given, τx z0 [1 + λ2 + (h  − λ)2 ] = (σx0 λ + c)[1 + λ2 − (h  − λ)2 ] √ In particular, if τx z0 = 0 is true, h  = λ ± 1 + λ2 will be true. = 0 will Moreover, when it is symmetrical in terms of the vertical axis x = xc , τx z √ necessarily be true on this symmetry axis and correspondingly h  = λ ± 1 + λ2 will be true. Similar methods will provide some assistance in selecting a suitable slip surface. Additionally, the smoothness of the foundation bottom that is commonly referred to has no influence on the ground bearing capacity. Because the ground bearing capacity is involved with the stability along the slip surface (family), the boundary condition of the foundation bottom refers to the condition when any point on the foundation bottom is buried along the slip surface, instead of the condition along the foundation bottom. The stability along the foundation bottom is involved with horizontal slip and will be discussed separately.

2.9 Limit Equilibrium and Variation

45

2.9.3 Limit Equilibrium of the Load Boundary Condition From the above, Eqs. (2.23) or (2.24) can be used to determine both the stress field in the limit state and the load boundary condition for the limit analysis. In other words, Eq. (2.23) and the equilibrium equations (2) together constitute a complete limit equilibrium problem (boundary value), which may be used to provide a solution for the stress field and slip surface family under the limit state and the limit load required for the engineering. In the above limit equilibrium, the geometric equation and the flow rule (or velocity equation) are not introduced. Evidently, for the limit equilibrium of the load boundary condition, if corresponding solution methods can be found, no geometric equation or flow rule will be required. In fact, in the limit analysis theory based on the simultaneous solution to the equilibrium equation and the yield criterion, because the stress field and slip surface family under the conditions of the limit state will meet the equilibrium equation, the yield condition and the extremum condition of the yield function, the stress field, the slip surface family, and the corresponding limit load will be unique. Therefore, as long as the same yield criterion and yield function are adopted, no matter what relationship or rule (certainly it shall be correct) is introduced, the results obtained (stress field, slip surface family, and corresponding limit load) will be the same as long as the boundary condition is coordinated (load boundary condition is equivalent to the kinematic boundary condition). Otherwise, it only shows that the limit analysis solution is not unique or that the limit analysis theories including or excluding the geometric equation and flow rule are different. Therefore, consideration of the geometric equation and the flow rule is not critical for the limit equilibrium of the load boundary condition, but it is critical for how to solve it. Because the yield condition and the extremum condition of the yield function have a nonlinear equation, actually one of the main tasks for the limit analysis theory is to solve the limit equilibrium problem. Certainly, during the solution of the limit equilibrium problem, virtual work principle-based or variational principle-based methods may also be adopted. At this time, the velocity equation will be established; moreover, the introduction of the geometric equation and the flow rule is only aimed at obtaining the velocity equation.

2.9.4 The Limit Equilibrium of the Kinematic Boundary Condition and the Mixed Boundary Condition For problems including the velocity boundary, a geometric equation and the flow rule will be introduced for the treatment of the kinematic boundary condition. Thus, the equilibrium equation, the yield condition, the extremum condition of the yield function, the velocity equation, and the corresponding boundary condition constitute a complete limit equilibrium problem.

46

2 Failure Mode and the Fundamental Equation

Even for the problems including a kinematic boundary condition, here the geometric equation and the flow rule are not regarded as a stress–stress relationship for consideration. In other words, only the velocity equation showing the relationship between the velocity field and the slip surface is considered; thus, it is not necessary to consider the continuity condition (deformation). Similarly, because the yield condition, the extremum condition of the yield function, and the velocity equation are nonlinear equations, it is very difficult to obtain the theoretical solution; generally, only the numerical solution (characteristic line method) can be obtained. Moreover, generally, the theoretical solution can only be obtained under a certain assumption or simplification, such as the form and motion mode of the assumed slip surface. In other words, for the exact solution under the limit state condition, one of the primary functions to introduce the geometric equation and flow rule is to deal with the kinematic boundary condition and the mixed boundary condition, but this will not reduce the difficulty of the solution. If the virtual work principle-based method is adopted, vx , vz will be the velocity fields and μ, η (or γm ) will not have any physical meaning; only the Lagrange multiplier introduced when the limit equilibrium solution is obtained according to the virtual work principle is used. The above limit equilibrium requires the soil mass to be in the limit state in a certain area, but in the stable state in other areas; therefore, it is involved with the field failure mode. For the problems including a kinematic boundary condition, here we do not intend to discuss whether the stress–stress relationship needs to be considered because the ground bearing capacity, the soil pressure, and the slope stability discussed here may be considered according to the load boundary condition. As mentioned above, though the load boundary condition has no velocity equation, it has been completed. In this sense, it is not necessary to introduce the geometric equation and the flow rule; moreover, by the virtual work principle-based methods, when the Eqs. (2.33) and (2.38) are only used to eliminate the unknown stress field in the virtual work equation, equations that vx , vz will meet may be obtained according to the solution process of the virtual work equation (Chap. 6).

2.9.5 Variation in the Surface Failure Mode If the soil mass meets the equilibrium equation, the yield criterion, the yield condition along the slip surface, and the corresponding boundary condition as well as allowing the extremum (minimum) to be used for the limit load or the safety factor, this will be the conditional extremum in the surface failure mode variation. Theoretically, it should be possible for the stress field to be expressed as the function of the slip surface according to the equilibrium equation and the yield condition. In this way, the functional of the limit load or safety factor with the independent variable of the slip surface can be obtained; because the extremum condition of the yield function is not applied, the minimization process needs to be implemented for that functional,

2.9 Limit Equilibrium and Variation

47

which is in terms of the variation. However, in the common limit equilibrium method, the equilibrium equation needs to be established on the slip surface for the surface failure mode and the functional thus obtained often inevitably contains unknown stress, which can be eliminated only under a certain assumption. If the unknown stress in the functional can be eliminated with stress Eqs. (2.58) or (2.65) along the slip surface and it is observed that the stress equations are obtained according to the yield condition and the extremum condition of the yield function, they should also meet either of the conditions; thus, it is still necessary for the minimization process to be implemented for that functional, which is in terms of the variation as well.

2.10 The Simplest Velocity Field The velocity component has the following relationship (Fig. 2.8): 1 vt = √1+h (vx + h  vz ) 2 1 (vz − h  vx ) vn = √1+h 2

 (2.70)

(1) Rotational motion around one point and the logarithmic helical slip surface: When the soil mass has a rotational motion around the point (x R , z R ), it will only be true for any point (x, z) (Fig. 2.9) on the slip surface: vz x − xR = tan α = − , namely, vx (x − x R ) + vz (z − z R ) = 0 vx z − zR

(2.71)

A simultaneous solution is made with the velocity equation of the associated flow rule and Eq. (2.71) is substituted in the velocity equation; it will be as follows: ∂vx ∂x ∂vx ∂x

⎫      ∂vx x−x R  1  ⎬ h  − λ + vx z−z 1 + h − h λ = 0 ∂z z−z R R  1 x ⎭ =0 + h  ∂v − vx z−z ∂z R



Fig. 2.8 Schematic diagram for the velocity component

Slip surface

(2.72)

48

2 Failure Mode and the Fundamental Equation

Fig. 2.9 Schematic diagram for rotational motion

Hence,  ∂vx  x − xR h −λ+ (1 + h  λ) = 0 ∂x z − zR When

∂vx ∂x

(2.73)

= 0 is true, it will be easy to obtain the following equation: vx = Av (z − z R ) vz = −Av (x − x R )

 (2.74)

where Av is a constant, and the velocity component in Eq. (2.74) is true for any slip surface.    R (1 + h λ) = 0 is true, the slip surface is a logarithmic When h  − λ + x−x z−z R helical curved surface with the following polar form: x − x R = R exp(−λθ ) cos θ z − z R = R exp(−λθ ) sin θ

 (2.75)

It is well-known that a general logarithmic helical curved surface is as follows: x − x R = R exp(−kθ ) cos θ z − z R = R exp(−kθ ) sin θ

 (2.76)

where k is a parameter. When k is constant, Eq. (2.76) describes a surface that is referred to as a general helicoid hereafter. In Eq. (2.75), k = λ = tan ϕ; when the soil angle ϕ is determined, it is a specific value and relatively common at present. Thus, Eq. (2.75) is a common helicoid or hereinafter referred to as the helicoid. Thus, 

1 + h 2 (vn + vt λ) = −vx (h  − λ) + vz (1 + h  λ) = 0

(2.77)

2.10 The Simplest Velocity Field

49

This is the same as Eq. (2.38). If Eq. (2.47) is satisfied, Eq. (2.33) must be satisfied. In other words, the velocity component may be obtained as Eq. (2.75) according to the velocity equation corresponding to the generalized flow rule as well, which shows that Eq. (2.47) corresponding to the associated flow rule is equivalent to the velocity equation corresponding to the generalized flow rule for the rotational motion. If the soil mass on the slip surface is identified to have a rotational motion around one point (x R , z R ), the slip surface will certainly be a common helicoid with the center of (x R , z R ) and vice versa. Additionally, the soil mass on the slip surface has a rotational motion around one point (x R , z R ), which refers to a motion mode of any point on the slip surface, instead of the motion along the slip surface. When any point (xi , z i ) on the common helicoid has a rotational motion around the point (x R , z R ), its motion curve and direction are an arc and the tangential direction of the arc at an angle of ϕ with respect to the tangential direction of the common helicoid. (2) Translational motion and planar slip surface: When the soil mass on the slip surface has a translational motion, vx and vz will be constants. Obviously, Eq. (2.47) holds, and the slip surface obtained according  to Eq. (2.33) will be a plane. surface √ Conversely, if the slip √ is a plane and h is a   2 2 constant, vx = vt (1 + h λ)/ 1 + h and vz = vt (h − λ)/ 1 + h will be obtained according to Eqs. (2.47) and (2.33). Namely, it certainly has a translational motion. For Eq. (2.47) corresponding to the associated flow rule, it still cannot be determined that the slip surface must be a plane. The slip surface can be determined only if vn + vt λ = 0 is true along the slip surface.  For the translational motion, the maximum velocity is vmax = vx2 + vz2 = vt / cos ϕ. Namely, the direction of the maximum velocity is at an angle of ϕ with respect to the tangential direction of the slip surface. Of course, the velocity field here is not necessarily the real velocity field because it is still not yet known whether the plane or common helicoid (including its combination) is a real slip surface. However, they may be regarded as the approximate value of the velocity field.

2.11 Solution Using a Limit Analysis 2.11.1 Examples Example 1 The limit load of the simplest purely cohesive material is solved (Fig. 2.10). For known loads (σz = 0, τx z = 0) on the boundary b , the vertical load σz that the foundation bottom may bear when the vertical load τx z = 0 on the boundary a is true.

50

2 Failure Mode and the Fundamental Equation

x

Γa (A)

Γb

z

Γb

Γa

Γb

(C )

Γ

(B)

Γ

Γ (a) One-way Failure

(b) Two-way Failure Γb

Γa

Γb

(A)

(B)

Γ

Γ

(c) Pressure Failure Fig. 2.10 Schematic diagrams for the calculation of limit load

Because λ = tan ϕ = 0, Eq. (2.24) may be changed to this: ⎫ 2ch  ⎪ σx = 21 (σx + σz ) − 1+h 2 ⎬ 2ch  σz = 21 (σx + σz ) + 1+h 2 ⎪ 2 ⎭ τx z = 1−h c 2 1+h

(2.78)

σe = 21 (σx + σz ) is presumed. Correspondingly, Eq. (2.28) is changed to: ∂σe ∂x

e −h  ∂σ + ∂x





 2c ∂h  = γ h + h  ∂h 1+h 2 ∂x ∂z  ∂σe 2c ∂h   ∂h  −h =γ + + 2 ∂z 1+h ∂x ∂z

e + h  ∂σ − ∂z

⎫ ⎬ ⎭

(2.79)

It is easy to obtain the following solution of Eq. (2.79): σe = γ z + 2c arctan h  + C0

(2.80)

h  = −x/z, or h  = k0

(2.81)

where C0 , k0 are constants to be determined. Equation (2.80) is substituted in Eq. (2.78) to get:   ⎫ h + C0 ⎪ σx = γ z + 2c arctan h  − 1+h ⎪ 2 ⎬   h  σz = γ z + 2c arctan h + 1+h 2 + C0 ⎪ ⎪ 2 ⎭ τx z = 1−h c 1+h 2

(2.82)

From the load boundary condition on b : on b , h  = −1 and C0 = (π + 2)c/2.

2.11 Solution Using a Limit Analysis

51

From the load boundary condition on a : on a , h  = 1; and the vertical limit load on a is obtained as follows:  σz a = (π + 2)c

(2.83)

This is the exact solution (Plandel’s solution) of the limit load. According to smooth (continuity of the first derivative) requirements for the slip surface (family), the slip surface family satisfies the following conditions: ⎧ (A) ⎨1 h  = −x/z (B) ⎩ −1 (C)

(2.84)

According to Eq. (2.84), a and b in Fig. 2.10 are possible in the soil mass area in the limit state, but the above two conditions are corresponding to different failure modes. In case of a one-way failure, Fig. 2.10a will be the obtained limit state area; in case of a two-way failure, Fig. 2.10b will be the obtained limit state area. Because the partial area in (A) has two slip surfaces according to the slip surface analysis and the interface of (A) and (B) has two motion directions according to the motion analysis, Fig. 2.10c is not a limit state area. Example 2 There is an attempt to solve the ground ultimate load of the rationally weightless soil (γ = 0) as shown in Fig. 2.10. With known loads (σz = q, τx z = 0) on the surface boundary b of the soil mass and the horizontal load τx z = 0 on the vertical load σz that cannot foundation bottom (namely boundary a ), the √  be borne is calculated. According to Eq. (2.24), h  = λ + 1 + λ2 = tan π4 + ϕ2 is true for the √   foundation bottom; for the soil mass surface, h  = λ − 1 + λ2 = − tan π4 − ϕ2 √  1+λ2  are true. and σe = q + λc √1+λ 2 −λ Equation (2.28) is simplified as follows: ∂σe ∂x ∂σe ∂x

+





  2λ =0 σ ∂h + h  ∂h 1+h 2 e ∂ x ∂z  1+λh  ∂σe 2λ ∂h  1+λh  ∂h  + 1+h 2 σe ∂ x + λ−h  ∂z λ−h  ∂z

e + h  ∂σ − ∂z

⎫ ⎬ = 0⎭

(2.85)

According to the first equation in Eq. (2.85), it will be as follows: σe = C0 exp(2λ arctan h  )

(2.86)

It is substituted in the second equation in Eq. (2.85) and it will be as follows: ∂h  1 + λh  ∂h  + =0 ∂x λ − h  ∂z

(2.87)

52

2 Failure Mode and the Fundamental Equation

Fig. 2.11 Schematic diagram for the slip surface

−B

xb

x

(A)

θ1

z

θ2

(C ) (B)

If  h  = g(s) and s = x/z are true, Eq. (2.87) will be changed as  λ−s λ−s = 0 and thus h  is a constant or h  = 1+sλ is true. According to the g (s) h  − 1+sλ boundary condition and the smooth (continuity of the first derivative) requirement for the slip surface (family), the slip surface family satisfies the following condition: 

√ ⎧ (A) ⎨ λ + 1 + λ2 h  = (zλ − x)/(z + λx) (B) √ ⎩ (C) λ − 1 + λ2

(2.88)

Namely, the slip surface family is plane–common helicoid–plane (Fig. 2.11). From the surface boundary condition of the soil mass, √  π  ϕ  1 + λ2 c exp 2λ − C0 = q + √ λ 4 2 1 + λ2 − λ

(2.89)

Thus, on the foundation bottom, √  1 + λ2 c σe = q + exp(λπ ) √ λ 1 + λ2 − λ

(2.90)

According to Eq. (2.24), the following limit load is obtained:  ϕ c c  2 π σz = q + tan + exp(λπ ) − λ 4 2 λ

(2.91)

This is the solution known as Plandel’s solution [9] of the limit load.

2.11.2 Solution Method In general, the limit load or safety factor and the corresponding slip surface (family) in engineering may be obtained by carrying out a limit analysis (limit equilibrium or variation). However, because the above three boundary conditions contain nonlinear differential equations, except for a few simple ground conditions, it is extremely

2.11 Solution Using a Limit Analysis

53

difficult to obtain an exact theoretical solution. Therefore, a numerical or approximate solution is obtained. At present, there are generally two types of limit analysis methods. One type includes numerical methods such as the characteristic line method; such a method aims to find a direct solution for the limit equilibrium. Because the derived stress field, slip line field, and velocity field satisfy the corresponding limit equilibrium, such a method is intended to obtain the solution under field failure mode conditions when the soil mass is in the limit state. Because the field failure mode requires the soil mass to be in the limit state for a certain slip surface family and the slip surface is throughout a specific area, it may not be applicable when the surface boundary condition is the determined load boundary condition. For example, it is not possible for a cohesive soil slope without the surface load effect to be present in the slip mass in the limit state. Therefore, the field failure mode cannot be applied to determine the safety factor of the slope stability. The other type includes methods such as the limit equilibrium method and upper bound solution method (limit analysis method). Such methods aim to find a solution with an assumed slip surface or motion mode of the soil mass (functional form of the velocity field) by the moment equation or virtual work equation. When the slip surface is known (if the motion mode is known, the slip surface will be known), it will become a linear problem. Because the assumed slip surface generally cannot satisfy the yield criterion when the maximum yield function is zero, an approximate theoretical solution under field failure mode conditions or an approximate solution of the surface failure mode may be calculated with different specific solution methods. Because the obtained theoretical solution is convenient for application, a higher calculation accuracy in accordance with requirements has been achieved for some engineering problems according to many studies in the field of soil mechanics; thus, it is extensively valued. When a slip surface is artificially assumed for the solution, the extremum condition of the yield function in the above limit analysis is replaced with the assumed slip surface, which is also part of the corresponding definite solution. However, the approximate solution of the limit analysis obtained using the definite solution should be further discussed to ascertain the reliability of the calculated results (e.g., whether the calculated values of the safety factor or limit load are greater or smaller) and determine which to use. The theoretical solution generally includes a limit solution, stable solution, and failure solution.

2.11.3 Solutions in the Field Failure Mode (1) Limit solution (the exact solution): The limit analysis under the above boundary conditions is involved with the definite solution that the soil mass in the limit state must meet. Therefore, the solution in accordance with the limit analysis is referred to as the solution under the limit

54

2 Failure Mode and the Fundamental Equation

state condition, which is hereinafter referred to as the limit solution. Moreover, its corresponding stress field is referred to as the limit stress field. If the limit solution in the field failure mode exists, it must be unique. Because it is a complete limit equilibrium problem, the theoretical meaning of the limit solution (as an exact solution) will be very important. Just because of its uniqueness, it is restricted to being directly applied in practical engineering problems. For example, for the ground bearing capacity, because the distribution of the design load along the foundation bottom presents restriction (eccentricity) and the distribution form of the limit load in accordance with the limit equilibrium is unique, the limit load in the field failure mode cannot be directly regarded as the ground bearing capacity. (2) Failure solution (the upper bound solution and the movable solution): The solution that meets the equilibrium equation, the motion equation, and the boundary condition and puts the soil mass in a failure state in a certain specific area is hereinafter referred to as the failure solution, and the corresponding stress field is referred to as the failure stress field (movable stress field). When it satisfies the extremum condition of the yield function, but not the yield condition ( f > 0) or vice versa, it will be a failure solution; however, it only satisfies the yield condition along a certain curved surface (family), but does not satisfy f ≤ 0 for other curved surfaces. Generally, the failure solution is required to satisfy the yield condition along the slip surface; otherwise, the solution obtained may deviate greatly from the exact solution. In terms of its basic features, the failure stress field of the failure solution satisfies f ≥ 0 for the given slip surface; the failure solution of the limit load (including the passive soil pressure) that puts the soil mass in the failure state is necessarily greater than the limit solution of the limit load that puts the soil mass in the limit state. Therefore, the less the calculated value of the limit load obtained according to the failure solution is, the higher the accuracy of the solution is. In other words, for the same slip surface family, the stress field that gets the minimum calculated value of the limit load (the maximum active soil pressure) will be considered as the calculated result; for different slip surface families, the slip surface that gets the minimum calculated value of the limit load (the maximum active soil pressure) will be considered as the calculated result. (3) Stable solution (the lower bound solution and the static solution): The solution that satisfies the equilibrium equation, the motion equation, and the boundary condition and puts the soil mass in the stable state in the whole is hereinafter referred to as the stable solution, and the corresponding stress field is referred to as the stable stress field (static stress field). The stable solution is not required to satisfy the yield condition (only f ≤ 0 is required) because it will be a limit solution if it satisfies the yield condition. In terms of its basic features, the stable stress field of the stable solution satisfies f ≤ 0 for the given slip surface. For the limit load (including the passive soil pressure), it is easy to understand that the stable solution of the limit load that puts the soil mass in the stable state is necessarily less than the limit solution of the limit

2.11 Solution Using a Limit Analysis

55

load that puts the soil mass in the limit state. Therefore, the less the calculated value of the limit load obtained according to the stable solution is, the higher the accuracy of the solution is. In other words, for the same slip surface family, the stress field that obtains the maximum calculated value of the limit load will be considered the calculated result; for different slip surface families, the slip surface that obtains the minimum calculated value of the limit load will be considered the calculated result. In terms of the active soil pressure, for the same slip surface family, the stress field that obtains the minimum calculated value of the active soil pressure will be considered the calculated result; for different slip surface families, the slip surface that obtains the maximum calculated active soil pressure will be considered the calculated result. The stable solution and the failure solution are both approximate solutions and they are naturally not unique. Because the stress field does not satisfy the yield condition or the extremum condition of the yield function, the quantity of the equations is less than that of the unknown functions, and this belongs to the statically indeterminate problem. The solution needs to be carried out under the assumed slip surface form or motion mode condition. For each assumed slip surface form or motion mode, one kind of solution may be obtained. Due to the difficulty in the solution of the limit solution, the stable solution and the failure solution have to be changed to calculate the approximate solution. The conclusion that the stable solution of the limit load is not greater than the limit solution and the failure solution is not less than the limit solution is the extremum theorem of the limit analysis, which will be strictly proved later.

2.11.4 Solutions in the Surface Failure Mode The basic feature of the surface failure mode lies in that the stress field must satisfy f ≤ 0 and that f = 0 has to be true along the slip surface. For the limit load, the stress field in accordance with the above condition is not unique and the corresponding limit load is not unique; as shown in Fig. (2.12), the limit loads are distributed as p1 (x) and p2 (x), and they satisfy the equilibrium equation and the above conditions because it is doubtful that the stress field will satisfy f ≤ 0. In fact, the solution in accordance with the above condition is the stable solution, so the solution in the surface failure Fig. 2.12 Schematic diagram for the non-uniqueness of the limit Load

p2 ( x ) p1 (x )

56

2 Failure Mode and the Fundamental Equation

mode is the same as the stable solution in the field failure mode: For the same slip surface, the stress field that obtains the maximum calculated value of the limit load or the safety factor will be considered the calculated result; for different slip surfaces, the slip surface that obtains the minimum calculated value of the limit load or the safety factor will be considered the calculated result [10]. According to the definition of the yield function in Eq. (2.59), if f 0 (≤ 0) is the maximum to satisfy the yield condition ( f = 0) and f 0 = 0 is true along the slip surface, the corresponding limit load or safety factor will be the solution in the surface failure mode. For the limit load, the limit load in the surface failure mode defined according to the above actually is no different from the limit load in the field failure mode; due to different methods of solution of two failure modes, the total limit loads obtained will have some differences. Because the surface failure mode generally comes down to the variation and the real slip surface is very difficult to obtain, it is difficult to obtain the solution in the surface failure mode in a strict sense. At present, when the limit load is solved, usually only the conformity with the yield condition along the slip surface is considered, but no attention is paid to the condition f ≤ 0 that shall be met in the slip mass, so that the limit load obtained (ultimate bearing capacity) is usually on the large side. The discussion about the calculation of the limit load under the uniform load of the foundation surface is one example. In fact, as long as γ = 0 and ϕ = 0 are true, it will be impossible for the limit load to be a uniform load. The uniform load obtained after the total limit load obtained according to the slip surface through the foundation rear toe is divided by the foundation width is regarded as the distribution of the limit load, and the ground soil near the foundation front toe is necessarily in the failure state (Fig. 2.13). In most cases, only an approximate solution in the surface failure mode can be found if the state of the soil mass on both sides of the slip surface is not considered when the solution in the surface failure mode is calculated. In the actual solution process, the minimum solution in the surface failure mode is often calculated for different assumed slip surfaces. Most of the existing calculation equations including those that have been extensively applied in engineering are approximate solutions. In the book, when the limit equilibrium method and generalized limit equilibrium method for the surface failure mode are applied to discuss the limit load in the surface failure mode, the solution process is the same as that of the derived calculation Fig. 2.13 Schematic diagram for partial failure

Real limit load

−B

−b

0 Failure area

2.11 Solution Using a Limit Analysis

57

equation. Both of them are discussed on the basis of the failure mode in consideration of the integral foundation action. Therefore, the derived equation is actually an approximate solution in the field failure mode. It is emphasized that the obtained solution in any mode or state is not merely the limit load or safety factor. This solution also includes the slip surface or slip surface family, which is very important to determine whether the calculated value of the limit load or safety factor is correct because the slip surface or slip surface family is one of the major bases for analyzing whether the soil mass is in the limit state; this is an important result required for engineering. It is no exaggeration to say that the slip surface or slip surface family is one of the main difficulties in limit analyses of the soil mass. If the slip surface is able to be determined, the fundamental equations in the limit analysis will be linear equations that are easy to solve.

References 1. Huan CZ (2002) Stress field and ultimate load for soil mass in limit state. Chin J Geotech Eng 24(3):389–391 2. Timoshenko SP, Goodier JN (1964) Theory of elasticity (trans: Xu Z, Wu Y). Higher Education Press, Beijing 3. Huang CZ (2003) Extremum condition of the yield function—one of fundamental equations for the limit analysis on the soil mass. In: 9th conference proceedings on soil mechanics and geotechnical engineering. Tsinghua University Press, Beijing, pp 343–346 4. Huang CZ (2006) Verification of the theorem of the limit analysis of the soil mass and the generalized flow rule. Chin J Geotech Eng 28(6):700–704 5. Shen ZJ (2000) Theoretical soil mechanics. China Water Power Press, Beijing 6. Chen HF (1995) Limit analysis and soil plasticity (trans: Zhan S, Proofread by Han D). China Communications Press, Beijing 7. Zheng YR (2003) New developments in geotechnical plastic mechanics—generalized plastic mechanics. Chin J Geotech Eng 25(1):1–10 8. Qian WC (1980) Variational method and finite element, vol 1. Science Press, Beijing 9. Tianjin University (1980) Soil mechanics and foundation. China Communications Press, Beijing 10. Pan JZ (1980) Analysis on stability and coast of the structure. Hydraulic Press, Beijing

Chapter 3

Characteristic Line Method for Limit Analysis

3.1 Characteristic Line Equation and Stress Equation The characteristic line method is a kind of numerical method for the solution of limit equilibrium; Kotter (1903) established the limit equilibrium equations through the simultaneousness of equilibrium equation and Mohr–Coulomb yield condition. Sokolovskii (1956) first proposed the numerical method for the solution of the limit equilibrium equations—characteristic line method and applied it in the study on the limit load and limit soil pressure. In China, scholars also have done enormous beneficial work [1, 2] with regard to the calculation of the ground limit load by the characteristic line method. In this chapter, calculation of the ground limit load is taken as an example to discuss the characteristics of the calculation process and calculated results by characteristic line method. For the application, the stress equation given in Chap. 2 is the same as the characteristic line method equation derived according to conventional method [3]. Here, the characteristic line method equation derived according to conventional method is first introduced. The conventional characteristic line method is used for the direct solution of the equilibrium equation and Mohr–Coulomb yield condition. Two new functions σe , θ are introduced and converted as below: ⎫ σz = σe (1 − sin ϕ cos 2θ ) − c/ tan ϕ ⎬ σx = σe (1 + sin ϕ cos 2θ ) − c/ tan ϕ ⎭ τx z = σe sin ϕ sin 2θ

(3.1)

where σe =

1 (σz + σx ) + c/ tan ϕ 2

(3.2)

It is easy to prove that Eq. (3.1) is in accordance with the Mohr–Coulomb yield condition. © Springer Nature Singapore Pte Ltd. and Zhejiang University Press, Hangzhou, China 2020 C. Huang, Limit Analysis Theory of the Soil Mass and Its Application, https://doi.org/10.1007/978-981-15-1572-9_3

59

60

3 Characteristic Line Method for Limit Analysis

After the conversion of Eq. (3.1), the equilibrium equation is as below:    ∂σe ∂θ ∂θ e = 0 + sin ϕ sin 2θ − 2σ sin ϕ sin 2θ − cos 2θ (1 + sin ϕ cos 2θ ) ∂σ e ∂x ∂z ∂x ∂z   e e sin ϕ sin 2θ ∂σ =γ + (1 − sin ϕ cos 2θ ) ∂σ + 2σe sin ϕ cos 2θ ∂∂θx + sin 2θ ∂θ ∂x ∂z ∂z (3.3) Further, the following equation may be obtained: ∂σe ∂x ∂σe ∂x

  ∂θ ∂θ e = γ − + tan(θ − μ) ∂σ − 2σ λ + tan(θ − μ) μ) − λ] [(tan(θ e ∂z ∂z  ∂x (3.4) e = γ [(tan(θ + μ) + λ] + tan(θ + μ) ∂σ + 2σe λ ∂∂θx + tan(θ + μ) ∂θ ∂z ∂z

where μ = π4 − ϕ2 . According to characteristic line theory of the partial differential equation, Eq. (3.4) has two groups of intersecting characteristic lines and may be solved by the characteristic line method. Thus, it will be as below: ⎫ dz ⎪ ⎪ = tan(θ − μ) ⎬ dx

 (3.5) dz dσe dθ ⎪ − 2λσe =γ −λ ⎪ ⎭ dx dx dx ⎫ dz ⎪ ⎪ = tan(θ + μ) ⎬ dx

 (3.6) dz dσe dθ ⎪ + 2λσe =γ +λ ⎪ ⎭ dx dx dx Equations (3.5) and (3.6) are true along the characteristic line (family) determined according to the first equation in the equations, which are referred to as the characteristic line equation and stress equation along the characteristic line, respectively. It will be seen below that it is also very easy to derive the characteristic line equation and stress equation along the characteristic line according to the equations given in Chap. 2. In fact, it is supposed as below: dz = h  = tan(θ − μ) dx

(3.7)

In Chap. 2, it has derived that Eqs. (3.1) and (2.17) are the same. Moreover, the curve (family) determined according to Eq. (3.7) is the commonly referred slip line field and consistent with the slip surface family. According to Eq. (3.7), the following is obtained:

3.1 Characteristic Line Equation and Stress Equation

61

1 + λh  = tan(θ + μ) λ − h

(3.8)

It is observed that: 

 1 + λh  ∂ 1 ∂h  ∂   = arctan h = arctan = 1 + h 2 ∂ x ∂x ∂x λ − h 

 ∂  1 + λh  1 ∂h  ∂  = = arctan h = arctan 1 + h 2 ∂z ∂z ∂z λ − h

∂θ ∂x ∂θ ∂z

Then, it is immediately obtained that Eq. (2.20) is completely the same as Eq. (3.4). In other words, the stress field and slip line field in accordance with Eq. (3.1), characteristic line equation and stress equation along the characteristic line are the same as the stress field and slip surface family in accordance with the equilibrium equation, yield condition, and extremum condition of the yield function. For the limit equilibrium of the load boundary condition, slip line (family) and stress component at the intersection of the slip line and characteristic line are solved through simultaneousness of Eqs. (3.5), (3.6), and (3.1). Thus, the limit load (ultimate bearing capacity and limit soil pressure) may be obtained because the limit load at this time is the stress component on the boundary. Additionally, the two groups of curves (families) determined according to the dz dz = tan(θ − μ) and dx = tan(θ + μ) are collectively referred to as the equations dx slip lines in some references. In fact, the slip line is of explicitly physical meaning; dz = tan(θ − μ) is the slip the curve family determined according to the equation dx line because it makes the maximum taken for the yield function and be in accordance with max{ f } = 0. In other words, the slip line is the slip surface. If the curve family dz = tan(θ + μ) is not in accordance with determined according to the equation dx the yield condition, it at least cannot prove that all the curves in this curve family are in accordance with f = 0. In this sense, it is not accurate for the two groups of curves (families) to be collectively referred to as two groups of slip lines (families), dz = so it is suggested that the curve family determined according to the equation dx tan(θ + μ) shall still be referred to as the characteristic line. Moreover, the two equations in Eq. (3.5) also can be referred to as the slip line equation and stress equation along the slip line, respectively. Both Eqs. (3.5) and (3.6) are nonlinear (mathematically referred to as quasi-linear equation) and only true on the slip line family and characteristic line family; thus, it is difficult to obtain the theoretical solution for the general problems in engineering, and only the numerical method can be adopted to get the numerical solution. Fortunately, for such quasi-linear problem, the relatively mature numerical method—the characteristic line method—has been mathematically available for direct application.

62

3 Characteristic Line Method for Limit Analysis

3.2 Calculation Process of Limit Load Calculation of the limit load by the characteristic line method has been studied by many people and introduced in many references. However, its calculation process may be improved further.

3.2.1 Differential Equation For the difference scheme shown in Fig. 3.1, when σe , θ, z, x of all the lattice points on the ith slip line are known, σe , θ, z, x of all the lattice points on the i +1th slip line is calculated, and the differential equation may be derived according to Eqs. (3.5) and (3.6):  ⎫    z i+1, j+1 − z i+1, j = tan 21 θi+1, j+1 + θi+1, j −μ xi+1, j+1 − xi+1, j ⎪ ⎪  ⎬ 1 σ θ − σ − 2λ + σ − θ σe i+1, j+1 ei+1, j ei+1, j+1 ei+1, j i+1, j+1 i+1, j 2    (3.9) ⎪ = γ z i+1, j+1 − z i+1, j − λ xi+1, j+1 − xi+1, j ⎪ ⎭ j =1∼n+1   ⎫    z i+1, j+1 − z i, j = tan 21 θi+1, j+1 + θi j + μ xi+1, j+1 − xi, j ⎪ ⎪ ⎬ 1 σe i+1,  j+1 − σei, j + 2λ 2 σei+1, j+1 + σei, j θi+1, j+1 − θi, j (3.10) ⎪ = γ z i+1, j+1 − z i, j + λ xi+1, j+1 − xi, j ⎪ ⎭ j =1∼n The characteristic line equations are two less than the slip line equations, and thus, there are 4n + 2 differential equations in total.

3.2.2 Boundary Condition For the calculation of the limit load by characteristic line method, the main difficulty lies in the treatment of the boundary condition. There are two boundaries (Fig. 3.1),

Fig. 3.1 Schematic diagram for the difference scheme

3.2 Calculation Process of Limit Load

63

one is the foundation bottom boundary: starting boundary −B ≤ x < 0 of the slip line; the other is the soil mass surface boundary: the stopping boundary 0 < x ≤ x B of the slip line. In Eqs. (3.5) and (3.6), the four equations are first-order ordinary differential equations with four unknowns σe , θ, z, and x. In theory, only four boundary conditions need to be given because differential equations along the characteristic line are too less, and six boundary conditions are needed to obtain the unique solution by the characteristic line method. Generally, the following boundary conditions are given. Soil mass surface boundary: generally, the load value also may be given: σz = q, τx z = 0 (when z = 0 and x ≥ 0),

(3.11)

According to Eq. (3.1), it may be converted into the following boundary condition applicable to the characteristic line method:

c  θ = 0, σe = q + (1 − sin ϕ), z = 0, x ≥ 0. λ

(3.12)

Foundation bottom boundary: the calculation position of the limit load or the soil entry point x of the slip line is also known. Generally, the relationship between the 0 total horizontal force and the total vertical force also may be given: −B (τx z )z=0 dx = 0   tan δ −B σz + λc z=0 dx (when z = 0 and x ≤ 0).   If this relationship is met, τx z = tan δx σz + λc shall be true; for the correct method, tan δx is regarded as a variable, and the distribution of tan δx along the foundation bottom is determined with the minimum total limit load on the foundation bottom. With relatively great difficulty in this way, tan δx is generally regarded as the constant tan δ for the treatment; obviously, such simplification treatment may result in a litter larger total limit load. Similarly, according to Eq. (3.1), the boundary condition of the foundation bottom may be converted into the following boundary condition applicable to the characteristic line method: θ = θ0 =

 sin δ 1 π − δ − arcsin 2 sin ϕ

(3.13)

Or  

1 − tan2 δ/ tan2 ϕ − tan δ φ α = θ0 − μ, where tan α − = 2 1 + tan δ/ sin ϕ

(3.14)

The above are all the same. In this way, six boundary conditions are determined. In general, all the necessary boundary conditions cannot be simultaneously given on the foundation bottom (or soil mass surface) while that on the soil mass surface (or foundation surface) is a completely free boundary. Therefore, the characteristic line method is involved with the solution for the boundary value of the ordinary differential equations, which is

64

3 Characteristic Line Method for Limit Analysis

very important for the specific calculation process. The calculation process generally shall be as follows: under known σe , θ, z, x of all the lattice points on the ith slip line, σe , θ, z, x of all the lattice points on the i + 1th slip line are calculated. So, there are 4n + 2 differential equations in total; for unknowns, there are n + 1 σe (with (i + 1, n + 2) point known), n θ (with (i + 1, n + 2) and (i + 1, 1) points known), n z (with (i + 1, n + 2) and (i + 1, 1) points known), and n + 1 x (with (i + 1, 1) point known), 4n + 2 in total. Thus, the unique solution may be obtained. Note that σe at the (i + 1, 1) and x at the (i + 1, n + 2) point are unknown, and generally, the solution shall be carried out through the simultaneousness of the equations of the 4n + 2 equations. Presently, most solution of the differential equation along the characteristic line is approximately calculated according to the following method: σe at the (i + 1, 1) point is assumed, and the estimation is carried out forward along the slip line from the foundation bottom to the soil mass surface. If the boundary condition of the soil mass surface is met, the calculation for the i + 1th slip line will come to an end; otherwise, σe at the (i + 1, 1) point is reassumed for the above calculation until the boundary condition of the soil mass surface is met. Of course, x at the (i + 1, n + 2) point also may be assumed, and the estimation is carried out backward along the slip line from the soil mass surface until the boundary condition of the foundation bottom is met. Such calculation process is relatively inconvenient; according to characteristics of the differential equation, it is necessary to discuss about the specific calculation method. Below is a two-way recursive calculation process for the solution of slip line one by one.

3.2.3 σ e and θ at the Foundation Edge Point For the calculation according to the above process, σe and θ on the foundation edge point (1, 1) need to be given. It is a key for the characteristic line method to determine σe and θ at this point. Because this is a singularity and σe and θ are different along different directions (soil entry directions), namely different characteristic lines, the value cannot be taken simply. For example, for σe , either edge load of the soil mass surface or the limit load at the edge point of the foundation bottom is incorrect; instead, σe along different characteristic lines is between the edge load and limit load. Such problem containing a singularity is the difficulty that needs to be specially treated for all the numerical methods and will be discussed below. σe and θ at the foundation edge point shall be irrelevant to the soil weight (γ = 0), namely the solution of the weightless soil. According to the calculation example in Chap. 2, it is easy to obtain:   σe = C0 exp 2λ arctan h 

(3.15)

According to different entry directions, the slip surface family will be as follows in different areas (Fig. 3.2):

3.2 Calculation Process of Limit Load

65

Fig. 3.2 Slip surfaces in different areas

⎧  (A) ⎨ h0 h  = (zλ − x)/(z + λx) (B) √ ⎩ λ − 1 + λ2 (C)

(3.16)

where h 0 =

   1 + λ2 λ2 − tan2 δ   λ + tan δ 1 + 2λ2

λ(λ − tan δ) +

(3.17)

Equation (3.17) also may be written in the same form of Eq. (3.14). According to h  = tan(θ − μ), the following equation is obtained by utilizing the boundary condition of the soil mass surface: √  π

ϕ  1 + λ2 c − exp 2λ C0 = q + √ λ 4 2 1 + λ2 − λ Thus, the following equation is true at the foundation edge point: √

1 + λ2 c σe = q + exp(2λθ ) √ λ 1 + λ2 − λ ⎧ (A) ⎨ θ0 θ = θ0 ≥ θ ≥ 0 (B) ⎩ 0 (C)

(3.18)

(3.19)

So, the following equation will be true at the foundation edge point: √

1 + λ2 c c exp(2λθ0 )(1 − sin ϕ cos 2θ0 ) − σz = q + √ 2 λ λ 1+λ −λ = q Nq + cNc

(3.20)

where Nq = exp(2λθ0 )

1 − sin ϕ cos 2θ0 1 − sin ϕ

(3.21)

66

3 Characteristic Line Method for Limit Analysis

px

pz

x

o

D

A

z C B

Fig. 3.3 Difference grid of characteristic line method

Nc =

 1 Nq − 1 λ

(3.22)

It is completely the same as [2]  

π  π   1 + sin ϕ sin(2α − ϕ) 1 ϕ 2 tan exp(π λ) Nc = + exp − + ϕ − 2α λ − 1 λ 4 2 1 + sin ϕ 2

Nq = Nc λ + 1 where α is determined according to Eq. (3.14). The division principle of the characteristic line is determined in Eq. (3.19), for example, the limit load; the difference grid is generally as shown in Fig. 3.3.

3.2.4 Two-Way Recursive Method for Calculation of Differential Equation Specific calculation method for the differential equation is discussed below: (1) Linearization and iterative calculation: Differential equation is nonlinear, and the following format may be adopted for the calculation for the purpose of the linearization and iterative calculation of the differential equation, with the following steps: Step I, (θi+1, j+1 + θi+1, j )/2, (σei+1, j+1 + σei+1, j )/2, (θi+1, j+1 + θi, j )/2, and (σei+1, j+1 + σei, j )/2 on the (i + 1)th slip line are replaced by θi, j , σei, j on the known ith slip line to make the differential equation become a linear equation and calculate the first approximation of σe , θ, z, x on the (i + 1)th slip line. Step II, σe , θ, z, x on the (i + 1)th slip line is replaced by the first approximation, and the second approximation on the (i + 1)th slip line is calculated. Generally, σe , θ, z, x on the (i + 1)th slip line is replaced by Kth approximation, and (K + 1)th approximation of σe , θ, z, x on the (i + 1)th slip line is calculated

3.2 Calculation Process of Limit Load

67

until the two approximations are fully close to each other (the difference between them is in accordance with the given calculation accuracy). If only the first approximation or second approximation is calculated, the difference grid needs to be divided very densely; otherwise, larger error will be caused. (2) Two-way recursive calculation method: For the above steps, the equations of 4n + 2 differential equations need to be calculated; it is noted that (θi+1, j+1 + θi+1, j )/2, (θi+1, j+1 + θi, j )/2, x, z on the foundation bottom and σe , θ on the soil surface are known. In order to take full advantage of such boundary condition and characteristics of differential equations and avoid the solution of 4n + 2order equations or repeated trial calculation, the following calculation method may be adopted. As x, z of the foundation bottom is known, by slip line equation and characteristic line equation, values of all the lattice points of the slip line may be firstly calculated through forward recurrence from the foundation bottom along the slip line. Moreover, σe , θ of the soil mass surface and the slip line are known, stress equation on the slip line and stress equation on the characteristic line may be applied, and σe , θ at all the lattice points on the slip line may be calculated through the backward recurrence from the soil mass surface along slip line, which is the two-way recursive method and characterized in that higher-order numerical solution of very high accuracy may be calculated slip line by slip line.

3.3 Calculated Results of Limit Load After σe , θ of the foundation bottom is obtained, the limit load pz = σz may be derived according to the first equation in Eq. (3.1). It is not difficult to write a calculation procedure for the characteristic line method according to the above calculation process. Because the differential equation is nonlinear, higher-order approximate solution needs to be calculated; according to the actual calculation, generally, the approximate solution needs to be calculated 4–5 times; when ϕ is relatively large, approximate solutions need to be calculated 7–8 times so that the difference between two calculated values of σe of the foundation bottom reaches the accuracy of four significant figures. Total limit load equation generally may be written as below: 

1 Pz = B q Nq + cNc + γ B Nγ 2 It is rewritten as below:

Pz = γ B 2

q c 1 Nq + Nc + Nγ γB γB 2

 = γ B 2 P0

(3.23)

68

3 Characteristic Line Method for Limit Analysis

Calculated results of the total limit loads P0 and P0 are detailed in Table 3.1. Maximum depth and width of the slip surface are supposed as z m /B = {z i /B}max and xm /B = {xi /B}max , respectively. The maximum depth and width of the slip surface are detailed in Table 3.2. Eccentricity e/B = e B /B of the limit load may be determined according to 0 −B pz (x)(x + B/2 − e B )dx = 0, see Table (3.3). Distribution of the limit load along the foundation bottom is in a closely linear convex as shown in Fig. 3.4 (ϕ = 25◦ , q/(γ B) = 0.1, and c/(γ B) = 0.2). For different q/(γ B) and c/(γ B), it is easy to calculate corresponding limit loads. Obviously, the above calculation process also applies to the soil pressure.

3.4 Comparison with Existing Calculation Equations 3.4.1 Comparison with Total Limit Load Equation Among existing calculation equations for ultimate bearing capacity (such as Teraghi’s equation (Hansen 61), Vesic’s equation, Meyerhof’s equation, and Chen’s upper bound solution), the total limit load equation may be written as Eq. (3.23). In theory, any calculation equation for the limit load derived through application of the surface failure mode is the limit load in surface failure mode. However, these equations are not derived in strict accordance with the surface failure mode; most of the obtained calculation equations for the limit load, in consideration of the failure mode under the integral action of the foundation, are derived on the basis of Plandtl’s weightless soil equation, namely the distribution of the limit load at front toe of the calculated surface is the calculated value of Plandtl’s weightless soil equation. Because the limit load in field failure mode is close to the linear distribution, the front toe of the calculated surface is also the calculated value of Plandtl’s weightless soil equation, and these equations are actually the approximate equations in the field failure mode. In order to distinguish them from those derived in strict accordance with field failure mode, they are still referred to as the calculation equation in the surface failure mode. Nc , Nq of these equations are identical, while Nγ of them is different. Calculated results by the characteristic line method and Nγ in several common equations are detailed in Table 3.4. where   Teraghi’s equation(Hansen 61) : Nγ = 1.8 Nq − 1 tan ϕ;   Meyerhof : Nγ = Nq − 1 tan(1.4ϕ);   Vesic : Nγ = 2 Nq + 1 tan ϕ;

1.221

1.669

2.501

3.859

6.196

10.442

18.693

36.072

76.605

184.39

10

15

20

25

30

35

40

45

288.25

110.25

47.035

21.649

10.463

5.178

2.540

1.169

0.429

0.072

127.25

54.807

26.565

14.079

7.990

4.772

2.942

1.796

P0

5

0.1

P0

Nγ

0.0

tan δ

1

ϕ(◦ )

192.06

75.464

32.793

15.241

7.362

3.583

1.676

0.669

Nγ

Table 3.1 Total limit loads P0 and Nγ (q/(γ B) = 0.1, c/(γ B) = 0.2)

85.435

38.042

18.899

10.176

5.802

3.408

1.945

P0

0.2

123.45

49.582

21.783

10.107

4.790

2.209

0.882

Nγ

56.196

25.744

13.018

7.041

3.937

2.117

P0

0.3

76.986

31.358

13.771

6.265

2.803

1.090

Nγ

36.433

17.047

8.674

4.604

2.350

P0

0.4

46.786

19.109

8.239

3.541

1.335

Nγ

3.4 Comparison with Existing Calculation Equations 69

0.642

0.662

0.718

0.801

0.909

1.055

1.244

1.502

1.867

5

10

15

20

25

30

35

40

45

6.803

4.829

3.555

2.695

2.085

1.655

1.342

1.121

0.989

0.977

1.599

1.287

1.061

0.891

0.756

0.644

0.542

0.420

z m /B

0.680

0.1

z m /B

xm /B

0.0

tan δ

1

ϕ(◦ )

5.838

4.148

3.042

2.286

1.743

1.340

1.021

0.719

xm /B

1.345

1.078

0.880

0.725

0.593

0.469

0.318

z m /B

0.2

4.926

3.487

2.536

1.869

1.377

0.984

0.608

xm /B

Table 3.2 Maximum depth and width of the slip surface (q/(γ B) = 0.1, c/(γ B) = 0.2)

1.113

0.883

0.707

0.560

0.423

0.262

z m /B

0.3

4.090

2.870

2.047

1.454

0.990

0.556

xm /B

0.905

0.705

0.543

0.396

0.232

z m /B

0.4

3.339

2.300

1.582

1.037

0.551

xm /B

70 3 Characteristic Line Method for Limit Analysis

3.4 Comparison with Existing Calculation Equations

71

Table 3.3 Eccentricity of the limit load e/B (q/(γ B) = 0.1, c/(γ B) = 0.2) ϕ(◦ )

tan δ 0.0

0.1

0.2

0.3

1

– 0.0048

5

– 0.0202

10

– 0.0359

– 0.0293

15

– 0.0501

– 0.0439

20

– 0.0635

– 0.0575

– 0.0502

– 0.0408

25

– 0.0762

– 0.0704

– 0.0635

– 0.0553

0.4

– 0.0358 – 0.0451

30

– 0.0883

– 0.0827

– 0.0761

– 0.0686

– 0.0599

35

– 0.0997

– 0.0945

– 0.0884

– 0.0813

– 0.0734

40

– 0.1105

– 0.1058

– 0.1001

– 0.0937

– 0.0864

45

– 0.1207

– 0.1165

– 0.1114

– 0.1056

– 0.0990

Note The negative indicates backward eccentricity

pz /(γ B )

Fig. 3.4 Distribution of limit load along foundation bottom

tan δ = tan δ = tan δ = tan δ = tan δ =

0 1 2 3 4

−x/B

π   ϕ + . Chen W.F : Nγ = 2 Nq + 1 tan ϕ tan 4 5 According to the calculated results, it clearly shows that the result by Sokolovskii’s characteristic line method is slightly smaller than the above-calculated result, which may be because the division of the difference grid is relatively rough or the higherorder approximation is not calculated [1]; however, it is certain that an extremely small value is taken for η = (qλ + c)/(γ B). When ϕ in Terzghi’s equation is relatively large, it is very close to the calculated result by the characteristic line method when q/(γ B) = 0.05 and c/(γ B) = 0.01. For Meyerhof’s equation, Vesic’s equation, and Chen W.F’s equation, calculated results by characteristic line method of a group of q/(γ B) and c/(γ B) may be found very close to them. Of course, these are approximate equations (if the selected slip surface is approximate), and it is no wonder that they have some difference with the calculated results by the characteristic line method in some cases.

94.70

65.81

39.15

23.90

17.39

12.74

8.07

5.14

3.80

2.80

1.75

1.07

0.75

0.51

0.33

0.19

0.09

q = 0.05 c = 0.01 (4)

130.0

60.00

26.00

12.00

5.50

2.50

Meyerhof (5)

126.5

53.37

24.22

11.49

5.55

2.63

q = 1.1 c = 0.0 (6)

Note 1 (1), (5), and (9) are excerpted from Ref. [4], (3) from Ref. [5], and (7) from Ref. [6]

67.41

95.45

90.74

40

86.50

38

40.71

36.64

35

35.20

24.94

15.95

17.68

15.30

30

32

13.13

28

4.961

8.110

7.25

25

6.90

22

2.490

3.537

3.34

1.418

15

3.16

0.755

12

20

0.467

10

18

0.268

8

1.49

0.135

1.40

0.055

6

Terzghi (3)

4

q = 0.01 c = 0.0 (2)

0.012

Sokolovskii (1)

2

ϕ (◦ )

109.4

78.03

48.03

30.22

22.40

16.72

10.88

7.13

5.39

4.07

2.65

1.69

1.22

0.86

0.57

0.34

0.15

Vesic (7)

110.2

77.54

47.04

29.31

21.65

16.12

10.46

6.85

5.18

3.91

2.54

1.62

1.17

0.82

0.55

0.33

0.15

q = 0.1 c = 0.2 (8)

145.3

102.1

61.49

37.86

27.67

20.36

12.97

8.318

6.198

4.614

2.941

1.837

1.313

0.909

0.595

0.350

0.156

Chen W.F (9)

Table 3.4 Comparison between calculated results by characteristic line method and Nγ in several equations (q = q/(γ B) and c = c/(γ B))

142.4

99.93

60.36

37.46

27.61

20.45

13.20

8.581

6.446

4.832

3.014

1.945

1.388

0.956

0.622

0.361

0.158

q = 1.5 c = 1.5 (10)

72 3 Characteristic Line Method for Limit Analysis

3.4 Comparison with Existing Calculation Equations

73

Therefore, just for the calculated result of the total limit load, Nγ of these equations is relatively close to that by the characteristic line method and only restricted to specific q/(γ B) and c/(γ B). However, accurate Nγ varies from η = (qλ+c)/(γ B), which has been discussed by Xiao et al. [1]. It shall be indicated that, according to the stress field requirements of two kinds of failure modes, for the total limit load, calculated value in the surface failure mode shall be less than or equal to that in the field failure mode, which also shows that any calculation equation in which calculated total limit load is greater than the calculated value by the characteristic line method cannot be a genuine solution in the surface failure mode. However, the calculation equation in which the calculated total limit load is less than the calculated value by the characteristic line method is not necessarily the genuine solution in the surface failure mode.

3.4.2 Comparison of Limit Load Distribution Generally, only total limit load can be obtained in the surface failure mode; however, for the slip surface for which any point (−b) through the foundation bottom is taken, the total limit load within partial width [−b, 0] of the foundation may be obtained. Thus, distribution approximation of the limit load also may be obtained: pz = q Nq + cNc + γ bNγ and 0 ≤ b ≤ B Namely, the limit load along the foundation bottom is in a linear distribution, while the limit load by the characteristic line method is in a convex distribution (Fig. 3.5) which is monotonically increasing with increasing distance from the front toe. After analysis on the calculation process of this two kinds of methods (surface and field failure modes), it is easy to get different reasons for their distribution forms. For the surface failure mode, limit load of [−bk−1 , 0] is not considered when upper limit load of [−bk , −bk−1 ] is calculated, while the upper limit load of [−bk , −bk−1 ] calculated according to the characteristic line method is obtained based on the calculated upper limit load of [−bk−1 , 0]. Fig. 3.5 Schematic diagram for distribution of limit load

Limit load in surface failure mode Limit load in field failure mode

−B

− bk − bk −1

0

74

3 Characteristic Line Method for Limit Analysis

The surface failure mode only requires the soil mass on the slip surface through foundation rear toe to be in the limit state, while the soil mass in the slip mass is only required to be in the stable state or in the limit state; under the same total load, the more the backward eccentricity of the load distribution is relatively large, the easier it is to meet the above requirements for the surface failure mode, which is also because the limit load in surface failure mode is relatively small at the front toe (compared with that in the field failure mode) and relatively large at the rear toe. If it is approximately considered according to linear distribution of the foundation edge points pz (0), pz (−B), it will be necessarily a stable solution of the limit load and more convenient for the application.

3.5 Discussion on Characteristic Line Method 3.5.1 Treatment for Boundary Condition of Foundation Bottom As stated before, the foundation bottom is subject to the combined action of the horizontal and vertical forces. For the characteristic line method, it needs to determine tan δx so that the relationship between the total horizontal force and the total vertical 0 0   force, −B (τx z )z=0 dx = tan δ −B σz + λc z=0 dx (when z = 0 and x ≤ 0), is met, and the total limit load is the minimum. The following calculation examples will show that it is objective. If,  tan δx =

− Bx aλ when − Bx a ≤ 0.9 0.9λ when − Bx a  0.9

(3.24)

When tan δx is a constant and a variable of Eq. (3.24), the comparison between calculated results of the total limit load is detailed in Table 3.5. Obviously, when tan δ is relatively large, the total limit load calculated according to Eq. (3.24) is relatively small. When tan δx is a constant and a variable of Eq. (3.24), comparison between calculated results of limit load distribution is detailed in Table 3.6. Table 3.5 Comparison of total limit load P0 in different distributions of horizontal force (ϕ = 25◦ , q/(γ B) = 0.1, and c/(γ B) = 0.2) tan δx tan δ Equation (3.24)

tan δ 0.096

0.174

0.226

0.258

0.279

8.086 8.298

6.343 6.379

5.288 4.939

4.683 4.223

4.304 3.805

3.5 Discussion on Characteristic Line Method

75

Table 3.6 Comparison of limit load distribution in different distributions of horizontal force (ϕ = 25◦ , q/(γ B) = 0.1, and c/(γ B) = 0.2) −x/B

tan δ 0.174

0.258

tan δ

tan δx

tan δ

tan δx

1.0

8.706

5.025

6.258

3.329

0.9

8.264

5.561

5.964

3.238

0.8

7.826

6.029

5.664

3.147

0.7

7.359

6.425

5.359

3.055

0.6

6.893

6.735

5.048

2.962

0.5

6.415

6.940

4.729

3.821

0.4

5.921

7.021

4.400

4.654

0.3

5.409

6.948

4.058

5.324

0.2

4.869

6.685

3.700

5.760

0.1

4.289

6.175

3.318

5.845

0.0

3.639

5.210

2.897

5.210

The distribution of the limit load along the foundation bottom is completely different. Considering tan δx along the foundation bottom is distributed as described in Table 3.6 (3.24), when tan δ is relatively large, the limit load has been eccentric forward. If tan δx = tan δ is required to be true, the calculated limit load only can be eccentric backward. Of course, the above result is only a calculation example. However, it is enough to show that it is necessary for the solution of engineering problems by characteristic line method to take different forms of distribution of tan δx along the foundation bottom into consideration. Generally, the design load is eccentric forward and necessarily accompanied with relatively large horizontal force; if tan δx cannot be determined, it is safe to regard tan δx as a variable and make the minimum total limit load (ground bearing capacity) on the foundation bottom to determine the distribution of tan δx along the foundation bottom.

3.5.2 Calculation for Limit Load of Ground with Heterogeneous Soil For the general ground with heterogeneous soil, the adaptability of the characteristic line method is relatively poor. The characteristic line method actually is a kind of difference methods (difference along slip line and characteristic line). However, these methods are inconvenient to deal with the interface between soils with different strength indexes, so only some approximate treatment can be carried out, which will cause relatively large error of the calculated results.

76

3 Characteristic Line Method for Limit Analysis

Fig. 3.6 High strength of topsoil and low strength of subsoil

Hard soil Soft soil

Characteristic line method cannot be applicable to the general ground with heterogeneous soil; what is more important, correct limit load may not be obtained at all in certain cases, such as high strength of topsoil and low strength of subsoil (Fig. 3.6). According to the characteristic line method, possible slip surface (the maximum surface of slip line) is only in the hard soil, while the slip surface corresponding to the real limit load may pass through the soft soil in lower layer. Especially, when tan δ is relatively large, it is more easy that such slip surface only passes through the topsoil; if maximum depth of the slip surface is close to the bottom of the topsoil, the calculated limit load is necessarily on a large side, which may be the reason for that the characteristic line method has been applied few for calculation of the ground with heterogeneous soil by far.

3.5.3 Determination of Ground Bearing Capacity Limit load calculated by the characteristic line method is obtained in the limit state at each point of the soil mass in the slip mass, which means that the strength of each point has reached the extremum and is the maximum load that the ground can bear—ultimate bearing capacity; moreover, its distribution form is unique and definite. Distribution form of actual engineering design load generally cannot be just corresponding to that of the calculated limit load; distributions of the design load and limit load along the foundation bottom are supposed as pv (x) and pz (x) (−B ≤ x ≤ 0) with Pz , Pv of the total limit load and total design load, respectively. When their total loads are the same, according to the design load, one part of soil mass is necessarily in the stable state while the other part of soil mass in the failure state (Fig. 3.7). Therefore, when the limit load calculated by the characteristic line method is used for the determination of the engineering ground bearing capacity, distribution forms of actual engineering design load and limit load shall be considered simultaneously. Generally, it is reasonable to consider the eccentricity included in the calculated value of the limit load. In some cases, the calculated value of the ground bearing capacity may be relatively small, for example, when the eccentricity of the design load is the same as that for the calculated value of the limit load, eccentricity of the design load will be used to correct the calculated value of the limit load, which obviously reduces the ground bearing capacity. In some other cases, calculated value of the ground bearing capacity may be relatively large; for example, when the eccentricity

3.5 Discussion on Characteristic Line Method

77

p v*1

p z1

p v*2

p z1

pv*1

pz2

pz2

−B

−b

(a) Design Loadwith Forward Eccentricity

0

p v*2

−B

−b

0

(b) Design Load with Backward Eccentricity

Fig. 3.7 Schematic diagrams for limit load and design load

of the design load is zero, if the determined bearing capacity amounts to the maximum limit load when the eccentricity is the calculated value, it is obvious that the ground bearing capacity is overestimated.   If the calculation is carried out according to the effective width B  = B − 2|ev | of the foundation in replace of its actual width, the ground bearing capacity will be greatly reduced in case of forward eccentricity of the design load. Considering an extreme example, q = 0, c = 0, and ev = B/6, if it is in accordance with the boundary condition of the routine foundation bottom according to the characteristic line method, e ≈ −B/6 shall be true; here, the effective width is B  = B − 2|ev − e| = B/3, which means only about 1/3 of the foundation bears the load, which will make the ground bearing capacity become 1/9 of the calculated total limit load. Additionally, according to its solution process, the characteristic line method requires it always meet the limit equilibrium in the solution area, so the result shall be the calculated result of the limit state solution in the field failure mode. For the surface failure mode, it may be powerless, and the calculated result maybe cannot be directly obtained; for example, for the slope stability with known load on the soil mass surface (including foundation bottom). Certainly, among the above, some have already exceeded the range of the characteristic line method, for example, the ground bearing capacity determined according to the limit load, which will be specially discussed later. It is undeniable that the characteristic line method is an effective method for the solution of the limit equilibrium; by it, not only the limit load (ultimate bearing capacity and soil pressure) on the boundary surface can be obtained but also the area (slip line field) in the limit state and the limit stress field in this area can be obtained; in theory, it is an exact solution of the limit equilibrium (excluding numerical calculation error). If the difference grid along the slip line and characteristic line is divided relatively dense, the obtained results will be of very high accuracy. In addition, by the characteristic line method, on one hand, boundary surface of unknown limit load may be regarded as the load boundary condition (limit load to be determined) for

78

3 Characteristic Line Method for Limit Analysis

treatment; on the other hand, it also provides theoretical basis for other solution methods.

References 1. Xiao DP, Zhu WY, Chen H (1998) Progress in slip lines method to solve the bearing capacity problem. Chin J Geotech Eng 20(4):25–29 2. Xiao DP (1995) Ultimate bearing capacity equation for solution of exact value based on slip line method. Master Dissertation of Tianjin University 3. Zhang MY, Huang CZ (2005) Ultimate loads in characteristic line method and ultimate bearing capacity of foundation. China Harbour Eng 3:8–12 4. Chen HF (1995) Limit analysis and soil plasticity (trans: Zhan S, Proofread: Han D). China Communications Press, Beijing 5. Shen ZJ (2000) Theoretical Soil mechanics. China Water Power Press, Beijing 6. Soil mechanics Staff Room of Huadong Hydraulic Faculty (1979) Principle and calculation of geotechnical, vol 1. Hydraulic Press, Beijing

Chapter 4

Stress Field Method for Limit Analysis

For the limit analysis on the soil mass, it is the most effective method to obtain the stress field of the soil mass in the limit equilibrium state. When the stress component is known, the limit load (limit soil pressure and ultimate bearing capacity) is only the value of the stress component at corresponding boundary. Moreover, the states of the stress field and slip surface family may be judged. However, under general conditions, it is very difficult to obtain the stress field in accordance with the static equilibrium equation, yield condition, corresponding extremum condition, and boundary condition. In this chapter, the simplest stress field is first discussed and then stress field [1] in accordance with the static equilibrium equation, yield condition, and corresponding boundary condition is given under the slip surface (family) (fracture plane and slip line) of the plane and logarithmic helical curved surface by the stress function method, and thereby, critical height of slope excavation and calculation of limit load are discussed. Lastly, the stress field is solved by stress equation along the slip surface, yield condition, and extremum condition of the yield function and calculation of the limit load is discussed.

4.1 The Simplest Stress Field 4.1.1 Plane Slip Surface If the slip surface family is a plane, h  will be a constant, and Eq. (2.21) will be changed as below: ∂σe ∂x ∂σe ∂x

e + h  ∂σ = γ (h  − λ) ∂z

+

1+λh  ∂σe λ−h  ∂z

=γ

1+λh  λ−h 

 +λ

© Springer Nature Singapore Pte Ltd. and Zhejiang University Press, Hangzhou, China 2020 C. Huang, Limit Analysis Theory of the Soil Mass and Its Application, https://doi.org/10.1007/978-981-15-1572-9_4

 (4.1)

79

80

4 Stress Field Method for Limit Analysis

A solution in accordance with both equations in Eq. (4.1) is as below:    2  γ z  2 1 + λ + h − λ 1 + h 2    2  γ λx  2 1 + λ − h − λ − 1 + h 2

σe = A +

(4.2)

where A is a constant to be determined. Therefore, the stress component has the following form: 1+λ2 +(h  −λ) σ (1+λ2 )(1+h 2 ) e 2 2 2 1+λ )h +(1+λh  ) σe σz + λc = ( 1+λ ( 2 )(1+h 2 ) 2 2  1+λ − h −λ τx z = λ 1+λ2 ( 1+h 2) σe ( )( )

σx +

c λ

=

2

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ (4.3)

⎪ ⎪ ⎪ ⎪ ⎭

If boundary condition is precisely proper to the h  (a constant), A may be determined according to the boundary condition and the stress field is derived. For example, if the boundary condition is σz = q, τx z = 0, (z = 0), exact stress field in the limit state is derived as below: ⎫ σz = q + γ z ⎬ (4.4) τx z = 0 ⎭ σx = (q + γ z)/ h a2 − 2c/ h a where 

h =

h a

=

√ λ + √1 + λ2 (active failure) λ − 1 + λ2 (passive failure)

Equation (4.4) is only true in the area shown in Fig. 4.1. q

45 0 +

ϕ 2

(a) Active Failure Area Fig. 4.1 Schematic diagram for limit state area

q 45 0 −

ϕ 2

(b) Passive Failure Area

(4.5)

4.1 The Simplest Stress Field

81

2 h  −h  Equation (4.4) is substituted in the yield function; hence, f = − (1+h 2 a )h 2 [(γ z + ( )a q)λ + c]; for any slip surface (h  ), the stress field in Eq. (4.4) is the stable stress field while only the slip surface determined according to Eq. (4.5) is the limit stress field; in other words, the stress field in the limit state inevitably accompanies the slip surface family in the limit state. If the slip surface is not discussed, stress field in the limit state cannot be discussed either.   If the boundary condition is τx z = tan δ σz + λc , (z = 0), the following equation will be derived according to Eq. (4.3):

h  = h a =

   1 + λ2 λ2 − tan2 δ   λ + tan δ 1 + 2λ2

λ(λ − tan δ) ±

(4.6)

After it is substituted in Eqs. (4.2) and (4.3), stress field will be obtained, when the specific stress form of the surface boundary may be determined; in other words, only when the surface boundary condition is consistent with the specific stress forms of the surface boundary determined according to Eq. (4.6), the stress field is that in the limit state. Similarly, it also may be discussed when the boundary is a vertical surface; however, except in some special cases, it is very difficult to meet the horizontal and vertical boundary conditions.

4.1.2 Purely Cohesive Material (ϕ = 0) Considering the limit load of the purely cohesive material, with the load σz = q, τx z = 0 known on the ground surface boundary b , the vertical load σz that the foundation bottom, namely boundary a , may bear is solved; thereinto, the horizontal load is an uniformly distributed load τx z = ph . Because tan ϕ = 0, the following solutions may be easily obtained according to those discussed in Chap. 2:   h + (π + 2)c/2 + q σx = γ z + 2c arctan h  − 1+h 2    h σz = γ z + 2c arctan h  + 1+h + (π + 2)c/2 + q 2 τx z =

2

1−h c 1+h 2

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(4.7)

And, h  = −(x − x R )/(z − z R ) or h  = k0 . The following is obtained according to the boundary condition of the load on the a :  h a

=

1 − ph /c 1 + ph /c

(4.8)

82

4 Stress Field Method for Limit Analysis

Obviously, ph /c ≤ 1.0 shall be true and vertical limit load on the a is a uniformly distributed load:  σz a = pz = cNc + q

(4.9)

where  Nc = 2

arctan h a

h a + 1 + h a2

 +

π +2 2

(4.10)

According to smooth requirements (continuity of first derivative) of the slip surface (family), the slip surface family will meet the following condition (Fig. 4.2): ⎧  (A) ⎨ ha  h = −x/z (B) ⎩ −1 (C) It is noted that ph /c = Hk /(Bc) and Hk are the total horizontal forces within the foundation width B. Nc in Eq. (4.10) is very close to the calculated results of the current common equations [2] (see Table 4.1).

Fig. 4.2 Schematic diagram for slip surface

Table 4.1 Comparison with common equations

ph /c

Ns

Nc

0.0

5.142

5.142

0.1

5.010

5.036

0.2

4.870

4.920

0.3

4.722

4.791

0.4

4.562

4.647

0.5

4.389

4.484

0.6

4.197

4.298

0.7

3.979

4.080

0.8

3.720

3.814

0.9

3.384

3.458

1.0

2.571

2.571

4.1 The Simplest Stress Field

83

Ns = (π + 2)

  1 1 + 1 − ph /c 2

(4.11)

It is worth noting that Eq. (4.9) is derived under the actual load condition of the horizontal load; namely, the vertical load is a limit load while the horizontal load is an actual load. Therefore, during the application of the ground bearing capacity, certain safety factor shall be provided for the vertical and horizontal design loads, compared with the vertical and horizontal limit loads, respectively. Generally, if the following equations are required to be true, ph ≤  h a =

1 c Kh

1 − K h ph /c 1 + K h ph /c

(4.12)

(4.13)

After the above are substituted in Eqs. (4.9) and (4.10), the calculation equation for the limit load will be derived so that the limit load is calculated with the safety factor for the horizontal design load of K h compared with the horizontal limit load. In general, it is very difficult to obtain the limit stress field which can satisfy the static equilibrium equation, yield condition, extremum condition of the yield function, and boundary condition.

4.1.3 Approximation of Limit Stress Field by Stress Function Due to difficulty in solution of the limit stress field, the approximation of the limit stress field has to be solved. Generally, any stress components σx , σz , τx z in accordance with the equilibrium equation constitute a stress field and the state of the stress field may be judged by the yield function according to the definition of stress field state in Chap. 2. Stress function is taken as φ0 (x, z) = φ(z−z R , s) and when s = (x −x R )/(z−z R ) is in accordance with the equilibrium equation, the stress components will be shown in the following forms: 1 ∂ 2φ ∂ 2 φ0 c c = + γ − z + γ (z − z R ) − − (z ) R 2 2 2 ∂x λ λ (z − z R ) ∂s   2 2 ∂ 1 ∂φ s ∂ φ ∂ φ0 τx z = − =− + 2 ∂ x∂z ∂z z − z R ∂s (z − z R ) ∂s 2   ∂ 2φ 1 ∂φ s2 ∂ 2φ ∂ 2 φ0 c ∂ c σx = = + − − 2s − 2 2 2 2 ∂z λ ∂z ∂z z − z R ∂s λ (z − z R ) ∂s σz =

(4.14) (4.15) (4.16)

84

4 Stress Field Method for Limit Analysis

In other words, stress function φ0 (x, z) or φ(z − z R , s) may be properly selected; after the stress component is determined according to Eqs. (4.14)–(4.16), it is substituted in the yield function for the discussion of the stress field state and determines whether to be selected, which is obviously a kind of the simplest solution methods. It is the key for this method to select the stress function so that the obtained stress field is close to the limit stress field. Equations (4.14)–(4.16) are substituted in the yield function; hence,        1 ∂φ  ∂ 2 1 + h 2 − 2h  h  − λ + s 1 + h  λ 1+h f =− ∂z z − z R ∂s    ∂ 2φ   1 h − λ + s 1 + h  λ 1 − sh  + 2 ∂s 2 (z − z R )   ∂ 2φ   − h 1 + hλ + γ (z − z R ) h  − λ (4.17) 2 ∂z

4.2 Stress Field of Plane Slip Surface The solution by stress function requires the stress field to meet the yield condition; namely under the condition of the selected slip surface (family) (fracture surface and slip line), the stress field in accordance with the static equilibrium equation and yield condition is solved and state of the obtained stress field is discussed. If the slip surface is a plane, namely (x R , z R ) is a point on the plane, h  = 1/s, the yield condition f = 0 will be changed as below:   ∂  1 ∂φ ∂ 2φ − (s + λ) 2 = −γ (z − z R )s(1 − sλ) 1 + s2 ∂z z − z R ∂s ∂z

(4.18)

Both sides of the equation are integrated for z; hence,    2 1+s

1 ∂φ z − z R ∂s

 − (s + λ)

∂φ 1 = − γ (z − z R )2 s(1 − sλ) + f u (s) ∂z 2

where f u (s) is an arbitrary function, and further, the stress function is derived as follows:   (z − z R )3   1 + 3sλ 1 − 3sλ + s 2 + g1 (v) + (z − z R )g2 (s) φ= γ  2 6 1 + 9λ  v = (z − z R ) 1 + s 2 exp(λ arctan s) where g1 (v), g2 (s) are arbitrary functions.

(4.19) (4.20)

4.2 Stress Field of Plane Slip Surface

85

Equation (4.19) is substituted back to Eqs. (4.14)–(4.16), and the following stress field will be derived:   s−λ 1 c + gb (s) σz = − + γ (z − z R ) 1 + 3λ 2 λ 1 + 9λ (z − z R )   1 + λ2 1 + ga (v) + (s + λ)2 ga (v) exp(2λ arctan s) (4.21) v 1 + s2 s λ(3λs − 1) τx z = γ (z − z R ) gb (s) + 2 1 + 9λ (z − z R )   1 + λ2 1  sga (v) − (1 − sλ)(s + λ)ga (v) + exp(2λ arctan s) (4.22) v 1 + s2 c s2 1 + sλ σx = − + γ (z − z R ) gb (s) + λ 1 + 9λ2 (z − z R )   1 + λ2 2 1 + s ga (v) + (1 − sλ)2 ga (v) exp(2λ arctan s) v 1 + s2

(4.23)

where gb (s) and ga (v) are arbitrary functions and ga (v) is the first derivative of ga (v). In order to discuss this stress field state, for any h  , it is substituted in the yield function; hence, f =

    1   2  −K 1 − sh 1 − sh + K 0 1 1 + h 2

(4.24)

γ (z − z R ) 2 1 + λ2   4λ + 2λ K0 =  g exp(2λ arctan s) 1 + s2 a 1 + 9λ2 s 

  1 1 + λ2 1 1 + sλ  g gb (s) + (v) − g (v) exp(2λ arctan s) + a a s z − zR 1 + s2 v (4.25)  γ (z − z R )   2λ 2λ − s − 3sλ2 K1 =  2 1 + 9λ s 

  1 1 + λ2 1 1 + s2  g gb (s) + (v) − g (v) exp(2λ arctan s) + a a s z − zR 1 + s2 v (4.26) Obviously, if K 0 ≥ 0, K 1 = 0, f ≤ 0 must be true. For the weightless soil (γ = 0), when gb (s) = 0, ga (v) = Av are taken, if A ≥ 0, for any h  , f ≤ 0 will be true. When the ground with horizontal surface is subject to the uniformly distributed vertical load and when gb (s) = 0, ga (v) = A0 v + A1 v2 are taken, s, A0 , A1 may be determined according to the boundary conditions σz = q, τx z = 0, (z = 0) and f ≤ 0. Thus, Eq. (4.4) in the limit state is derived.

86

4 Stress Field Method for Limit Analysis

Except in cases shown in Fig. 4.1, when the surface boundary is under general load and weight soil (γ = 0) conditions, according to Eqs. (4.24), (4.21)–(4.23) only meet f = 0 on the plane family h  = 1/s, but cannot meet f ≤ 0 for any h  . Therefore, it is the stress field in the failure state while the plane family may be the slip surface family in the failure state. It needs to further study how to divide the solved areas into several subareas and properly select functions to be determined so that Eqs. (4.21)–(4.23) are close to the limit stress field as much as possible.

4.3 Stress Field of Common Helicoid Slip Surface   If the slip surface meets the equation h  − λ + s 1 + h  λ = 0, it may be known that the slip surface is a common helicoid, and its function expression is as below: x − x R = R exp(−λθ ) cos θ z − z R = R exp(−λθ ) sin θ

 (4.27)

And the yield condition f = 0 is true; Eq. (4.17) will be changed as below:     1 ∂φ ∂ 2φ 2 ∂ + (s − λ) 2 = γ (z − z R )s(1 + sλ) − 1+s ∂z z − z R ∂s ∂z

(4.28)

Both sides of the equation are integrated for z; hence,    − 1 + s2

1 ∂φ z − z R ∂s

 + (s − λ)

∂φ 1 = γ (z − z R )2 s(1 + sλ) + f u (s) ∂z 2

where f u (s) is an arbitrary function and the stress function may be derived:   (z − z R )3   1 − 3sλ 1 + 3sλ + s 2 + f 1 (v) + (z − z R ) f 2 (s) φ=γ  6 1 + 9λ2  v = (z − z R ) 1 + s 2 exp(−λ arctan s)

(4.29) (4.30)

where f 1 (v) and f 2 (s) are arbitrary functions. Equation (4.29) is substituted back to Eqs. (4.14)–(4.16), and general forms of the stress component will be as follows:   s+λ 1 c + f b (s) σz = − + γ (z − z R ) 1 − 3λ 2 λ 1 + 9λ z − zR

4.3 Stress Field of Common Helicoid Slip Surface

 1 1 + λ2  2 f a (v) + f a (v)(s − λ) + exp(−2λ arctan s) v 1 + s2

87



(4.31)

s 1 + 3sλ τx z = γ (z − z R )λ + f b (s) 2 1 + 9λ z − zR   1 + λ2 1 s f a (v) − f a (v)(s − λ)(1 + sλ) + exp(−2λ arctan s) (4.32) v 1 + s2 c s2 1 − sλ σx = − + γ (z − z R ) + f b (s) 2 λ 1 + 9λ z − zR   1 1 + λ2 2 s f a (v) + f a (v)(1 + sλ)2 + exp( − 2λ arctan s) v 1 + s2

(4.33)

where f b (s) and f a (v) are arbitrary functions and f a (v) is the first derivative of f a (v). For any h  , this stress field is substituted in the yield function; hence,  2      1  −K (1 + sλ) + s − λ + K (1 + sλ) + s − λ h h f = 3 4 1 + h 2 (1 + sλ) (4.34) γ (z − z R ) s 2λ + f b (s) 2 1 + 9λ z − zR   1 + λ2 1  f a (v) − f a (v) exp(−2λ arctan s) + s 1 + s2 v 2λ(1 + sλ)  + f a (v) exp(−2λ arctan s) 1 + s2

K3 =

 1 + s2 γ (z − z R )  2 + 2λ 2λ + s + 3sλ f b (s) 1 + 9λ2 z − zR     1 f a (v) − f a (v) exp(−2λ arctan s) + 1 + λ2 v

(4.35)

K4 =

(4.36)

Apparently, as long as γ = 0 and ϕ = 0 are true, no such f b (s), f a (v) can make the K 4 = 0 true. In other words, Eqs. (4.31)–(4.33) meet f = 0 only along the logarithmic helical curved surface family, but cannot meet f ≤ 0 for any h  . Therefore, it is the stress field in the failure state and it may be the slip surface family in the failure state for the logarithmic helical curved surface. This may be a better stress field given in analytical expression up to now, because results close to real stress field may be obtained within several subareas. Ideas to obtain critical excavation height and limit load by applying the stress field are stated in examples below:

88

4 Stress Field Method for Limit Analysis

4.4 Application Example: Critical Excavation Height 4.4.1 Vertical Critical Excavation Height of Plane Slip Surface When slip surface is a plane, the stress field may be expressed in the form of Eqs. (4.2) and (4.3). According to the boundary condition, when the horizontal surface, z = 0, σz = τx z = 0; when the vertical slope, x = 0, σx = τx z = 0; thus, it is easy to know that the slip surface that Eqs. (4.2) and (4.3) meet the boundary condition is the plane through the z = Hc on the vertical surface; thereinto, Hc is as below: Hc =

π ϕ 2c tan + γ 4 2

(4.37)

Namely, meaning of Hc in Eq. (4.37) is as follows (Fig. 4.3a): The stress field meets both the boundary condition of the horizontal surface and τx z = 0 on the vertical boundary surface; however, for σx , the following equation is true: ⎧ ⎨ < 0 z < Hc σx = = 0 z = Hc ⎩ > 0 z > Hc In other words, the soil mass is in the limit state only when a tension is applied to the vertical slope surface; because no tension is applied to the vertical slope surface, the soil mass is stable; when z = Hc is true, the soil mass is in the limit state. Therefore, Hc in Eq. (4.37) is referred to as the vertical critical excavation height in the field failure mode. σs

σx Hc Hc

(a) Vertical Excavation Fig. 4.3 Schematic diagram for critical excavation height

(b) Slope Excavation

4.4 Application Example: Critical Excavation Height

89

4.4.2 Plane–Helicoid Slip Surface Soil mass enclosed by the slip surface and the slope surface is divided into (A) and (B) (Fig. 4.4); according to Eqs. (4.2) and (4.3) and surface boundary condition, the vertical force and horizontal force on the side of (A) at the interface of (A) and (B) are as below, respectively: ⎫ pz = −γ h  x ⎬ −γ x − 2h  c px =√ ⎭ h  = 1 + λ2 + λ

(4.38)

The stress field when the slip surface family in (B) is a logarithmic helical curved surface has been given as Eqs. (4.31)–(4.33); thereinto, the arbitrary functions f b (s) and f a (v) shall be determined according to the boundary condition. f a (v) = A0 v + A1 v2 is taken; thus on the interface h B = x/s0 of (A) and (B), according to the continuity condition of the stress field, the horizontal force and vertical force determined according to the stress component at the interface of (A) and (B) shall be equal, namely: 

 (B)  (A) 1 = σx h B − τx z σx h B − τx z 1 + h 2B 1 + h 2B

(4.39)

 (B)  (A) 1 σz − h B τx z σz − h B τx z = 1 + h 2B 1 + h 2B

(4.40)



1

1

Thus,  1 + s2 c 1 0 q + + γ zR exp(2λ arctan s0 ) 2 λ 1 + λ2  2 γ λ(s0 + λ) 1 + s0 A1 = exp(3λ arctan s0 ) 1 + 9λ2 1 + λ2

A0 =

Fig. 4.4 Schematic diagram for calculation of critical excavation height

(4.41)

(4.42)

90

4 Stress Field Method for Limit Analysis

In order to make the slip surface smooth (continuity of first derivative at the interface), the following equation shall be true: s0 = λ −



1 + λ2

Hence, in (B):     exp(−2λ arctan s) s+λ c + γ (z − z R ) 1 − 3λ + A0 1 + λ2 + (s − λ)2 2 λ 1 + 9λ 1 + s2  exp(−3λ arctan s)  1  + A1 (z − z R ) 1 + λ2 + 2(s − λ)2 + f b (s) (4.43) 2 z − zR 1+s

σz = −

λ(1 + 3sλ) 1 + 9λ2    exp(−2λ arctan s) + A0 1 + λ2 s − (s − λ)(1 + sλ) 1 + s2   exp( − 3λ arctan s)  + A1 (z − z R ) 1 + λ2 s − 2(s − λ)(1 + sλ) √ 1 + s2 s + f b (s) (4.44) z − zR

τx z = γ (z − z R )

c 1 − sλ + γ (z − z R ) λ 1 + 9λ2    exp(−2λ arctan s) + A0 1 + λ2 s 2 + (1 + sλ)2 1 + s2   exp( − 3λ arctan s)  + A1 (z − z R ) 1 + λ2 s 2 + 2(1 + sλ)2 √ 1 + s2 s2 + f b (s) z − zR

σx = −

(4.45)

f b (s) = z R f b1 (s) + γ z 2R f b2 (s) is taken, and thus, on the slope surface h s = x tan β; it shall meet the boundary condition (τs = 0, σs = 0). According to the following condition (see Fig. 4.3b):     τs = (σz − τx z tan β) tan β − (σx tan β − τx z ) / 1 + tan2 β = 0 f b (u) may be solved. According to the following condition: ⎧ ⎨ < 0 z < Hc     σs = (σz − τx z tan β) + tan β(σx tan β − τx z ) / 1 + tan2 β = 0 z = Hc ⎩ > 0 z > Hc Critical excavation height in the field failure mode is obtained:

4.4 Application Example: Critical Excavation Height

c c Hc = Ns = γ γ

91

tan β(s0 − s1 )(1 − A2 g(s1 ))      λg(s1 ) A2 (1 − s1 tan β) − 1 − 3s1 λ + A3 (1 + s12 ) / 1 + 9λ2

 min

(4.46) where g(s1 ) = 1 − λ(1 − s1 tan β)/(s1 + tan β)

(4.47)

1 1 + s02 exp[2λ(arctan s0 − arctan s1 )] 2 1 + λ2  λ(s0 + λ) 1 + s02 A3 = exp[3λ(arctan s0 − arctan s1 )] 1 + λ2 1 + s12 A2 =

(4.48)

(4.49)

Stability coefficient Ns is the minimum s1 (or θ1 ) in Eq. (4.46). The calculated results are detailed in Table 4.2. It is easy to know that when the slop surface is a vertical surface, the calculated results (β = 90◦ ) in this table are less than the results calculated according to Eq. (4.46); in other words, for the vertical side slope, plane–common helicoid slip surface is better than the planar slip surface. Table 4.2 Stability coefficient Ns under the field failure condition of plane–common helicoid slip surface β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

2.179

2.379

2.603

2.853

3.137

3.462

3.841

4.288

85.0

2.393

2.636

2.912

3.227

3.592

4.022

4.538

5.169

80.0

2.608

2.901

3.239

3.633

4.099

4.663

5.359

6.243

75.0

2.828

3.180

3.591

4.081

4.675

5.413

6.353

7.595

70.0

3.056

3.476

3.976

4.585

5.343

6.311

7.593

9.361

65.0

3.292

3.793

4.402

5.161

6.132

7.418

9.191

11.775

60.0

3.539

4.137

4.880

5.832

7.091

8.825

11.344

15.272

55.0

3.799

4.515

5.428

6.634

8.292

10.690

14.406

20.742

50.0

4.076

4.938

6.071

7.622

9.857

13.294

19.087

30.284

45.0

4.376

5.422

6.846

8.885

11.999

17.189

27.022

49.994

40.0

4.705

5.991

7.816

10.581

15.138

23.620

42.774

105.097

35.0

5.076

6.684

9.091

13.016

20.202

35.967

84.065

30.0

5.510

7.572

10.887

16.868

29.679

66.835

25.0

6.041

8.795

13.684

23.964

52.717

20.0

6.747

10.680

18.829

41.190

15.0

7.818

14.215

31.841

92

4 Stress Field Method for Limit Analysis

It needs to be noted that when the slip surface plane, the vertical excavation height obtained by limit equilibrium method (Chap. 5) is twice those obtained by stress field method. Moreover, the general slope excavation height of the common helicoid slip surface by the limit equilibrium method and upper bound solution method [3] (limit analysis method) is also almost twice the results of plane–common helicoid slip surface by stress field method, which is the difference between the surface failure mode and the field failure mode (see Chap. 7).

4.5 Limit Load in Field Failure Mode 4.5.1 Stress of Foundation Bottom Similarly, possible failure area may be divided into (A), (B), and (C) (Fig. 4.5). (A): common helicoid slip surface:h  = [λ(z − z R ) − (x − x R )]/[z − z R + λ(x − x R )]. (B): common helicoid slip surface: h  = (λz − x)/(z  + λx). (C): plane slip surface: h  = h c = − tan θ2 = − tan π4 − ϕ2 . Interface of (A) and (B): z = h A = x tan θ1 or h A = x/s 1 ; and (x R , z R ) is located in the extension line of this interface so that the first derivatives of the slip surface are equal to each other. Interface of (B) and (C): z = h B = −h c x = x tan θ2 or h B = x/s2 . Generally, when (x R , z R ) is above the soil mass surface (Fig. 4.5a), it is relatively suitable to retroversion failure mode; when (x R , z R ) is below the soil mass surface (Fig. 4.5b), it is relatively suitable to anteversion failure mode. However, the retroversion failure and anteversion failure also are only different at the center point of the common helicoid in (A); when (A) is a plane slip surface, it may be regarded as a special case of common helicoid slip surface; thus, general forms of the calculation equation may be considered. (C) (passive failure area): The stress field is shown as Eq. (4.4) and vertical stress on the interface of (B) and (C) is obtained:

(A)

(C ) (B)

(a) ( xR , z R ) above the Soil Mass Surface Fig. 4.5 Possible failure areas

(A)

(B)

(C )

(b) ( xR , z R ) below the Soil Mass Surface

4.5 Limit Load in Field Failure Mode



93

 (C)   1 σz − h B τx z q − γ h c x = 2 1 + hc 1 + h 2 1

(4.50)

B

 (C)     1 1  σx h B − τx z − q − γ h c x / h c + 2c = 1 + h 2 c 1 + h 2

(4.51)

B

(B): the stress field is shown as Eqs. (4.31)–(4.33); x R = 0, z R = 0, and f a (v) = A0 v + A1 v2 , f b = 0 are taken. Thus, on the interface h B = x/s2 of (B) and (C), according to the continuity condition of the stress field, horizontal and vertical forces determined by the stress components in (B) and (C) shall be equal to each other; namely 

 (B)  (C) 1 σx h B − τx z σx h B − τx z = 1 + h 2B 1 + h 2B 1

 (B)  (C) 1 1  σz − h B τx z σz − h B τx z = 1 + h 2B 1 + h 2B where h  B is the slope of the interface, h B = 1/s2 . Hence, c  1 + s22 1 q+ exp(2λ arctan s2 ) 2 λ 1 + λ2  2 γ λ(s2 + λ) 1 + s2 A1 = exp(3λ arctan s2 ) 1 + 9λ2 1 + λ2 A0 =

(4.52)

(4.53)

In order to make the slip surface smooth (continuity of first derivative), the following equation shall be true: s2 = λ +



1 + λ2 = tan

π 4

+

ϕ 2

(4.54)

Thus, stress field in (B) may be determined as below:   s+λ c σz = − + γ z 1 − 3λ λ 1 + 9λ2   exp(−2λ arctan s) + A0 1 + λ2 + (s − λ)2 1 + s2   exp(−3λ arctan s) + A1 z 1 + λ2 + 2(s − λ)2 √ 1 + s2

(4.55)

94

4 Stress Field Method for Limit Analysis

  λ(1 + 3sλ) + A 0 [ 1 + λ2 s 2 1 + 9λ exp(−2λ arctan s) − (s − λ)(1 + sλ)] 1 + s2   exp(−3λ arctan s)  + A1 z 1 + λ2 s − 2(s − λ)(1 + sλ) √ 1 + s2

τx z = γ z

1 − sλ c +γz λ 1 + 9λ2    exp(−2λ arctan s) + A0 1 + λ2 s 2 + (1 + sλ)2 1 + s2    exp(−3λ arctan s) + A1 z 1 + λ2 s 2 + 2(1 + sλ)2 √ 1 + s2

(4.56)

σx = −

(4.57)

For any h  , the following equation will be derived after the substitution in the yield function:      2  1  −K h h (1 + sλ) + s − λ + K (1 + sλ) + s − λ f = 1 2 1 + h 2 (1 + sλ) (4.58) γz 2 A0 λ 2λ + (1 + sλ) exp(−2λ arctan s) 2 1 + 9λ 1 + s2 A1 z (4λ − s + 3sλ2 ) exp(−3λ arctan s) +√ 1 + s2     γz K2 = 2λ 2λ + s + 3sλ2 − A1 z 1 + λ2 1 + s 2 exp(−3λ arctan s) 2 1 + 9λ K1 =

(A): f a ( ) = A2 v + A3 v2 is taken in Eqs. (4.31)–(4.33). According to the continuity condition of the interface of (A) and (B), the following equation is true: A3 = A1

(4.59)



γ (1 − 3s1 λ) exp(2λ arctan s1 ) 1 + 9λ2   +2 A1 1 + s12 exp(−λ arctan s1 )

A2 = A0 + z R

In order to get smooth slip surface, s1 = x R /z R shall be true. Therefore, the stress field in (A) is shown as below:   s+λ c σz = − + γ (z − z R ) 1 − 3λ λ 1 + 9λ2

(4.60)

4.5 Limit Load in Field Failure Mode

95

  exp(−2λ arctan s) + A2 1 + λ2 + (s − λ)2 1 + s2  exp(−3λ arctan s)  1 + A1 (z − z R ) 1 + λ2 + 2(s − λ)2 + f b (s) (4.61) √ 2 z − zR 1+s λ(1 + 3sλ) 1 + 9λ2    exp(−2λ arctan s) + A2 1 + λ2 s − (s − λ)(1 + sλ) 1 + s2   exp(−3λ arctan s)  + A1 (z − z R ) 1 + λ2 s − 2(s − λ)(1 + sλ) √ 1 + s2 s + f b (s ) (4.62) z − zR

τx z = γ (z − z R )

   exp(−2λ arctan s) c 1 − sλ + γ (z − z R ) + A2 1 + λ2 s 2 + (1 + sλ)2 2 λ 1 + 9λ 1 + u2   exp(−3λ arctan s)  + A1 (z − z R ) 1 + λ2 s 2 + 2(1 + sλ)2 √ 1 + s2 2 s + f b (s) (4.63) z − zR

σx = −

If the following equation is taken: ! f b (s) =

−z 2R

"   2γ λ s + 2λ + 3sλ2 1 + λ2    − A1 √ exp(−3λ arctan s) 1 + 9λ2 1 + s 2 1 + s2

(4.64)

For any h  , the following equation will be derived after the substitution in the yield function:    2  1  −K 3 h  (1 + sλ) + s − λ + K 4 h  (1 + sλ) + s − λ f = 2 1 + h (1 + sλ) (4.65) 1  γ 2λ (z − z R )2 − z 2R 1 + 9λ2 z − z R 1 + sλ  + z R (z − z R )(1 − 3s1 λ) exp(2λ(arctan s1 − arctan s)) 1 + s2 2 A0 λ (1 + sλ) exp(−2λ arctan s) + z R (1 − 3sλ)]} + 1 + s2 ⎧ ⎡  1 + s12 1 ⎨ A1 4λ(1 + sλ)(z − z R )⎣z − z R + z R exp(λ(arctan s − arctan s1 ) +√ 1 + s2 1 + s2 z − z R ⎩    − s 1 + λ2 [(z − z R )2 − z 2R exp(−3λ arctan s)

K3 =

K4 =

      1 γ 2λ  2 2 2 exp(−3λ arctan s) (z − z )2 − z 2 1 + s 2λ + s + 3sλ − A 1 + λ 1 R R z − z R 1 + 9λ2

96

4 Stress Field Method for Limit Analysis

Thus, on the foundation surface, because z = 0, K 4 = 0; if K 3 ≥ 0, the stress component will be the extremum condition in accordance with the yield function. The slip surface determined in this way is expected to be close to the most dangerous slip surface so that the obtained limit load is close to a genuine solution. Thus, on the surface of (A), it is supposed as below: s0 = −(x − x R )/z R = s1 − x/z R

(4.66)

    exp(−2λ arctan s0 ) s0 + λ c + A2 1 + λ2 + (s0 − λ)2 − γ z R 1 − 3λ 2 λ 1 + 9λ 1 + s02   exp(−3λ arctan s0 ) 1  − A1 z R 1 + λ2 + 2(s0 − λ)2 − f b (s0 ) (4.67) zR 1 + s02

σz0 = −

   exp(−2λ arctan s0 ) 1 + 3s0 λ + A2 1 + λ2 s0 − (s0 − λ)(1 + s0 λ) 2 1 + 9λ 1 + s02    exp(−3λ arctan s0 ) s0  − A1 z R 1 + λ2 s0 − 2(s0 − λ)(1 + s0 λ) − f b (s0 ) z 2 R 1 + s0

τx z0 = −γ z R

(4.68) After further rearrangement: c c  + q+ A zq + γ x A zγ λ λ  c Atq + γ x Atγ = q+ λ

σz0 = − τx z0

(4.69) (4.70)

where

Ar =

  A zq = Aq 1 + λ2 + (s0 − λ)2

(4.71)

   Atq = Aq 1 + λ2 s0 − (s0 − λ)(1 + s0 λ)

(4.72)

  A zγ = Ar 1 + λ2 + (s0 − λ)2

(4.73)

   Atγ = Ar 1 + λ2 s0 − (s0 − λ)(1 + s0 λ)

(4.74)

1 + s22   exp[2λ(arctan s2 − arctan s0 )] Aq =  2 1 + λ2 1 + s02

(4.75)

1 1    {(1 − 3s1 λ) exp[2λ(arctan s1 − arctan s0 )] 2 s1 − s 0 1 + 9λ 1 + s02

4.5 Limit Load in Field Failure Mode

−(1 − 3s0 λ) + A

97

   1 + s12 exp(−λ(arctan s1 − arctan s0 )) − 1 + s02 (4.76)



2λ(s2 + λ)  1 + s22 exp[3λ(arctan s2 − arctan s0 )] A=  1 + λ2

(4.77)

Equations (4.69) and (4.70) are general forms for distribution of the limit load on the foundation bottom (−B, 0), where parameters s0 , s 1 are undetermined. Determination of 2 parameters is discussed, and calculation equation for the limit load is given below.

4.5.2 Calculation Mode 1 The slip surface (family) is shown in Fig. 4.6. (A): θ0 ≤ θ ≤ θ1 ; slip surface is as below: x − x R = −R A exp(−λ θ ) cos θ z − z R = −R A exp(−λ θ ) sin θ

 (4.78)

It is noted that (x R , z R ) is a point on the interface (extension line) of (A) and (B) and the slip surface is through the point (−b, 0); hence, ⎫ x R = b sin θ0 cos θ1 / sin(θ1 − θ0 ) ⎬ z R = b sin θ0 sin θ1 / sin(θ1 − θ0 ) ⎭ R A = b exp(λθ0 ) sin θ1 / sin(θ1 − θ0 ) After it is substituted in Eq. (4.78), the slip surface will be as below:

Fig. 4.6 Schematic diagram for calculation mode 1

(4.79)

98

4 Stress Field Method for Limit Analysis

x = sin(θb1 −θ0 ) {sin θ0 cos θ1 − exp[λ(θ0 − θ )] sin θ1 cos θ } z = sin(θb1 −θ0 ) {sin θ0 sin θ1 − exp[λ(θ0 − θ )] sin θ1 sin θ }

 (4.80)

Because the slip surface must be in the foundation, z ≥ 0 shall be true; namely, sin θ0 − exp[λ(θ0 − θ )] sin θ ≥ 0 must be true; considering monotone increasing characteristics, only π2 − ϕ ≤ θ0 ≤ θ ≤ θ1 ≤ π − ϕ needs to be true: (B): θ1 ≥ θ ≥ θ2 ; the slip surface will be as below: x = R B exp[λ(θ1 − θ )] cos θ z = R B exp[λ(θ1 − θ )] sin θ

 (4.81)

where RB =

b {sin θ0 − exp[λ(θ0 − θ1 )] sin θ1 } sin(θ1 − θ0 ) π ϕ θ2 = − 4 2

(4.82)

(C): the slip surface will be as below: z = z 2 − tan θ2 (x − x2 )

(4.83)

In the above slip surface, another parameter b is the distance from any point on the foundation bottom to the front toe; for each b, a slip surface is always provided with; when all of values within 0 ∼ B are taken for b, a slip surface family will be constituted. If the boundary condition of the known foundation bottom is known as below:  c (4.84) τx z0 = tan δb σz0 + λ According to Eqs. (4.69) and (4.70) and considering h  |z=0 ≥ 0 , thus, only the following equation needs to be true:  % &  λ − tan δ π b θ0 = − arctan λ − 1 + λ2 2 λ + tan δb

(4.85)

  √1−(tan δb /λ)2 −tan δb If it is written as tan α − ϕ2 = , θ0 = α + π2 − ϕ will be true. 1+tan δb / sin ϕ It is noted that arctan s = π2 − θ is true, thus Eq. (4.69) may be written as below: pz = σz0 = − where

c c  + q+ Nq + γ bNγ λ λ

(4.86)

4.5 Limit Load in Field Failure Mode



Nq = 

99

sin2 θ0 λ exp[2λ(θ0 − θ2 )] λ + tan δb sin2 θ2

(4.87)

2λ 1 + λ2 sin2 θ0  {sin θ0 (sin θ1 − 3λ cos θ1 ) exp[2λ(θ0 − θ1 )] Nγ =  2 1 + 9λ (λ + tan δb ) sin(θ1 − θ0 )   (4.88) − sin θ1 (sin θ0 − 3λ cos θ0 ) + A sin θ0 exp(λ(θ1 − θ0 )) − sin θ1

A=

2λ(cos θ2 + λ sin θ2 )   exp[3λ(θ0 − θ2 )] 1 + λ2 sin2 θ2

(4.89)

This is the calculation equation for the limit load in the calculation mode 1, where Nq is same as the common equation [2], but irrelevant of θ1 . Because θ0 and θ2 have been determined, another parameter θ1 in the equation is to be determined.

4.5.3 Calculation Mode 2 The slip surface (family) is shown in Fig. 4.7. (A): θ0 ≥ θ ≥ θ1 ; the slip surface will be shown as below: x = sin(θb0 −θ1 ) {− sin θ0 cos θ1 + exp[λ(θ0 − θ )] sin θ1 cos θ } z = sin(θb0 −θ1 ) {− sin θ0 sin θ1 + exp[λ(θ0 − θ )] sin θ1 sin θ }

 (4.90)

And θ2 ≤ θ1 ≤ θ ≤ θ0 ≤ π − ϕ must be met. (B): θ1 ≥ θ ≥ θ2 ; the slip surface will be shown as below: x = R B exp[λ(θ1 − θ )] cos θ z = R B exp[λ(θ1 − θ )] sin θ

Fig. 4.7 Schematic diagram for calculation mode 2

 (4.91)

100

4 Stress Field Method for Limit Analysis

Table 4.3 Distribution of limit load along bottom surface of foundation bed (q/(γ B) = 0.1, c/(γ B) = 0.2, ϕ = 25◦ ) b/B

tan δ 0.0

0.1

0.2

0.3

0.4

0.0

5.210

4.309

3.407

2.536

1.683

0.1

6.233

5.047

3.915

2.864

1.869

0.2

7.255

5.784

4.423

3.193

2.056

0.3

8.263

6.495

4.899

3.488

2.216

0.4

9.213

7.145

5.316

3.734

2.339

0.5

10.135

7.770

5.715

3.968

2.455

0.6

11.041

8.384

6.105

4.196

2.568

0.7

11.940

8.992

6.491

4.421

2.679

0.8

12.833

9.595

6.874

4.644

2.790

0.9

13.723

10.196

7.255

4.866

2.899

1.0

14.611

10.795

7.634

5.087

3.009

10.055

7.696

5.651

3.919

2.422

  Pz / γ B 2

where RB =

b {− sin θ0 + exp[λ(θ0 − θ1 )] sin θ1 } sin(θ0 − θ1 )

sin θ0 − exp[λ(θ0 − θ1 )] sin θ1 ≤ 0 must be met. (C): Slip surface is the same as the anteversion failure. Similarly, Eqs. (4.86)–(4.89) may be derived; attention shall be paid to difference of θ1 . This calculation mode is similar to the slip surface family by characteristic line method; when tan δb = tan δ, the limit load very close to the characteristic line method may be obtained by properly selecting θ1 ; for example, θ1 = π2 − 0.5(1 − η)[δ + (1 − η)ϕ] ≤ θ1 ≤ θ0 , where η = (qλ + c)/(γ b) and distribution pz /(γ B) of the calculated limit load is detailed in Table 4.3.

4.5.4 Calculation Mode 3 In the above two calculation modes, if θ1 = θ0 is true, the slip surface in (A) will be retrograded into a plane (Fig. 4.8). Corresponding calculation equations for Nq , Nc keep constant; Nγ is the same at different positions (b) of the foundation bottom, and it shall meet the following equation:

4.5 Limit Load in Field Failure Mode

101

Fig. 4.8 Schematic diagram for calculation mode 3

   2λ 1 + λ2 sin2 θ0   Nγ =  3λ − 2λ sin2 θ0 + 6λ2 cos θ0 sin θ0 + A(λ sin θ0 − cos θ0 ) 2 1 + 9λ (λ + tan δ)

(4.92) According to the calculation mode 3, when tan δb = tan δ, the calculated results in case of filed failure are generally as follows: When ϕ is relatively large, the calculated limit load will be apparently larger. If without the action of the horizontal force, namely tan δb = tan δ = 0, the equation will be retrograded as below: pz (x) = q Nq + cNc + γ bNγ

(4.93)

ϕ 2

(4.94)

where Nq = exp(π λ) tan2

π 4

+

  Nc = Nq − 1 /λ

(4.95)

     π ϕ  ϕ 3π λ 2 π  tan Nγ =  + 3 tan + − 1 exp λ 4 2 4 2 2 1 + 9λ2 π ϕ  + (4.96) +3 − tan2 4 2 If two-way failure is considered, the limit load must be symmetric around the center point b = B/2 of the foundation:

p(x) =

0 ≤ b ≤ B/2 q Nq + cNc + γ bNr q Nq + cNc + γ (B − b)Nr B/2 ≤ b ≤ B

(4.97)

102

4 Stress Field Method for Limit Analysis

4.5.5 Total Limit Load Because it is difficult to give the distribution of tan δb along the foundation bottom (b), in actual problems, the relationship between total horizontal force and total vertical force on the foundation bottom is considered for most boundary condition of the foundation bottom. And the total limit load on the foundation bottom is calculated. When the action of the horizontal force is considered, generally tan δ can be given as below: '0

'0 τx z0 dx =

−B

−B

c dx = tan δ tan δb σz + λ 

'0  c dx σz0 + λ

(4.98)

−B

When tan δb = tan δ, total load on the foundation bottom is as below: '0 Pz = −B

  c  γ c Nq + B Nγ σz0 dx = B − + q + λ λ 2

(4.99)

For calculation mode 1, the below is taken: !

" 3δ  /(1 + η), where η = (qλ + c)/(γ B).  θ1 = θ0 − 0.5 θ0 − θ2 − (1 + tan δ) 1 + 4λ2

Calculated result of the total limit load may be obtained; see Table 4.4 (Pm =   Pz / γ B 2 , where P0 is the calculated result by characteristic line method. It shows that the result is very consistent with that by the characteristic line method. For different q/(γ B), c/(γ B), the calculated total limit load is also very consistent with that by the characteristic line method (Tables 4.5 and 4.6). For calculation mode 2, θ1 also can be properly selected so that the calculated total limit load is very close to the result by characteristic line method. Moreover, the slip surface in the calculation mode 2 will be discussed by other method, but its calculated result is not given here.

4.5.6 Limit Load in Surface Failure Mode (0   If the moment equation −B σz0 (x − x R ) + τx z0 z R dx showing the relationship between the force on the foundation bottom and the point (x R , z R ) is applied, calculation equation for the total limit load identical to that by limit equilibrium method may be derived. Actually, according to Eqs. (4.69) and (4.70), the following equation will be derived:

2.422

3.731

5.991

10.11

18.14

35.12

74.94

181.6

10

15

20

25

30

35

40

45

184.4

76.60

36.07

18.69

10.44

6.196

3.859

2.501

1.669

Note The maximum relative error is 5.4%

1.628

5

1.221

123.4

52.96

25.63

13.58

7.704

4.601

2.837

1.736

Pm

P0

Pm

1.212

0.1

0.0

tan δ

1

ϕ(◦ )

127.2

54.81

26.56

14.08

7.990

4.772

2.942

1.796

P0

81.88

36.46

18.14

9.796

5.599

3.297

1.888

Pm

0.2

Table 4.4 Calculated result of total limit load (q/(γ B) = 0.1, c/(γ B) = 0.2)

85.43

38.04

18.90

10.18

5.802

3.408

1.945

P0

53.41

24.56

12.48

6.788

3.817

2.065

Pm

0.3

56.20

25.74

13.02

7.041

3.937

2.117

P0

34.45

16.23

8.325

4.456

2.295

Pm

0.4

36.43

17.05

8.674

4.604

2.350

P0

4.5 Limit Load in Field Failure Mode 103

25.61

56.85

142.4

35

40

45

145.1

57.40

25.73

12.67

6.719

3.776

2.223

1.356

0.844

Note The maximum relative error is 5.0%

6.697

3.746

20

12.64

2.186

15

30

1.319

10

25

0.816

5

0.573

94.54

39.13

18.14

9.156

4.919

2.761

1.585

0.890

Pm

P0

Pm

0.563

0.1

0.0

tan δ

1

ϕ(◦ )

97.67

39.99

18.41

9.255

4.977

2.811

1.634

0.931

P0

61.07

26.12

12.41

6.358

3.426

1.884

0.990

Pm

0.2

Table 4.5 Calculated result of total limit load (q/(γ B) = 0.0, c/(γ B) = 0.1)

63.69

26.88

12.65

6.448

3.475

1.924

1.023

P0

38.60

16.96

8.193

4.208

2.215

1.100

Pm

0.3

40.50

17.53

8.371

4.271

2.247

1.124

P0

24.00

10.75

5.208

2.611

1.235

Pm

0.4

25.26

11.12

5.323

2.651

1.255

P0

104 4 Stress Field Method for Limit Analysis

235.9

102.7

50.68

27.54

16.14

10.04

6.565

4.465

3.137

Note The maximum relative error is 3.9%

235.2

26.78

30

45

15.67

25

49.49

9.759

20

101.0

6.394

15

40

4.369

10

35

3.093

5

2.407

163.2

73.11

37.02

20.55

12.24

7.684

4.985

3.213

Pm

P0

Pm

2.398

0.1

0.0

tan δ

1

ϕ(◦ )

166.0

75.03

38.16

21.22

12.63

7.907

5.112

3.277

P0

110.9

51.65

26.92

15.24

9.142

5.656

3.410

Pm

0.2

Table 4.6 Calculated result of total limit load (q/(γ B) = 0.2, c/(γ B) = 0.4)

114.1

53.37

27.84

15.73

9.408

5.796

3.474

P0

74.32

35.78

19.07

10.88

6.421

3.657

Pm

0.3

77.00

37.11

19.72

11.20

6.575

3.719

P0

49.35

24.39

13.13

7.378

4.002

Pm

0.4

51.36

25.31

13.55

7.563

4.068

P0

4.5 Limit Load in Field Failure Mode 105

106

4 Stress Field Method for Limit Analysis

'0

  σz0 (x − x R ) + τx z0 z R dx =

−B

'0 γ zR c − (x − x R ) − (1 − 3s0 λ)(x − x R + λz R ) λ 1 + 9λ2

−B

⎫ ⎤ ⎬ 1 2z R exp(−2λ arctan s0 ) − A1  exp(−3λ arctan s0 )⎦(s0 − λ)[s0 (x − x R ) − z R ] dx + ⎣ A2 2 ⎭ 1 + s0 1 + s2 ⎡

0

It is noted that x −x R = s0 z R and A0 , A1 are unrelated to x; the following equation will be derived after A1 , A2 are substituted and rearranged: '0 −b

  σz0 (x − x R ) + τx z0 z R dx =

 c   B2 sin2 θ0 − sin2 θ1 − 2 2λ sin (θ1 − θ0 )

 c  1 + s22  2 1 q+ sin θ0 exp(2λ(θ1 − θ2 )) − sin2 θ1 exp(2λ(θ0 − θ2 )) 4 λ 1 + λ2      γB  3 sin3 θ0 2λ cos θ1 − 1 + 3λ2 sin θ1 +  2 6 1 + 9λ sin(θ1 − θ0 )     − 3 sin3 θ1 2λ cos θ0 − 1 + 3λ2 sin θ0   + 3 sin θ0 (sin θ1 − 3λ cos θ1 ) sin2 θ0 − sin2 θ1 exp(2λ(θ0 − θ1 ))   + 2 A sin3 θ0 − 3 sin θ0 sin2 θ1 exp(2λ(θ0 − θ1 )) + 2sin3 θ1 exp(3λ(θ0 − θ1 )) (4.100) +

where A=

 λ(s2 + λ) 1 + s22 exp[3λ(θ1 − θ2 )] 1 + λ2

Equation (4.100) is the same as the equation derived by the limit equilibrium method, and the calculation of the limit load in the surface failure mode by this equation will be discussed in the limit equilibrium method section.

4.6 Stress Field Along Slip Surface (Family) and Limit Load 4.6.1 General Form of Stress Field According to those described in Chap. 2, if equilibrium equation, yield condition, and extremum condition of the yield function are met, the following equation will be derived:

4.6 Stress Field Along Slip Surface (Family) and Limit Load ∂σe ∂x ∂σe ∂x







  2λ = γ h − λ σ ∂h + h  ∂h 1+h 2 e ∂ x ∂z    ∂σ  1+λh 2λ ∂h 1+λh  ∂h  1+λh  e = γ + σ + e  2  λ−h ∂z 1+h ∂x λ−h ∂z λ−h 

e + h  ∂σ − ∂z

+



107

⎫ ⎬

 +λ ⎭

(4.101)

If the slip surface family is known, any equation in the above equation may be applied for the solution of σe . Because along the slip surface, the first equation in Eq. (4.101) is as below:   2λ dσe dh  − = γ h − λ σ e 2 dx 1+h dx

(4.102)

This is an ordinary differential equation, and it is easy to obtain any slip surface  along the slip surface family: ⎧ ⎫ 'xb ⎨ ⎬        σe = exp 2λ arctan h  C0 − γ h  − λ exp −2λ arctan h   dx (4.103) ⎩ ⎭ x

where C0 is a constant to be determined. After it is substituted in Eq. (4.3), general form of the stress field will be obtained. If the slip surface family h  in Eq. (4.103) also meets the second equation in Eq. (4.101), namely the slip surface family is real, the stress field will be a real stress field. Under the assumed stress field, if the slip surface family and Eq. (4.103) cannot meet the second equation in Eq. (4.101), the stress field will be an approximate stress field.

4.6.2 General Forms of Limit Load According to Eq. (4.3), the following equation will be derived: ⎫ ⎧   2  'xb ⎬     ⎨   1 + λ2 h 2 + 1 + h  λ c      σz + = C0 − [γ h − λ exp −2λ arctan h ] dx exp 2λ arctan h 2 2 ⎭ ⎩ λ 1+λ 1+h x

According to the boundary condition of the soil mass surface (when x = xb and σz = q, τx z = 0 will be true), h  = h c ; hence,       1 + λ2 1 + h 2 c c  h  = h c , C0 = q +    2 exp −2λ arctan h c λ 1 + λ2 h 2 + 1 + h  λ c c According to the boundary condition of the foundation bottom, when   x = −b, where −b is any point on the foundation bottom; τx z0 = tan δb σz0 + λc will be true, where σz0 is the limit load pz of the foundation bottom; hence,

108

4 Stress Field Method for Limit Analysis

⎫ ⎧   2  'xb ⎬     ⎨   1 + λ2 h a2 + 1 + h a λ c      pz + = − [γ h − λ exp −2λ arctan h dx C ] exp 2λ arctan h 0   a ⎭ ⎩ λ 1 + λ2 1 + h a2 −b

And h a =

   1 + λ2 λ2 − tan2 δb   , λ + tan δb 1 + 2λ2

λ(λ − tan δb ) +

Thus, general equation for the limit load is derived: pz = q Nq + cNc + γ bNγ 0 ≤ b ≤ B

(4.104)

where      h a 1 − λ/ h c   exp 2λ arctan h a − arctan h c Nq =   h a − λ + tan δb 1 + λh a Nc =

 1 Nq − 1 λ

  −h a /b   exp 2λ arctan h a Nγ =  h a − λ + tan δb 1 + λh a

'xb

(4.105) (4.106)

    [ h  − λ exp −2λ arctan h  ] dx

−b

(4.107) As long as the slip surface is determined, the equation also will be determined uniquely. As long as the boundary condition is determined, Nq , Nc can be determined, which are unrelated to the selected slip surface. If tan δb = tan δ, it is easy to prove that Nq , Nc are weightless solutions. According to this equation, it is very easy to derive the calculation equation for the limit load; see the following examples.

4.6.3 Example: Plane–Common Helicoid–Plane This example hasbeen given in the above; for the boundary when x = −b,    condition, h a = tan π4 + ϕ2 will be true; when x = xb , h c = − tan π4 − ϕ2 and tan δ = 0 will be true. The slip surface will be as below: Within [−b, −b/2], h  = h a ; within [−b/2, xb /2], h  = (λz − x)/(z + λx); within [xb /2, xb ], h  = −1/ h a ; and xb = bh a exp 21 π λ . After they are substituted in Eq. (4.92), calculated, and rearranged to derive as below: Nq = h a2 exp(λπ )

4.6 Stress Field Along Slip Surface (Family) and Limit Load

Nγ =

109

      3 2λ 2   π λ + 1 − λh h + λ exp h a 1 + 9λ2 a a 2

This is the same as Eqs. (4.94) and (4.96) derived under the same slip surface family condition. Corresponding stress field is obtained according to the stress equation along the slip surface, and its solution method is actually similar to the characteristic line method because the characteristic line method is actually used to obtain σe and the slip surface family according to Eq. (4.101). Moreover, if the slip surface family is assumed, one equation will be solved less. If the slip surface family is close to the real slip surface family, the limit load close to the real one may be obtained. For the advantage of this method, any slip surface family may be arbitrarily selected; as along as the slip surface family meets the boundary condition, only one integral needs to be calculated for the solution; compared with other solution methods, it is easier. This method is also applicable to the limit soil pressure (it will be discussed later).

4.7 Limit Load in the Plane–General Helicoid–Plane Calculation Mode The slip surface is shown in Fig. 4.9. When −b ≤ x ≤ x1 , h = (x + b)h a ;  x − x R = R exp[k(θ1 − θ)] cos θ x1 ≤ x ≤ x2 , , k = tan ψ, h  = tan(θ + ψ − π/2), h − z R = R exp[θ1 − θ)] sin θ x2 ≤ x ≤ xb = 2x2 , h = (x − xb )h c ,

Because the slip surface is smooth, the following equation shall be true: θ1 =

π π ϕ − ψ + arctan h a , θ2 = + − ψ, 2 4 2

Fig. 4.9 Schematic diagram for the plane–general helicoid–plane calculation mode

(4.108)

110

4 Stress Field Method for Limit Analysis

and R = b A0   1 + x R∗ h a + x R∗ h c   (4.109a) A0 = sin θ1 − h a cos θ1 − sin θ2 + h c cos θ2 exp[k(θ1 − θ2 )]       1 + x R∗ h a sin θ2 + h c cos θ2 exp[k(θ1 − θ2 )] − x R∗ h c sin θ1 − h a cos θ1 ∗   zR = sin θ1 − h a cos θ1 − sin θ2 + h c cos θ2 exp[k(θ1 − θ2 )] (4.109b)    ∗  ∗ (4.109c) x1 = b x R + A0 cos θ1 , x2 = b x R + A0 exp[k(θ1 − θ2 )] cos θ2 where x R∗ = x R /b, and z ∗R = z R /b. Integral in Eq. (4.107) is calculated, and the following equation is easily derived: 'xb −

[(h  − λ) exp(−2λ arctan h  )] dx

−b 







= −(b + x1 )(h a − λ) exp(−2λ arctan h a ) − x2 (h c − λ) exp(−2λ arctan h c ) 

+ R{(Ak sin θ1 − Bk cos θ1 ) exp(−2λ arctan h a ) 

− (Ak sin θ2 − Bk cos θ2 ) exp[k(θ1 − θ2 )] exp(−2λ arctan h c )}

(4.110)

where   λ 3 + k 2 + 2kλ 1 + k 2 + 2kλ − 2λ2 , Bk = Ak = 1 + (k + 2λ)2 1 + (k + 2λ)2

(4.111)

Hence,      h a   x R∗ λ − h a + λ − h c exp(2λ(θ1 − θ2 ))   h a − λ + tan δb 1 + λh a     + λ − h a + A0 Ak sin θ1 − Bk − λ + h a cos θ1      +A0 −Ak sin θ2 + Bk + λ − h c cos θ2 exp[(k + 2λ)(θ1 − θ2 )] (4.112)

Nγ =

  h a 1 − λ/ h c   exp[2λ(θ1 − θ2 )] Nq =  h a − λ + tan δb 1 + λh a

(4.113)

In the distribution equation for the limit load in this calculation mode, parameters k, x R are to be determined; if they are selected properly, the calculated results may be close to those by the characteristic line method. For example, when k = λ, x R = −b(0.4 + 0.075λ + 0.15 tan δ), calculated value of Nγ is detailed in Table 4.7. This calculated result is very close (generally speaking, slightly smaller than that by the characteristic line method) to Nγ when q/(γ B) = 0.01, c/(γ B) = 0.02 are true by the characteristic line method. When q/(γ B), c/(γ B) increase, calculated

4.7 Limit Load in the Plane–General Helicoid–Plane Calculation Mode

111

Table 4.7 Calculated result of Nγ in plane–general helicoid–plane calculation mode ϕ(◦ )

tan δ 0.0

1

0.037

5

0.251

0.1

0.2

10

0.760

0.461

15

1.767

1.196

0.639

20

3.785

2.647

1.641

0.3

0.4

0.809

25

7.947

5.599

3.642

2.124

1.002

30

16.949

11.879

7.842

4.831

2.710

35

37.785

26.102

17.187

10.773

6.389

40

90.627

61.189

39.698

24.814

14.958

45

242.186

158.384

100.116

61.488

36.854

value of Nγ by characteristic line method will also increase and Nq , Nc are identical to each other; therefore, the limit load when q/(γ B) ≥ 0.01, c/(γ B) ≥ 0.02 are calculated according to Nγ in Table 4.6 is not greater than the calculated value by the characteristic line method.

4.8 Limit Load in Helicoid–Helicoid–Plane Calculation Mode Considering the slip surface is in the helicoid–helicoid–plane shown in Fig. 4.10.

Fig. 4.10 Schematic diagram for helicoid–helicoid–plane calculation mode

112

4 Stress Field Method for Limit Analysis

x − x R = R exp(−λθ ) cos θ z − z R = R exp(−λθ ) sin θ x − x0 = R0 exp(−λθ ) cos θ (B): the slip surface is as below: z − z 0 = R0 exp(−λθ ) sin θ (C): the slip surface is as below: z = (x − xb )h c , h c = − tan θ2 , xb = 2x2 Supposing x0 = −kb, z 0 = −kb tan(θ2 ) will be true; because the slip surface is smooth, the following equations shall be true: (A): the slip surface is as below:

π π ϕ − ϕ + arctan h a , θ2 = − 2 4 2

(4.114)

R = b exp(λθ0 )A, R0 = b exp(λθ0 )A0

(4.115)

θ0 =

where A= A0 = A −

1 [(1 − k) sin θ1 + k tan θ2 cos θ1 ] sin(θ0 − θ1 )

(4.116a)

1 [(1 − k) sin θ0 + k tan θ2 cos θ0 ] exp[−λ(θ0 − θ1 )] sin(θ0 − θ1 ) (4.116b)

The integral in Eq. (4.107) is calculated to immediately obtain Nγ : Nγ =

2λ sin2 θ0 λ + tan δb

 1 + λ2  A sin θ0 − 3λ cos θ0 − (sin θ1 − 3λ cos θ1 ) exp(3λ(θ0 − θ1 )) 1 + 9λ2

  1 + λ2 A0 (sin θ1 − 3λ cos θ1 ) exp(3λ(θ0 − θ1 )) − (sin θ2 − 3λ cos θ2 ) exp(3λ(θ0 − θ2 )) 1 + 9λ2   (4.117) +(tan θ2 + λ) A0 cos θ2 exp(3λ(θ0 − θ2 )) − k exp(2λ(θ0 − θ2 )) +

Nq =

sin2 θ0 λ exp[2λ(θ0 − θ2 )] λ + tan δb sin2 θ2

(4.118)

As long as k, θ1 are determined, the slip surface may be determined. In other words, parameters k, θ1 to be determined in the equation may be selected. If they are properly selected, the calculated result may be consistent with the calculated value by the characteristic line method. For example, δ) , θ1 = π2 − 0.25ϕ − 0.5δ, the calculated value of Nγ is very k = 0.1+0.5(λ+tan 1+2.5λ2 close to that specified in Table 4.6. tan δ , the calculated total limit load Pm = k = 0, θ1 = θ0 − 2θ2 0.12+1.2λ−1.4 1+λ2  2 Pz / γ B and the depth of the slip surface Z m /B are detailed in Table 4.8. Eccentricity e/B of the limit load is detailed in Table 4.9. The calculated total limit load and depth of the slip surface are very close to those by the characteristic line method; eccentricity of the limit load is slightly greater than that (backward) by the characteristic line method because the distribution of

6.046

18.040

35.116

76.364

30.0

35.0

40.0

3.805

15.0

10.101

2.485

10.0

25.0

1.663

5.0

20.0

1.219

Pm

0.0

tan δ

1.0

ϕ(◦ )

Z m /B

1.570

1.259

1.049

0.905

0.806

0.738

0.692

0.661

0.644

53.870

25.629

13.518

7.707

4.645

2.891

1.774

Pm

0.1

1.316

1.053

0.870

0.737

0.633

0.537

0.408

Z m /B

37.121

18.214

9.807

5.632

3.340

1.919

Pm

0.2

1.089

0.867

0.706

0.579

0.461

0.309

Z m /B

25.104

12.621

6.857

3.870

2.097

Pm

0.3

0.889

0.699

0.553

0.421

0.261

Z m /B

Table 4.8 Total limit load and depth of slip surface in helicoid–helicoid–plane calculation mode (q/(γ B) = 0.1, c/(γ B) = 0.2)

16.713

8.509

4.552

2.342

Pm

0.4

0.713

0.547

0.405

0.243

Z m /B

4.8 Limit Load in Helicoid–Helicoid–Plane Calculation Mode 113

114

4 Stress Field Method for Limit Analysis

Table 4.9 Eccentricity e/B 0.1, c/(γ B) = 0.2) ϕ(◦ )

in helicoid–helicoid–plane calculation mode (q/(γ B)

=

tan δ 0.0

0.1

0.2

0.3

0.4

1.0

−0.0046

5.0

−0.0209

10.0

−0.0381

−0.0294

15.0

−0.0533

−0.0454

−0.0361

20.0

−0.0672

−0.0598

−0.0517

−0.0417

25.0

−0.0807

−0.0735

−0.0658

−0.0574

−0.0469

30.0

−0.0940

−0.0870

−0.0796

−0.0717

−0.0629

35.0

−0.1071

−0.1005

−0.0934

−0.0857

−0.0774

40.0

−0.1198

−0.1138

−0.1072

−0.0998

−0.0920

the limit load is linear. According to the calculated result by the characteristic line method, the slip surface is related to (qλ + c)/(γ b). It needs to further discuss how to select k, θ1 so that any (qλ + c)/(γ b) is very close to the calculated value by the characteristic line method.

References 1. Huang CZ, Zhang J, Sun WH (2002) Stress field and ultimate load for soil mass in limit state. Chin J Geotech Eng 24(3):389–391 2. JTJ 250-98 (1998) Code for foundation in port engineering. China Communications Press, Beijing 3. Winterkorn HF, Fang XY (1983) Foundation engineering handbook (translated by Qian H, Ye S). China Architecture and Building Press, Beijing

Chapter 5

Limit Equilibrium Method

5.1 Basic Idea of Limit Equilibrium Method One of main differences between the limit equilibrium method and the characteristic line method is as follow: for the former, the yield condition is first adopted and then expression of the safety factor or limit load is derived according to the force and moment equilibrium equations, the corresponding extremum is solved for this expression including the slip surface and finally, the required result is obtained; for the latter, yield condition and extremum condition of the yield function are directly applied during the solution. If it can prove that, the stress field or slip surface, which can guarantee that an extremum is taken for the safety factor or limit load, also meets the extremum condition of the yield function, the solution obtained by the limit equilibrium method will be a rigorous solution. However, actually, on the one hand, the expression of the safety factor or limit load derived based on the force and moment equilibrium generally contains unknown stress field, which can be eliminated only after necessary simplification; on the other hand, it is very difficult to determine a real slip surface; for the purpose of simple calculation process, slip surfaces with parameters to be determined (such as center and radius of the arc slip surface) are generally selected earlier and then corresponding extremum solution is carried out for the expression with parameters to be determined so as to obtain the required result. Obviously, according to the above aspects, generally only approximate solution can be obtained. Many scholars indicate that if the slip surface is properly selected, the result will be of sufficient accuracy; practically, many common calculations (for example, slope stability analysis) in engineering are derived by the limit equilibrium method. For the conventional limit equilibrium method, after the soil mass is divided into several soil strips or soil blocks and then force analysis is carried out for the soil strip or soil block; finally, the required result is obtained based on the force equilibrium and moment equilibrium. It must be sure that historical function of the conventional limit equilibrium method is indelible. From the practical applications, the established © Springer Nature Singapore Pte Ltd. and Zhejiang University Press, Hangzhou, China 2020 C. Huang, Limit Analysis Theory of the Soil Mass and Its Application, https://doi.org/10.1007/978-981-15-1572-9_5

115

116

5 Limit Equilibrium Method

slope stability analysis methods [such as simple slice method (Fellenius), simplified Bishop method (Bishop)], calculation equations for the limit load (such as Terzaghi’s, Meyerhof’s, Hansen’s and Vesic’s equations) and limit soil pressure equations (such as Coulomb’s equation) are still extensively applied in engineering. From the theoretical perspective, if this solution method has relatively large limitation, for example, in the slope stability and simplification or assumption has to carry out on the soil mass force before approximate results are obtained even though the slip surface is selected. Therefore, it is already very difficult to make new development under the framework of the slice method [1]. In addition, more study will not bring no material effect on the understanding of this field [2]. Here, the solution method is discussed from the equilibrium equation, yield condition, and stress relationship; moreover, such limit equilibrium method shares the same starting point as the characteristic line method and upper bound solution method for the convenience of comparison and mutual verification.

5.2 Equilibrium Equations for Force and Moment on Slip Surface For the basic idea of limit equilibrium method, force and moment equilibrium equations are applied for the solution. For any two curved surfaces in the soil mass:  A : z = h A (x),  B : z = h B (x), where h A (x) ≥ h B (x); according to the equilibrium equation, the following equations shall be true: h A  hB y A 

yB

h A  hB

 ∂σx ∂τx z + dz = 0 ∂x ∂z  ∂τx z ∂σz + − γ dz = 0 ∂z ∂x  ∂σx ∂τx z + (h A − z)dz = 0 ∂x ∂z

The above equations are calculated and the following equations are derived, respectively, according to the stress relation: (σ h  − τ ) A =

d E(h B , h A ) + (σ h  − τ ) B dx

(σ + τ h  ) A = w(h B , h A ) + (σ + τ h  ) B −

d T (h B , h A ) dx

(5.1) (5.2)

5.2 Equilibrium Equations for Force and Moment on Slip Surface

h A E(h B , h A ) − T (h B , h A ) =

d M(h B , h A ) + (h A − h B )(σ h  − τ )B dx

117

(5.3)

where, h A E(h B , h A ) =

σx dz

(5.4)

τx z dz

(5.5)

(h A − z)σx dz

(5.6)

hB

h A T (h B , h A ) = hB

h A M(h B , h A ) = hB

h A w(h B , h A ) =

γ dz

(5.7)

hB

Equations (5.1) and (5.2) are differential equations of the horizontal force and vertical force on the vertical surface h B ≤ z ≤ h A , while Eq. (5.3) is the moment differential equation of the horizontal force against z = h A on the vertical surface. Equations (5.1)–(5.3) generally may be derived according to the force analysis on the soil strip [3, 4]; however, the derived equations are differential equations, which are the same as the above differential equations essentially. Equations (5.1)–(5.3) and the yield condition are generally considered as the fundamental equations for the limit equilibrium method. However, it shall be pointed out that Eqs. (5.1) and (5.3) are derived according to the equilibrium equation of the horizontal force; thus, they are not mutually independent, which shall be noted in the process of the solution (it will be discussed in Chap. 10). For any point (x R , z R ), both sides of Eqs. (5.1) and (5.2) are multiplied by (h A −z R ) and (x − x R ), respectively; their products are added and the following equation is derived according to Eq. (5.3): σ A [h A (h A − z R ) + x − x R ] − τ A [h A − z R − h A (x − x R )] d [(h A − z R )E(h B , h A ) − (x − x R )T (h B , h A ) − M(h B , h A )] = dx + (x − x R )w(h B , h A ) + (x − x R )(σ + τ h  ) B + (h B − z R )(σ h  − τ )B (5.8) Equation (5.8) is the general form of the moment equation of the force against the point (x R , z R ) on the vertical surface h B ≤ z ≤ h A . If h B = h s , h A = h are true for the soil mass surface and slip surface, respectively (), the following equation is derived according to Eqs. (5.1) and (5.2):

118

(1 + h  λ)

5 Limit Equilibrium Method

  dE dT + (h  − λ) = (h  − λ) w(h s , h) + (σ + τ h  )Γs dx dx   − 1 + h 2 ( f + c) − (1 + h  λ)(σ h  − τ )Γs (5.9)

Equation (5.9) is the equation of the force on the vertical surface h s ≤ z ≤ h. According to the yield condition, Eq. (5.8) will be as below: d [(h − z R )E(h s , h) − (x − x R )T (h s , h) − M(h s , h)] dx   = −( f + c)[h − z R − h  (x − x R )] − (x − x R ) w(h s , h) + (σ + τ h  )s − (h s − z R )(σ h  − τ )s + σ [(h  − λ)(h − z R ) + (1 + h  λ)(x − x R )] (5.10) Because σ =

1 (w 1+h  λ

+ pz −

dT dx

− h  c), Eq. (5.10) also may be rewritten as:

d [(h − z R )E(h s , h) − (x − x R )T (h s , h) − M(h s , h)] dx 

h − z R  w(h s , h) + (σ + τ h  )s (h  − λ) − ( f + c)(1 + h 2 ) =  1+h λ 1 dT [(h  − λ)(h − z R ) + (1 + h  λ)(x − x R )] − (h s − z R )(σ h  − τ )s − 1 + hλ dx (5.11)

5.3 Calculation of Limit Load 5.3.1 Force Equilibrium-Based Limit Load Expression When h s = 0, pz (x), px (x) are limit loads to be determined within the foundation width [−B, 0] and q is the uniformly distributed vertical load on the soil mass surface (−b, 0). : z = h is the potential slip surface and it is used to derive the expression of the limit load (functional). By applying the force equilibrium equation, according to the yield condition, Eqs. (5.1) and (5.2) are rewritten as below, respectively: [σ (h  − λ) − c] =

d E(0, h) − px dx

[σ (1 + h  λ) + ch  ] = γ h + pz −

d T (0, h) dx

(5.12) (5.13)

5.3 Calculation of Limit Load

119

For the integral within the horizontal area from any point at the foundation bottom to the soil exit point (xb , 0), it is noted that E = 0, T = 0 are true at the soil entry point and soil exit point, 0

xb pz dx = −

−b

xb qdx −

xb γ hdx +

−b

0

[σ (1 + h  λ) + h  c] dx

(5.14)

−b

0

xb px dx = −

−b

[σ (h  − λ) − c] dx

(5.15)

−b

This is the expression of the limit load, thereinto, it contains unknown slip surface 0 0 and normal stress. Supposing Jz = −b pz dx and Jx = −b px dx, it will be the functional of the limit load.

5.3.2 Force and Moment Equilibrium-Based Limit Load Expression For the integral in Eq. (5.10) within the horizontal area from any point (−b, 0) within the foundation width to the soil exit point (xb , 0), it is noted that E = 0, T = 0, M = 0 are true at the soil entry point and soil exit point; thus, the following equation is derived: 0

xb (−x pz + h s px )dx =

xb xqdx +

−b

xb γ xhdx −

−b

0

[σ (h  h + x) − τ (h − h  x)] dx

−b

Or, 0

xb (−x pz + h s px )dx =

−b

xb xqdx +

xb



c(h − h x)dx + −b

0

xb −



γ xhdx −b



xb

σ [h h + x − λ(h − h x)] dx + −b

f (h − h  x) dx

−b

(5.16) where, x = x − x R and h = h − z R 0 Supposing J = −B (−x pz + h s px )dx, the following equation is derived:

120

5 Limit Equilibrium Method

xb J=

xb xqdx +

0



xb

c(h − h x)dx + −b

xb γ xhdx −

−b

σ [h  h + x − λ(h − h  x)] dx

−b

(5.17) This is the limit load expression based on the force and moment equilibrium. In theory, Eq. (5.17) may be applied to calculate the limit loads pz , px . If the slip surface  in the equation is that in the limit state and the stress σ ( f = 0) on the slip surface is that in the limit state, the calculated values of pz , px are the exact solutions of the limit load. In practical problems, it is difficult to obtain the stress field and slip surface in accordance with the equilibrium equation, yield condition, and extremum condition of the yield function. Generally, the approximation of the limit load can be calculated only when the slip surface has been selected. Therefore, how to select a proper slip surface to make the calculated approximation of the limit load much closer to the actual limit load is one of the main tasks with regard to the limit equilibrium method. In order to make the selected slip surface much closer to the real slip surface, the possible failure area is generally divided into several subareas; in order to eliminate unknown normal stress in the limit load expression, corresponding slip surface in each subarea is selected for the discussion and it is determined force equilibrium or force and moment equilibrium in different areas is applied to calculate the approximation of the limit load according to different slip surfaces selected in different areas. Below are examples for this solution thought.

5.3.3 Example: Limit Load of Slip Surface in Plane–Helicoid–Plane Form The possible failure area is divide into (A), (B), and (C) (Fig. 5.1) √ and slip surface in (A) is supposed as the plane h = h  (b + x), h  = h a = λ + 1 + λ2 , that in (B) as the helicoid slip surface h  = (λz − x)/(z √ + λx) and that in (C) as the plane slip surface h = h  (x − x2 ), h  = h c = λ − 1 + λ2 .

Fig. 5.1 Schematic diagram for calculation mode of plane–helicoid–plane

5.3 Calculation of Limit Load

121

Interface of (A) and (B): z = h a = −h a x, h a = tan θ1 = tan( π4 + ϕ2 ); interface of (B) and (C): z = h c = −h c x, h c = − tan θ2 = − tan( π2 − ϕ2 ). Additionally, the boundary condition of the ground surface is given as the load boundary condition: in the foundation bottom, σz = pz (x), τx z = 0; in other areas, σz = q, τx z = 0. Thereinto, the foundation bottom load pz is the required limit load. Under the above conditions, calculation equation for the bearing capacity is derived. (A): within [−b, x1 ], the following equations are derived according to the equilibrium equation: (σ h a − τ )1 =

dE dx

(5.18)

(σ + τ h a )1 = γ h a (x + b) + pz −

dT dx

(5.19)

According to the above equations, noting that 1 + 2h a λ − h a2 = 0, the following equation will be derived: 2ch a = γ h a (x + b) + pz − h a

dT dE − dx dx

(5.20)

Similarly, within [x1 , 0], the following equation is true: σn (1 + h a2 ) = −γ h a x + pz − h a

dT dE − dx dx

(5.21)

where, σn is the nominal stress on the interface of (A) and (B). Equations (5.20) and (5.21) are integrated and added within [−b, x1 ] and [x1 , 0], hence: 0

x1 pz dx = −

−b

σn (1 +

h a2 )dx

0

¨ −

x1 γ dzdx +

A

2ch a dx

(5.22)

−b

(B): because the slip surface meets the equation h  h + x = (h − h  x)λ, the following equation will be true: Within [x1 , 0], −c(h − h  x) = σn (1 + h a2 )x + xw(h a , h) +

d [h E(h a , h) − x T (h a , h) − M(h a, h)] dx

(5.23)

Within [0, x2 ],

122

5 Limit Equilibrium Method

−c(h − h  x) = σm (1 + h 2 c )x + xw(h c , h) +

d [h E(h c , h) − x T (h c , h) − M(h c, h)] dx

(5.24)

where, σm is the normal stress on the interface of (B) and (C). When x = 0, h a = h c = 0. Hence, the following equation is derived according to Eqs. (5.23) and (5.24): 0 σn (1 +

h a2 )xdx

x2 =−

x1

σm (1 +

h 2 c )xdx−

0

¨

x2 γ xdzdx −

c(h − h  x)dx

x1

B

(5.25) Equation (5.25) is the moment equation of the force of the soil mass in (B) against the point (0, 0). (C): according to similar discussion in (A), the following equation is derived: 0 σm (1 + x2

h 2 c )dx

¨

xb =−

qdx + 0

xb γ dzdx +

C

2ch c dx

(5.26)

x2

Thus, σm may be obtained according to Eq. (5.26) and σn is obtained after it is substituted in Eq. (5.25) and then the calculation equation for limit load is derived after it is substituted in Eq. (5.22). The limit load is calculated below according to these steps. ) and xb = Considering the following relationship: x1 = −b/2, x2 = b2 h a exp( πλ 2 2x2 .   According to Eq. (5.26), it is easy to get σm (1 + h 2 c ) = 2(q − ch c − γ h c x 2 ); because x2 ≤ x B /2 is arbitrary, the following equation is true:   σm (1 + h 2 c ) = 2(q − ch c − γ h c x)

(5.27)

After it is substituted in Eq. (5.25), through calculation and rearrangement, the following equation will be true:    c σn 1 + h a2 = 2qh a2 exp(π λ) + 2h a2 exp(π λ) − 1 − h a2 λ

     2  3 λ  2  h 3h a − 1 exp π λ + 3 − h a + 2h a x + 2γ 1 + 9λ2 a 2 (5.28) Thus, the total limit load within the foundation width [−b, 0] is obtained after it is substituted in Eq. (5.22):

5.3 Calculation of Limit Load

0 −b

123

1 pz dx = bq Nq + bcNc + b2 γ Nγ 2

(5.29)

where, Nq = h a2 exp(λπ )

(5.30)

1 (Nq − 1) λ

   2λ 3 2   Nγ = h (h + λ) exp π λ + 1 − h a λ 1 + 9λ2 a a 2 Nc =

(5.31) (5.32)

Derivation (for b) on both sides of the integral equation is carried out and it is noted that b is any point in the foundation bottom; the limit load in the one-way failure mode is obtained as below: pz = q Nq + cNc + γ bNγ 0 ≤ b ≤ B

(5.33)

The limit load in the two-way failure mode must be symmetrical with respect to the foundation center point b = B/2, hence: pz =

0 ≤ b ≤ B/2 q Nq + cNc + γ bNγ q Nq + cNc + γ (B − x)Nγ B/2 ≤ b ≤ B

(5.34)

Distribution of the limit load is shown in Fig. 5.2. Distribution of the limit load given in Eqs. (5.33) and (5.34) is the approximate result in the field failure mode, which is the same as the calculation equation derived by the stress field method. pz

pz

(a) One-way Failure Fig. 5.2 Schematic diagram for distribution of the limit load

(b) Two-way Failure

124

5 Limit Equilibrium Method

5.4 Slope Stability 5.4.1 Equivalence of the Extremum Condition of the Yield Function and the Extremum Condition of the Safety Factor The yield function of the slope stability is as below: f =τ−

1 (σ λ + c) Fs

(5.35)

When the yield condition is met, the following equation is true: Fs =

1 (σ λ + c) τ

(5.36)

The yield function is rewritten as below: f =

1 [(σz − h  τx z )(h  − λ F ) − (σx h  − τx z )(1 + h  λ F ) − c F (1 + h 2 )] 1 + h 2 (5.37)

The necessary condition to take the extremum for the safety factor is as below: fh =

2 1 [ (σz − σx )(1 + 2h  λ F − h 2 ) − τx z (2h  − λ F + h 2 λ F )] = 0 2 2 (1 + h ) 2 (5.38)

where, λ F = λ/Fs and c F = c/Fs . It must be noted that the yield condition and extremum condition of the yield function here are only true on the slip surface, while f = f 0 ≤ 0 shall be true in areas except the slip surface. Similarly, (1 + h  λ F )

  dE dT + (h  − λ F ) = (h  − λ F ) w(h s , h) + (σ + τ h  )Γs dx dx − (1 + h 2 )( f + c F ) − (1 + h  λ F )(σ h  − τ )Γs (5.39)

d [(h − z R )E(h s , h) − (x − x R )T (h s , h) − M(h s , h)] dx   = −( f + c F )[h − z R − h  (x − x R )] − (x − x R ) w(h s , h) + (σ + τ h  )s − (h s − z R )(σ h  − τ )s + σ [(h  − λ F )(h − z R ) + (1 + h  λ F )(x − x R )] (5.40)

5.4 Slope Stability

125

Or, d [(h − z R )E(h s , h) − (x − x R )T (h s , h) − M(h s , h)] dx  h − zR  = { w(h s , h) + (σ + τ h  )s (h  − λ F ) − ( f + c F )(1 + h 2 )} 1 + hλF 1 dT − (h s − z R )(σ h  − τ )s − [(h  − λ F )(h − z R ) + (1 + h  λ F )(x − x R )] 1 + hλF dx

(5.41)

5.4.2 Expression of Safety Factor General slope is shown in Fig. 5.3; for the integral in Eq. (5.10) within the horizontal area from the slope entry point (a, h a ) to the slope exit point (b, h b ), it is noted that E = 0, T = 0, M = 0 are true at the slope entry point and the slope exit point. Hence: b 0=

{−( f + c F )[h − z R − h  (x − x R )] − (x − x R )[w(h s , h) + (σ + τ h  )s ]

a

− (h s − z R )(σ h  − τ )s + σ [(h  − λ F )(h − z R ) + (1 + h  λ F )(x − x R )]}dx (5.42) Fig. 5.3 Schematic diagram for general slope and slip surface

126

5 Limit Equilibrium Method

b a

1 [ f (h − h x)] dx = − Fs 

b −

b

[(σ λ + c)(h − h  x)] dx

a

{x[w(h s , h) + pz ] − h s px − σ (h  h + x)}dx (5.43)

a

If f = 0 is true, the safety factor is solved as below: b

Fs =  b a

(σ tan ϕ + c) (h − h  x)dx MR = b M0 {−x[ pz + w(h s , h)] + h s px }dx + a σ (h  h + x)dx a

(5.44)

where, pz = (σz − h s τx z )s and px = (τx z − σx h s )s are known surface loads; x = x − xR, h = h − zR. If the slip surface  in the equation is a real slip surface, the stress on the slip surface will be that σ ( f = 0) in the limit state and the safety factor Fs may be calculated according to Eq. (5.44). Similarly, if it meets the equilibrium equation in the slip mass, it does not violate the yield criterion; however, it is very difficult to get the stress field and slip surface in accordance with the yield condition and extremum condition of the yield function on the slip surface and generally only the approximation of the safety factor can be obtained. How to eliminate the unknown σ in the equation according to the extremum condition of the yield function and select suitable slip surface so that the approximation of the calculated safety factor is much closer to the real safety factor, is one of main tasks for the limit equilibrium method.

5.4.3 Examples Below are examples for the simplest slope with homogeneous soil. Example I: Safety Factor and Vertical Critical Excavation Height of Plane Slip Surface For the simplest slope in Fig. 5.4a, the slip surface is taken as a plane with the slope entry point and slope exit point of x0 and x N , respectively; according to Eq. (5.39) as well as E = 0 and T = 0 at x0 and x N , respectively, the following equation will be derived: x N {(h  − λ F )[w(h s , h) + pz ] − (1 + h 2 )c F + (1 + h  λ F ) px }d x = 0 x0

Hence, the calculation equation for safety factor may be derived:

(5.45)

5.4 Slope Stability

127

Fig. 5.4 Schematic diagram for slip surface

 xN Fs =

x0

{[w(h s , h) + pz ] tan ϕ + (1 + h 2 )c}dx  xN  x0 {[w(h s , h) + pz ]h + px }dx

(5.46)

If Fs = 1.0, pz = px = 0 are true and the slope is a vertical surface, supposing the excavation height H = h N − z 0 , hence: H=

c 2(1 + h 2 ) γ h  − tan ϕ

(5.47)

The minimum of h  is calculated according to the above equation, hence: Hc =

π ϕ 4c tan( + ) γ 4 2

(5.48)

For the slip surface through any point on the vertical surface, Fs ≥ 1.0 are true; and thus this is the vertical critical excavation height in the surface failure mode. Example II: Safety Factor and Vertical Critical Excavation Height of Helicoid Slip Surface (Fig. 5.4b) The slip surface in accordance with the following equation is taken: h  (h − z R ) + (x − x R ) = [h − z R − h  (x − x R )]λ F

(5.49)

This is an ordinary differential equation and its general solution is a logarithm helical curve with relatively simple polar form: x − x R = R exp(−λ F θ ) cos θ h − z R = R exp(−λ F θ ) sin θ



where, R is an arbitrary constant and x R , z R are helical center points.

(5.50)

128

5 Limit Equilibrium Method

According to Eq. (5.42), as along as the slip surface is taken as the logarithm helically curved surface determined by any three undetermined constants x R , z R and R, the calculation equation for the safety factor may be derived: Fs = M R /M0 x N MR =

{[(h − h s )γ + pz ] tan ϕ + c}[h − z R − h  (x − x R )]dx

(5.51)

(5.52)

x0

x N M0 =

{[(h − h s )γ + pz ]h  (h − z R ) + (h s − z R ) px }dx

(5.53)

x0

M R , M0 also may be rewritten in other forms, for example: x N

c[h − z R − (x − x R )h  ]dx

(5.54)

{[(h − h s )γ + pz ](x R − x) + (h s − z R ) px }dx

(5.55)

MR = x0

x N M0 = x0

According to the derivation process, except that the slip surface is required to be a logarithm helically curved surface, any assumption or simplification is not required to be carried out for any force on the slip mass. During the actual calculation, the minimum safety factor is required to be calculated for which the calculation process may be absolutely similar to that by common arc slip method; different x R , z R and R are selected for trial calculation, respectively, so as to obtain the minimum safety factor and the corresponding slip surface. In addition, because both the slip surface equation and right side of Eq. (5.51) contain Fs , functional equation solution needs to be adopted during the calculation of Fs , for example, iterative method, which is the same as the calculation process by simplified Bishop method. Under some simple conditions, the function expression may be calculated according to the integral in the equation. For example, for the simple slope shown in Fig. 5.5, if external force is not considered, it is easy to calculate: Fs =

c f1 γ R · f2

(5.56)

where, f 1 and f 2 may be expressed in the polar form as below:   f 1 = s12 + c12 − s02 − c02 )/(2λ F

(5.57)

5.4 Slope Stability

129

Fig. 5.5 Schematic diagram for stability calculation of slope with homogeneous soil

 1   1 3 s1 − s03 + s0 c12 − c02 3 2     λ F  2 + s0 + c02 (3λ F s0 + c0 ) − s12 + c12 (3λ F s1 + c1 ) 2 1 + 9λ F   1 1 2 1 (5.58) c − s tan β − − s + ) ) (s (s 1 0 1 1 0 tan2 β 2 6

f2 =

where, s1 = exp(−λ F θ1 ) sin θ1 , c1 = exp(−λ F θ1 ) cos θ1 s0 = exp(−λ F θ0 ) sin θ0 , c0 = exp(−λ F θ0 ) cos θ0

 (5.59)

According to H = R(s1 − s0 ), hence: Fs =

c (s1 − s0 ) f 1 γH · f2

(5.60)

The calculated results are detailed in Table 5.1a–c. Similarly, the so-called rigorous solution [5] given based on the upper bound method may be given here. In fact, according to H = R(s1 − s0 ) and Fs = 1.0, supposing the stability factor and critical slope height as Ns and Hc , respectively, hence: Hc =

c Ns γ

(5.61)

where, N = (s1 − s0 ) f 1 / f 2 Ns = min{N (θ0 , θ1 )} Hc = min{H }

(5.62)  (5.63)

130

5 Limit Equilibrium Method

Table 5.1 a Safety factor of helicoid slip surface Fs (c/(γ H ) = 0.05), b Safety factor of helicoid slip surface Fs (c/(γ H ) = 0.1), c Safety factor of helicoid slip surface Fs (c/(γ H ) = 0.2) a β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

0.266

0.324

0.375

0.422

0.466

0.510

0.555

0.601

80.0

0.307

0.380

0.446

0.508

0.568

0.630

0.694

0.762

70.0

0.349

0.439

0.521

0.600

0.679

0.759

0.844

0.936

60.0

0.395

0.505

0.606

0.705

0.805

0.908

1.018

1.138

50.0

0.446

0.581

0.708

0.832

0.959

1.091

1.232

1.388

40.0

0.508

0.679

0.840

1.000

1.163

1.336

1.521

30.0

0.592

0.818

1.032

1.248

1.469

20.0

0.731

1.059

1.376

b β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

0.464

0.533

0.596

0.656

0.714

0.771

0.831

0.893

80.0

0.530

0.615

0.693

0.769

0.843

0.919

0.998

1.082

70.0

0.597

0.700

0.796

0.889

0.982

1.077

1.178

1.287

60.0

0.667

0.792

0.908

1.022

1.137

1.256

1.382

1.520

50.0

0.742

0.895

1.038

1.179

1.322

1.471

1.629

1.804

40.0

0.828

1.019

1.199

1.378

1.560

1.750

1.964

30.0

0.936

1.187

1.426

1.662

1.907

20.0

1.104

1.466

1.812

c β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

0.851

0.929

1.003

1.075

1.146

1.219

1.295

1.375

80.0

0.967

1.061

1.152

1.240

1.329

1.420

1.516

1.619

70.0

1.083

1.196

1.305

1.413

1.521

1.633

1.752

1.883

60.0

1.199

1.336

1.468

1.598

1.730

1.867

2.016

2.182

50.0

1.318

1.487

1.648

1.808

1.970

2.140

2.324

2.535

40.0

1.447

1.658

1.861

2.062

2.266

2.479

2.775

30.0

1.598

1.877

2.143

2.407

2.716

20.0

1.813

2.215

2.598

5.4 Slope Stability

131

The calculated results are detailed in Table 5.2. They are all the same as the results [5] by the limit analysis method (small difference lies in the error of the numerical calculation). Example III: Safety Factor and Vertical Critical Excavation Height of Plane– Helicoid Slip Surface The soil mass enclosed by the slip surface and slope surface is divide into (A) and (B) (Fig. 5.6); in (A), the slip surface is a plane; in (B), it is a helicoid slip surface; moreover, the interface of (A) and (B) are z − z R = h s (x − x R ) or z = h s x. (A): the following equation is true: 0 − x1

=

{σ [h a − λ F − h s (1 + λ F h a )] + τ [1 + λ F h a + h s (h a − λ F )]}s dx γ c (1 + h a2 )(x1 − a) − (h a − λ F )[h a (x1 − a)2 − h s x12 ] Fs 2

(5.64)

Because h a (x1 − a) = h s x1 is true and it is noted that it is correct when x1 in the equation is changed as any x(x1 ≤ x ≤ 0), thus, the following equation is true on s : Table 5.2 Stability factor Ns of helicoid slip surface β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

4.190

4.583

5.018

5.505

6.056

6.687

7.421

85.0

4.502

4.971

5.498

6.099

6.793

7.607

8.581

8.288 9.771

80.0

4.818

5.375

6.012

6.751

7.624

8.675

9.969

11.608

75.0

5.140

5.799

6.567

7.477

8.577

9.940

11.677

13.970

70.0

5.469

6.249

7.176

8.299

9.696

11.485

13.857

17.152

65.0

5.807

6.732

7.854

9.253

11.048

13.439

16.774

21.724

60.0

6.158

7.258

8.629

10.391

12.743

16.035

20.939

28.915

55.0

6.526

7.844

9.537

11.799

14.972

19.712

27.448

41.887

50.0

6.920

8.515

10.642

13.628

18.098

25.413

39.109

71.486

45.0

7.351

9.310

12.053

16.161

22.896

35.540

65.520

185.499

40.0

7.839

10.298

13.972

19.999

31.333

58.275

166.387

35.0

8.414

11.606

16.828

26.655

50.059

144.201

30.0

9.136

13.499

21.690

41.215

119.920

25.0

10.124

16.630

32.108

94.608

20.0

11.667

23.128

69.384

15.0

14.677

45.488

132

5 Limit Equilibrium Method

Fig. 5.6 Schematic diagram for plane–helicoid

σs [h a − λ F − h s (1 + λ F h a )] + τs [1 + λ F h a + h s (h a − λ F )] c h h = (1 + h a2 ) s − γ (h a − λ F )(h s − h a ) s x Fs ha ha

(5.65)

Assuming h s = −(1 + λ F h a )/(h a − λ F ). It is easy to prove that the slip surface is smooth at the interface of (A) and (B), namely h  = h a . Thus, Eq. (5.65) is as below: σs (1 +

h 2 s )

   c 1 + λ F h a  =− +γx   1 + h a2 Fs h a (h a − λ F )2

(5.66)

(B): the following equation is true. 0 [σ (x − x R )(1 +

h 2 s )]Γs dx

b =−

x1

x1

¨ −

 c [z − z R − h  (x − x R )] dx Fs Γ2

γ (x − x R )dzdx

(5.67)

B

It is substituted in Eq. (5.66) and the integral is calculated and rearranged, hence: Fs =

 c (s1 − c1 tan θ0 ) tan β f 1 γH tan β − tan θ0 f 2 min

(5.68)

where,   f 1 = s12 + c12 − s02 − c02 )/(2λ F ) − As (k 2 − c02 /2 f2 =

(5.69)

     λ F  2 1 3 s1 − s03 + s0 + c02 (3λ F s0 + c0 ) − s12 + c12 (3λ F s1 + c1 ) 2 3 1 + 9λ F

5.4 Slope Stability

133

    1 1 1 − k tan θ0 c12 − c02 + tan β kc12 + c13 − k 3 2 3 6   1 1 1 − (As + tan θ0 ) kc02 + c03 − k 3 2 3 6 s1 = exp(−λ F θ1 ) sin θ1 , c1 = exp(−λ F θ1 ) cos θ1 s0 = exp(−λ F θ0 ) sin θ0 , c0 = exp(−λ F θ0 ) cos θ0

(5.70)  (5.71)

k = (s1 − c1 tan β)/(tan β − tan θ0 ) As =

(5.72)

1 + λ F h a (1 + h a2 ) − λ F )2

(5.73)

h a (h a

Calculated results are detailed in Tables 5.3a–b. Compared with safety factor of the helicoid slip surface, they only have small difference. Table 5.3 a Safety factor of plane–helicoid Fs (c/(γ H ) = 0.1), b safety factor of plane–helicoid slip surface Fs (c/(γ H ) = 0.2) a β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

0.447

0.518

0.581

0.641

0.699

0.757

0.816

0.878

80.0

0.511

0.598

0.678

0.753

0.829

0.905

0.984

1.069

70.0

0.577

0.683

0.780

0.875

0.968

1.065

1.166

1.276

60.0

0.648

0.776

0.895

1.011

1.127

1.247

1.374

1.512

50.0

0.725

0.881

1.027

1.170

1.314

1.464

1.624

1.799

40.0

0.814

1.009

1.192

1.372

1.556

1.746

1.951

30.0

0.926

1.182

1.422

1.660

1.902

20.0

1.099

1.464

1.812

b β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

0.816

0.896

0.971

1.045

1.117

1.190

1.267

1.347

80.0

0.926

1.024

1.116

1.206

1.297

1.389

1.486

1.590

70.0

1.038

1.157

1.269

1.379

1.489

1.603

1.723

1.853

60.0

1.155

1.298

1.434

1.568

1.701

1.841

1.988

2.148

50.0

1.277

1.453

1.619

1.781

1.946

2.118

2.300

2.498

40.0

1.412

1.631

1.838

2.041

2.248

2.464

2.693

30.0

1.571

1.856

2.127

2.393

2.664

20.0

1.795

2.203

2.589

134

5 Limit Equilibrium Method

Table 5.4 Stability factor Ns of plane–helicoid slip surface under plane failure condition β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

3.995

4.387

4.819

5.302

5.847

6.470

7.193

85.0

4.286

4.755

5.253

5.877

6.565

7.372

8.335

8.047 9.511

80.0

4.586

5.144

5.780

6.517

7.314

8.430

9.716

11.346

75.0

4.895

5.559

6.328

7.238

8.336

9.697

11.430

13.720

70.0

5.217

6.005

6.936

8.063

9.463

11.254

13.630

16.930

65.0

5.552

6.489

7.620

9.027

10.830

13.231

16.578

21.545

60.0

5.905

7.022

8.406

10.182

12.550

15.859

20.786

28.793

55.0

6.281

7.621

9.333

11.615

14.812

19.576

27.348

41.825

50.0

6.686

8.309

10.462

13.476

17.976

25.323

39.057

71.490

45.0

7.134

9.128

11.902

16.044

22.813

35.497

65.528

185.694

40.0

7.642

10.142

13.854

19.918

31.294

58.277

166.513

35.0

8.241

11.481

16.745

26.615

50.064

144.255

30.0

8.990

13.408

21.646

41.220

120.054

25.0

10.009

16.576

32.112

94.675

20.0

11.587

23.115

69.433

15.0

14.638

45.546

Similarly, critical height of the slope excavation may be obtained as below:  c c (s1 − c1 tan θ0 ) tan β f 1 Hc = Ns = γ γ tan β − tan θ0 f 2 min

(5.74)

The calculated results are detailed in Table 5.4. Similarly, they are very close to the stability factor of the helicoid slip surface.

5.5 Calculation of Soil Pressure 5.5.1 Active Soil Pressure (Fig. 5.7a) Supposing that the straight wall is located at b : x = 0 and 0 ≤ z ≤ H , for the integral in Eq. (5.10) within the horizontal area from the rear wall (−a, 0) to the straight wall, it is noted that E = 0, T = 0, M = 0 are true at the soil entry point, hence:

5.5 Calculation of Soil Pressure

135

a

b

H

H

(a) Active Soil Pressure

(b) Passive Soil Pressure

Fig. 5.7 Schematic diagram for calculation of soil pressure

H

H (z − z 0 )σx1 dz + x0

0

0 τx z1 dz = −

0 x pz dx −

−a

0

0 +



0

c(h − h x)dx − −a

xwdx −a

σ [h  h + x − λ(h − h  x)]dx

−a

0 −

f (h − h  x)dx

(5.75)

−a

If the slip surface is in the limit state, the stress on the slip surface shall be the stress σ, f = 0 in the limit state and the limit state solutions (exact) σx1 , τx z1 of the active soil pressure on the wall surface shall meet the requirements of Eq. (5.75). Example When c = 0 and an uniformly distributed load q is applied on the horizontal surface, the slip surface is taken as a plane, according to Eq. (5.11), it is easy to derive the following equation: (1 + λh  )E + (h  − λ)T = (h  − λ)



 1 γ H 2 + q H / h 2

(5.76)

If T = E tan δ, the following equation will be true:   1 h − λ 2 γ H + qH E=  h [1 + λh  + (h  − λ) tan δ] 2

(5.77)

  The  maximum of h is calculated according to the above equation and h = 2) λ + λ(1+λ is derived; thus, the total soil pressure on the straight wall is as below: λ+tan δ

  1 1 2 γ H + qH E= √ √ [ 1 + λ2 + λ(λ + tan δ)]2 2

(5.78)

136

5 Limit Equilibrium Method

This is the total soil pressure in the plane failure mode and the soil pressure distributed along the straight wall in the field failure mode is easy to be obtained below: 1 (γ z + q) σx1 = √ √ 2 [ 1 + λ + λ(λ + tan δ)]2

(5.79)

Apparently, the total soil pressure in the field failure mode is the same as that in the surface failure mode; this is Rankine’s soil pressure equation.

5.5.2 Passive Soil Pressure (Fig. 5.7b) Similar to those described in the active soil pressure, the following equation is true: H

H (z − z 0 )σx2 dz + x0

0

b τx z2 dz =

0

b x pz dx +

0

−

b

c(h − h x)dx + 0

b



xwdx 0

σ [h  h + x − λ(h − h  x)]dx

0

b +

f (h − h  x)dx

(5.80)

0

If the slip surface is in the limit state and the stress on the slip surface is the stress σ ( f = 0) in the limit state, the (exact) solutions σx2 , τx z2 of the passive soil pressure in the limit state on the wall surface shall be in accordance with Eq. (5.80). Calculation equation for the soil pressure is derived similarly to that for the limit load, in order to make the selected slip surface much closer to the real slip surface, the possible failure area is divided into multiple subareas and the corresponding slip surface is selected for discussion of every subarea.

5.6 Calculation Equation for Limit Load 5.6.1 Calculation Mode 1 The calculation mode [6] shown in Fig. 5.8 is discussed below with specific helicoid– specific helicoid–plane slip surface.

5.6 Calculation Equation for Limit Load

137

Fig. 5.8 Schematic diagram for calculation mode 1

(1) Moment equation Moment equation in (A) (A): θ0 ≤ θ ≤ θ1 ; within [−b, x1 ], according to the moment equation, the following equation is true: (σ h  − τ )(h − z R ) + (σ + τ h  )(x − x R ) d [(z − z R )E(h s , h) − (x − x R )T (h s , h) − M(h s, h)] = dx + [w(h s , h) + pz ](x − x R ) − (h s − z R ) px If, h  (z − z R ) + (x − x R ) = [z − z R − h  (x − x R )]λ

(5.81)

The following equation will be derived: − ( f + c)(z − z R − h  (x − x R )) d [(z − z R )E(h s , h) − (x − x R )T (h s , h) − M(h s, h)] = dx + [w(h s , h) + pz ](x − x R ) − (h s − z R ) px

(5.82)

And the slip surface in accordance with Eq. (5.81) is the following specific helicoid:  x − x R = −R A exp[λ(θ1 − θ )] cos θ (5.83) z − z R = −R A exp[λ(θ1 − θ )] sin θ

138

5 Limit Equilibrium Method

It is noted that (x R , z R ) is a point on the extension line of the interface of (A) and (B) and the slip surface passes through the point (−b, 0), hence: z R = x R tan θ1 z R = (b + x R ) tan θ0

 (5.84)

If x R , z R are given, θ0 , θ1 may be solved, vice versa. In other words, there are two parameters to be determined. And, RA =

 (b + x R )2 + z 2R exp[−λ(θ1 − θ0 )] x1 = x R − R A cos θ1 z 1 = z R − R A sin θ1

(5.85)

 (5.86)

Within [x1 , 0]: (σa h a − τa )(h a − z R ) + (σa + τa h a )(x − x R ) d [(h a − z R )E(h s , h a ) − (x − x R )T (h s , h a ) − M(h s, h a )] = dx + [w(h s , h a ) + pz ](x − x R ) − (h s − z R ) px where, σa and τa are normal stress and tangential stress on the interface of (A) and (B), respectively. The interface of (A) and (B) is h a − z R = h a (x − x R ) (or h a = x tan θ1 ), hence: d [(h a − z R )E(h s , h a ) − (x − x R )T (h s , h a ) − M(h s, h a )] dx + [w(h s , h a ) + pz ](x − x R ) − (h s − z R ) px (5.87)

σa (1 + h a2 )(x − x R ) =

When x = x1 , h a = h will be true. Both sides of Eqs. (5.82) and (5.87) are integrated within [−b, x1 ] and [x1 , 0], added and rearranged, hence: x1 −

c[h − z R − h  (x − x R )]dx +

−b

0

σa (1 + h a2 )(x − x R )dx

x1

x1

0 (x − x R )w(h s , h)dx +

= −b

(x − x R )w(h s , h a )dx x1

0 [(x − x R ) pz − (h s − z R ) px ]dx

+ −b

(5.88)

5.6 Calculation Equation for Limit Load

139

It is easy to observe that it is the moment equation showing the relationship between the force on the soil mass in (A) and the point (x R , z R ). Moment equation in (B) (B): θ2 ≤ θ ≤ θ1 ; similarly, if the slip surface meets the following equation: h  z + x = (z − h  x)λ

(5.89)

The slip surface will be derived as below: x = R B exp[λ(θ1 − θ )] cos θ z = R B exp[λ(θ1 − θ )] sin θ

 (5.90)

where, RB =



x R2 + z 2R − R A

(5.91)

And: x2 = R B exp[λ(θ1 − θ2 )] cos θ2 z 2 = R B exp[λ(θ1 − θ2 )] sin θ2

 (5.92)

Apparently, the slip surface is smooth at the interface of (A) and (B). Similar to those discussed in (A), the following equation is true: x2



0

c(z − h x)dx =

− x1

σa (1 +

h a2 )xdx

0 +

x1

xw(h a , h)dx x1

x2 σb (1 +

+ 0

h 2 b )xdx

x2 +

xw(h b , h)dx

(5.93)

0

This is the moment equation showing the force of the soil mass in (B) and the point. Force equation in (C) The slip surface is a plane: z = h c = z 2 + h c (x − x2 )

(5.94)

If h c = − tan θ2 = −h b and θ2 = π4 − ϕ2 are true, it will be easy to verify that the slip surface determined according to the above equation is also smooth (continuity of first derivative).

140

5 Limit Equilibrium Method

Corresponding force equation will be as below: x2

σb (1 + h 2 b )dx =

xb

c(h b − h c )dx +

x2

0

x2

xb [w(h s , h b ) + q]dx +

0

[w(h s , h c ) + q]dx x2

(5.95) Thus, when the slip surface is subject to Eqs. (5.94), (5.90) and (5.83), σb may be calculated according to Eq. (5.95); σa and the limit load can be calculated after σb and σa are substituted in Eqs. (5.93) and (5.88), respectively. Thereinto, when different values are taken for the two parameters to be determined x R and z R (or θ0 and θ1 ), the slip surface will be different. According to the above derivation process, only the slip surface is selected and no any assumption or simplification is made. (2) Calculation Equation for Limit Load Equations (5.95), (5.93), and (5.88) are calculated, respectively, below. It is noted that Eq. (5.95) will be correct when x2 is changed as any x(0 ≤ x ≤ x2 ), hence:   σb (1 + h 2 b ) = 2(ch b + q) + 2γ h b x

(5.96)

After it is substituted in Eq. (5.93) and calculated, hence: 0

σa (1 + h a2 )xdx = −

x1

x12 c {exp[2λ(θ1 − θ2 )] − 1} 2λ cos2 θ1

γ x13 {exp[3λ(θ1 − θ2 )](3λ cos θ2 − sin θ2 ) 3(1 + 9λ2 ) cos3 θ1 x2 − (3λ cos θ1 − sin θ1 )} − (ch b + q) 21 exp[2λ(θ1 − θ2 )] cos2 θ2 cos θ1 2γ h b x13 − exp[3λ(θ1 − θ2 )] cos3 θ2 3 cos3 θ1

−

Similarly, the above equation will be correct when x1 is changed as any x(x1 ≤ x ≤ 0), hence: σa (1 + h a2 ) = A0 (θ1 , θ2 ) + A1 (θ1 , θ2 )γ x A0 (θ1 , θ2 ) = A1 (θ1 , θ2 ) =

c   1  c 2 + 2 + q exp[2λ(θ − − θ )] cos θ 1 2 2 cos2 θ1 λ λ 1 (1 +

9λ2 ) cos3

θ1

(5.97) (5.98)

{8λ exp[3λ(θ1 − θ2 )] cos3 θ2 (1 + λtgθ2 )

5.6 Calculation Equation for Limit Load

141

− (3λ cos θ1 − sin θ1 )}

(5.99)

Finally, the calculation equation for the limit load is derived. Equation (5.97) is substituted in Eq. (5.88), hence: 0

x1 [(x − x R ) pz + z R px ]dx = −

−b

c[h − z R − h  (x − x R )]dx

−b

0 (x − x R )[A0 (θ1 , θ2 ) + γ x A1 (θ1 , θ2 )]dx

+ x1

x1

0 γ (x − x R )hdx −

− −b

γ (x − x R )h a dx

(5.100)

x1

The integral on the right side is calculated, hence: 0 −b

2 b 1 [q Aq (θ0 , θ1 , θ2 ) + c Ac (θ0 , θ1 , θ2 )] [(x − x R ) pz + z R px ]dx = sin(θ1 − θ0 ) 2  b3 γ Aγ (θ0 , θ1 , θ2 ) (5.101) + 6

where, Aq (θ0 , θ1 , θ2 ) =

2 cos2 θ2 {exp[2λ(θ1 − θ2 )] sin2 θ0 − exp[2λ(θ0 − θ2 )] sin2 θ1 } sin(θ1 − θ0 ) (5.102)

Ac (θ0 , θ1 , θ2 ) =

1 1 [Aq (θ0 , θ1 , θ2 ) − (sin2 θ0 − sin2 θ1 )] λ sin(θ1 − θ0 )

(5.103)

1 {A1 (θ1 , θ2 ) cos3 θ1 [sin θ0 − sin θ1 exp(λ(θ0 − θ1 ))]2 sin (θ1 − θ0 ) [sin θ0 + 2 sin θ1 exp(λ(θ0 − θ1 ))]

Aγ (θ0 , θ1 , θ2 ) =

2

2 sin3 θ1 [(sin θ1 − 3λ cos θ1 ) exp(3λ(θ0 − θ1 )) 1 + 9λ2 − (sin θ0 − 3λ cos θ0 )] − sin θ0 sin θ1 (sin2 θ0 − sin2 θ1 )} (5.104)

−

Equation (5.101) is the same as that derived by the stress field method. Left side of Eq. (5.101) is considered below; if the ratio tan δb of the total horizontal limit load within the foundation width [−b, 0] to the total vertical limit load and the eccentricity eb are known:

142

5 Limit Equilibrium Method

0 px dx = tan δb −b

0 

c dx λ

pz +

−b

0

 x+

−b

 b − eb pz dx = 0 2

(5.105)

(5.106)

Thus, 0 −b

b [(x − x R ) pz + z R px ]dx = [A z (θ0 , θ1 ) 2 sin(θ1 − θ0 )

0

−b

c pz d x + 2 tan δ sin θ0 sin θ1 ] λ

A z (θ0 , θ1 ) = − sin(θ1 − θ0 )(1 − 2eb /b) − 2 sin θ0 (cos θ1 − tan δb sin θ1 ) (5.107) Thus, total limit load within the foundation width [−b, 0] is obtained: 0 Pzb = −b

  bγ Nγ b pz dx = b q Nqb + cNcb + 2

(5.108)

where, ⎫ ⎪ Nqb = Aq (θ0 , θ1 , θ2 )/A z (θ0 , θ1 ) ⎬ 2 θ0 −sin2 θ1 − 2 tan δ sin θ sin θ ]/A (θ , θ ) Ncb = λ1 [Aq (θ0 , θ1 , θ2 ) − sinsin(θ 0 1 z 0 1 −θ ) 1 0 ⎪ ⎭ Nγ b = 23 Aγ (θ0 , θ1 , θ2 )/A z (θ0 , θ1 ) (5.109) The two parameters θ0 and θ1 (θ2 is known) in the equation are determined by the minimum of Eq. (5.108) within the range of π/2 − ϕ ≤ θ0 ≤ π − ϕ and θ0 ≤ θ1 ≤ π − ϕ. The subscript b of Nqb , Ncb , and Nγ b is related to b due to the minimization. Total limit load P B in the plane failure mode may be directly calculated according to Eq. (5.108). It is easy to observe that if the limit load is eccentric forward, the larger eccentricity e, the larger A z ; therefore, the total load when e ≤ B/6 is the minimum.

5.6.2 Calculation Mode 2 For the calculation mode shown in Fig. 5.9, (A): θ0 ≥ θ ≥ θ1 ; the slip surface will be as below:

5.6 Calculation Equation for Limit Load

143

Fig. 5.9 Schematic diagram for calculation mode 2

x = sin(θb0 −θ1 ) {− sin θ0 cos θ1 + exp[λ(θ0 − θ )] sin θ1 cos θ } z = sin(θb0 −θ1 ) {− sin θ0 sin θ1 + exp[λ(θ0 − θ )] sin θ1 sin θ }

 (5.110)

(B): θ1 ≥ θ ≥ θ2 ; the slip surface will be as below: x = R B exp[λ(θ1 − θ )] cos θ z = R B exp[λ(θ1 − θ )] sin θ

 (5.111)

where, RB =

b {− sin θ0 + exp[λ(θ0 − θ1 )] sin θ1 } sin(θ0 − θ1 )

(5.112)

(C): the slip surface is the same as that in the mode 1. For the calculation mode 2, Eq. (5.108) may be similarly derived; however, attention shall be paid to different values of θ1 , hence: θ2 ≤ θ1 ≤ θ0 ≤ π − ϕ and − sin θ0 + exp[λ(θ0 − θ1 )] sin θ1 ≥ 0 If the limit load is eccentric backward (e ≤ 0), the smaller e, the larger A z ; therefore, the total limit load when e ≥ −B/6 will be the minimum.

5.6.3 Calculation Mode 3 For the slip surface family in the mode 1 or 2, if θ1 = θ0 , it will be retrograded into the plane–helicoid–plane slip surface (calculation mode 3). Correspondingly, Nqb , Ncb , Nγ b in Eq. (5.108) are, respectively, simplified as below: Nqb = 2 cos2 θ2 exp[2λ(θ1 − θ2 )]

1 − λ tan θ1 1 − tan δ tan θ1

(5.113)

144

5 Limit Equilibrium Method

Fig. 5.10 Schematic diagram for calculation mode 3

Ncb = (Nqb − 1.0)

1 λ

Nγ b = [A1 (λ sin θ1 − cos θ1 ) + sin θ1 cos θ1 ]

(5.114) 1 − λ tan θ1 1 − tan δ tan θ1

(5.115)

where, A1 =

1 {8λ exp[3λ(θ1 − θ2 )] cos3 θ2 (1 + λ tan θ2 ) − (3λ cos θ1 − sin θ1 )} (1 + 9λ2 ) (5.116)

θ1 is determined by the minimum of Eq. (5.108). For the second method, according to the current common slip surface, the slip surface in (A) is still taken as a plane (Fig. 5.10); however, θ1 = θ0 = π − ( π4 + ϕ2 ) is selected, which is the same as the results obtained in the field failure mode. It shall be noted that, according to the derivation process, the eccentricity in the equation is that of the limit load. The eccentricity shall be determined according to the distribution of the limit load; with known field failure mode of the same slip surface, the distribution of the limit load may be approximately calculated according B Nγ /Pz . For the purpose of the calculation to linear distribution, namely e = −γ 12 of the total limit load, the following steps may be adopted: Step I, e = 0 is taken to obtain the first approximation of Pz , Nγ ; Step II, the approximation of e is calculated and the second approximation of Pz , Nγ are obtained; the calculation process in Step II is repeated until the difference between Pz , e is very small. According to the actual calculation, the calculated second approximation has reached adequate accuracy. Calculation equations for the limit load in three modes are given above and they may be directly used for the calculation of the total limit load Pz on the foundation width B. Differences among the above three modes lie in those among the slip surfaces and calculated results. Generally, much smaller the calculated value of the total limit load is, it shows that the slip surface is much closer to the real slip surface, the better calculation mode. It is noted that the limit load is eccentric backward and the slip surface in the calculation mode 2 is relatively close to that by the characteristic line

5.6 Calculation Equation for Limit Load

145

Table 5.5 Calculated results of total limit load (q/(γ B) = 0.1, c/(γ B) = 0.2) ϕ(◦ )

tan δ 0.0

0.1

0.2

0.3

0.4

1

1.222

5

1.671

10

2.496

1.795

15

3.841

2.939

1.943

20

6.171

4.767

3.404

2.113

25

10.468

8.009

5.805

3.932

2.345

30

18.998

14.239

10.240

7.055

4.600

35

37.474

27.301

19.231

13.140

8.705

57.748

39.460

26.350

17.269

91.218

58.865

37.561

40 45

82.095 205.97

138.85

method. For the calculation description in three modes, the total limit load in the calculation mode 2 is the minimum.

5.6.4 Calculated Results The calculated results in the mode 2 are detailed in Table 5.5. Pm = Pz /(γ B 2 ) and Pz are total limit loads within the foundation bottom width. When ϕ ≤ 30◦ , the total limit load is quite consistent with the results by the characteristic line method and, in general, it is slightly larger than the results by the characteristic line method with increasing ϕ.

5.6.5 Corrected Calculation Equation After the calculation equation is slightly corrected, the calculated results very consistent with those by the characteristic line method may be obtained. For example, if Nγ is divided by 1 + |e | (λ − tan δ), the calculated results in the calculation mode 2 will be quite consistent with those by the characteristic line method: when q/(γ B) = 0.1 and c/(γ B) = 0.2, the relative error is less than 2%; if ϕ ≤ 40◦ , the relative error is less than 1.1%. If the strength index is slightly reduced: ck = c/F0 and tan ϕk = tan ϕ/F0 ; when, F0 = 1 +

tan ϕ − tan δ 0.035 1 + 2(q tan ϕ + c)/(γ B) 1 + 2 tan δ tan ϕ

(5.117)

146

5 Limit Equilibrium Method

c and tan ϕ are replaced by ck and tan ϕk respectively; comparison between calculated results in the calculation mode 2 and those by the characteristic line method is given in Tables 5.6, 5.7 and 5.8. According to those described in Tables 5.6, 5.7 and 5.8, c and tan ϕ are replaced by ck and tan ϕk ; the total limit load of the ground is limited with the calculation mode 2, which is very correspondent with the total limit load calculated by the characteristic line method. If the slip surface family is a specific plane or helicoid, any assumption or simplification is not required to be made for any force on the soil mass by the limit equilibrium method before the limit load or safety factor is calculated. In addition, as long as the slip surface families are identical to each other, the obtained results will be the same by the stress field method, limit equilibrium method as well as the upper bound solution method and generalized limit equilibrium as described later. It shall be stated that the calculated results according to the calculation equation by the limit equilibrium method are generally the safety factor or limit load in the surface failure mode . For example, in the above, only the limit load when the slip surface passes through the foundation rear toe (there are many such potential slip surfaces) is calculated and it is the limit load in the surface failure mode. For example, although slip circles with different centers and radiuses may be selected for the calculation of the common arc slip surface according to the slope stability, each calculation of the slip surface is only independent to each other; thus, the obtained results are those in the surface failure mode. Certainly, the limit load in field failure mode can also be obtained by the limit equilibrium method as long as the limit load is calculated for any slip surface through the foundation bottom; however, the results obtained in this way are only approximate solutions. Secondly, because the extremum condition of the yield function is not considered for the limit equilibrium method, even for the homogeneous soil, it only can be called as an approximate solution method because the safety factor or limit load only in accordance with the equilibrium equation and yield condition may be calculated according to corresponding equation; on the contrary, it can not be proved that the safety factor or limit load calculated by the limit equilibrium method is the solution of limit equilibrium solution. It is certainly the approximate solution of the limit equilibrium. The approximation method described here means that the solution method is not rigorous, which is entirely different from that only the approximate solution can be obtained due to great complicity of rigorous solution method. However, when the surface boundary condition is the known load boundary, the safety factor calculated by the limit equilibrium method will be the approximate solution in the surface failure mode; therefore, it is also an effective method for the slope stability. For the limit load, when the slip surface is a common plane and helicoid, those by the limit equilibrium method and upper bound solution method will be the same. In addition, the ground conditions in the practical problems are usually diverse, such as the slope and ground with heterogeneous soil. If the calculation equation for the limit load or safety factor is intended to be obtained without any other assumption or simplification, the selection of the slip surface will be restricted (generally, it is

2.499

3.850

6.168

10.381

18.592

35.909

76.446

184.69

10

15

20

25

30

35

40

45

184.39

76.605

36.072

18.693

10.442

6.196

3.859

2.501

1.669

Note The relative error is less than 1.6%

1.670

5

1.221

125.94

53.999

26.147

13.882

7.905

4.739

2.932

1.793

Pm

P0

Pm

1.221

0.1

0.0

tan δ

1

ϕ(◦ )

127.25

54.807

26.565

14.079

7.990

4.772

2.942

1.796

P0

85.162

37.694

18.698

10.078

5.759

3.392

1.940

Pm

0.2

85.435

38.042

18.899

10.176

5.802

3.408

1.945

P0

56.117

25.560

12.904

6.986

3.914

2.109

Pm

0.3

56.196

25.744

13.018

7.041

3.937

2.117

P0

Table 5.6 Comparison between total limit load Pm and P0 by the characteristic line method (q/(γ B) = 0.1, c/(γ B) = 0.2)

36.352

16.935

8.609

4.574

2.340

Pm

0.4

36.433

17.047

8.674

4.604

2.350

P0

5.6 Calculation Equation for Limit Load 147

25.644

57.553

145.63

35

40

45

145.09

57.403

25.728

12.673

6.719

3.776

2.223

1.356

0.844

Note The relative error is less than 2.8%

6.634

3.726

20

12.557

2.201

15

30

1.349

10

25

0.845

5

0.573

96.435

39.162

17.920

8.997

4.852

2.758

1.616

0.927

Pm

P0

Pm

0.573

0.1

0.0

tan δ

1

ϕ(◦ )

97.667

39.986

18.406

9.255

4.977

2.811

1.634

0.931

P0

63.590

26.545

12.411

6.316

3.413

1.899

1.016

Pm

0.2

63.694

26.884

12.651

6.448

3.475

1.924

1.023

P0

40.547

17.346

8.236

4.199

2.215

1.112

Pm

0.3

40.501

17.528

8.371

4.271

2.247

1.124

P0

Table 5.7 Comparison between total limit load Pm and P0 by the characteristic line method (q/(γ B) = 0.0, c/(γ B) = 0.1)

25.254

11.002

5.245

2.612

1.240

Pm

0.4

25.259

11.121

5.323

2.651

1.255

P0

148 5 Limit Equilibrium Method

235.86

102.67

50.682

27.545

16.141

10.045

6.565

4.465

3.137

Note The relative error is less than 0.8%

236.64

27.426

30

45

16.079

25

50.485

10.016

20

102.50

6.553

15

40

4.461

10

35

3.135

5

2.407

165.56

74.514

37.862

21.069

12.561

7.879

5.101

3.273

Pm

P0

Pm

2.407

0.1

0.0

tan δ

1

ϕ(◦ )

166.02

75.035

38.159

21.217

12.628

7.907

5.112

3.277

P0

114.13

53.140

27.684

15.651

9.367

5.778

3.468

Pm

0.2

114.08

53.367

27.841

15.735

9.408

5.796

3.474

P0

77.044

36.966

19.623

11.150

6.551

3.711

Pm

0.3

77.005

37.109

19.723

11.203

6.575

3.719

P0

Table 5.8 Comparison between total limit load Pm and P0 by the characteristic line method (q/(γ B) = 0.2, c/(γ B) = 0.4)

51.317

25.208

13.488

7.532

4.057

Pm

0.4

51.361

25.315

13.553

7.563

4.068

P0

5.6 Calculation Equation for Limit Load 149

150

5 Limit Equilibrium Method

only applicable to the logarithm helical slip surface). Moreover, for the surface failure mode, only the overall failure of the ground can be considered; for the ground bearing capacity, if the partial failure of the ground needs to be considered, corresponding calculation modes shall be established according to the field failure mode. Certainly, the limit load in field failure mode may also be calculated by the limit equilibrium method, which will be specially discussed in Chap. 8.

References 1. Yin ZZ, Lv QF (2005) Finite element analysis of soil slope based on circular slip surface assumption. Rock Soil Mech 26(10):1525–1529 2. Chen ZY (2003) Soil slope stability analysis—theory, methods and programs. China Water Power Press, Beijing 3. Zhu BL, Shen ZJ (1990) Computation of soil mechanics. Shanghai Science and Technology Press, Shanghai 4. Soil mechanics Staff Room of Huadong Hydraulic Faculty (1979) Principle and calculation of geotechnical, vol 1. Hydraulic Press, Beijing 5. Winterkorn HF, Fang XY (1983) Foundation engineering handbook (trans: Qian H, Ye S). China Architecture and Building Press, Beijing 6. Huang CZ, Jiang MB, Zhang J (2003) Computation method for bearing capacity of ground under action of eccentric and inclined loads. Chin J Geotech Eng 25(1):96–100

Chapter 6

Virtual Work Equation-Based Generalized Limit Equilibrium Method

It has been decades since the upper and lower bound theorems for limit analysis on the soil mass were proposed and the upper and lower bound solution methods (limit analysis method) have also received extensive attention, in which respect Chen [1], Shen [2] have made introduction in their works. Moreover, the theorem is always established under a certain condition, so when the condition required for the theorem proving is carefully analyzed, some problems still exist in the current given proving. The upper and lower bound theorems are normally seen as one of the main theoretical foundations for limit analysis; therefore, it is doubtlessly important to have an exact understanding on the strict proving and actual meaning of the theorem. In addition, how to correctly apply the theorem to obtain the approximate calculation equation of the limit load and how to compare the approximate calculation equation with the actual limit load also need to be further discussed.

6.1 Approximate Solution of Limit Analysis As stated earlier, the exact solution of the limit analysis is difficult to obtain, and one of the major reasons is that the real slip surface (family) is difficult to obtain; therefore, generally only the approximate solution can be obtained. In addition, the scope of the approximate solution is quite extensive, so when different slip surfaces are selected, different solutions may be obtained. Generally, the yield condition is required to be met along the selected slip surface, while for the real slip surface, it cannot be met and in this case, it may be expressed as f = f 0 . First, the yield function in three states may be uniformly written as: f = τ − (σ λ + c + f 0 )

© Springer Nature Singapore Pte Ltd. and Zhejiang University Press, Hangzhou, China 2020 C. Huang, Limit Analysis Theory of the Soil Mass and Its Application, https://doi.org/10.1007/978-981-15-1572-9_6

(6.1)

151

152

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

Namely, f =

       1  σz − h  τx z h  − λ − σx h  − τx z 1 + h  λ − (c + f 0 ) 1 + h 2 2 1+h (6.2)

where f 0 = f 0 (x, z), the yield condition is f = 0. As for the field failure mode, the limit state is f 0 = 0, failure state f 0 ≥ 0 and stable state f 0 ≤ 0. As for surface failure mode, f 0 shall be determined according to the limit load or safety factor; but when f 0 = 0 is on the slip surface, considering the approximate solution, the limit state will be also f 0 = 0, failure state f 0 ≥ 0, and stable state f 0 ≤ 0. Thus, it may be clearly seen that: After c is changed to c + f 0 , the failure state and stable state actually are in the limit state; namely, it requires to increase/decrease the strength of soil mass before it is in the limit state. The extremum condition for the corresponding yield function is as below:        1 2 ∂f  2 2 = f h =  2 (σz − σx ) 1 + 2h λ − h − τx z 2h − λ + h λ = 0 ∂h  2 1 + h 2 (6.3) Under suitable boundary conditions, the equilibrium equation, Eqs. (6.2) and (6.3), constitutes the limit equilibrium. According to Eqs. (6.2) and (6.3), the following is obtained:      ⎫   1 1  ⎪ ⎪ + σ − λ − f + c) 1 + h λ h σx h  − τx z = ( ) (σ x z 0 ⎬ 1 + λ2 2   (6.4)     ⎪  1 1 ⎪  ⎭ + σ λ + f + c) h − λ 1 + h σz − h  τ x z = ( ) (σ x z 0 1 + λ2 2 The following may be derived: σx +

f 0 +c λ

=

σz +

f 0 +c λ

=

τx z = λ

2

1+λ −(h −λ) (1+λ2 )(1+h 2 ) 2



⎫

1+λ2 +(h  −λ) 1 ⎪ ⎪ + σz ) + f0λ+c ⎪ (1+λ2 )(1+2 ) 2 (σx ⎬

⎪ 2 2  2 (1+λ )h +(1+λh ) 1 (σ + σ ) + f0 +c z 2 x λ (1+λ2 )(1+h 2 ) ⎪ 

2



1 2 (σx

+ σz ) +

f 0 +c λ

⎪ ⎪ ⎪ ⎭

(6.5)

The above is substituted in the equilibrium equation and further rearranged to obtain:   ⎫     1   ∂ f0    ∂ f0 ⎪ ∂h  ∂h 2λ ⎪ = γ h − λ + 1 + h λ σf + h + h −λ ⎪ 2 ∂x ∂z ∂x ∂z λ ∂x ∂z ⎬ 1+h       ∂σ f ∂h  1  ∂ f0 1 + λh  ∂σ f 2λ 1 + λh  ∂h  1 + λh  ∂ f0 ⎪ ⎪ ⎪ ⎭ σf +λ γ − + + + = h − ∂x λ − h  ∂z ∂x λ − h  ∂z λ − h λ ∂x ∂z 1 + h 2 ∂σ f

+ h

∂σ f

−

(6.6)

6.1 Approximate Solution of Limit Analysis

153

where σf =

1 f0 + c (σx + σz ) + 2 λ

(6.7)

Likewise, for the previously given equation under the limit state, the solutions in failure state and stable state may be obtained as long as c is changed to c + f 0 . It is very obvious that Eqs. (6.6) and (2.20) are entirely the same under the condition of f 0 = 0. As for the above equation, if only the strength index c, tan ϕ is changed to c/Fs = c F , tan ϕ/Fs = λ F , it may also be used to discuss slope stability.

6.2 Virtual Work Equation Under Yield Criterion Condition Generally, the virtual work equation is always given when the equilibrium equation condition is met, and if the yield condition and extremum condition of the yield function are also met, how about the virtual work equation? Here it will be discussed. Any point x0 on the foundation bottom (−B ≤ x ≤ 0) is supposed as: pz , px at the vertical load and horizontal load of a : (x0 ≤ x ≤ 0), the vertical load and horizontal load of the soil mass surface b : (0 ≤ x ≤ xn ) is q, 0; the area enclosed by the surface and curved face : (x0 ≤ x ≤ xn ) is D (Fig. 6.1). If the velocity fields vx (x, z) and vz (x, z) are independent variants, it is equivalent for the stress component to be in accordance with the equilibrium equation and the following equation: ¨  D

    ∂σx ∂τx z ∂τx z ∂σz + vx + + − γ vz dzdx = 0 ∂x ∂z ∂z ∂x

(6.8)



Tangent slope of the boundary surface is supposed as h s , and it is easy to obtain Eq. (6.8) as below:

Γa

x0

x

−B

Γ

z

(D)

Fig. 6.1 Schematic diagram for limit load

Γb

xn

154

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method 0

xn ( pz vz + px vx )a dx = −

x0

x N (qvz )b dx −

     σx h  − τx z vx + τx z h  − σz vz  dx

x0

0

  ¨  ∂vz ∂vx ∂vx ∂vz + σx + τx z + dzdx γ vz + σz − ∂z ∂x ∂x ∂z

(6.9)

D

Equation (6.5) is substituted to obtain: 0

xn ( pz vz + px vx )a dx = −

x0

(qvz ) dx b

0 x N 

+

  (c + f 0 ) vx + h  vz +

x0

−



  σf     c + f0  λv  dx − λ v − 1 + h − h x z λ 1 + λ2 

  ¨  c + f 0 ∂vx ∂vz γ vz − + λ ∂x ∂z D

  σf  2  ∂vz   1 + λ2 h 2 + 1 + h  λ + 2 2 ∂z 1+λ 1+h      ∂v    2  ∂vx ∂vz x 2  2 2 dz d x + 1+λ + h −λ +λ 1+λ − h −λ + ∂x ∂z ∂x

(6.10)

According to Chap. 2, vx and vz are in accordance with the following equation: 

   h  − λ v x − 1 + h  λ vz = 0

   2 ∂vz  2 ∂vx 1 + λ2 h 2 + 1 + h  λ + 1 + λ2 + h  − λ ∂z ∂x 

 ∂v   ∂v 2 x z + =0 + λ 1 + λ2 − h  − λ ∂z ∂x

(6.11)

(6.12)

Then the following is derived: 0

xn ( pz vz + px vx )a dx = −

x0

x N (qvz )b dx +

0

   (c + f 0 ) vx + h  vz  dx

x0

  ¨  ∂vz c + f 0 ∂vx + dzdx γ vz − − λ ∂x ∂z

(6.13)

D

In fact, if the stress fields σx , σz , and τx z as well as the slip surface family h  are in accordance with the equilibrium equation, yield condition, and extremum condition of the yield function, as for any functions of vx (x, z), vz (x, z), η(x, z), and κ(x, z), thereinto, μ, η, and κ are Lagrange multipliers to be determined, it shall be:

6.2 Virtual Work Equation Under Yield Criterion Condition

¨  D

    ∂σx ∂τx z ∂τx z ∂σz + vx + + − γ vz dzdx ∂x ∂z ∂z ∂x

¨ +

155

xn (μf + η f h  )dzdx +

(κ f )dx = 0 x0

D

Namely, x N

     σz − h s τx z vz − σx h s − τx z vx a +b dx

x0

¨

x N +

[(c + f 0 )κ] dx + x0 x N 

  γ vz + (c + f 0 )μ dzdx

D

    1 + hλ  σz − h τ x z − σx h − τx z vx + κ + dx 1 + h 2  x0  ¨  ∂vz h − λ 1 + 2h  λ − h 2 −μ + −η  σz dzdx 2 ∂z 1 + h 2 1 + h 2 D    ¨  h 1 + hλ 1 + 2h  λ − h 2 ∂vx +μ +η  + σx dzdx 2 ∂x 1 + h 2 1 + h 2 D  ¨  ∂vx ∂vz 1 + 2h  λ − h 2 2h  − λ + h 2 λ + −μ + 2η  + τx z dzdx = 0 2 ∂z ∂x 1 + h 2 1 + h 2 



 

h − λ vz + κ 1 + h 2



D

(6.14) Considering when the boundary condition is the load boundary condition, for vx , vz and μ, η, and κ are all unknown functions to be determined, in order to simplify Eq. (6.14), the following are taken:   h 1 + hλ 1 + 2h  λ − h 2 ∂vx = −μ − η  2 ∂x 1 + h 2 1 + h 2 h − λ ∂vz 1 + 2h  λ − h 2 =μ +η  2 2 ∂z 1+h 1 + h 2 ∂vz 1 + 2h  λ − h 2 ∂vx 2h  − λ + h 2 λ + =μ − 2η  2 2 ∂z ∂x 1+h 1 + h 2 and:

(6.15)

156

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method 

λ vx = −κ 1+h 2 1+h  h −λ vz = −κ 1+h 2

 (6.16)

Equation (6.15) is the generalized flow rule, and it is easy to obtain:

1  η=  2 1 + λ2



∂vx ∂vz − ∂z ∂x

  ∂vz 1 ∂vx + μ=− λ ∂ x  ∂z  



1 + 2h  λ − h 2 −

  ∂vz   ∂vx + 2h − λ + h 2 λ ∂z ∂x

(6.17) (6.18)

  z z x x and Eq. (6.12). or η = − ∂v + h  ∂v + h  ∂v + h  ∂v ∂x ∂z ∂x ∂z According to Eq. (6.16), the following are obtained:   κ = − v x + h  vz

(6.19)

and Eq. (6.11). When the Eq. (6.14) is substituted, Eq. (6.13) is immediately derived. The Eq. (6.13) is obtained on the basis that the yield condition and extremum condition of the yield function are considered with regard to the virtual work equation, so it is referred to as the virtual work equation under yield criterion condition. According to the above derivation process, the following items may be got: First, when the stress field and slip surface family are in accordance with the equilibrium equation, yield condition (6.2) and extremum condition of the yield function (6.3) and the velocity field are in accordance with Eqs. (6.11) and (6.12), and load on the boundary surface must be in accordance with Eq. (6.13). And the virtual work equation is simplified when the equation is free from any unknown stress component. However, it shall be noted that when the stress field, velocity field, and slip surface family are in accordance with the limit equilibrium, it is only the sufficient condition for load on the boundary surface to be in accordance with Eq. (6.13); while the necessary condition is that the stress field velocity field and slip surface family must be in accordance with the limit equilibrium, when load on the boundary surface is in accordance with Eq. (6.13) and it still remains to be proven. Second, the limit load (under the condition of f 0 = 0) determined according to Eq. (6.13) is used to calculate both the limit load in field failure mode and the limit load in surface failure mode. When every slip surface of the slip surface family is taken for , it is the solution in field failure mode, and when only a certain slip surface of the slip surface family is taken for , it is the solution in surface failure mode. For instance, in the respect of the ultimate bearing capacity, for the field failure mode, every point (−B ≤ x0 ≤ 0) on the foundation bottom shall be taken for x0 ; while for the surface failure mode,  is the slip surface through foundation rear toe (x0 = −B). In addition, Eq. (6.13) provides a simple method to calculate the approximation of the limit load without the solution of the stress field: When the slip surface family h  is selected, the velocity field may be solved according to the velocity equation,

6.2 Virtual Work Equation Under Yield Criterion Condition

157

and with the substitution of Eq. (6.13), the approximation of the limit load may be obtained. However, it still remains to be further demonstrated to judge whether the obtained result is stable solution or failure solution; moreover, the upper bound theorem and lower bound theorem are required to be proven.

6.3 Upper and Lower Bound Theorems of Limit Load 6.3.1 Proving for Upper and Lower Bound Theorems of Limit Load First of all, the velocity equation isn’t related to the stress field, but is only related to the slip surface family. In other words, if the slip surface family is determined, the velocity field is determined as function of the slip surface family according to the velocity equation; if the slip surface family is that in the limit state, the velocity field must be that in the limit state. In this case: If the stress field on (D) is the stress field f 0 = 0 in the limit state, exact solutions pz , px of the limit load shall be in accordance with the following equation: 0

xn ( pz vz + px vx )a dx = −

x0

x N (qvz )b dx +

  c vx + h  vz  dx

x0

0

  ¨  ∂vz c + f 0 ∂vx γ vz − + dzdx − λ ∂x ∂z

(6.20)

D

If the stress field on (D) is the stress field (movable stress field): f 0 = f ∗ ≥ 0 in the failure state, failure solutions pz∗ , px∗ (or may be referred as movable solution and upper bound solution) of the limit load shall be in accordance with the following equation: 0 x0



pz∗ vz

+



px∗ vx a dx

xn =−

x N (qvz )b dx +

0

   c + f ∗ vx + h  vz  dx

x0

  ¨  ∂vz c + f ∗ ∂vx γ vz − + dzdx (6.21) − λ ∂x ∂z D

If the stress field on (D) is the stress field (static stress field): f 0 = f # ≤ 0 in the stable state, stable solutions pz# , px# (or may be referred as static solution and lower bound solution) of the limit load shall be in accordance with the following equation:

158

0

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method



pz# vz

+

px# vx )a dx



xn =−

x0

x N (qvz )b dx +

0

¨  D

Since

∂vx ∂x

+

∂vz ∂z

  ∂vz c + f # ∂vx + dzdx (6.22) λ ∂x ∂z

≥ 0 and vx + h  vz ≥ 0 are true, it must be: 

 ( pz# vz + px# vx )d ≤ a

x0

γ vz −

−

   c + f # vx + h  vz  dx

 ( pz vz + px vx )d ≤

a

( pz∗ vz + px∗ vx )d

(6.23)

a

The above is the inequation required to be proved by the upper and lower bound theorems of limit load. It needs to be noted that the failure solution and stable solution mentioned in the above upper and lower bound theorems are especially referred to the approximate solutions calculated according to Eqs. (6.21) and (6.22). In fact, the scope of the approximate solution is extensive and the approximate solution is not limited to be obtained according to a certain method. For instance, when the slip surface limit isn’t the slip surface of the limit solution, the approximate solution of the limit load may also be obtained. However,  and in the equation of the approximate solution are different from the limit solution, so how to prove the upper and lower bound theorems in this case still needs to be discussed.

6.3.2 Proving for Upper and Lower Bound Theorems of Limit Soil Pressure (1) Active soil pressure (Fig. 6.2) If a is the known load boundary and b unknown limit load boundary, because vertical stress pz and horizontal stress px at any point of the boundary surface are as below: ⎫ 1 h s ⎪ ⎪ pz = σz  − τx z  ⎪ ⎬ 1 + h 2 1 + h 2 s s (6.24)  ⎪ h 1 ⎪ ⎪ px = σx  s − τx z  ⎭ 1 + h 2 1 + h 2 s s It is entirely similar to the previous derivation, so the virtual work equation under yield criterion condition may be obtained as below:

6.3 Upper and Lower Bound Theorems of Limit Load

159

x

Γa

x0

z Γ

Γb

xN Fig. 6.2 Active soil pressure





¨

( px vx − pz vz )d = b

( pz vz − px vx )d + a



−

¨ (c + f 0 )vt d −



D

γ vz dzdx D



∂vx ∂vz + (c + f 0 ) ∂x ∂z

 λdzdx (6.25)

Under the condition that the slip surface family is that in the limit state: If the stress field is that in the limit state f 0 = 0, Eq. (6.25) is the equation that the exact solutions pz , px of active soil pressure on b shall be in accordance with. If the stress field is the stress field ( f 0 = f ∗ ≥ 0) in the failure state, Eq. (6.25) is the equation that failure solutions pz∗ , px∗ (movable solution and upper bound with. solution) of active soil pressure on b shall be in accordance  If the stress field is the stress field f 0 = f # ≤ 0 in the stable state, Eq. (6.25) is the equation that stable solutions pz# , px# (static solution and lower bound solution) of the active soil pressure on b shall be in accordance with. It is easy to prove the upper and lower bound theorems of active soil pressure, namely the inequation:  b



 px∗ vx − pz∗ vz d ≤



 ( px vx − pz vz )d ≤

b



 px∗ vx − pz# vz d

(6.26)

b

The upper bound solution of active soil pressure is no greater than its exact solution, and the lower bound solution is no less than exact solution, which is just opposite to the previously proven limit load. In particular, if the surface boundary a is a horizontal surface: −B ≤ x ≤ 0 and bears vertical load q on the surface, b is a vertical straight wall surface: 0 ≤ z ≤ H and the horizontal force and vertical force on the wall surface are supposed as σxb , τx zb , respectively, (6.25) is as below:

160

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

x

Fig. 6.3 Passive soil pressure

Γb

xN

z Γa

Γ

x0

H

¨

0 (σxb vxb + τx zb vzb )dz =

qvza dx +

γ vz dzdx

−B

0

D



− ¨

(c + f 0 )vt d− 



(c + f 0 ) D

∂vx ∂vz + ∂x ∂z

 λdzdx

(6.27)

Obviously, it will be: H

∗ (σxb vxb

+

τx∗zb vzb )dz

H ≤

0

H (σxb vxb + τx zb vzb )dz ≤

0

# (σxb vxb + τx#zb vzb )dz 0

(6.28) (2) Passive soil pressure (Fig. 6.3): if b is a known load boundary and a an unknown limit load boundary, the virtual work equation under yield criterion condition is: 



¨

( px vx − pz vz )d = a

( pz vz − px vx )d − b

¨

 (c + f 0 )vt d +

+ 

D

γ vz dzdx D



∂vx ∂vz + (c + f 0 ) ∂x ∂z

 λdzdx (6.29)

Here, the upper and lower bound theorems are similar to the limit load. It is easy to be seen that the ultimate bearing capacity of the horizontal ground surface is actually a special case in which an unknown load boundary is taken as the horizontal ground surface with regard to passive soil pressure.

6.3 Upper and Lower Bound Theorems of Limit Load

161

It must be clearly realized that the upper and lower bound theorems proven here are provided with two conditions: I. the slip surfaces of the stable solution and failure solution are all slip surfaces of the limit solution; II. f 0 = f ∗ ≥ 0 or (σi j −σi∗j )εi∗j ≥ 0 is required for the upper bound solution and f 0 = f # ≤ 0 or (σi j − σi#j )εi#j ≤ 0 for the lower bound solution. However, the two conditions bring a great limitation to the theoretical significance and application value of the theorem. In fact, proving this theorem is nothing more than to obtain limit load (approximation) when the solution of the stress field is avoided. Without the stress field, neither f 0 = f ∗ ≥ 0 and f 0 = f # ≤ 0 nor (σi j − σi∗j )εi∗j ≥ 0 and (σi j − σi#j )εi#j ≤ 0 can be verified. Besides, determination of the slip surface (family) cannot be avoided in any case and the real slip surface is very difficult to obtain (by means of ordinary functional form). Generally, the stable solution and failure solution are not restricted to the slip surface (family) as the real slip surface, f 0 = f ∗ ≥ 0 and f 0 = f # ≤ 0 in the equation are also unable to be applied, and actual calculation is able to be carried out only under the condition of assumed slip surface family and f 0 = 0. From this respect, the proven theorem is only said to be a narrow upper and lower bound theorems.

6.4 Discontinuity of Velocity Field In practical problems, the slip surface may be relatively complicated. For the purpose that the selected slip surface is close to the real slip surface as much as possible; generally, the possible soil mass failure area is divided into several subareas for which different slip surfaces are selected respectively. Thus, it is relatively difficult to construct a continuous velocity field and the condition where the velocity discontinuity surface exists may be directly considered. It is worthy to state that considering the velocity discontinuity surface is only an approximate treatment method to obtain the approximate solution and it cannot show that such a motion mode exists in practical problems. It is also unscientific to regard some divided subareas as rigid areas, because this is a phenomenon generated when the solution has to be obtained by some approximate treatment methods; on the contrary, it will cause any unnecessary misunderstanding when they are endowed with physical interpretation. Below, two velocity discontinuity surfaces are taken as examples for discussion and several velocity discontinuity surfaces may also be similarly discussed. n , m are supposed as two velocity discontinuity surfaces, and the area (D) is divided into (A), (B), and (C) (Fig. 6.4). Here, virtual work equation under yield criterion condition in the whole area (D) is not true anymore, but is always true for every subarea. Slopes of n , m are supposed as h n , h m , respectively, and vx A , vx A are values of the velocity field of (A) on n . In (A), it will be:

162

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

Fig. 6.4 Schematic diagram for slip mass subarea 0

     σz − h s τx z vz − σx h s − τx z vx  dx = − a

x0

xa

     σz − h n τx z vz A − σx h n − τx z vx A  dx n

0

¨



−

 γ vz + (c + f 0 )μ dz dx +

xa



x0

A

 (c + f 0 )κ  dx 1

(6.30)

In (B), it will be: 0

xb

     σz − h n τx z vz B − σx h n − τx z vx B  dx = −



n

    σz − h m τx z vz B − σx h m − τx z vx B  dx m

xa

0

¨ −



xb

 γ vz + (c + f 0 )μ dz dx +



xa

B

 (c + f 0 )κ  dx 2

(6.31)

In (C), it will be: 0



 (σz − h m τx z )vzC − (σx h m − τx z )vxC  dx = − m

xb

x N

     σz − h s τx z vz − σx h s − τx z vx  dx b

0

¨ − C

  γ vz + (c + f 0 )μ dz dx +

x N



 (c + f 0 )κ  dx 3

(6.32)

xb

With regard to proving for upper and lower bound theorems, it may be like this. As for limit load, first it is derived that the upper and lower bound theorems of limit load on m are true according to Eq. (6.32), then it is derived that the upper and lower bound theorems of limit load on n are true according to Eq. (6.31), and finally the upper and lower bound theorems of limit load on a are obtained according to Eq. (6.30). Calculation of limit load may be considered according to the following steps: Eq. (6.32) is used to obtain σz − h m τx z and σx h m − τx z on m , then the results are substituted in Eq. (6.31) to obtain σz − h n τx z and σx h n − τx z on n , then the results are also substituted in Eq. (6.30), and the limit load on the boundary surface may be calculated. So it is obvious that the solution process of limit load is relatively complicated in case of velocity discontinuity surface.

6.5 Upper and Lower Bound Theorems Under Associated Flow …

163

6.5 Upper and Lower Bound Theorems Under Associated Flow Rule Condition Here, the application of associated flow rule on existing proving of the theorem is briefly discussed.

6.5.1 Required Condition for Theorem Proving The existing proving needs to introduce the geometric equation and associated flow rule and may be combined as below: ⎫ ∂f ∂vx ⎪ ⎪ εx = − = γm ⎪ ⎪ ⎪ ∂x ∂σx ⎪ ⎪ ⎬ ∂f ∂vz = γm εz = − ∂z ∂σz ⎪ ⎪ ⎪   ⎪ ∂vx ∂f ⎪ ∂vz ⎪ ⎪ + = γm γx z = − ⎭ ∂x ∂z ∂τx z

(6.33)

where vx , vz are horizontal and vertical velocities, respectively, with γm ≥ 0. The yield condition is Mohr–Coulomb yield condition; likewise, the corresponding yield conditions of the exact solution and approximate solution may be uniformly rewritten as below:    1 f = 41 (σz − σx )2 + τx2z − (σz + σx ) sin ϕ + (c + f 0 ) cos ϕ = 0 (6.34) 2 The common signs are applied: σi j εi j = σx εx + σz εz + τx z γx z      σz − τx z h  vz − σx h  − τx z vx m

  ∂vx ∂vz ∂vx ∂vz − σz − τx z + , = −σx ∂x ∂z ∂z ∂x      = σ vz − h  vx + τ vx + h  vz m  = 1 + h 2 (σ vn + τ vt )m

Equation (6.9) is easy to write as: 

 ( pz vz + px vx )d + a

b

¨

 qvz d − m

(σ vn + τ vt )d − D



 σi j εi j − γ vz dzdx = 0

(6.35)

164

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

where a is the whole foundation bottom [−B, 0] and m is the slip surface through foundation rear toe (−B, 0). It is the same as the virtual work equation given in Ref. [2] (vt = 0, vn = 0 are true for outer side of the slip surface, so vt = vt , vn = vn are true); and it is the same as the virtual work equation given in Ref. [1] (the integral path on m is opposite, so difference of this integral is just a minus sign). The existing theorem proving is carried out according to Eq. (6.35). And different references provide different provings; for instance, some provings are given under condition of vt = 0, vn = 0, which is obviously wrong. The followings are included in the conditions for the true theorem: (I) the slip surface is real slip surface; (II) m shall be provided with [2] vn + vt tan ϕ = 0, which is actually Eq. (6.11). Thus, the yield function is applied and the virtual work equation is as below: 

 ( pz vz + px vx )d + a



¨ (c + f 0 )vt d −

qvz d −

b

m



 σi j εi j − γ vz dzdx = 0

(6.36)

D

(III) Equation (6.33) of the associated flow rule is adopted. According to the yield condition, it is: ⎫  ∂vx γm 1 ⎪ σm sin ϕ + (σz − σx ) ⎪ = ⎪ ⎪ ⎪ ∂x 2σm 2 ⎪ ⎪  ⎬ γm ∂vz 1 (6.37) σm sin ϕ − (σz − σx ) = ⎪ ∂z 2σm 2 ⎪ ⎪ ⎪ ⎪ ∂vz ∂vx γm ⎪ ⎪ ⎭ + = − τx z ∂x ∂z σm where σm =



1 4 (σz

− σx )2 + τx2z

There shall be: ∂vz ∂vx + = γm sin ϕ ∂x ∂z   1 σi j εi j = γm σm − (σz + σx ) sin ϕ = γm ( f 0 + c) cos ϕ 2

(6.38) (6.39)

If f 0 = 0 is the energy dissipation condition, it is the same as that obtained by applying Coulomb yield condition in Chap. 2. Thus, Eq. (6.36) may also be simplified as: 0

x B ( pz vz + px vx )a dx = −

−B

x B (qvz )b dx +

0

−B

   (c + f 0 ) vx + h  vz m dx

6.5 Upper and Lower Bound Theorems Under Associated Flow …

−

165

  ¨  ∂vz c + f 0 ∂vx γ vz − + dzdx λ ∂x ∂z

(6.40)

D

  Besides, conditions f 0 = f ∗ ≥ 0 and σi j − σi∗j εi∗j ≥ 0 of the upper bound   solution are equivalent and conditions f 0 = f # ≤ 0 and σi j − σi#j εi#j ≤ 0 of the lower bound solution are equivalent. So there is: 0 −B



pz# vz

+



px# vx a dx

0 ≤

0 ( pz vz + px vx )a dx ≤

−B



pz∗ vz + px∗ vx

 a

dx (6.41)

−B

The first condition for both the above proving and the existing proving is the same; the second condition is required to be true on D for the above proving while it is only required to be true on m for the existing proving; and the third condition is adopted with the generalized flow rule [namely Eq. (6.12)] for the above proving while it is adopted with the associated flow rule for the existing proving. The analysis with regard to this respect is analyzed below.

6.5.2 Difference in Actual Content of the Theorem It is noted that Eqs. (6.41) and (6.23) are different. The inequation for the existing proving is only true on the whole foundation bottom [−B, 0]; in Chap. 2, it has been proved that vn + vt tan ϕ = 0 and the associated flow rule cannot be true simultaneously inside the slip mass, which means that only the total limit load can be calculated. While the above-proven inequation is true at any [x0 , 0] of the foundation bottom, the distribution of the limit load may be calculated. Therefore, the obvious reason is that different second conditions generate different results and the two provings of the theorem reflect the difference in the actual content of the theorem.

6.5.3 Difference in the Corresponding Limit Solution As for the problem including kinematic boundary condition, the equilibrium equation, yield condition, extremum condition of the yield function, and motion equation as well as the corresponding boundary condition also constitute a limit equilibrium problem. However, it has been obtained in Chap. 2 that this boundary value is different from that corresponding with the above-proven theorem and the difference lies in the different motion equations. Unless the two kinds of motion equations are able to be proven as equivalent, their corresponding limit solutions are different.

166

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

6.5.4 Incompletion of the Existing Proving In order to simplify the virtual work equation, it must also be in accordance with the following on m (the maximum slip line): 

    h  − λ vx − 1 + h  λ vz m = 0

(6.42)

The virtual work equation is not only corresponding with the aforementioned limit equilibrium but also with Eq. (6.42), which produces a problem: Will the slip surface family and velocity field of the limit solution be in accordance with this condition on m ? And this is a proposition to be proven and shall continue to be completed with regard to the present proving; otherwise, proving of the theorem is incomplete. In addition, if this proposition fails to be true, the theorem cannot be true either.  Moreover, some provings are free from the treatment of m σ (vn + vt tan ϕ)d and the following is only proven with regard to proving of the theorem: 



 pz∗ vz0 + px∗ vx0 d ≥



 ( pz vz0 + px vx0 )d +

a

a

m









a

 pz# vz0 + px# vx0 d ≤

( pz vz0 + px vx0 )d + a

 ∗  σ − σ (vn + vt tan ϕ)d (6.43)  #  σ − σ (vn + vt tan ϕ)d

m

(6.44) As for the inequation that needs to be proven with regard to the limit theorem, the following remains to be proven:  m

(σ ∗ − σ )(vn + vt tan ϕ)d ≥ 0,

 (σ # − σ )(vn + vt tan ϕ)d ≤ 0

(6.45)

m

Namely, proving of the theorem is not completed. In contrast, the above proving only requires that the velocity field be in accordance with Eqs. (6.11) and (6.12) within D and the condition on m is not required, so the above problem does not exist.

6.6 Generalized Limit Equilibrium Method for Limit Load As stated earlier, one of the main conditions to prove the upper and lower bound theorems is true is that the slip surface is that of the limit solution and it brings a great limitation to the theoretical significance and application value of this theorem. Moreover, scope of the approximate solution is extensive, the slip surface may be the

6.6 Generalized Limit Equilibrium Method for Limit Load

167

approximate slip surface, and the approximate solution is not limited to be obtained according to a certain method. As for another main condition f 0 ≥ 0 (or f 0 ≤ 0), according to the yield condition, it is observed that the upper bound solution is actually the solution that is changed from the soil strength index c under condition of c + f 0 ≥ c and the lower bound solution is actually the solution that is changed from the soil strength index c under condition of c + f 0 ≤ c. In addition, the slip surface is that of the limit solution, so the theorem is naturally true. However, such a theorem cannot be overestimated with regard to the theoretical significance and application value. When the slip surface family is the real slip surface family, the velocity field determined according to the velocity equation is the real velocity field and the limit load calculated according to the virtual work equation under yield criterion condition ( f 0 = 0 is taken) of Eq. (6.13) is the actual limit load. However, the real slip surface family is difficult to determine. Thus, the limit analysis theorem provides a simple method to calculate the approximation of the limit load without the solution of the stress field: When the slip surface family h  is selected, the corresponding velocity field may be solved according to the velocity equation. Additionally, when the slip surface family h  is determined, the velocity equation is a linear equation, so it is not difficult to solve the velocity field. After the virtual work equation under yield criterion condition is substituted, the approximation of the limit load may be obtained. Whether the approximate solution is close to the actual limit load depends on the selection of the slip surface family. If the selected slip surface family is close to the real slip surface family, the calculated limit load is then close to the actual limit load. As far as the solution method is concerned, on one hand, application of the virtual work equation for solution is the popularization of the limit equilibrium method with application of force and moment equation for solution; on the other hand, the slip surface is artificially selected and the real slip surface is very difficult to obtain (in the common functional form). In this case, the upper and lower bound theorems are no longer true. Besides, it is unable to verify whether f 0 = f ∗ ≥ 0 or (σi j − σi∗j )εi∗j ≥ 0 is the condition for the upper bound theorem and whether f 0 = f # ≤ 0 or (σi j − σi#j )εi#j ≤ 0 is the condition for the lower bound theorem. Therefore, it cannot be guaranteed whether the calculated limit load is the upper bound solution or the lower bound solution. Hence, the above solution method is referred to as the generalized limit equilibrium method. And the basic steps to calculate the limit load by the generalized limit equilibrium method as below: Step I: the soil mass is divided into several areas and the assumed slip surface family h  (x, z) may be plane, general helicoid (including arc surface and common helicoid) and their combination; Step II: the boundary condition is determined; Step III: the velocity fields vx , vz are determined by applying Eqs. (6.11) and (6.12); Step IV: the approximation of the limit load is calculated according to Eq. (6.13).

168

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

In case of discontinuity of velocity field, the approximation of the limit load is calculated by applying similar Eqs. (6.30)–(6.32).

6.7 Common Slip Surface Family and Velocity Field The slip surface and velocity field are included in the Eq. (6.13) of the limit load. Theoretically, the slip surface shall be obtained according to the equilibrium equation, yield condition, and the extremum condition of the yield function. However, as stated earlier, due to the solution difficulty, the real slip surface is difficult to obtain and generally, the approximate solution is obtained under the condition of the assumed slip surface. After the slip surface family is selected, the velocity field is determined according to Eqs. (6.11) and (6.12). At present, the common slip surface is plane, logarithmic helicoid, and arc surface, etc., and their corresponding velocity fields will be discussed in the following text.

6.7.1 Velocity Field of Plane Slip Surface When h  is a constant, according to Eq. (6.11), Eq. (6.12) may be changed to: ∂vx ∂vx + h =0 ∂x ∂z Thus, it is easy to obtain: ⎫ 1 + hλ  ⎪ ⎪ (z − h x) ⎬ 1 + h 2  ⎪ h −λ ⎭ vz = A v (z − h  x) ⎪ 2 1+h

vx = Av

(6.46)

Because the slip surface is in accordance with z − h  x = −h  x0 , thereinto, (x0 , 0) is one point in the foundation bottom, only translational motion is possible for the soil mass on the planar slip surface. And the velocity along any determined slip surface is certainly the same; while (x0 , 0) is different with regard to different slip surfaces, so the velocity along different √ slip surfaces is different. Especially, when h  = λ ± 1 + λ2 is true,  ⎫ 1  Av z − h  x ⎪ ⎬ 2  ⎪ 1 vz =  A v z − h  x ⎭ 2h

vx =

(6.47)

6.7 Common Slip Surface Family and Velocity Field

169

Besides Eq. (6.46), other forms also are in accordance with the velocity equation, for instance: vx = v0 (1 + h  λ) and vz = v0 (h  − λ) where v0 , Av are any constants. The same velocity field may also be obtained by applying the velocity equation corresponding with the associated flow rule.

6.7.2 Velocity Field of Common Helicoid Slip Surface In case of the common helicoid, the velocity field obtained in Chap. 2 is as below: vx = Av (z − z R ) vz = −Av (x − x R )

 (6.48)

The same velocity field may also be obtained by applying the velocity equation corresponding with the associated flow rule. After Eq. (6.48) is substituted in Eq. (6.11), it is known that the motion mode is rotational motion.

6.7.3 Velocity Field of Arc Slip Surface 0 As for the arc surface, h  = − x−x = −s is supposed, thereinto, (x0 , z 0 ) is the circle z−z 0 center coordinate. Let:  vx = (z − z 0 )(1 − sλ)g(s)

vz = −(z − z 0 )(s + λ)g(s) be in accordance with Eq. (6.11); after the above equations are substituted in Eq. (6.12), it shall be: 



  (1 − sλ)g  − λg 1 + λ2 + (s + λ)2 + s(s + λ)g  − λg (1 + λ2 )s 2 + (1 − sλ)2

− [s(1 − sλ) + s + λ]g  λ 1 + λ2 − (s + λ)2 = 0

It is rearranged to get: g −

2λ g=0 1 + s2

170

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

Hence, g(s) = Av exp(2λ arctan s) 

vx = Av (z − z 0 )(1 − sλ) exp(2λ arctan s)

(6.49)

vz = −Av (z − z 0 )(s + λ) exp(2λ arctan s)

The same velocity field cannot be obtained by applying the velocity equation corresponding with the associated flow rule.

6.7.4 Velocity Field of General Helicoid Slip Surface s−k R h  = − 1+sk is taken as the slip surface family; thereinto, x−x = s is true, and it is z−z R easy to obtain the polar form of the slip surface as below:

x − x R = R exp(−kθ ) cos θ

 (6.50)

z − z R = R exp(−kθ ) sin θ

where k is any constant. Obviously, it is the general form of the logarithmic helicoid. When k = λ is true, the slip surface is the same as the above common helicoid; while k = 0 is true, the slip surface is the arc surface and k = tan ψ is supposed hereafter. If the following form is taken for the velocity field;  vx = [(1 + kλ)(z − z R ) + (k − λ)(x − x R )]g(s) vz = [(k − λ)(z − z R ) − (1 + kλ)(x − x R )]g(s) It must be in accordance with Eq. (6.11); after the above equations are substituted in Eq. (6.12) and rearranged to get: g  (s) + g(s)

2(k − λ) =0 1 + s2

The following may be solved: g(s) = Av exp[2(λ − k) arctan s]

(6.51)

So, when the slip surface is the general logarithmic helicoid, the velocity field is: vx = Av [(1 + kλ)(z − z R ) + (k − λ)(x − x R )] exp[2(λ − k) arctan s] vz = Av [(k − λ)(z − z R ) − (1 + kλ)(x − x R )] exp(2(λ − k) arctan s]

 (6.52)

The same velocity field cannot be obtained by applying the velocity equation corresponding with the associated flow rule.

6.7 Common Slip Surface Family and Velocity Field

171

It is worthy to state that when the slip surface is plane, common logarithmic z x + ∂v = 0 is true, which provides convenience for helicoid, and arc surface, ∂v ∂x ∂z calculating the limit load according to Eq. (6.13). Because f 0 of the generalized limit equilibrium method has no longer existed in the slip mass at this time; however, f 0 = 0 shall be true on the slip surface.

6.7.5 Combination Form of Slip Surface and Velocity Field In the practical problem, generally, the combination form of slip surfaces is taken for the slip surface. Take examples as slip surface in (A) is plane. √  in Fig. 6.5: The  = −h a as the interface of (A) and (B); h a = λ + 1 + λ2 = tan π4 + ϕ2 with h ab the slip surface in (B) is common helicoid h b = (λz − x)/(z + λx); the slip surface √  in (C) is plane h c = λ − 1 + λ2 = −1/ h a with h bc = −h c as the interface of (A) and (B), the velocity field is:    vx = Av z − h a x 2      (A) vz = z − h a x 2h a  vx = Av z (B) vz = −Av x    vx = Av z − h c x 2      (C) vz = Av z − h c x 2h c

(6.53a)

(6.53b)

(6.53c)

It is easy to verify that the slip surface is smooth and the velocity field is continuous.

6.8 Calculation Process of Limit Load The ultimate bearing capacity under the simplest vertical load needs to be solved: With the ground surface as horizontal surface, when the vertical load to be solved

Fig. 6.5 Schematic diagram for plane–common helicoid–plane calculation mode

172

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

in the foundation bottom a : [−B, 0] and horizontal load as well as the vertical evenly distributed load in other area b : 0 < x are supposed as pz = σz |z=0 , px = τx z |z=0 = 0 and q, respectively, the limit load pz needs to be solved.

6.8.1 Example I: Slip Surface in Plane–Common Helicoid–Plane Form (1) Continuity of velocity field (Fig. 6.5)

√   Obviously, the slip surface family h a = λ + 1 + λ2 = tan π4 + ϕ2 in (A) and √   the slip surface family h c = λ − 1 + λ2 = −1/ h a in (C) are in accordance with the surface boundary condition px = τx z |z=0 = 0. Equation (6.53) and the corresponding slip surface family are substituted in the virtual work equation, and it will be: 0 −b

x

b   x x pz − d x = − q − d x 2 2 0

⎡

−b/2 

⎢ + c⎣

(z − h a x) A dx +

⎡ ⎢ − γ⎣

−b

¨ A

xb /2 

−b/2

z − h a x dz d x + 2h a

⎤  xb  ⎥  λz − x  z−x dx + z − h c x dx ⎦ z + λx  B

¨

xb /2

¨ (−x)dz dx +

B

C

⎤

z − h c x ⎥ dz d x ⎦ 2h c

(6.54)

The integral in the equation is calculated to obtain: 0 −b

 2    x  b 2 b 1  2 2 pz − dx = qh a exp(λπ ) + c h a exp(λπ ) − 1 2 2 λ 2  3    b 4λ 3  2 2 h a (3h a − 1) exp π λ + 3 − h a +γ 2 3(1 + 9λ ) 2 2

Derivatives (as for b) on both sides of the integral equation are calculated, and it is noticed that x = −b is any point of the foundation bottom; the limit load in the one-way failure mode is obtained as below: pz = q Nq + cNc + γ bNγ 0 ≤ b ≤ B

(6.55)

The limit load in two-way failure mode is:  pz =

0 ≤ b ≤ B/2 q Nq + cNc + γ bNγ q Nq + cNc + γ (B − b)Nγ B/2 ≤ x ≤ B

(6.56)

6.8 Calculation Process of Limit Load

173

where Nq = h a2 exp(π λ)

(6.57)

1 (Nq − 1) λ     2λ 3 2   Nγ = h (h + λ) exp π λ + 1 − λh a 1 + 9λ2 a a 2 Nc =

(6.58) (6.59)

(2) Discontinuity of velocity field Continuity of the slip surface is the same as that of the velocity field, but the velocity field may be: vx = h a vz vz = A a vx = Av z

 (A) 

vz = −Av x  vx = h c vz vz = A c

(6.60a)

(B)

(6.60b)

(C)

(6.60c)

         + τx z vz A − σx h ab − τx z vx A = σab vz A − vx A h ab According to  σz − h ab    and h ab τab vx A + vz A h ab = −h a , it will be: In (A), 0

−b/2 

σab 1 +

pz dx = − −b



0

2 h ab

 dx −

¨

−b/2 

2ch a dx

γ dzdx + A

(6.61)

−b

Likewise, in (B), 0

xb

2 σab (1 + h ab )xdx = −

− b2

2 0

¨ − B

In (C),

  σbc 1 + h 2 bc xdx  xb /2  λz − x γ xdzdx− c z−x dx z + λx 2 −b/2

(6.62)

174

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

0 xb 2

  σbc 1 + h 2 bc dx = −

¨

xb qdx + 0

xb γ dzdx +

C

2ch c dx

xb 2

  The following equation may be derived: σbc (1 + h 2 bc ) = 2(q − ch c − γ h c x) After the above equation is substituted in Eq. (6.62), it will be:

   c 2 = 2qh a2 exp(π λ) + 2h a2 exp(π λ) − 1 − h a2 σab 1 + h ab  λ      3 λ  2 2  h + 2h π λ + 3 − h (3h − 1) exp + 2γ a a a x 1 + 9λ2 a 2 After it is substituted in Eq. (6.61), the limit load whose continuity is the same as that of the velocity field is obtained. Through this example, it is seen that the result obtained by limit equilibrium method, stress field method, and generalized limit equilibrium method (including upper bound solution method applied at present) is the same, when the slip surface is the same.

6.8.2 Example II: Slip Surface in Plane–Arc Surface–Plane Form The slip surface is changed to plane–arc surface–plane (Fig. 6.6), and the limit load needs to be solved. √ The slip surface in (A) is plane h a = λ + 1 + λ2 = tan θ and θ = π4 + ϕ2 ; the slip surface in (B) is arc surface h b = −x/z and the slip surface in (C) is plane √  h c = λ− 1 + λ2 = −1/ h a . −b is supposed as any point on the surface a : [−B, 0]. In order to get a smooth slip surface, it shall be: x1 = −b

h a2 h a h a h a2 , h = b , x = b , h = b and xb = bh a . 1 2 2 1 + h a2 1 + h a2 1 + h a2 1 + h a2

Fig. 6.6 Schematic diagram for plane–arc surface–plane calculation mode

6.8 Calculation Process of Limit Load

175

(1) Continuity of velocity field # # Since arctan s # BC = π2 − θ¯ and arctan s # AB = −θ are true respectively for the interface  BC of (B) and (C) and interface  AB of (A) and (B), the velocity field will be:     vx = exp −2λθ¯ z − h a x 2 (6.63a)       (A) 2h a vz = exp −2λθ¯ z − h a x  vx = (z − xλ) exp(2λ arctan s) (6.63b) (B) vz = −(x + zλ) exp(2λ arctan s)      vx = exp 2λ π/2 − θ¯ z − h c x 2 (6.63c)        (C) 2h c vz = exp 2λ π/2 − θ¯ z − h c x continuous. The slip surface and velocity field are substituted in Eq. (6.25), since: ¨ D

  xb 0 xb  ∂vz c ∂vx c c c  + dzdx = vz − h vx Γ dx − (vz )a dx − (vz )b dx λ ∂x ∂z λ λ λ −b

−b

0

(6.64) ¯ is supposed, it will be: Aλ = exp(−2λθ) 0 −b

b  x x Aλ dx = q Aλ exp(λπ )dx pz − 2 2 x

0

⎡

c⎢ + ⎣ λ

0

−b

1 x Aλ dx + 2

xb

⎤ 1 ⎥ x Aλ exp(λ(π )dx ⎦ 2

0

⎡   ¨ ¨ z − h a x x dzdx Aλ dzdx − (x + zλ) exp 2λ arctan − γ⎣  z 2h a A

¨ + C

B

⎤

z − h c x Aλ exp(π λ)dzdx ⎦ 2h c

(6.65)

The integral in Eq. (6.65) is calculated; after rearrangement, the limit load is still Eq. (5.55) [or Eq. (5.56)], only Nγ is as below: Nγ =

2λ [h 3 exp(π λ) + 1] 1 + 4λ2 a

(6.66)

176

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

(2) Discontinuity of velocity field If the velocity field is discontinuous:   vx = Aa 1 + h a λ   (A) vz = Aa h a − λ

(6.67a)



vx = (z − xλ) exp(2λ arctan s) vz = −(x + zλ) exp(2λ arctan s)

(B)

(6.67b)

  vx = Ac 1 + h c λ   (C) vz = Ac h c − λ

(6.67c)

Calculation shall be carried out in subareas.  In (A), for the interface slope of (A) and (B) h ab = −1/ h a is true, so it will be: 

        τx z vz A − σx h ab − τx z vx A = σab vz A − vx A h ab σz − h ab    + τab vx A + vz A h ab    2 = −(σab + λτab ) 1 + h ab h ab

(6.68)

The above equation is substituted in the virtual work equation to get: 0



pz h a

−b

 − λ dx =

 x1   1  2 dx (σab + λτab ) 1 + h ab  h ab 0

¨ −

γ



h a

 − λ dzdx +

x1

  c 1 + h a2 ddx

(6.69)

−b

A

In (B), x = R cos θ, z = R sin θ is true for the slip surface. According to, ¨ B

  x2  ∂vz c ∂vx c + dzdx = vz − h  vx  dx λ ∂x ∂z λ x1

0 − x1

 c  vz − h ab vx ab dx − λ

x2 0

 c vz − h bc vx bc dx λ (6.70)

It will be: 0 x1

     2 −(σab + λτab ) 1 + h ab x exp −2λθ¯ dx

6.8 Calculation Process of Limit Load

177

x2 $ π  %   = − θ¯ dx (σbc + λτbc ) 1 + h 2 bc x exp 2λ 2 0 ¨ [γ (x + zλ) exp(2λ arctan s)dzdx + B

0 + x1

   c 2 1 + h ab exp −2λθ¯ xdx + λ

x2 0

π   c 1 + h 2 − θ¯ xddx bc exp 2λ λ 2 (6.71)

In (C), it is:  ¨ 0  xb  1       2 −(σbc + λτbc ) 1 + h bc  dx = − q h c − λ dx + γ h c − λ dzdx h bc

x2

0

xb +

C

  c 1 + h 2 c dx

(6.72)

x2

Calculation is carried out for Eq. (6.72), and it may be obtained:          2     (6.73) (σbc + λτbc ) 1 + h 2 bc = 1 + h a q 1 + h a λ + ch a + γ h a 1 + h a λ x The above is substituted in Eq. (6.75); after calculation and rearrangement, it is obtained as below:   $  2 = 1 1 + h 2 q 1 + h  λ exp(π λ) + c h  λ + 1 exp(π λ) − 1 (σab + λτab ) 1 + h ab a a a 2 λ ha 

  2 γ x 1 + ha 2 2 2  − (6.74) 4λ 1 + λ h a exp(π λ) + 1 − 2λ + 3h a λ 1 + 4λ2 h a3

Then the above is substituted in Eq. (6.69); after calculation and rearrangement, the equation that is consistent with continuity of the velocity field is derived. From the two examples abovementioned, it is concluded that the velocity field is not unique and if only the slip surface is the same, the limit load will be the same.

6.8.3 Example III: Slip Surface in Plane–General Helicoid–Plane Form Equation of the general helicoid is:

178

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

x − x R = R exp(−kθ ) cos θ



z − z R = R exp(−kθ ) sin θ

(6.75)

Similarly, if the combination form of the plane–general helicoid–plane (Fig. 6.7) is taken as the slip surface, the smooth slip surface and continuous velocity field may also be obtained. √   The slip surface in (A) is plane h  = h a = λ + 1 + λ2 = tan π4 + ϕ2 ; the slip  surface in (B) is general helicoid of Eq. (6.75) h  = h b = (kz − x) (z + kx), and √ the slip surface in (C) is plane h  = h c = λ − 1 + λ2 = −1/ h a . In order to obtain a smooth slip surface, as for the interface  AB of (A) and (B), it shall be:  π ϕ π π + + −ψ = +θ 2 4 2 2 √ h a 1 + k 2 R=b  exp(kθ1 ) 1 + h a2

θ1 =

(6.76) (6.77)

For the interface  BC of (B) and (C), it will be: ϕ π + −ψ =θ 4 2 π  xb = b exp k h a 2

θ2 =

(6.78) (6.79)

The velocity field is: ⎫   1  2 ⎪ vx = z − h a x 1 + k g(s1 ) ⎪ ⎬ 2 (A)   1  ⎪ vz =  z − h a x 1 + k 2 g(s1 )⎪ ⎭ 2h a  vx = [(1 + kλ)z + (k − λ)x]g(s) (B) vz = [(k − λ)z − (1 + kλ)x]g(s)

Fig. 6.7 Schematic diagram for plane–general helicoid–plane calculation mode

(6.80a)

(6.80b)

6.8 Calculation Process of Limit Load

179

⎫   1 ⎪ z − h c x 1 + k 2 g(s2 ) ⎪ ⎬ 2 (C)   1  ⎪ vz =  z − h c x 1 + k 2 g(s2 )⎪ ⎭ 2h c

vx =

(6.80c)

It is easy to verify # that the velocity field is continuous. Since# arctan s # BC = π2 − θ is true for the interface  BC of (B) and (C) and arctan s # AB = −θ is true for the interface  AB of (A) and (B), the slip surface and velocity field are substituted in Eq. (6.25), since  ˜ c  ∂vx xb c  0 c xb c  ∂vz  dzdx = v + − h v dx − (v ) dx − (v ) dx, it z x z  a λ ∂x ∂z λ λ λ z b  −b

D

will be: 0 −b

−b

0

x

b  x x pz − A dx = q Aλ exp[(λ − k)π ]dx 2 λ 2 0

⎧   x ⎨¨ z − h  x b 1 a A dz d x x Aλ d x + x Aλ exp[(λ − k)π ]dx − γ λ ⎩ 2 2h a 0 −b 2 A  ¨  x (k − λ)z − (1 + kλ)x] exp[2(λ − k) arctan dz d x + z c + λ

B

¨ + C

&

0 1

⎫ ⎪ ⎬ z − h c x Aλ exp[(λ − k)π ]dz dx ⎪ 2h c ⎭

(6.81)

where  π − θ1 . Aλ = (1 + k 2 ) exp 2(λ − k) 2 Each integral in the equation is calculated, ¨ A

¨

  z − h a x b3 h a 1 + kh a dzdx = 2h a 6 1 + h a2

(6.82)

[(k − λ)z − (1 + kλ)x] exp[2(λ − k) arctan s]dzdx B

   

 $  h a3 1 + k 2 Aλ 1 3 2 − kλ h  − 3λ − k exp π (k + 2λ) A 1 − 2λ = b  2  λ a  3 2 1 + (k + 2λ)2 1 + h a2

% − 1 − 2λ2 − kλ + (3λ + k)h a (6.83)

¨ C

    z − h c x 1 3 h a3 h a − k 3 dzdx = − b exp kπ 2h c 6 1 + h a2 2

(6.84)

180

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

Table 6.1 Nγ in plane–general helicoid–plane calculation mode ϕ(◦ )

r0 0.0

0.25

0.5

0.75

1.0

1

0.074

0.074

0.074

0.074

0.075

5

0.460

0.469

0.477

0.486

0.495

10

1.237

1.286

1.336

1.390

1.447

15

2.555

2.713

2.885

3.074

3.283

20

4.824

5.248

5.724

6.268

6.905

25

8.847

9.877

11.079

12.525

14.327

30

16.267

18.679

21.612

25.347

30.382

35

30.722

36.358

43.528

53.280

67.740

40

60.977

74.531

92.653

119.233

163.501

45

130.646

165.282

214.168

292.370

442.751

After the above is substituted in Eq. (6.81), the limit load is still Eq. (5.55) [or Eq. (5.56)]. Where Nq and Nc are still the same as the equations obtained above. Nγ =

$ % π 2λ 2   (k + 2λ) + 1 − kh h (h + k) exp a 1 + (k + 2λ)2 a a 2

(6.85)

Nγ contains an arbitrary constant k. It is easy to verify that the obtained plane–common helicoid (k = λ)–plane equation and plane–arc surface (k = 0)–plane equation are all special cases of Eq. (6.85). And the calculation results (ψ = r0 ϕ) of Nγ are detailed in Table 6.1. Obviously, with different ψ = r0 ϕ, Nγ shows a great difference, which indicates that if ψ is properly selected, the total limit  load may be sufficiently close to the  π − 21 tan δ (1 + λ2 ) is true, the calculated total actual limit load. If r0 = λ − tan 12 limit load is detailed in Table 6.2. The calculated result is much close to the characteristic line method. Proper adjustment of ψ = r0 ϕ so as to make it be applicable to any q, c is a problem to be further discussed.

6.9 Limit Load in Helicoid–Helicoid–Plane Calculation Mode The generalized limit equilibrium method is applied to solve the limit load (Fig. 6.1) under combined action of horizontal force and vertical force: With the ground surface as horizontal surface, when the vertical load to be solved in the foundation bottom a : [−B, 0] and horizontal load as well as the distribution relationship of

6.9 Limit Load in Helicoid–Helicoid–Plane Calculation Mode

181

Table 6.2 Total limit load Pm = Pz /(γ B 2 ) in plane–general helicoid–plane calculation mode (q/(γ B) = 0.1 and c/(γ B) = 0.2) ϕ(◦ )

tan δ 0.0

1

1.222

5

1.679

10

2.518

0.1

0.2

0.3

0.4

1.808

15

3.867

2.960

1.957

20

6.160

4.774

3.424

2.128

25

10.321

7.944

5.801

3.953

2.362

30

18.443

13.941

10.126

7.039

4.620

35

35.704

26.280

18.737

12.955

8.669

40

76.210

54.217

37.577

25.468

16.923

45

182.842

124.895

83.477

54.892

35.687

the horizontal force and vertical force and vertical evenly distributed load on the soil mass surface b: 0 < x are supposed as pz = σz |z=0 = σz0 , px = τx z |z=0 = τx z0 ,  τx z0 = σz0 + λc tan δb and q, respectively, the limit load pz needs to be solved. The combination form of general helicoid–general helicoid–plane is taken as the slip surface family to derive the calculation equation for limit load.

6.9.1 Calculation Mode 1 The slip surface (family) is shown in Fig. 6.8. In (A), the slip surface is general helicoid h  = h a = [k(z − z R ) − (x − x R )]/[z − z R + k(x − x R )]; namely,

Fig. 6.8 Schematic diagram for general helicoid–general helicoid–plane calculation mode 1

182

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

⎫ b ⎪ exp[k(θ0 − θ )] sin θ1 cos θ ⎪ ⎬ sin(θ0 − θ1 ) θ0 ≥ θ ≥ θ1 b ⎪ ⎪ z − zR = exp[k(θ0 − θ )] sin θ1 sin θ ⎭ sin(θ0 − θ1 ) x − xR =

(6.86)

where  x R = −b sin θ0 cos θ1 / sin(θ0 − θ1 ) z R = −b sin θ0 sin θ1 / sin(θ0 − θ1 )

(6.87)

The boundary condition for the slip surface family in the foundation bottom is: h  = h 0 =

   1 + λ2 λ2 − tan2 δb   λ + tan δb 1 + 2λ2

λ(λ − tan δb ) +

(6.88)

1+kh 

It shall be tan θ0 = − h  −k0 . Thus, when the entry point (−b) on the slip surface 0 is different, (x R , z R ) also varies. In (B), the slip surface is general helicoid h  = h b = (kz − x)/(z + kx), namely, x = R B exp[k(θ1 − θ )] cos θ



z = R B exp[k(θ1 − θ )] sin θ where R B =

θ1 ≥ θ ≥ θ2

b {− sin θ0 + exp[k(θ0 − θ1 )] sin θ1 } sin(θ0 − θ1 )

(6.89) (6.90)

The following inequation must be met: sin θ0 − exp(k(θ0 − θ1 )) sin θ1 ≤ 0 It is easy to verify that the slip surface is smooth with regard to the interface  AB of (A) and (B). √ In (C), the slip surface is plane h  = h c = λ − 1 + λ2 . In order to the smooth slip surface in the interface  BC of (B) and (C), it shall be: θ2 =

ϕ π + −ψ 4 2

 R2 = {exp[k(θ0 − θ2 )]sinθ1 − exp[k(θ1 − θ2 )] sin θ0 } sin(θ0 − θ1 )

6.9.2 Velocity Field The velocity field is:

(6.91) (6.92)

6.9 Limit Load in Helicoid–Helicoid–Plane Calculation Mode

vx = [(1 + kλ)(z − z R ) + (k − λ)(x − x R )]g(¯s )

183



vz = [(k − λ)(z − z R ) − (1 + kλ)(x − x R )]g(¯s )  vx = [(1 + kλ)z + (k − λ)x]g(s) (B) vz = [(k − λ)z − (1 + kλ)x]g(s) ⎫   1 + h c λ   2 ⎪ z − h g(s x 1 + k vx = ) 2 ⎪ c ⎬ 1 + h 2 c (C)   ⎪ h − λ  ⎭ vz = c 2 z − h c x 1 + k 2 g(s2 ) ⎪ 1 + hc

(A)

(6.93)

(6.94)

(6.95)

where  s¯ = (x − x R ) (z − z R )

(6.96)

g(s) = Av exp[2(λ − k) arctan s]

(6.97)

It is easy to verify that the velocity field is discontinuous at the interface of (A) and (B).

6.9.3 Calculation Equation for Limit Load In (A), the velocity field is substituted in the virtual work equation to get: 0

+ * −σz0 [(k − λ)z R + (1 + kλ)(x − x R )] − τx z0 [(1 + kλ)z R − (k − λ)(x − x R )]

−b

   x − xR dx exp 2(λ − k) arctan −z R =−

x1 $ 0

¨ − A

+

  x − xR dzdx γ [(k − λ)(z − z R ) − (1 + kλ)(x − x R )] exp 2(λ − k) arctan z − zR

0

−b x1

+ 0

π    % − θ1 dx σ (1 + kλ) − τ (k − λ) 1 + tan2 θ1 (x − x R ) exp 2(λ − k) 2  AB

  

c x − xR dx [(k − λ)z R + (1 + kλ)(x − x R ) exp 2(λ − k) arctan λ −z R

  π  c (1 + kλ) 1 + tan2 θ1 (x − x R ) exp 2(λ − k) − θ1 dx λ 2

(6.98)

184

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

Since the velocity fields in (B) and (C) are continuous, the virtual work equation may be combined as below: 0 $

 % π − θ1 [σ (1 + kλ) − τ (k − λ)](1 + tan2 θ1 )x exp 2(λ − k) dx 2  AB x1   ¨ x dzdx γ [(k − λ)z − (1 + kλ)x] exp 2(λ − k) arctan =− z B ¨  π h − λ − − θ2 dzdx γ c  2 (1 + k 2 )(z − h c x) exp 2(λ − k) 1 + hc 2 C

xb + 0

x1 − 0

 π c h c − λ 2  − θ2 dx  2 (1 + k )h c x exp 2(λ − k) λ 1 + hc 2  π c (1 + kλ)(1 + tan2 θ1 )x exp 2(λ − k)( − θ1 dx λ 2

xb +

q 0

 π h c − λ 2  − θ2 dx  2 (1 + k )h c x exp 2(λ − k) 1 + hc 2

(6.99)

The integral in the Eq. (6.99) is calculated, ¨

1 [(k − λ)z − (1 + kλ)x] exp[2(λ − k) arctan s]dzdx = − x13 Ak f B 3

(6.100)

B

where *    exp(3kθ1 ) exp[−(k + 2λ)θ1 ] sin θ1 1 + k 2 − 2λ2 + 2kλ cos3 θ1 (1 + (k + 2λ)2 )      − λ cos θ1 3 + k 2 + 2kλ − exp[−(k + 2λ)θ2 ] sin θ2 1 + k 2 − 2λ2 + 2kλ  + − λ cos θ2 3 + k 2 + 2kλ (6.101)

fB =

Ak = exp[π(λ − k)] ¨ C

 π   1 h c − λ  2  − θ dzdx = − x13 Ak f C 1 + k z − h x exp 2(λ − k) 2 c 2 1 + hc 2 3 (6.102)

6.9 Limit Load in Helicoid–Helicoid–Plane Calculation Mode

185

 2  1 1 2 exp(3kθ1 ) fC = 1 + k exp[−(k + 2λ)θ2 ] sin θ2 cos θ2 −  sin θ2 2 cos3 θ1 hc (6.103) Supposing, fq =

 2  exp(2kθ1 ) 1 1 1 + k2 exp(−2λθ − sin θ cos θ ) 2 2 2 2 cos2 θ1 h c

(6.104)

The above is substituted in Eq. (6.99), at the interface of (A) and (B), it will be:     − σ (1 + kλ) − τ (k − λ)] 1 + tan2 θ1 exp[−2(λ − k)θ1    c c f q − (1 + kλ) 1 + tan2 θ1 exp[−2(λ − k)θ1 ] + γ x( f B + f C ) = q+ λ λ (6.105) The above is substituted in Eq. (6.98) to get:

−

0  c {[(k − λ)z R + (1 + kλ)(x − x R )] σz0 + λ

−b

   x − xR dx + tan δx [(1 + kλ)z R − (k − λ)(x − x R )]} exp 2(λ − k) arctan −z R x1

= 0

% $ c dx f q + γ x( f B + f C ) exp[π(λ − k)] (x − x R ) q + λ  AB

¨

− A

  x − xR dzdx γ [(k − λ)(z − z R ) − (1 + kλ)(x − x R )] exp 2(λ − k) arctan z − zR

(6.106)

Since, ¨ A

  b3 x − xR dzdx = Ak f A [(k − λ)(z − z R ) − (1 + kλ)(x − x R )] exp 2(λ − k) arctan z − zR 3

(6.107)

where

fA =

sin2 θ1 exp(3kθ0 ) sin3 (θ0 − θ1 )

θ0 [(k − λ) sin θ − (1 + kλ) cos θ ]{sin θ1 − exp[k(θ − θ0 ) ] θ1

sin θ0 [(sin θ1 − k cos θ1 ) sin θ + (cos θ1 + k sin θ1 ) cos θ ]} exp[−(k + 2λ)θ ]dθ

(6.108)

After the above is substituted in Eq. (6.106), the limit load in one-way failure mode may be obtained as below:

186

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

pz = q Nq + cNc + γ bNγ 0 ≤ b ≤ B

(6.109)

  1 + k2 sin θ2 2 exp[2λ(θ0 − θ2 )] E 10 cosθ2 − 2A h c

(6.110)

 1 Nq − 1 λ

(6.111)

where Nq =

Nc = Nγ =

  1 sin θ1 2 g E 10 (g B + gC ) − A A sin2 (θ0 − θ1 )

A = [k − λ + (1 + kλ) tan δx ] sin θ0 − [1 + kλ − (k − λ) tan δx ] cos θ0   E 10 = sin θ1 − exp(k(θ1 − θ0 )) sin θ0 sin(θ0 − θ1 ) Ak1 = 1 + k 2 − 2λ2 + 2kλ and Ak2 = λ(3 + k 2 + 2kλ)

(6.112) (6.113) (6.114) (6.115)

1 {exp[(k + 2λ)(θ0 − θ1 )](Ak1 sin θ1 − Ak2 cos θ1 ) 1 + (k + 2λ)2 (6.116) − exp[(k + 2λ)(θ0 − θ2 )](Ak1 sin θ2 − Ak2 cos θ2 )}

gB =

 2  1 1 2 gC = 1 + k exp[(k + 2λ)(θ0 − θ2 )] sin θ2 cos θ2 −  sin θ2 2 hc

(6.117)

sin θ1 {Ak1 sin θ0 − Ak2 cos θ0 1 + (k + 2λ)2 − exp[(k + 2λ)(θ0 − θ1 )](Ak1 sin θ1 − Ak2 cos θ1 )}     sin θ0 *  1 + λ2 1 + k 2 (cos θ1 + λ sin θ1 ) 1 − exp(2λ(θ1 − θ0 )) λ −  2 4 1+λ   + (Ac1 cos θ1 + Ac2 sin θ1 ) sin 2θ1 exp(2λ(θ1 − θ0 )) − sin 2θ0 +  + (Ac2 cos θ1 − Ac1 sin θ1 ) cos 2θ1 exp(2λ(θ1 − θ0 )) − cos 2θ0 (6.118)

gA =

Ac1 = (1 + kλ)2 − (k − λ)2 and Ac2 = 2(1 + kλ)(k − λ)

(6.119)

Nq , Nc , and Nγ all contain the parameter k = tan ψ. In addition, θ1 is also an alternative parameter. It is easy to verify that when θ0 = θ1 is true, the equation is retrograded into the plane–general helicoid–plane equation obtained above; and when ψ = ϕ is true, the slip surface is in common helicoid–common helicoid–plane form.

6.9 Limit Load in Helicoid–Helicoid–Plane Calculation Mode

187

Fig. 6.9 Schematic diagram for general helicoid–general helicoid–plane calculation mode 2

6.9.4 Calculation Mode 2 If the slip surface family is in the form shown in Fig. 6.9, which is only different from the above slip surface family with regard to the position of center (x R , z R ) of the helicoid in (A), just as the stress field method or limit equilibrium method, Eqs. (6.109)–(6.119) may also be obtained.

6.9.5 Calculated Results ψ = r0 ϕ and qc = (qλ + c)/(γ B) are supposed, when r0 = 1.0 and θ1 = θ0 − 0.5(ϕ − δ) exp[1.6(1 + (λ − tan δ)(1 − λ))/(1 + 18qc )]/(1 + 1.5qc ) are true, the calculated total limit load Pm = Pz /(γ B 2 ) is detailed in Tables 6.3, 6.4 and 6.5. It is quite close to the limit load calculated by the characteristic line method. The chief differences between the generalized limit equilibrium method and limit equilibrium method are as follows: Firstly, the virtual work equation replaces the moment equation, which is a doubtless improvement compared with the conventional limit equilibrium method. And its main reflections are as below: Without any assumption and simplification for the force to the soil mass, the alternative slip surface is widened, and as long as the slip surface family of the velocity field may be determined according to the velocity equation, they all may serve as the alternative slip surface family, which shows a big improvement compared with the limit equilibrium method by which only the plane and logarithmic helicoid (as for other slip surfaces, assumption, and simplification for the force to the slip surface must be carried out to derive the equation) are taken. Secondly, the above-calculated result indicates: when k is relatively small while ϕ is relatively large, the calculated Nγ is obviously less than the calculated result by the characteristic line method; thus, one of the main differences between the solution process of the generalized limit equilibrium method and that of the limit

188

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

Table 6.3 Total limit load Pm (q/(γ B) = 0.1 and c/(γ B) = 0.2) in general helicoid–general helicoid–plane calculation mode ϕ(◦ )

tan δ 0.0

0.1

0.2

0.3

0.4

1

1.162

5

1.602

10

2.432

1.728

15

3.788

2.849

1.883

20

6.107

4.641

3.299

25

10.299

7.779

5.615

3.810

30

18.405

13.689

9.832

6.792

4.453

35

35.465

25.783

18.212

12.513

8.348

2.058 2.290

40

75.452

53.231

36.632

24.690

16.343

45

183.308

124.455

82.657

54.021

34.928

Table 6.4 Total limit load Pm (q/(γ B) = 0.0 and c/(γ B) = 0.1) in general helicoid–general helicoid–plane calculation mode ϕ(◦ )

tan δ 0.0

0.1

0.2

0.3

0.4

1

0.532

5

0.817

10

1.365

0.913

15

2.276

1.639

1.018

20

3.862

2.838

1.934

1.129

25

6.785

4.990

3.489

2.268

1.271

30

12.584

9.156

6.410

4.286

2.687

35

25.217

17.987

12.430

8.315

5.360

40

56.284

39.011

26.311

17.313

11.129

45

145.777

97.245

63.301

40.415

25.418

equilibrium method (including the present upper bound solution method) is that the minimization process is not required anymore. Because the extremum condition of the yield function is considered in the derivation process of the calculation equation for the limit load by means of the generalized limit equilibrium method. However, the extremum condition of the yield function isn’t considered in the derivation process by the limit equilibrium method and upper bound solution method. In addition, when the slip surface is in the combination form of common helicoid and plane, the generalized limit equilibrium method in the field failure mode is different from the limit equilibrium method and upper bound solution method with regard to the specific solution method. As for the generalized limit equilibrium method, the

6.9 Limit Load in Helicoid–Helicoid–Plane Calculation Mode

189

Table 6.5 Total limit load Pm (q/(γ B) = 0.2 and c/(γ B) = 0.4) in general helicoid–general helicoid–plane calculation mode ϕ(◦ )

tan δ 0.0

0.1

0.2

0.3

0.4

1

2.324

5

3.026

10

4.313

3.158

15

6.350

4.915

3.357

20

9.729

7.598

5.564

25

15.648

12.130

9.004

6.306

3.941

30

26.747

20.384

15.031

10.690

7.245

35

49.374

36.717

26.582

18.766

12.901

40

100.686

72.524

51.060

35.291

24.019

45

234.283

162.076

109.906

73.513

48.767

3.600

solution is carried out according to the field failure mode and distribution of the limit load may be obtained; but as for the upper bound solution method, even though the soil mass in the slip mass is required to meet the yield criterion, the solution is carried out according to the surface failure mode, and because the slip surface family isn’t considered, distribution of the limit load cannot be obtained but only the total limit load. In fact, if the slip surface is only in the combination form of common helicoid and plane, the upper bound solution method by which the solution is carried out according to the surface failure mode is the same as the limit equilibrium method. As for heterogeneous soil under general condition, it is relatively difficult for the virtual work equation-based generalized limit equilibrium method to obtain the continuous velocity field, because the velocity equation is related to the strength index. And this is a problem remains to be solved to apply the virtual work equationbased generalized limit equilibrium method to heterogeneous soil.

6.10 Discussion on Variational Principle for Limit Load The above-generalized limit equilibrium method is obtained on the basis that the stress field, slip surface family, and velocity field are in accordance with the limit equilibrium. The following is the problem to be discussed now: If the selected slip surface family by the generalized limit equilibrium method is the real slip surface family, is it the calculated exact solution (the same as the obtained limit load according to the limit equilibrium) of the limit load?

190

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

6.10.1 Variation of Limit Load D is supposed as the area (Fig. 6.10) where the soil mass is in the limit state; thereinto, s : a +b + is the boundary surface in D, a is the foundation bottom (the known velocity boundary), x0 is any point on the foundation bottom, b is the ground surface (known load boundary), and  is the slip surface (enveloping surface). As for any stress field, potential slip surface family and velocity field, it will be: 

     σz − h s τx z vz − σx h s − τx z vx a +b d x x0 ⎛     ¨  ∂σx ∂τx z ∂τx z ∂σz + vx + + − γ vz dzdx = −⎝ ∂x ∂z ∂z ∂x D   ¨  ∂vx ∂vz ∂vx ∂vz γ vz + σz dzdx − + σx + τx z + ∂z ∂x ∂z ∂x D  xN      + σz − h  τx z vz − σx h  − τx z vx  dx xN

(6.120)

x0

Supposing, 0

     σz − h s τx z vz − σx h s − τx z vx a dx = J0

(6.121)

x0

J0 = −

  ¨  ∂vz ∂vx ∂vz ∂vx + σx + τx z + dzdx γ vz + σz ∂z ∂x ∂z ∂x D

x N + x0 x N

−

     σz − h  τx z vz − σx h  − τx z vx  dx    #  σz − h s τx z vz − σx h s − τx z vx #b dx

0

Fig. 6.10 Schematic diagram for limit load

(6.122)

6.10 Discussion on Variational Principle for Limit Load

¨  − D

191

    ∂σz ∂σx ∂τx z ∂τx z + + − γ vz dzdx vx + ∂x ∂z ∂z ∂x

In Eq. (6.122), a and b have already been supposed as the corresponding boundary velocity and boundary load. Since the stress on a is the load to be solved, J0 is the functional regarding the limit load. In addition, when the functional J0 is in accordance with the equilibrium equation, yield condition, extremum condition of the yield function, and conditional extremum under the corresponding boundary condition, it is the variation of the limit load. Generally, the conditional extremum of the functional may be changed to the unconditional extremum by applying the Lagrange multiplier method. When μ, η, and κ are supposed as the Lagrange multipliers, only the extremum of the following functional needs to be discussed: x N

¨ Gdzdx +

x N (Fb )b dx

(6.123)

     Fb = − σz − h b τx z vz − σx h b − τx z vx b

(6.124)

J=

(F) dx + x0

D

0

where

       F = F x, h, h  , σz , . . . = σz − h  τx z vz − σx h  − τx z vx + κ f 

(6.125)

     ∂vz ∂vx ∂vx ∂vz + σx + τx z + + μf + η f h  (6.126) G = G x, z, h  , σz , . . . = − γ vz + σz ∂z ∂x ∂z ∂x           1 f x, h  , σx , σz , τx z = σz − h  τx z h  − λ − σx h  − τx z 1 + h  λ − c 1 + h 2 2 1+h        1 2  f h  x, h , σx , σz , τx z =  (σz − σx ) 1 + 2h  λ − h 2 − τx z 2h  − λ + h 2 λ 2 2 1 + h 2

The reason why the equilibrium equation is not included in the equation is that this functional is derived according to the equilibrium equation. In Eq. (6.123), there exist many unknown functions such as the stress components σx , σz , τx z , slip surface family h  , velocity fields vx and vz as well as Lagrange multipliers μ, η, and κ, and the slip surface  is unknown, so the area D of the surface integral and  x  and upper limit of the line integral are variable. As for Jb = 0 N (Fb )b dx, since x N is variable and vx and vz are unknown, the first-order variation δ Jb will be:      δ Jb = − σz − h b τx z vz − σx h b − τx z vx x N δx N x N − 0



    σz − h b τx z δvz − σx h b − τx z δvx b dx

(6.127)

192

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

x As for J1 = x0N [F(x, h, h  , σz , . . .)] dx, since x N ,  is variable and the stress and velocity components are unknown, the increments of the stress and velocity components not only exists on  but also takes place in that  is variable, namely x N+δx N

   F x, h, h  , σz (x, h) + δσz . . . +δ dx

x0 x N+δx N

=

  F x, h + δh, h  + δh  , σz (x, h + δh) + δσz . . . dx

x0

Thus, the first-order variation will be: x N+δx N

δ J1 =

  F x, h + δh, h  + δh  , σz (x, h + δh) + δσz . . . dx −

x0

x N

  F x, h, h  , σz . . . dx

x0

x N+δx N

=

  F x, h + δh, h  + δh  , σz (x, h + δh) + δσz . . . dx

x0

x N −

F(x, h + δh, h  + δh  , σz (x, h + δh) + δσz . . .)dx

x0

x N +

F(x, h + δh, h  + δh  , σz (x, h + δh) + δσz · · · )dx −

x0

x N

F(x, h, h  , σz · · · )dx

x0

 # Hence, δ J1 = F x, h, h  , σz , . . . #x=x N δx N    x N  ∂F ∂F ∂ F ∂σz δh +  δh  + δh + δσz + · · · dx + ∂h ∂h ∂σz ∂z x0

It is noted that (x N , h N ) is variable along the boundary surface, so it will be [3]:   #     ∂   , σ , . . . # δ J1 = F x, h, h  , σz , . . . + h s − h  F x, h, h z x=x N δx N ∂h  x N  + δ J1 =



x0

d ∂F − ∂z dx



∂F ∂h 

 +

    ∂ F ∂σz ∂F + · · · δh + dx δσz + · · · ∂σz ∂z ∂σz  

    σz − h b τx z vz − σx h b − τx z vx + κ f x δx N N

x N  + x0

− vx h 

  ∂vz   ∂vx ∂σz d − σx h  − τx z + vz (τx z vz + σx vx ) + σz − h  τx z dx ∂z ∂z ∂z

     ∂σx ∂τx z + (vx − h  vz ) δh + σz − h  τx z δvz − σx h  − τx z δvx ∂z ∂z 

(6.128)

6.10 Discussion on Variational Principle for Limit Load

193

     + δσz − h  δτx z vz − h  δσx − δτx z vx      ∂f ∂f ∂κ ∂f ∂f ∂f dx δh + k δh +  δh  + + f δσz + δσx + δτx z ∂z ∂z ∂h ∂σz ∂σx ∂τx z 

Likewise, under condition of J2 =

˜

(6.129)

G(x, z, h  , σz , . . .)dzdx, the first-order

D

variation is: ¨ ¨     δ J2 = G x, z, h  + δh  , σz + δσz , . . . dzdx − G x, z, h  , σz , . . . dzdx D+δ D

D

¨



 G x, z, h  + δh  , σz + δσz , . . . dzdx

= D+δ D

¨

  G x, z, h  + δh  , σz + δσz , . . . dzdx

− D

¨



 G x, z, h + δh , σz + δσz , . . . dzdx −

+





D

¨

  G x, z, h  , σz , . . . dzdx

D

Hence, x N δ J2 =

  G x, z, h  , σz , . . . | δhdx +

x0

¨  D

 ∂G  ∂G δh + δσ + · · · dzdx z ∂h  ∂σz (6.130)

x N δ J2 = x0

  G x, z, h  , σz , . . . | δhdx

¨ 

+ D

   ∂σx ∂τx z ∂τx z ∂σz + δvx + + − γ δvz ∂x ∂z ∂z ∂x

∂ (μf + η f h  )δh  + f δμ + f h  δη  ∂h     ∂ ∂ ∂vz ∂vx δσz + δσx + (μf + η f h  ) − (μf + η f h  ) − ∂σz ∂z ∂σx ∂x   ∂vz ∂ ∂ ∂vx − δτx z − + (μf + η f h  ) − (σx δvx + τx z δvz ) ∂τx z ∂z ∂x ∂x  ∂ (6.131) − (σz δvz + τx z δvx ) dzdx ∂z +

Since the velocity on a is known, namely δvx = δvz = 0, it will be:

194

6 Virtual Work Equation-Based Generalized Limit Equilibrium Method

¨  D

=

 ∂ ∂ (σx δvx + τx z δvz ) + (σz δvz + τx z δvx ) dzdx ∂x ∂z

.

(σz δvz + τx z δvx )ddx − (σx δvx + τx z δvz )dz  xN      =− σz − h b τx z δvz − σx h b − τx z δvx b dx  0x N      + σz − h  τx z δvz − σx h  − τx z δvx  dx s

(6.132)

x0

Therefore, the first-order variation of the functional J is: δ J = δ Jb + δ J1 + δ J2 = (k f )x N δx N x N  + x0

x N + x0

       h − λ  1 + h λ   f δκ + κ f h  δh  + vz + κ δσz − h  δτx z − vx + κ h δσx − δτx z dx 2 2 1+h 1+h 

¨ 

+ D

      ∂σx ∂σz ∂κ ∂f ∂τx z ∂τx z vx + f + η fh + κ δh dx + + − γ vz + μ + ∂x ∂z ∂z ∂x ∂z ∂z 

     ∂f  ∂σx ∂σz ∂f ∂τx z ∂τx z + δvx + + − γ δvz + μ  + η h δh  + f δμ + f h  δη ∂x ∂z ∂z ∂x ∂h ∂h

    ∂f  ∂f  ∂f ∂vz ∂f ∂vx + μ +η h − +η h − δσz + μ δσx ∂σz ∂σz ∂z ∂σx ∂σx ∂x    ∂f  ∂f ∂vx ∂vz + μ +η h − − δτx z dz dx ∂τx z ∂τx z ∂z ∂x

(6.133)

6.10.2 Equivalence Between Variation and Limit Equilibrium If the equilibrium equation, yield condition, and extremum condition of the yield function are true ( ∂∂zf = 0 must be true), the Lagrange multipliers vx , vz , μ, and η are in accordance with the generalized flow rule, and Eq. (6.11) is true, and then: ¨

∂ fh  δh dzd x ∂h 

(6.134)

4λσe ∂ fh =− 1.0 is true, since z = H = Hc , Fs = 1.0 is true, and since z > H , Fs < 1.0 is true. And in these cases, the calculated result in the field failure mode is obtained (H in Fig. 7.4), and it may be explained as the result when the soil mass can’t bear the tensile stress.

σs

σx

H

H

Hc Hc Fig. 7.4 Excavation heights in surface failure and field failure modes

7.5 Discussion on Surface Failure Mode and Field Failure Mode

213

However, the calculated result (Hc shown in Fig. 7.4) in the surface failure mode is obtained under the boundary condition that the total stress on the slope surface is zero. Since the stress field is unknown, it is incapable of discussing whether the boundary condition is met everywhere.

7.6 Moment Equilibrium Equation-Based Generalized Limit Equilibrium Method The boundary condition of the ground surface is the load boundary condition as below: The limit load to be solved σz = pz (x), τx z = px (x) is borne within the width [−B, 0] of the foundation bottom and σz = q, τx z = 0 is borne in other areas. In this case, calculation of the limit load in surface failure discussed. From the discussion on the limit equilibrium method, the following is derived:

0

xb (−x pz + h s px )dx =

−b

xb xqdx +

−









γ x hdx −b

xb

σ h h + x − λ(h − h x)  dx + −b

xb

c(h − h x)dx + −b

0

xb



f (h − h  x) dx

(7.29)

−b

where x = x − x R , h = h − z R . During derivation of Eq. (7.29), only the equilibrium equation and yield condition of the force are applied. If the extremum condition of the yield function is also applied, along the slip surface (Chap. 2), it shall be: 2λ h − λ c dh  dσ − − γ σ + =0 dx 1 + h 2 λ dx 1 + λ2

(7.30)

According to the boundary condition of the soil mass surface, it will be:   c (1 + h 2 ) c q+ = σ+ 2 2  2 λ x=xb (1 + λ )h + (1 + h λ) x=xb λ

π  ϕ h x=xb = h c = λ − 1 + λ2 = − tan − 4 2

(7.31) (7.32)

Thus, σ may be solved, and after it is substituted in Eq. (7.29), it is able to calculate the limit load as below: The following method may also be adopted, for any function, it will be:

214

7 Generalized Limit Equilibrium Method in Plane Failure Mode



xb  c dh  dσ 2λ h − λ dx = 0 σ+ v − −γ dx 1 + h 2 λ dx 1 + λ2

−b

Hence,

0

xb (−x pz + h s px )dx =

−b

xb xqdx +

−

c(h − h x)dx + −b

0

xb

xb



γ x hdx −b

    σ h h + x − λ(h − h  x)

−b



2λ h − λ c dh  dσ − −γ σ + + v(x, h, h ) dx 1 + h 2 λ dx 1 + λ2 



 dx



In case of dv dx

 2λ v dh 1+h 2 dx

+

= h  h + x − λ(h − h  x)



v|x=−b = 0

(7.33)

Namely, 

x

v = exp(−2λ arctan h )

  exp(2λ arctan h  ) h  h + x − λ(h − h  x)  dx

(7.34)

−b

Then, it is supposed as below:  A0 =

(1 + h 2 )v 2 (1 + λ )h 2 + (1 + h  λ)2

 (7.35) x=xb

It will be:

0 −b

  −x pz + h s px dx =

xb

xb xqdx +

0

−b

 c  h h + x dx λ



xb  h − λ c A0 + γ xh+v dx − q + 1 + λ2  λ

(7.36)

−b

In theory, Eqs. (7.29) or (7.36) may be applied to calculate the limit loads pz , px . If the slip surface  in the equation is that in the limit state and stress σ ( f = 0) on the slip surface is that in the limit state, the calculated pz , px are the exact solutions of the limit load in the surface failure mode.

7.6 Moment Equilibrium Equation-Based Generalized Limit Equilibrium Method

215

In practical problems, the stress field and slip surface that are in accordance with the equilibrium equation, yield condition, and extremum condition of the yield function are difficult to obtain. Generally, the approximation of the limit load can be calculated only when the slip surface has been selected. Therefore, how to select a proper slip surface to make the calculated approximation of the limit load much closer to the actual limit load is one of the main tasks with regard to the limit equilibrium method. In order to make the selected slip surface much closer to the real slip surface, in general, the possible failure area is divided into multiple subareas and the corresponding slip surface is selected for every subarea to discuss. Moreover, examples for this solution idea are discussed in the following text.

7.7 Limit Load in Plane–Helicoid–Plane Calculation Mode See the calculation mode in Fig. 7.5. According to the potential slip mass and subarea form, the calculation equation for limit load is derived as below: In (A), for the slip surface, h = (x + b)h a is true and according to Eq. (7.34), it will be: va =

  1  (1 + h a2 )(x + b)2 − (x + b) (b + x R )(1 + h a λ) + z R (h a − λ) 2

(7.37)

In (B), for the slip surface, x − x R = R exp(−λθ ) cos θ and z − z R = R exp(−λθ ) sin θ are true: vb = va (x1 ) exp[2λ(θ1 − θ )]

(7.38)

In (C), for the slip surface, h = (x − xb )h c is true and since z R = −h c x R is true, it will be: vc = va (x1 ) exp[2λ(θ1 − θ2 )] + (x − x2 )2 (1 + h c λ)

Fig. 7.5 Schematic diagram for plane–helicoid–plane calculation mode

(7.39)

216

7 Generalized Limit Equilibrium Method in Plane Failure Mode

Under the smooth slip surface, it will be: θ1 =

π − ϕ + arctan(h a ) 2 π ϕ θ2 = − 4 2

R = exp(λθ1 )

bh a + x R (h a + h c ) sin θ1 − h a cos θ1

(7.40a) (7.40b) (7.40c)

The above are substituted in Eq. (7.36), and the following will be derived, namely:

0   c − z R px dx −(x − x R ) pz + λ −b ⎡ x ⎤

b

 c ⎣ (h c − λ) = q+ xdx − vc (xb )⎦ λ (1 + λ2 )h c 0

xb  +γ −b

h − λ xh + v 1 + λ2



c Aq + γ Aγ dx = q + λ 

(7.41)

The integral in the equation is calculated, and it will be: (1 − λ/ h c ) va (x1 ) exp[2λ(θ1 − θ2 )] 1 + λ2   1 h a − λ   2 Aγ = 2h a + (1 + h a ) (x1 + b)3 6 1 + λ2    1 h a − λ   b ha + − (1 + h a λ) 2 1 + λ2   h a − λ     (1 + h a λ − h c (h a − λ)) (x1 + b)2 + x R ha + 1 + λ2  1 1 3 λ 2 + − x 2 h c x R − h + (x 2 + h )(x + 3λh) 2 2 3 1 + 9λ   va (x1 ) exp[2λ(θ1 − θ )](h − 3λx) xx21 + 1 + 9λ2 1 1 h − λ − h c x23 + h c x R x22 + c 2 va (x1 ) exp[2λ(θ1 − θ2 )]x2 3 2 1+λ

Aq = 2x2 (x2 − x R ) − x22 −

(7.42)

(7.43)

Here, the left side of Eq. (7.41) is considered. If the ratio of the total horizontal force to the total vertical force within the foundation width is tan δ and the eccentricity is e, it will be:

7.7 Limit Load in Plane–Helicoid–Plane Calculation Mode

0 px dx = tan δ −b

0

217

c dx λ

pz +

−b

0 



x+ −b

b − e pz dx = 0 2

Hence,

0 

0  c c − z R px dx = A z −(x − x R ) pz + pz dx + Ac λ λ

−b

(7.44)

−b

where b − e + x R − z R tan δ 2

(7.45a)

 1 (b + x R − z R tan δ)2 − (x R − z R tan δ)2 2

(7.45b)

Az = Ac =

Thus, the total load within the foundation width [−b, 0] is obtained as below:

0 Pz = −b

 1 pz dx = b q Nq + cNc + γ bNγ 2

(7.46)

where Nq = (Aq /b2 )/(A z /b)

(7.47a)

  Nc = (Aq − Ac )/b2 /(λA z /b)

(7.47b)

Nγ = 2(Aγ /b3 )/(A z /b)

(7.47c)

The two parameters h a and x R are included in the equations, and the eccentricity in Eq. (7.45a) shall be that of the limit load rather than that of the design load. In addition, the eccentricity shall be determined according to distribution of the limit load, which may be calculated according to the field failure mode of the same slip surface by b Nγ /Pz . The following steps may means of the linear distribution, namely e = −γ 12 be adopted to calculate the total limit load: Step I. e = 0 is taken, and the first approximation of Pz , Nγ is obtained; Step II. The approximation of e is calculated, the second approximation of Pz , Nγ is obtained, and then Step II is repeated until small difference of approximation between Pz , e is got. In addition, it is found in the

218

7 Generalized Limit Equilibrium Method in Plane Failure Mode

actual calculation that the adequate accuracy can be obtained only by calculating the second approximation. If h a and x R are determined according to the field failure mode and the minimum limit load, respectively, the calculated results of total limit load Pm = Pz /(γ B 2 ) and depth (Z m /B) of the slip surface are detailed in Table 7.5. It is quite close to the results in the field failure mode (by the characteristic line method); the total limit load is slightly larger, and depth of the slip surface is slightly smaller. If θ1 is slightly reduced, the total limit load that is very close to that by the characteristic line method may be obtained. For instance, when θ1 = π − ϕ + 0.95 arctan(h a ) is taken, depth of the corresponding slip surface will be 2 reduced.

7.8 Limit Load in Helicoid–Helicoid–Plane Calculation Mode See the calculation mode in Fig. 7.6. When the moment point is taken, it is not required to be the center point of the helicoid. Here, when the moment point (x R , z R ) is taken under any condition, the calculation equation is derived as follows. (A) x − x R1 = Ra exp[λ(θ1 − θ )] cos θ , h − z R1 = Ra exp[λ(θ1 − θ )] sin θ

va = Ra {exp[λ(θ1 − θ )][(z R1 − z R )(sin θ + λ cos θ ) + (x R1 − x R )(cos θ − λ sin θ )] − A0 exp[λ(θ1 + θ0 − 2θ )]}

(7.48)

A0 = (z R1 − z R )(sin θ0 + λ cos θ0 ) + (x R1 − x R )(cos θ0 − λ sin θ0 )

(7.49)

where

(B) x − x R2 = Rb exp[λ(θ1 − θ )] cos θ , z − z R2 = Rb exp[λ(θ1 − θ )] sin θ

vb = va (θ1 ) exp[2λ(θ1 − θ )] + Rb {exp[λ(θ1 − θ )][(z R2 − z R )(sin θ + λ cos θ ) + (x R2 − x R )(cos θ − λ sin θ )] − A1 exp[2λ(θ1 − θ )]}

(7.50)

where A1 = (z R2 − z R )(sin θ1 + λ cos θ1 ) + (x R2 − x R )(cos θ1 − λ sin θ1 ) (C) h = (x − xb )h c

(7.51)

6.219

18.756

36.364

77.970

190.927

30.0

35.0

40.0

45.0

3.883

15.0

10.469

2.520

10.0

25.0

1.681

5.0

20.0

1.224

Pm

0.0

tan δ

1.0

ϕ(◦ )

1.754

1.403

1.160

0.997

0.872

0.787

0.718

0.673

0.647

0.683

Z m /B

131.830

56.083

27.016

14.286

8.110

4.850

2.993

1.823

Pm

0.1

1.480

1.189

0.981

0.837

0.720

0.629

0.537

0.420

Z m /B

88.604

39.115

19.358

10.412

5.933

3.480

1.971

Pm

0.2

Table 7.5 Calculation results of Pm and Z m /B (q/(γ B) = 0.1 and c/(γ B) = 0.2)

1.230

0.987

0.820

0.677

0.570

0.454

0.316

Z m /B

58.345

26.569

13.394

7.228

4.024

2.136

Pm

0.3

1.006

0.802

0.654

0.527

0.403

0.258

Z m /B

37.853

17.624

8.931

4.708

2.363

Pm

0.4

0.821

0.644

0.506

0.375

0.228

Z m /B

7.8 Limit Load in Helicoid–Helicoid–Plane Calculation Mode 219

220

7 Generalized Limit Equilibrium Method in Plane Failure Mode

Fig. 7.6 Schematic diagram for helicoid–helicoid–plane calculation mode

vc = vb (θ2 ) + (x − x2 )2 (1 + h c λ) − (x − x2 )(h c − λ)(z R + h c x R )

(7.52)

The above are substituted in Eq. (7.36) to derive the following, namely: ⎡ x ⎤

0 

b  c ⎣ c (h c − λ) − z R px dx = q + −(x − x R ) pz + xdx − vc (xb )⎦ λ λ (1 + λ2 )h c

−b

0

xb  +γ

xh + v −b

h − λ 1 + λ2



c Aq + γ Aγ dx = q + λ 

(7.53)

The integral in the equation is calculated, and it will be: (h c − λ) vb (θ2 ) (1 + λ2 )h c 

 1 2 1 3 λ 2 2  Aγ = xa z R − h a + x + h xa + 3λh a a a 2 3 1 + 9λ2   + (x R1 − x R ) xa h a + z R1 xa 

 x1 1 2 A0 2  exp[λ(θ0 − θ1 )] xa + h a h a − 3λxa −b − Ra 1 + 9λ2  1 2 1 3 λ 2 + xb z R − h b + (xb 2 + h b )(xb + 3λh b ) 2 2 3 1 + 9λ   + (x R2 − x R ) xb h b + z R2 xb   

 x2 1 2 A1 1 2  va (θ1 ) − xb + h b h b − 3λxb x1 + Rb 1 + 9λ2 Rb2 1 1 h − λ − h c x23 − z R x22 + c 2 vb (θ2 )x2 3 2 1+λ Aq = x2 (x2 − x R ) + x2 z R / h c −

(7.54)

(7.55)

where xa = x − x R1 , h a = h − z R1 , xb = x − x R2 , h b = h − z R2

(7.56)

7.8 Limit Load in Helicoid–Helicoid–Plane Calculation Mode

221

⎫ sin θ0 cos θ1 sin θ0 sin θ1 ⎪ ⎪ , z R1 = −(b − b ) sin(θ0 − θ1 ) sin(θ0 − θ1 ) ⎬ (7.57a) (b − b ) sin θ1 ⎪ ⎪ ⎭ exp[λ(θ0 − θ1 )] Ra = sin(θ0 − θ1 ) ⎫ sin θ1 cos θ2 sin θ1 sin θ2 ⎪ ⎪ , z R2 = −b x R2 = −b ⎬ sin(θ1 − θ2 ) sin(θ1 − θ2 ) (7.57b) sin θ2 ⎪ b − b  ⎪ ⎭ {exp[λ(θ0 − θ1 )] sin θ1 − sin θ0 } + b Rb = sin(θ0 − θ1 ) sin(θ1 − θ2 ) ⎫ π ⎪ θ0 = − ϕ + arctan(h a )⎬ 2 (7.57c) ϕ π ⎪ ⎭ θ2 = − 4 2 x R1 = −b − (b − b )

Thus, within the foundation width [−b, 0], the total load is also Eq. (7.46); thereinto, Aq , Aγ are Eqs. (7.54) and (7.55), respectively. If h a is determined according to the field failure mode, the parameters b and θ1 included in the equation and the moment point (x R , z R ) still need to be determined. If (x R1 , z R1 ) is taken as the moment point, when b = 0 is true, it will be the discussed calculation mode (Fig. 7.7), and the calculation is carried out when the total limit load is the minimum according to θ1 . The result will be slightly larger than that in the field failure mode (by the characteristic line method) and is basically equivalent to those by the limit equilibrium method. If c, ϕ is replaced by ck = c/F0 , tan ϕk = tan ϕ/F0 , and F0 = 1.008 + 0.022(tan ϕ − tan δ), the calculated total limit load Pm and depth of the slip surface Z m /B are very close to those by the characteristic line method. See the calculated results in Tables 7.6, 7.7 and 7.8.  of the limit load. For Different values  may also be taken for b to discuss calculation  qc  instance, if b = 0.03 + 0.46 (1.0 − 1+q 2 )(λ − tan δ) B is taken, and the minimum c calculated under condition of θ2 ≤ θ1 ≤ θ0 and θ1 ≤ θ0 ≤ π − ϕ is the limit load, it is quite close to that in the field failure mode (by the characteristic line method).

Fig. 7.7 Schematic diagram for helicoid–helicoid–plane

6.086

18.186

35.082

74.803

181.69

30.0

35.0

40.0

45.0

3.818

15.0

10.197

2.491

10.0

25.0

1.671

5.0

20.0

1.220

Pm

0.0

tan δ

1.0

ϕ(◦ )

1.950

1.547

1.272

1.075

0.929

0.819

0.737

0.682

0.656

0.678

Z m /B

126.62

54.144

26.168

13.880

7.902

4.740

2.933

1.791

Pm

0.1

1.653

1.318

1.080

0.905

0.768

0.652

0.547

0.418

Z m /B

85.244

37.734

18.707

10.079

5.755

3.383

1.923

Pm

0.2

Table 7.6 Calculation results of Pm and Z m /B (q/(γ B) = 0.1 and c/(γ B) = 0.2)

1.378

1.096

0.891

0.730

0.595

0.469

0.313

Z m /B

55.915

25.496

12.870

6.958

3.886

2.076

Pm

0.3

1.129

0.887

0.707

0.558

0.417

0.254

Z m /B

36.032

16.799

8.532

4.517

2.287

Pm

0.4

0.908

0.702

0.538

0.390

0.224

Z m /B

222 7 Generalized Limit Equilibrium Method in Plane Failure Mode

0.573

0.841

1.332

2.154

3.621

6.407

12.100

24.784

56.225

145.32

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

Pm

0.0

tan δ

1.0

ϕ(◦ )

1.806

1.402

1.128

0.937

0.797

0.694

0.623

0.584

0.582

0.647

Z m /B

98.598

39.533

17.943

8.964

4.823

2.741

1.607

0.923

Pm

0.1

1.521

1.188

0.956

0.785

0.659

0.557

0.467

0.362

Z m /B

64.274

26.625

12.387

6.286

3.391

1.885

1.004

Pm

0.2

Table 7.7 Calculated results of Pm and Z m /B (q/(γ B) = 0.0 and c/(γ B) = 0.1)

1.258

0.980

0.780

0.632

0.510

0.397

0.267

Z m /B

40.602

17.292

8.185

4.164

2.190

1.091

Pm

0.3

1.021

0.788

0.616

0.479

0.354

0.213

Z m /B

25.064

10.893

5.179

2.570

1.207

Pm

0.4

0.812

0.615

0.462

0.331

0.186

Z m /B

7.8 Limit Load in Helicoid–Helicoid–Plane Calculation Mode 223

9.988

27.162

49.801

100.66

231.22

30.0

35.0

40.0

45.0

6.551

15.0

15.982

4.467

10.0

25.0

3.140

5.0

20.0

2.398

Pm

0.0

tan δ

1.0

ϕ(◦ )

2.091

1.680

1.396

1.190

1.033

0.913

0.821

0.750

0.704

0.698

Z m /B

164.49

74.347

37.871

21.106

12.592

7.898

5.107

3.263

Pm

0.1

1.779

1.429

1.181

0.996

0.848

0.722

0.605

0.458

Z m /B

113.39

53.037

27.688

15.660

9.364

5.760

3.434

Pm

0.2

Table 7.8 Calculated results of Pm and Z m /B (q/(γ B) = 0.2 and c/(γ B) = 0.4)

1.483

1.189

0.973

0.802

0.657

0.516

0.344

Z m /B

76.383

36.792

19.549

11.094

6.496

3.648

Pm

0.3

1.216

0.968

0.774

0.613

0.462

0.282

Z m /B

50.677

24.958

13.346

7.427

3.963

Pm

0.4

0.983

0.766

0.589

0.429

0.249

Z m /B

224 7 Generalized Limit Equilibrium Method in Plane Failure Mode

7.8 Limit Load in Helicoid–Helicoid–Plane Calculation Mode

225

If (x R2 , z R2 ) is taken for the moment point, when θ1 = θ0 is true, it is easy to see that it is the condition of the above-given plane–helicoid–plane. Obviously, under the same calculation condition, when θ1 < θ0 is true, the calculated value of the limit load will reduce with the reduction of θ1 . For instance, when θ1 = θ0 − 0.0333π/(1 + 4(qλ + c)/(γ b)) is taken, the calculation result is quite close to those in the field failure mode (by the characteristic line method).

7.9 Limit Load in Plane–General Helicoid–Plane Calculation Mode The calculation mode is shown in Fig. 7.8. (A) h = (x + b)h a is true and va is the same as Eq. (7.37). (B) As for the slip surface,  x − x R = R exp[k(θ1 − θ )] cos θ h − z R = R exp[k(θ1 − θ )] sin θ   1 1 2 R − va (x1 ) exp[2λ(θ1 − θ )] vb = R 2 exp[2k(θ1 − θ )] − 2 2 π  ϕ − , (C) h = (x − xb )h c , h c = λ − 1 + λ2 = − tan 4 2

vc = vb (θ2 ) + (x − x2 )2 (1 + h c λ) − (x − x2 )(h c − λ)(z R + h c x R ) Through calculation, the following is supposed:

Fig. 7.8 Schematic diagram for plane–general helicoid–plane calculation mode

(7.58) (7.59) (7.60)

(7.61)

226

7 Generalized Limit Equilibrium Method in Plane Failure Mode

   1 k Aγ = R 3 exp[3k(θ1 − θ )] − sin3 θ + (3k sin θ + cos θ ) 3 1 + 9k 2 ! θ2 + 0.5R 2 z R exp[2k(θ1 − θ )] cos2 θ θ1  1 3 exp[3k(θ1 − θ )] 1 R + [(1 − 2kλ + 3k 2 ) sin θ 2 1+λ 2 1 + 9k 2 − (λ + 2k + 3k 2 λ) cos θ ]   exp[(k + 2λ)(θ1 − θ )] 1 3 R − Rva (x1 ) [(1 + k 2 + 2kλ − 2λ2 ) sin θ − 2 1 + (k + 2λ)2 ! θ2 − λ(3 + 2kλ + k 2 ) cos θ ] θ1   1 h − λ  2h a + a 2 (1 + h a2 ) (x1 + b)3 + 6 1+λ    1 h − λ (b + x R ) h a + a 2 (1 + h a λ) − 2 1+λ  h a − λ  1  3 1 h c − λ 2 2 h z (h − λ) (x + b) − x − x + vb (θ2 )x2 + zR 1 R 2 1 + λ2 a 3 c 2 2 1 + λ2 (7.62) Aq = x2 (x2 − x R + z R / h c ) −

1 − λ/ h c vb (θ2 ) 1 + λ2

(7.63)

It will be:

0 pz dx = −b

 c 1  c Aq − Ac + γ Aγ q+ Az λ λ

(7.64)

Thereinto, ϕ π + −ψ 4 2

(7.65)

π − ψ + arctan h a 2

(7.66)

(b + x R )h a − z R sin θ1 − h a cos θ1

(7.67)

θ2 = θ1 = R= zR =

(b + x R )h a (sin θ2 + h c cos θ2 ) − x R h c exp[k(θ1 − θ2 )](sin θ1 − h a cos θ1 ) sin θ2 + h c cos θ2 + exp[k(θ1 − θ2 )](sin θ1 − h a cos θ1 ) (7.68)

If h a and x R are determined according to the field surface method and the minimum limit load, respectively, the total limit load that is quite close to that in the field failure mode (by the characteristic line method) may be obtained as long as k is slightly

7.9 Limit Load in Plane–General Helicoid–Plane Calculation Mode

227

smaller than λ, for example, when k = λ/(1.01 + 0.01λ + 0.02 tan δ) is true, see the calculated results in Table 7.9. However, depth of the slip surface is slightly smaller than that by the characteristic line method.

7.10 Discussion on Limit Load and Critical Load As is well known, at the present time, the critical load (critical plastic load) [4] determined according to depth of the plastic deformation area is still applied in the calculation of the ground bearing capacity and the equation ( p1/4 ) recommended by the “Code for Design of Building Foundation” [5] to calculate the characteristic value of the ground bearing capacity is just a kind of critical load. There exists great difference in the theoretical foundations of the limit load and the critical load, especially for their calculated values. If ϕ is relatively large, the total difference between the total limit load and the critical load will reach more than five times. Even though the former serves as the ultimate bearing capacity and the latter serves as the allowable bearing capacity, their application status in the engineering is also unsatisfactory by now. Therefore, it is of great significance to analyze the difference between the limit load and the critical load. The plane–common helicoid–plane calculation mode is applied for discussion (see Fig. 7.5). If x R = 0 and z R = 0 are true, it will be:

0 −b

b3 c b2 c b2 Aq − + γ Aγ (−x pz )dx = q + λ 2 λ 2 3

(7.69)

where Aq = 2 exp(2λ(θ1 − θ2 ))

cos2 θ2 (sin2 (θ1 − θm ) + sin2 θ1 ) sin2 θm

3  sin θ1 1 1 Aγ = 3 λ sin θ1 − cos θ1 3 2 sin2 θm   1 λ(sin2 θ1 − cos2 θ1 ) − cos θ1 sin θ1 sin2 θ1 − 2 λ sin3 (θ1 − θm ) − (3λ sin θ + cos θ ) 1 1 1 + 9λ2 sin3 θm  2  sin θ1 sin(θ1 − θm ) 1 3 sin θ1 − λ cos θ1 + 1 + 9λ2 2 2 sin3 θm  2 + 3 exp[3λ(θ1 − θ2 )] λ sin2 θ2 cos θ2 3

(7.70)

6.150

18.423

35.551

75.810

184.278

30.0

35.0

40.0

45.0

3.850

15.0

10.321

2.504

10.0

25.0

1.674

5.0

20.0

1.220

Pm

0.0

tan δ

1.0

ϕ(◦ )

1.709

1.378

1.152

0.989

0.870

0.779

0.713

0.668

0.653

0.681

Z m /B

126.927

54.408

26.361

14.008

7.984

4.792

2.967

1.812

Pm

0.1

1.442

1.167

0.974

0.830

0.716

0.622

0.534

0.419

Z m /B

85.136

37.875

18.858

10.194

5.834

3.435

1.953

Pm

0.2

1.202

0.972

0.804

0.673

0.561

0.453

0.314

Z m /B

55.969

25.686

13.032

7.069

3.954

2.110

Pm

0.3

Table 7.9 Calculated results of total limit load and depth of the slip surface (q/(γ B) = 0.1 and c/(γ B) = 0.2)

0.987

0.793

0.645

0.521

0.402

0.255

Z m /B

36.269

17.021

8.682

4.603

2.326

Pm

0.4

0.800

0.633

0.497

0.371

0.224

Z m /B

228 7 Generalized Limit Equilibrium Method in Plane Failure Mode

7.10 Discussion on Limit Load and Critical Load

 3 λ sin (θ1 − θm ) + (3λ sin θ2 + cos θ2 ) 1 + 9λ2 sin3 θm  2λ sin2 θ1 sin(θ1 − θm ) 2 + [2λ sin θ2 + (1 + 3λ ) cos θ2 ] 1 + 9λ2 sin θm π θ1 = − ϕ + arctan h a 2 π θm = − ϕ 2

229

(7.71) (7.72) (7.73)

In Eqs. (7.70) and (7.71), θ1 (or h a ) is a parameter to be determined. Equation (7.69) is solved, and Ac = λ1 (Aq − 1) is supposed to obtain: pz = q Aq + c A c + γ b A γ

(7.74)

This is the distribution of the limit load along the foundation bottom, on which the mean of the limit load is: 1 pz = q Aq + c A c + γ B A γ 2

(7.75)

It shall be stated that determination of the parameter θ1 (or h a ) to be determined shall make sure that the slip surface is close to the real slip surface; thus, the calculated result can be close to the actual limit load. However, the following result is worthy to be noticed: When h a is determined according to the field failure mode and θ1 = π2 − ϕ +arctan(0.185h a ) is true, the calculated result is very close to the equation ( p1/4 )— pz = q Mq + cMc + γ B Mγ recommended by the “Code for Design of Building Foundation” to calculate the characteristic value of the ground bearing capacity. And the coefficient comparison of the bearing capacity is given in Table 7.10. This equation may also be used for calculation under the joint action of the horizontal force and vertical force. For example, when tan δ = 0.2 is true, see the calculated result in Table 7.11. If it is considered according to the two-way failure mode, pz = q Aq + c Ac + 1 γ B Aγ is true. In order to consider the allowable bearing capacity, ck = c/Fs and 4 tan ϕk = tan ϕ/Fs are taken, and the foundation bottom is determined according to the field failure mode (ϕ is replaced by ϕk ); finally, the calculated result when Fs = 1.7 is true is very close to that according to p1/4 (Table 7.12). In addition, it is also worthy to further discuss whether the calculation equation which is used to obtain the calculated value closer to that calculated according to the equation recommended by the “Code for Design of Building Foundation” to calculate the characteristic value of the ground bearing capacity may be obtained when the slip surface in other form is adopted.

230

7 Generalized Limit Equilibrium Method in Plane Failure Mode

Table 7.10 Coefficient comparison of the bearing capacity (tan δ = 0.0) ϕ(◦ )

0.5 Aγ

Mγ

Z m /B

2

3.460

3.32

1.121

1.12

0.048

0.03

0.189

4

3.635

3.51

1.254

1.25

0.103

0.06

0.197

6

3.820

3.71

1.402

1.39

0.164

0.10

0.206

8

4.018

3.93

1.565

1.55

0.234

0.14

0.214

10

4.228

4.17

1.746

1.73

0.312

0.18

0.224

12

4.454

4.42

1.947

1.94

0.401

0.23

0.234

14

4.697

4.69

2.171

2.17

0.501

0.29

0.244

16

4.960

5.00

2.422

2.43

0.616

0.36

0.255

18

5.248

5.31

2.705

2.72

0.746

0.43

0.268

20

5.562

5.66

3.025

3.06

0.896

0.51

0.281

22

5.910

6.04

3.388

3.44

1.070

0.61

0.295

24

6.296

6.45

3.803

3.87

1.272

0.80

0.310

26

6.729

6.90

4.282

4.37

1.510

1.10

0.327

28

7.219

7.40

4.838

4.93

1.793

1.40

0.345

30

7.777

7.95

5.490

5.59

2.132

1.90

0.365

32

8.419

8.55

6.261

6.35

2.545

2.60

0.387

Ac

Mc

Aq

Mq

34

9.168

9.22

7.184

7.21

3.055

3.40

0.412

36

10.049

9.97

8.301

8.25

3.694

4.20

0.440

38

11.099

10.8

9.671

9.44

4.509

5.00

0.471

40

12.367

11.7

11.377

5.568

5.80

0.506

10.8

7.11 Calculation Example of Slope Stability In the previous part, the virtual work equation is applied to discuss the calculation method of the slope stability. Since the velocity field is defined in the slip mass and the velocity field is generally discontinuous with regard to heterogeneous soil, the stress at the interface of the two types of soils needs to be solved, which is difficult to carry out. When the limit load in the surface failure mode is discussed, as long as the σ or v on the slip surface is obtained, the limit load may be calculated. In the following text, the slope stability will be discussed according to this idea.

7.11.1 Expression of the Safety Factor In Chap. 5, the moment equilibrium equation is obtained as below: d [(h − z R )E(h s , h) − (x − x R )T (h s , h) − M(h s , h)] dx

7.11 Calculation Example of Slope Stability

231

Table 7.11 Coefficient of the bearing capacity (tan δ = 0.2) ϕ(◦ )

Ac

Aq

0.5 Aγ

Z m /B

12

3.099

1.659

0.225

0.033

14

3.383

1.844

0.297

0.066

16

3.611

2.036

0.371

0.087

18

3.827

2.243

0.452

0.104

20

4.042

2.471

0.540

0.119

22

4.262

2.722

0.638

0.133

24

4.491

3.000

0.746

0.146

26

4.734

3.309

0.867

0.160

28

4.994

3.655

1.003

0.173

30

5.275

4.045

1.157

0.186

32

5.582

4.488

1.333

0.200

34

5.921

4.993

1.536

0.214

36

6.299

5.576

1.772

0.230

38

6.725

6.254

2.051

0.246

40

7.211

7.051

2.384

0.263

Table 7.12 Coefficient of the bearing capacity (Fs = 1.7)

ϕ(◦ )

Ac

Aq

0.25 Aγ

2

3.190

1.111

0.030

4

3.368

1.235

0.066

6

3.559

1.374

0.107

8

3.767

1.529

0.156

10

3.992

1.704

0.212

12

4.237

1.901

0.279

14

4.506

2.123

0.358

16

4.801

2.377

0.450

18

5.126

2.666

0.560

20

5.487

2.997

0.691

22

5.888

3.379

0.848

24

6.338

3.822

1.036

26

6.845

4.339

1.264

28

7.419

4.945

1.542

30

8.075

5.662

1.885

32

8.828

6.516

2.309

34

9.700

7.543

2.840

36

10.718

8.787

3.512

38

11.919

10.312

4.373

40

13.350

12.202

5.492

232

7 Generalized Limit Equilibrium Method in Plane Failure Mode

= −( f + c F )[h − z R − h  (x − x R )] − (x − x R )[w(h s , h) + pz ] + (h s − z R ) px + σ [(h  − λ F )(h − z R ) + (1 + h  λ F )(x − x R )]

(7.76)

Generally, the slope is as shown in Fig. 7.9. As for the integral in Eq. (7.76) within the horizontal area from the slope entry point (a, h a ) to the slope exit point (b, h b ), it shall be noticed that E = 0, T = 0, M = 0 are true between the slope entry point and the slope exit point. The following is derived:

b 0=



−( f + c F )[h − z R − h  (x − x R )] − (x − x R )[γ (h − h s ) + pz ]

a

! + (h s − z R ) px + σ [(h  − λ F )(h − z R ) + (1 + h  λ F )(x − x R )] dx

(7.77)

where pz = (σz − h s τx z )s and px = (τx z − σx h s )s are the known surface loads. Along the slip surface, it will be:  c F dh  2λ F h − λF dσ σ + − −γ =0 2 dx 1+h λ F dx 1 + λ2F

(7.78)

Likewise, the function v may also be introduced, and along the slip surface, it shall be as below:

b



−c F [h − z R − h  (x − x R )] − (x − x R )[γ (h − h s ) + pz ] + (h s − z R ) px

a

+ σ [(h  − λ F )(h − z R ) + (1 + h  λ F )(x − x R )]    c F dh  2λ F h − λF dσ σ+ dx = 0 − −γ +v dx 1 + h 2 λ F dx 1 + λ2F

Fig. 7.9 Schematic diagram for calculation of slope stability

7.11 Calculation Example of Slope Stability

233



2λ F dv dh   If dx + 1+h 2 v dx = (h − λ F )(h − z R ) + (1 + h λ F )(x − x R ) is true, Namely,

v = exp(−2λ F arctan h  ){C0 

   + exp(2λ F arctan h )[(h − λ F )(h − z R ) + (1 + h λ F )(x − x R )]dx (7.79) where C0 is a constant to be determined. Then, it will be:

b  − a

cF  [h (h − z R ) + (x − x R )] − (x − x R )[γ (h − h s ) + pz ] λF

+ (h s − z R ) px − γ v

   b cF h − λF dx + v σ + =0 λF a 1 + λ2F

(7.80)

If the slip surface and slope surface load are given, the calculation equation of the safety factor may be obtained by applying Eqs. (7.79) and (7.80). According to Eq. (7.78) or Eq. (7.79), it is shown that σ or v is a constant to be determined, but for σ , two boundary conditions of the slope entry point and slope exit point are given, in which the requirement for the form of the slip surface is raised. According to Eq. (7.78), it may be shown that if (h  − λ F )(h − z R ) + (1 + h  λ F )(x − x R ) = 0 is true with regard to the common helicoid, the two boundary conditions will not have any restriction; if it is common helicoid nearby the slope entry point (or the slope exit point), the boundary condition of the slope entry point (or the slope exit point) will not have any restriction. Therefore, this analysis method for slope stability is applicable to the slip surface on condition that it is common helicoid nearby the slope entry point or the slope exit point. It is obvious that if the slip surface is common helicoid, C0 = 0 is completely identical with the limit equilibrium method.

7.11.2 Example I: For Plane–Common Helicoid Without Load on Slope Surface The slip surface is plane–common helicoid (Fig. 7.10). (A) h = (x − a)h a , and it will be:

va = (1 + h a λ F )[(x − a)2 + a(x − a)] + C0

(7.81)

234

7 Generalized Limit Equilibrium Method in Plane Failure Mode

Fig. 7.10 Schematic diagram for plane–common helicoid calculation mode

(B) x − x R = R exp(−λ F θ ) cos θ , h − z R = R exp(−λ F θ ) sin θ , and it will be:

vb = va (x1 ) exp[2λ F (θ1 − θ )]

(7.82)

The above is substituted in Eq. (7.80); the following is derived, namely:  

b  cF  h − λF − [h (h − z R ) + (x − x R )] − γ (x − x R )(h − h s ) + v dx λF 1 + λ2F a

  b cF + v σ+ =0 λF a

(7.83)

According to:  cF cF (1 + λ2F )h 2 + (1 + λ F h  )2 σ + , = λF 1 + h 2 λF  cF cF 1 + λ2F + (h  − λ F )2 σ + σx + = λF 1 + h 2 λF  cF 1 + λ2F − (h  − λ F )2 σ + τx z = λ F 1 + h 2 λF σz +

and the boundary condition of the slope entry point—σz |a = 0 and τx z |a = 0 , it shall be:

3 π 1 θ1 = π − ϕk and h a = tan θ1 − + ϕk 4 2 2    cF cF 1 + h 2 σ+ = λF a (1 + λ2F )h 2 + (1 + λ F h  )2 λ F a Thereinto, ϕk = arctan(tan ϕ/Fs ) According to the boundary condition of the slope exit point

(7.84)

7.11 Calculation Example of Slope Stability

235

1 [(σz − tan βτx z ) tan β − (σx tan β − τx z )]|b = 0 , 1 + tan2 β 1 [(σz − tan βτx z ) + tan β(σx tan β − τx z )]|b = 0 σs = 1 + tan2 β τs =

It shall be: θ2 = 41 π + β − 21 ϕk and h b = tan(θ2 −

π 2

+ ϕk )

   cF cF 1 + h 2 σ+ = , λF b 1 + λ2F + (h  − λ F )2 − λ F [1 + λ2F − (h  − λ F )2 ]/ tan β λ F b (7.85) The above is substituted in Eq. (7.83), and the integral in the equation is calculated; the following is derived:     1 1 2 cF  2 b 2 2 2 2 2  h (x1 − a) + x1 − a + 2(h a − 1)(x1 − a)x R + (x + h ) x1 2 a 2 λF     2 h − λF 1 (x1 − a)3 + (2a − x R )(x1 − a)2 + C0 a + γ h a (x − a) 1 3 2 1 + λ2F    1 2 1 1 3 1 x zR − h + γ − tan β b3 − x R b2 + 3 2 2 3   λF va (x1 ) 2 2 + (x + h )(x + 3λ F h) + exp(2λ F (θ1 − θ ))(h − 3λ F x) bx1 1 + 9λ2F 1 + 9λ2F   (1 + h 2 )va (x1 ) exp(2λ F (θ1 − θ2 )) = 1 + λ2F + (h  − λ F )2 − λ F (1 + λ2F − (h  − λ F )2 )/ tan β b   cF (1 + h 2 )C0 − (7.86) (1 + λ2F )h 2 + (1 + λ F h  )2 a λ F where h 1 = − a2 h a , h 2 = H = b tan β, and z R = −h a x R , (1 + h a / tan β) , xR sin θ2 + h a cos θ2 (cos θ2 − sin θ2 / tan β) , x1 = z R = H h a sin θ2 + h a cos θ2 R = H exp(λ F θ2 )

= −H

(cos θ2 − sin θ2 / tan β) , sin θ2 + h a cos θ2

1 a, a = 2x R + 2R exp(−λ F θ1 ) cos θ1 . 2

As for this calculation equation, it shall be stated that: I.

according to Eq. (7.84) of the boundary condition of the soil entry point, if 2c F c = 0 is true, (σ )a = − √ c F 2 < 0 and (σx )a = − √ < 0 are true, 2 1+λ F

λF +

1+λ F

which indicates that the soil mass is subject to the tension. In order to eliminate this unreasonable factor, v(a) = 0 shall be taken, namely C0 = 0;

236

7 Generalized Limit Equilibrium Method in Plane Failure Mode

II. if σ along the slip surface is solved by applying Eq. (7.78) and is substituted in the solution process of Eq. (7.77), the problem that the soil mass is subject to the tension may not be well avoided; III. as for the determined slope, the slip surface corresponding to the equation is definite. Since no indefinite parameter exists in Eq. (7.86), the minimization process is no longer required. Supposing 1 + h a / tan β sin θ2 + h a cos θ2

(7.87a)

cos θ2 − sin θ2 / tan β sin θ2 + h a cos θ2

(7.87b)

AR = BR =

C R = {−B R + A R exp[−λ F (θ1 − θ2 )] cos θ1 }

(7.87c)

Equation (7.86) is as below : # c "  F C R [C R (h a2 − 3) + 2(h a2 − 1)B R ] + A2R 1 − exp[ − 2λ F (θ1 − θ2 )] 2λ F    1 1 4 1 1 C R + BR − + BR + γ H h a C R2 3 2 tan β 3 tan β 2   h a B R A2R cos2 θ2 − exp(−2λ F (θ1 − θ2 )) cos2 θ1 + 2  1  − A3R sin3 θ2 − exp(−3λ F (θ1 − θ2 )) sin3 θ1 3 λF + A3 [(cos θ2 + 3λ F sin θ2 ) 1 + 9λ2F R  − exp(−3λ F (θ1 − θ2 ))(cos θ1 + 3λ F sin θ1 ) 1 + λ F h a 2 − C A R [exp(2λ F (θ1 − θ2 ))(sin θ2 − 3λ F cos θ2 ) 1 + 9λ2F R − exp(−λ F (θ1 − θ2 ))(sin θ1 − 3λ F cos θ1 )]} $    (1 + h b2 )(1 + h a2 ) exp[2λ F (θ1 − θ2 )] cF 2 (7.88) = −C R 1 + λ2F + (h b − λ F )2 − λ F [1 + λ2F − (h b − λ F )2 ]/ tan β 2λ F Equation (7.88) is a functional equation with the safety factor as an unknown number, so there is no difficulty in obtaining the solution (there are many numerical methods for solving the root in the functional equation). And the calculation result of the safety factor Fs of the plane–common helicoid is detailed in Table 7.13a, b. On the one hand, it is basically consistent with the result by the limit equilibrium method; on the other hand, it is quite close to the calculated result of the arc slip surface given in Table 7.1.

7.11 Calculation Example of Slope Stability

237

Table 7.13 a Safety factor Fs of plane–common helicoid (c/(γ H ) = 0.1), b Safety factor Fs of plane–common helicoid (c/(γ H ) = 0.2) (a) β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

0.460

0.528

0.590

0.648

0.705

0.762

0.819

0.879

80.0

0.517

0.600

0.677

0.751

0.824

0.898

0.975

1.058

70.0

0.570

0.671

0.765

0.856

0.948

1.042

1.141

1.248

60.0

0.634

0.756

0.871

0.984

1.098

1.216

1.341

1.479

50.0

0.710

0.859

1.001

1.142

1.284

1.434

1.594

1.770

40.0

0.801

0.988

1.167

1.346

1.530

1.723

1.932

2.163

30.0

0.920

1.164

1.402

1.640

1.889

2.152

2.440

2.762

20.0

1.100

1.451

1.800

2.156

2.530

2.930

(b) β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

0.845

0.922

0.994

1.065

1.135

1.206

1.280

1.359

80.0

0.943

1.035

1.124

1.210

1.297

1.387

1.481

1.582

70.0

1.031

1.142

1.249

1.354

1.460

1.570

1.687

1.813

60.0

1.137

1.270

1.399

1.527

1.657

1.792

1.937

2.093

50.0

1.259

1.422

1.579

1.736

1.896

2.063

2.242

2.439

40.0

1.403

1.605

1.801

1.998

2.200

2.413

2.640

2.892

30.0

1.582

1.844

2.100

2.358

2.624

2.907

20.0

1.833

2.205

2.573

2.947

7.11.3 Example II: For Arc Slip Surface Without Load on Slope Surface In the present slope stability analysis, the arc slip surface is widely applied in engineering. And for the arc slip surface (including slip surfaces in other forms), application of the above-mentioned method still needs further discussion in that (as stated earlier) σ or v is a constant to be determined, but for σ , two boundary conditions of the slope entry point and slope exit point are given. As a matter of fact, it is the integral failure of the slope that is discussed in the slope stability. The stress field is required to be all in the limit state at every point (including the slope entry point and slope exit point) of the slip surface, and the corresponding slip surface shall be the real slip surface. However, as discussed above, the real slip surface is difficult to obtain and only the approximate solution of the safety factor may be obtained. Therefore, as for the general slope stability of the slip surface, the following method may be used to obtain the calculation equation of the safety factor:

238

7 Generalized Limit Equilibrium Method in Plane Failure Mode

Fig. 7.11 Schematic diagram for arc slip surface calculation mode

First, σ is solved according to Eq. (7.78); thereinto, a constant to be determined is included; then, σ is substituted in the vertical force equilibrium equation and moment equilibrium equation, and it is integrated from the slope entry point to the slope exit point; thus, two equations of the constant to be determined and the safety factor may be derived; finally, the safety factor is obtained by solving the two equations. With the arc slip surface (Fig. 7.11) as the example, the problem will be discussed as below. If pz = 0 and px = 0 is true and x − x R = R cos θ , h − z R = R sin θ , h  = tan(θ − π2 ) are true with regard to the slip surface, according to Eq. (7.78), it will be: σ+

 cF γR (1 − 2Fϕ2 ) sin θ = C0 exp[2λ F (θ − θ0 )] + λF (1 + λ2F )(1 + 4λ2F )  ! − 3λ F cos θ ] − (1 − 2λ2F ) sin θ0 − 3λ F cos θ0 ] exp[2λ F (θ − θ0 ) (7.89)

where C0 is a constant to be determined and may be determined according to force equilibrium. According to the vertical force equilibrium equation σ (1 + h  λ F ) + h  c F = , it will be: w + pz − dT dx

b  a

cF cF σ+ (1 + h  λ F ) − − γ (h − h s ) dx = 0 λF λF

(7.90)

  R (1 − 2λ2F ) sin θ0 − 3λ F cos θ0 is supposed and substituted in A = (1+λ2 γ)(1+4λ 2 ) F F Eq. (7.89) to obtain:

θ1 −

R(sin θ − λ F cosθ ){(C0 − A) exp[2λ F (θ − θ0 )] θ0

+

γR [(1 − 2λ2F ) sin θ − 3λ F cos θ ]}dθ (1 + λ2F )(1 + 4λ2F )

7.11 Calculation Example of Slope Stability

θ1 +

γ R 2 sin2 θ dθ −

b

239



a

θ0

 cF + γ (z R − h s ) dx = 0 λF

(7.91)

The integral in the equation is calculated to obtain: (C0 − A)B0 + γ R B1 +

cF (cos θ1 − cos θ0 ) = 0 λF

(7.92)

where  1 exp[2λ F (θ1 − θ0 )][λ F sin θ1 − (1 + 2λ2F )cosθ1 ] 1 + 4λ2F  ! − λ F sin θ0 − (1 + 2λ2F )cosθ0

B0 =

(7.93)

 2λ F (1 + λ2F )(θ0 − θ1 ) + λ F (5 + 2λ2F )(sin θ1 cos θ1 (1 + +  − sin θ0 cos θ0 ) − (2 − λ2F )(sin2 θ1 − sin2 θ0 ) λF 2 λ F )(1

B1 =

4λ2F )

− sin θ0 (cos θ1 − cos θ0 ) −

(sin θ1 − sin θ0 )2 2 tan β

(7.94)

According to Eq. (7.77), it will be:

b

  ! γ (x − x R )(h − h s ) + (σ λ F + c F ) (h − z R ) − h  (x − x R ) dx = 0

(7.95)

a

Equation (7.89) is substituted, and the integral in the equation is calculated to obtain: (C0 − A)A0 + γ R A1 = 0

(7.96)

where A0 = A1 =

λF 2 λ F )(1

4λ2F )

1 {exp[−2λ F (θ0 − θ1 )] − 1} 2

(7.97)

  (1 − 2λ2F )(cos θ0 − cos θ1 ) + 3λ F (sin θ0 − sin θ1 )

(1 + + 1 1 + sin θ0 (cos2 θ1 − cos2 θ0 ) + (sin3 θ1 − sin3 θ0 ) 2 3 2 sin θ1 − sin θ0 (sin θ1 − sin θ0 ) cos θ1 − + 2 tan β 3 tan β

(7.98)

240

7 Generalized Limit Equilibrium Method in Plane Failure Mode

According to Eqs. (7.92) and (7.96), C0 is eliminated to obtain: γ H (A1 B0 − A0 B1 ) −

cF A0 (sin θ1 − sin θ0 )(cos θ1 − cos θ0 ) = 0 λ2F

(7.99)

or, Fs =

A0 c (sin θ1 − sin θ0 )(cos θ1 − cos θ0 ) γ H λ F (A1 B0 − A0 B1 )

(7.100)

θ0 and θ1 in the equation are determined according to the minimum safety factor, and the calculated result of the safety factor Fs of the arc slip surface is detailed in Table 7.14a, b. It is quite close to the result given in Table 7.1. Table 7.14 a Safety factor of the arc slip surface (c/(γ H ) = 0.1), b Safety factor of the arc slip surface (c/(γ H ) = 0.2) (a) β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

0.464

0.532

0.594

0.652

0.709

0.759

0.812

0.869

80.0

0.529

0.613

0.689

0.763

0.834

0.906

0.975

1.051

70.0

0.597

0.697

0.791

0.882

0.972

1.065

1.162

1.269

60.0

0.666

0.789

0.904

1.016

1.128

1.246

1.369

1.505

50.0

0.742

0.892

1.034

1.174

1.316

1.463

1.622

1.795

40.0

0.827

1.017

1.197

1.374

1.556

1.746

1.949

2.177

30.0

0.936

1.187

1.425

1.661

1.905

2.164

2.444

2.759

20.0

1.105

1.467

1.813

2.161

2.525

2.914

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

90.0

0.851

0.928

1.001

1.072

1.142

1.214

1.288

1.367

80.0

0.967

1.060

1.149

1.235

1.322

1.411

1.504

1.604

70.0

1.082

1.195

1.301

1.407

1.513

1.622

1.737

1.863

60.0

1.199

1.335

1.464

1.592

1.722

1.856

2.000

2.155

50.0

1.318

1.485

1.645

1.803

1.963

2.129

2.307

2.503

40.0

1.447

1.657

1.858

2.058

2.260

2.472

2.699

2.948

30.0

1.598

1.877

2.141

2.404

2.673

2.954

20.0

1.814

2.215

2.598

2.977

(b) β(◦ )

ϕ(◦ )

7.12 Variational Principle of Generalized Limit Equilibrium Method

241

7.12 Variational Principle of Generalized Limit Equilibrium Method 7.12.1 Variational Principle of Limit Load J=

0 −b

(−x pz + h s px ) dx is supposed; then, Eq. (7.29) is as below:

xb J=

xb xqdx +

0

xb



c(h − h x)dx + −b

xb γ x hdx −

−b

σ [h  h + x − λ(h − h  x)] dx

−b

(7.101) where x = x − x R , h = h − z R . The variational principle of the limit load, namely, if the stress field and slip surface are in accordance with the equilibrium equation, yield condition, extremum condition of the yield function, and the load boundary condition of the soil mass surface, the limit load calculated according to Eq. (7.101) is the minimum, is proved as below. It only needs to prove the first-order variation δ J = 0 of the functional is true. It is noticed that the above variational principle of the limit load is nothing but the sufficient proposition when the limit load is the minimum; the necessary proposition when the limit load is the minimum is that the stress field and slip surface must be in accordance with the equilibrium equation, yield condition, extremum condition of the yield function, and the load boundary condition of soil mass surface; and the necessary and sufficient propositions are that the stress field and slip surface that are in accordance with the equilibrium equation, yield condition, extremum condition of the yield function, and the load boundary condition of soil mass surface are the same as those when the limit load is the minimum. If the necessary and sufficient propositions have been proved, the solution of the limit equilibrium may be converted as below: The functional about the limit load is constructed, and according to the equilibrium equation and under the yield condition, extremum condition of the yield function, and the load boundary condition of soil mass surface, the conditional extremum (the variation) is solved, which has gone beyond the scope of the limit equilibrium method. As for the sufficient proposition, when the equilibrium equation, yield condition, and extremum condition of the yield function are met, it may be regarded that the stress field is expressed by the slip surface. According to Eq. (7.30), it is: ⎧ ⎫

xb ⎨ ⎬   c exp(−2λ arctan h  )γ (h  − λ)/(1 + λ2 )  dx σ + = exp(2λ arctan h  ) C − ⎩ ⎭ λ x

(7.102)

242

7 Generalized Limit Equilibrium Method in Plane Failure Mode

where C = [(σ + λc ) exp(−2λ arctan h  )]x=xb Only the slip surface is unknown; therefore, it is not necessary to regard the stress field as an independent unknown when the variation is solved. In addition, since xb is unknown, it is the stress field to be determined with regard to the boundary. F = c(h − h  x) + γ xh − σ [h  (h + λx) + x − λh]

(7.103)

According to the variational principle [6], F shall be proved to be in accordance with Euler’s equation: d ∂F − ∂h dx



∂F ∂h 

=0

(7.104)

and the supplementary condition (xb varies on the soil mass surface):  ∂F q x + F − h  =0 ∂h x=xb

(7.105)

It is still inconvenient for solving ∂σ by applying Eq. (7.102) directly. Actu∂h ally, when σ is known, all the stress components are known and according to 1    σ = 1+h 2 [σz − τ x z h + h (σ x h − τ x z )], it is easy to derive:  ∂τx z h  (h + λx) + x − λh ∂σz ∂F = c+γx − − h ∂h 1 + h 2 ∂z ∂z   ∂τx z ∂σx − + h h − σ (h  − λ) ∂z ∂z z=h According to the equilibrium equation, yield condition, and extremum condition of the yield function, it will be as follows:  ∂τx z dτx z ∂τx z ∂σx dσx ∂σz − h + h h − = γ + h − ∂z ∂z ∂z ∂z dx dx 

 2λ c dh dσ + σ+ = γ + (h  − λ) dx 1 + h 2 λ dx Hence,    ∂F 2λ c dh  1 dσ  γ + (h − λ) = c+γx − + σ+ ∂h 1 + h 2 dx 1 + h 2 λ dx [h  (h + λx) + x − λh] − σ (h  − λ) It is easy to derive:

(7.106)

7.12 Variational Principle of Generalized Limit Equilibrium Method

243

 ∂F c   2λ h (h + λx) + x − λh − σ (h + λx) (7.107) σ + = −cx +  2 ∂h 1+h λ  d ∂F 2λ dσ  = −c + [h (h + λx) + x − λh]  dx ∂h 1 + h 2 dx dσ − (h + λx) − σ (h  + λ) dx     2λ c dh 2 2 (1 + h (h + σ + ) + λx) + 1 + h (1 + h 2 )2 λ dx  dh − 2h  [h  (h + λx) + x − λh] dx Hence, ∂F d − ∂h dx



∂F ∂h 



 xh  − h dσ = γ (h  − λ) − (1 + λ2 ) 2 1+h dx  c dh  2λ(1 + λ2 ) σ+ + 1 + h 2 λ dx

(7.108)

According to Eqs. (7.30), (7.104) may be obtained. As for Eq. (7.105), firstly it is noticed that when x = xb is true, it is the intersection of the soil mass surface and the slip surface. Under the boundary condition of the soil mass surface, it will be: h = 0, 1 + 2h  λ − h 2 = 0, σx = q/ h 2 − 2c/ h      ∂F 2 q x + F − h  = q x + ch − (q − ch  ) h(h  − λ) + x(1 + h  λ) 2 ∂h x=xb 1+h     2ch 2h (q − ch  )(h + xλ) − h(h  − λ) + x(1 + h  λ) + 1 + h 2 1 + h 2     4h  λ   (q − ch ) h(h − λ) + x(1 + h λ) − (1 + h 2 )2 x=xb $   c(h ∂ F − h x) q x + F − h  = (1 + 2h  λ − h 2 ) ∂h x=xb 1 + h 2  (q − ch  )  2 4  2 + [2λh(1 + 2h λ − h ) + x(h − (1 + 2h λ) )] =0 (1 + h 2 )2 x=xb It is shown that when Eq. (7.101) is derived, the extremum condition of the yield function is not applied, whereas it is applied when the limit load calculated according to Eq. (7.101) is proved to be the minimum. Therefore, the variational principle of the limit load may be rephrased as below: When the stress field and slip surface

244

7 Generalized Limit Equilibrium Method in Plane Failure Mode

among those that are in accordance with the equilibrium equation, yield condition, and the load boundary condition of soil mass surface is simultaneously in accordance with the extremum condition of the yield function, the obtained limit load is the minimum. It actually proves that the extremum condition of the yield function is equivalent to the extremum condition of the limit load. It shall be stated that as for the functional of Eq. (7.101), if the stress field is regarded as an independent unknown, the discussion will be similar to that in Chap. 6; strict proving for the variational principle still cannot be given, and the difficulty is still the slip surface problem. Moreover, it shows that the slip surface problem is really a difficult one with regard to the limit analysis theory.

7.12.2 Variational Principle of Slope Stability The generalized limit equilibrium method of the slope stability is the one that Eqs. (7.77), (7.90), and (7.78) are used for simultaneous solution. As for the moment b equation, J = a [ f (h − h  x)] dx is supposed, and it will be: J = −M R /Fs + M0

b =−

  (σ λ F + c F )(h − h  x) − σ (h  h + x)  dx

a

b −



 x(w + pz ) − h s px dx

(7.109)

a

where x = x − x R , h = h − z R . b As for the vertical force equation, J1 = a (h  f ) dx is supposed, and it will be:

b  J1 = a

cF cF σ+ (1 + h  λ F ) − − w − pz dx = 0 λF λF

(7.110)

The variational principle of the slope stability, namely, if the stress field and slip surface are in accordance with the equilibrium equation, yield condition, and extremum condition of the yield function, the limit load calculated limit according to Eqs. (7.109) and (7.110) is the minimum, is proved as below. The variational principles of Eqs. (7.109) and (7.110) are discussed respectively as below. Since the first-order variation δ J of the functional J and the first-order variation δ Fs of the safety factor Fs are in accordance with [6]:

7.12 Variational Principle of Generalized Limit Equilibrium Method

245

( ( δ J = −δ M R Fs + δ M0 + M0 δ Fs Fs2 So, as long as −δ M R /Fs + δ M0 = 0 is true, δ J = 0 is equivalent to δ Fs = 0. Fm = (Fs )min is supposed, δ Fs = 0 is true, and Fm is a constant, so Eq. (7.109) of the functional is:

b J=

F(x, h, h  , σx , σz , τx z )dx

(7.111)

a

F = −[γ x(h − h s ) + x pz − h s px ] − c Fm (h − h  x) 1 + [σz − h  τx z + h  (h  σx − τx z )] [h  h + x − λ Fm (h − h  x)] (7.112) 1 + h 2 Fs in the equation is its minimum Fm . It is observed that a, b is the point to be determined on the soil mass surface z = h s (x), so Eq. (7.104) and supplementary condition shall be as below:    ∂F (F − (h  − h s ) ∂h  x=a = 0   ∂F F − (h  − h s ) ∂h  x=b = 0

(7.113)

It may be obtained that Eq. (7.112) is in accordance with the Euler equation by entirely imitating the condition of Eq. (7.101). Next, Eq. (7.113) will be proved as true. F is rewritten as below: F = −γ x(h − h s ) + x(− pz + σz − h  τx z ) + h s px + h(h  σx − τx z )

(7.114)

It will be: ∂F = −xτx z + hσx ∂h 

F − (h  − h s )

∂F = −γ x(h − h s ) + x(− pz + σz − h  τx z ) ∂h  + h s px + h(h  σx − τx z ) + (h  − h s )(xτx z − hσx ) = −γ x(h − h s ) + x(− pz + σz − h s τx z ) + h s px + h(h s σx − τx z )

Since (h − h s ) = 0, (h − h s ) = 0 and px = (τx z − h s σx )s , pz = (σz − h s τx z )s are true on the slope surface, Eq. (7.113) must be correct. As for Eq. (7.110), it is supposed as below: F1 = σ (1 + h  λ Fm ) − h  c Fm − w − pz It needs to be proved to be in accordance with the Euler equation:

(7.115)

246

7 Generalized Limit Equilibrium Method in Plane Failure Mode

d ∂ F1 − ∂h dx



∂ F1 ∂h 

=0

(7.116)

and the supplementary condition:    F1 − (h  − h s ) ∂∂hF1 x=a = 0   F1 − (h  − h s ) ∂∂hF1 x=b = 0

(7.117)

And it is easy to prove. In fact, it is easy to obtain: d ∂ F1 − ∂h dx



∂ F1 ∂h 



1  −γ h  (h  − λ Fm ) 1 + h 2    c Fm dh  2λ Fm dσ σ + + + (h  − λ Fm )(1 + h  λ Fm ) dx 1 + h 2 λ Fm dx dσ  + (1 + 2h  λ Fm − h 2 )λ Fm dx    c Fm dh  2h  − λ Fm + h 2 λ Fm 2λ Fm σ + − 1 + h 2 λ Fm dx

=

h  −γ (h  − λ Fm ) 1 + h 2    c Fm dh  2λ Fm dσ σ + − =0 + (1 + λ2Fm ) dx 1 + h 2 λ Fm dx

=

Thus, the Euler equation is proven. Since: F1 − (h  − h s )

∂ F1 = σ (1 + h  λ Fm ) − h  λ Fm − w − pz ∂h   c Fm 1 + 2h  λ Fm − h 2 σ + + (h  − h s ) F ϕm 1 + h 2 λ Fm    = σz − h τx z + (h − h s )τx z − w − pz = σz − h s τx z − w − pz

the supplementary condition equation is true. Similarly, the variational principle of the slope stability indicates that the extremum condition of the yield function is equivalent to the extremum condition of the safety factor. If the slip surface  in the equation is the real slip surface, Eqs. (7.77), (7.90), and (7.78) may be used for simultaneous solution to calculate the safety factor Fs and the calculated safety factor is the exact solution, which lays solid theoretical foundation for the generalized limit equilibrium method. As for the moment equilibrium equation-based generalized limit equilibrium method, it is unnecessary to introduce the velocity field and flow rule, and the

7.12 Variational Principle of Generalized Limit Equilibrium Method

247

unknown stress σ in the force and moment equilibrium equations may be eliminated only by applying Eqs. (7.30) or (7.78), which is much simpler than the virtual work equation-based generalized limit equilibrium method. Compared with the general limit equilibrium method, without assumption or simplification for the force on the soil mass, it is applicable to any slip surface (the limit equilibrium method is only applicable to the specific plane and common helicoid), which is no doubt a great improvement. Likewise, since the form of the slip surface needs to be selected in advance, in general, the result calculated according to the calculation equation given by the generalized limit equilibrium method is the approximate solution of the safety factor or limit load in the surface failure mode. The approximate solution of the limit load in the field failure mode may be obtained by the generalized limit equilibrium method in the plane failure mode. In order to obtain the calculated result under the field failure condition, it needs to select the slip surface family in the possible failure area and the corresponding calculation is carried out for every slip surface of the slip surface family simultaneously. As for the limit load, calculation of slip surface through any point −b ∈ [−B, 0] of the foundation bottom needs to be carried out to obtain distribution of the limit load (e.g., the above example); as for the soil pressure, calculation of slip surface through any point z ∈ [0, H ] of the straight wall surface needs to be carried out to obtain distribution of the limit soil pressure. Similarly, it also shall continue to study generalization of the moment equilibrium equation-based generalized limit equilibrium method to the general slope and ground of heterogeneous soil.

References 1. Shen ZJ (2000) Theoretical soil mechanics. China Water Power Press, Beijing 2. Pan JZ (1980) Analysis on stability and coast of the structure. Hydraulic Press, Beijing 3. Winterkorn HF, Fang XY (1983) Foundation engineering handbook, translated by Qian Hongjin and Ye Shulin etc. China Architecture and Building Press, Beijing 4. Zhujiang S (2000) Theoretical soil mechanics. China Water Power Press, Beijing 5. Tianjin University (1980) Soil mechanics and foundation. China Communications Press, Beijing 6. Qian WC (1980) Variational method and finite element (Volume 1). Science Press, Beijing

Chapter 8

Limit Load on Ground with Heterogeneous Soil

8.1 Basic Consideration on Calculation of Limit Load 8.1.1 Determination of Calculation Condition From Chaps. 3–7, calculation of the limit load on ground with homogeneous soil is discussed by applying the characteristic line method, stress field method, limit equilibrium method, and the generalized limit equilibrium method, respectively. As for ground with homogeneous soil, uniformly distributed edge load and horizontal soil mass surface, in theory, according to the above-mentioned calculation equations for the limit load and existing calculation equations, the limit load can be basically calculated (the ground bearing capacity needs to be further discussed). In practical problems, few are concerned about the ground with homogeneous soil, uniformly distributed edge load and horizontal soil mass surface, but most of them are concerned about the ground with heterogeneous soil and non-uniformly distributed edge load and the soil mass surface may not be horizontal. For a long time, most of the above practical problems are simplified into the simple problems for approximate calculation. For example, for the bearing capacity of the ground with heterogeneous soil, the calculation equation for the ground with homogeneous soil is mainly adopted for approximate calculation, which apparently has many problems. For example, in the case shown in Fig. 8.1, the following problems need to be considered: If the bottom surface of the rubble foundation bed is regarded as the foundation bottom, how the width of the load action surface on the base shall be determined? If the calculation equation for the ground with homogeneous soil is adopted for approximate calculation, is the soil strength index determined according to the index of sand cushion and soil layer 1 or index of the soft soil and soil layer 1? What treatment will be carried out in case of non-uniformly distributed edge load? As for ground and boundary conditions in some engineering, whatever method is adopted to convert the index of ground soil into the approximate index of homoge© Springer Nature Singapore Pte Ltd. and Zhejiang University Press, Hangzhou, China 2020 C. Huang, Limit Analysis Theory of the Soil Mass and Its Application, https://doi.org/10.1007/978-981-15-1572-9_8

249

250

8 Limit Load on Ground with Heterogeneous Soil

Rubble Rubble foundation bed Sand cushion

Soft soil

Soil layer 1

Fig. 8.1 Schematic diagram for ground with heterogeneous soil

neous soil, relatively large error in the calculated result even distortion is likely to be caused. In fact, the error in the calculated result caused by the approximate index of soil is possible to be far larger than that caused by the calculation equation itself. For instance, by the characteristic line method, when q/(γ B) = 0.1, c/(γ B) = 0.2 are true and 20◦ , 22◦ are taken for ϕ, respectively, the total limit load Pm = Pz /(γ B 2 ) is 6.196, 7.586, respectively, with about 20% difference, which may be very common for the error in ϕ after conversion into the average index in the case of relatively large ϕ difference between the soil layers. However, for an approximate calculation method, it is unacceptable when the error reaches 20%. Therefore, even the approximate calculation method for the bearing capacity of the ground with heterogeneous soil may be superior to that according to the equations of homogeneous soil and uniformly distributed edge load.

8.1.2 Study Method for Limit Load For the calculation of the limit load on the ground with heterogeneous soil, nonuniformly distributed edge load, and non-horizontal soil mass surface, “no reliable calculation methods are available at present [1],” and it remains to be a difficult problem for solution in the soil mechanics community and engineering. In the above, various methods and slip surfaces (families) have already be applied to discuss this problem; on the one hand, the calculation of the limit load on the ground with homogeneous soil is expected to be solved; on the other hand, one method (or several methods) is/are expected to be found for generalization to the ground with heterogeneous soil, non-uniformly distributed edge load, and non-horizontal soil mass surface. Moreover, it may be more difficult to apply the methods in the field failure mode (the characteristic line method and stress field method) to the ground with heterogeneous soil, while it is relatively less difficult to apply the methods in the surface failure mode (limit equilibrium method and generalized limit equilibrium method) to the ground with heterogeneous soil.

8.1 Basic Consideration on Calculation of Limit Load

251

8.1.3 Determination of Slip Surface (Family) As stated earlier, determination of the slip surface is one of the main difficulties in the limit analysis on the soil mass and it is more prominent when it comes to the ground with heterogeneous soil, non-uniformly distributed edge load, and non-horizontal soil mass surface. In theory, the slip surface is related to the solution method. If the extremum condition of the yield function (including the directly applied Mohr–Coulomb yield condition) is regarded as a fundamental equation and is solved (by the characteristic line method and the stress field method) according to the corresponding limit equilibrium, the slip surface family will be determined. However, every point of the soil mass is required to be in accordance with the extremum condition of the yield function, so the calculated value of the limit load obtained under this condition may not be the limit load (it has been discussed in Chap. 3) on the ground with heterogeneous soil. For another kind of the solution method for the limit load, the limit load is expressed in the functional of the slip surface and the slip surface and limit load (by limit equilibrium method and generalized limit equilibrium method) are obtained by solving the extremum of the functional. Its difficulty lies in that the functional form of the slip surface needs to be determined before the solution. The solution process of the extremum of the functional is actually the solution of the parameter in the function of the slip surface, and only approximate solution (including the slip surface and limit load) can be obtained in this case. Moreover, accuracy of the approximate solution mainly depends on whether the determined slip surface can be close to the real slip surface.

8.1.4 Basic Consideration on Calculation of Limit Load on Ground with Heterogeneous Soil As for the calculation mode, the calculation demands of practical problems shall be considered. As is well known, the limit load is the maximum load that the ground can bear in the limit state. In practical problems, the design load is not allowed to reach the limit load and is only allowed to be no greater than the allowable load. In the “Code for Design of Building Foundation” [2] in China, the characteristic value of the ground bearing capacity is actually a kind of allowable load. According to the calculation equation of the ground bearing capacity in the “Code for Foundation in Port Engineering” [3] in China, the limit load is calculated, and it is also a kind of allowable load after being divided by the safety factor (partial safety factor of resistance). Since the soil mass strength is one of the major influence factors of the ground limit load, for the two strength indexes, two coefficients Fsϕ ≥ 1.0 and Fsc ≥ 1.0 are defined and the allowable load is defined in the limit load under the characteristic values (ci /Fsc and tan ϕi /Fsϕ ) of the strength indexes. From the aspect

252

8 Limit Load on Ground with Heterogeneous Soil

of the probabilistic method (reliability), the variability of the soil mass strength is one of major influence factors of the stability of the ground bearing capacity. Apparently, introduction of the two coefficients will provide convenience for the reliability theory-based partial safety factors method. For example, in the Canadian design handbook of geotechnical engineering, when the ground bearing capacity is calculated, 2–2.5 and 1.2–1.5 are adopted as the partial safety factors of the cohesion is and the internal friction angle tan ϕ. In addition, when Fsϕ = Fsc = 1.0 is true, it is the calculation of limit load. Another main reason for the introduction of the two coefficients is that calculation of the limit load on the ground with heterogeneous soil, non-uniformly distributed edge load, and non-horizontal soil mass surface is very complicated and it is expected to obtain a relatively simple calculation mode under condition of assumed common slip surface (family) for the convenience of application. However, according to the existing study [4, 5], the ground limit load calculated under condition of the common slip surface is relatively large (compared with that by the characteristic line method). By applying the calculated results (including limit load, eccentricity, and slip surface depth) in the case of homogeneous soil and properly determining the two coefficients, the calculation mode may be calibrated so that the calculation method applicable to the limit load on the ground with heterogeneous is obtained. When the calculation modes (slip surfaces) are different, Fsϕ , Fsc will be different, too, and the closer the slip surface is to the real slip surface, the closer Fsϕ , Fsc are to 1.0. Therefore, only from the calculation of the limit load, the closer Fsϕ , Fsc are to 1.0, the more accurate the calculation mode is. Actually, according to the above discussion on ground with homogeneous soil, it is known that it is easy to select proper slip surface under condition of Fsϕ = Fsc = 1.0 so that the calculated limit load is consistent with that by the characteristic line method; however, it is difficult to make both the eccentricity of the limit load and slip surface depth be consistent with those by the characteristic line method.

8.2 Helicoid Calculation Mode in Surface Failure Mode According to discussions in the former chapters, the methods in the surface failure mode (limit equilibrium method and generalized limit equilibrium method) may be used for discussion according to the surface failure mode and that on the limit load in field failure mode. From the aspect of the slip surface, it is no doubt that the common helicoid is the simplest without simplification for the force on the soil mass. In the following text, calculation of the limit load on the ground with heterogeneous soil is discussed by applying the limit equilibrium method and the common helicoid.

8.2 Helicoid Calculation Mode in Surface Failure Mode

253

8.2.1 Common Helicoid Calculation Mode As for the general ground and load shown in Fig. 8.2, the calculation equation may be derived by imitating the idea of the slice method. Supposing strength indexes on the slip surface in the soil strip (xi−1 , xi ) as ϕ = ϕi , c = ci , the soil strip weight in unit width determined according to the ground soil is wγ i ; based on the above discussion, the moment equation is as below:      d h − z R  2 w 1 + h + p h − F − c [(h − z R )E − (x − x R )T − M] = z γ i ϕi si dx 1 + h  Fϕi   h − Fϕi dT − z R px − (h − z R ) + x − x R 1 + h  Fϕi dx x ∈ (xi−1 , xi ), i = 1, 2, . . . , N .

(8.1)

where csi = ci /Fsc and Fϕi = tan ϕi /Fsϕ ; for the ground surface (including foundation bottom and soil mass surface), z = h s is true; and for the potential slip surface, z = h is true. To make the last item on the right end of Eq. (8.1) be zero, the continuous logarithm helically curved surface is taken for the slip surface and it will be:   x − x R = Ri exp −Fϕi θ  cos θ h − z R = Ri exp

 −Fϕi θ sinθ  Ri+1 = Ri exp Fϕi+1 − Fϕi θ θi = θ0 − iθ

⎫ ⎪ ⎪ ⎬

i = 0, 1, 2, . . .

⎪ ⎪ ⎭

(8.2)

where R0 =

 (B + x R )2 + z 2R e Fϕ0 θ0

tan θ0 = z R /(B + x R )

(8.3a) (8.3b)

Equation (8.1) is integrated into (xi−1 , xi ), and all results of the soil strips are added; hence, Fig. 8.2 Schematic diagram for common helicoid calculation mode

pz q

px

−B

z z = h (x )

x i −1 x i

x

254

8 Limit Load on Ground with Heterogeneous Soil

0

0 (x − x R ) pz dx +

−B

z R px dx

−B

=−

xi 



  wγ i + q (x − x R ) + csi h − z R − (x − x R )h  dx

(8.4)

i x i−1

According to, 0

0 px dx = tan δ

−B



 pz + cs0 /Fϕ0 dx

(8.5)

−B

0 (x + B/2 − e) pz dx = 0

(8.6)

−B

where cs0 = c0 /Fsc and Fϕ0 = tan ϕ0 /Fsϕ ; c0 , ϕ0 are the soil mass strength indexes of the lower foundation bottom, and e is the eccentricity of the calculated value of the limit load. It is substituted in Eq. (8.1) and rearranged to derive the following expression of the total limit load: ⎧ ⎨ xi   1 wγ i + q (x − x R ) Pz = (0.5B + x R − e − z R tan δ) ⎩ i x i−1

+ csi



 cs0 h − z R − (x − x R )h dx + Bz R tan δ Fϕ0 

 (8.7)

The minimum of Eq. (8.7) on condition that Eq. (8.2) of the slip surface passes through (−B, 0) is the ground limit load. According to the derivation process, no assumption or simplification is made for the force on the ground, unless the logarithm helically curved surface is required to be a slip surface. The expression of the slip surface contains three parameters (x R , z R , R 1 ) to be determined. When Fsϕ , Fsc , and e are definite, it is easy to calculate the minimum of Eq. (8.7) on condition that the slip surface passes through the point (−B, 0). In order to investigate the actual effect of calculation equation, homogeneous soil is first analyzed so as to determine Fsϕ , Fsc , and e.

8.2 Helicoid Calculation Mode in Surface Failure Mode

255

8.2.2 Homogeneous Soil As for the ground with homogeneous soil, when the ground surface is horizontal and the strength action of the soil in the buried depth (h s < z < 0) of the ground is not considered, q = −γ h s is supposed, and it will be: 

  γ 0.5 Pz = (x N − x R )2 + z 2R 2 (0.5B + x R − e − z R tan δ) 1 + 9Fϕ       2Fϕ (x N − x R ) + 1 + 3Fϕ2 z R + (B + x R )2 + z 2R 2Fϕ (B + x R )   cs

   2Bz R tan δ + (x N − x R )2 − (B + x R )2 − 1 + 3Fϕ2 z R + Fϕ   2 2 (8.8) + q (x N − x R ) − x R Since the slip surface passes through two points (−B, 0), (x N , 0), it shall also be:      B + x R = −R exp −Fϕ θ0 cos θ0 , −z R = R exp −Fϕ θ0 sin θ0 x N − x R = R exp −Fϕ θ1 cos θ1 , − z R = R exp −Fϕ θ1 sin θ1

(8.9)

After further rearrangement, it will be:   1 Pz = B q Nq + cNc + γ B Nγ 2

(8.10)

⎫ Nq = Aq (θ0 , θ1 , Rb )/A(θ0 , θ1 , Rb ) ⎪ ⎬ Nc = Ac (θ0 , θ1 , Rb )/A(θ0 , θ1 , Rb ) ⎪ ⎭ Nγ = Aγ (θ0 , θ1 , Rb )/A(θ0 , θ1 , Rb )

(8.11)

where

Thereinto, ⎫      Aq = 0.5 Rb2 exp −2Fϕ θ1 cos2 θ1 − exp −2Fϕ θ0 ⎪ ⎪ ⎪     ⎪ 2 ⎪ ⎪ cos θ0 − 2Rb exp −Fϕ θ0 cos θ0 − 1 ⎪ ⎪ ⎪ ⎪  2   ⎪ 0.5  2  ⎪ ⎪ Ac = Rb exp −2Fϕ θ1 cos θ1 − exp −2Fϕ θ0 ⎪ ⎬ Fsc Fϕ     2 ⎪ cos θ0 − 2Rb tan δ exp −Fϕ θ0 sin θ0 ⎪ ⎪ ⎪ ⎪ 3 ⎪ 

   Rb   ⎪ 2 ⎪ exp −3F 2F sin θ θ cos θ − 1 + 3F Aγ = ⎪ ϕ 1 ϕ 1 1 ϕ ⎪ 2 ⎪ 1 + 9Fϕ ⎪ ⎪ ⎪   

  ⎭ 2 − exp −3Fϕ θ0 2Fϕ cos θ0 − 1 + 3Fϕ sin θ0   A = −(0.5 + e/B) − Rb exp −Fϕ θ0

(8.12)

256

8 Limit Load on Ground with Heterogeneous Soil

(cos θ0 − tan δ sin θ0 )     Rb = R/B, exp −Fϕ θ1 sin θ1 = exp −Fϕ θ0 sin θ0

(8.13) (8.14)

The equations contain two parameters θ0 , Rb [θ1 is determined according to (8.14)], and they shall be determined according to the minimum total limit load. As it has been stated before, when Fsϕ = Fsc = 1.0 is true, it will be inevitable that the limit load calculated according to this equation is relatively larger than that calculated according to the above given calculation equation of the ground with homogeneous soil [4, 5]. Proper Fsϕ , Fsc shall be selected to correct this equation and calibrate the calculated value of the limit load, and then, it is used for the ground with heterogeneous soil. Considering that the total limit load in the surface failure mode is no greater than the calculated value by the characteristic line method and according to the principle that it is, in general, slightly smaller than that by the characteristic line method, after calculation and comparison, if: Fsc = 1.08 + 0.05 tan δ, Fsϕ = 1.04 + 0.05 tan δ

(8.15)

Under the assumption that the limit load is in the linear distribution, the eccentricity may be approximately calculated according to the equation (e/B = 2 −γ B12 Nγ /Pz ). However, in general, the calculated eccentricity calculated in this way is relatively larger (generally, about 10% larger) than the calculated value by the characteristic line method. Because the surface failure mode is analyzed according to the slope stability and when the backward eccentricity of the load is relatively large under the same total load on the foundation bottom, it is unfavorable for the ground stability, which is a well-known fact. Actually, introduction of this equation is only aimed at calculating the total limit load; if distribution of the limit load cannot be calculated in the surface failure mode, the eccentricity of the limit load cannot be accurately calculated either. If it is rewritten as below: e/B = −γ

B2 Nγ /Pz /1.15 12

(8.16)

It will be quite close to the calculated value by the characteristic line method. Maximum depth of the slip surface: By the slip surface that makes the minimum total limit load be taken, namely x R , z R , R in Eq. (8.10) that make the minimum total limit load be taken, the maximum depth of the slip surface may be obtained as below: π   π   − arctan Fϕ sin − arctan Fϕ (8.17) Z m = z R + R exp −Fϕ 2 2

8.2 Helicoid Calculation Mode in Surface Failure Mode

257

Thus, the calculated total limit load Pm = Pz /(γ B 2 ) and slip surface depth Z m /B are quite consistent with those by the characteristic line method. Comparison of the calculated results is detailed in Table 8.1a, b. When the edge load q and cohesion c are different, the calculated result can be well consistent with that by the characteristic line method. As for homogeneous soil, generally the calculated result is slightly smaller than that by the characteristic line method; especially, the slip surface depth is also very close to that by the characteristic line method. In contrast, when the edge load q and cohesion c are relatively small, but tan δ is relatively large, the total limit load is more likely to be relatively small. Certainly, other Fsc and Fsϕ may also be selected so that the calculated result is much closer to that by the characteristic line method; however, in this case, Fsc and Fsϕ are relatively complex and will be related to c and ϕ, which will be discussed later with regard to the field failure mode.

8.2.3 Limit Load on Heterogeneous Soil Equation (8.7) is also applicable to the ground with heterogeneous soil, nonuniformly distributed edge load, and non-horizontal soil mass surface. Generally, Eq. (8.7) may be written as below: Pz = Pq + Pc + Pγ

(8.18)

⎫ Pq = Aq (x R , z R , R0 )/A(x R , z R ) ⎪ ⎬ Pc = Ac (x R , z R , R0 )/A(x R , z R ) ⎪ ⎭ Pγ = Aγ (x R , z R , R0 )/A(x R , z R )

(8.19)

A(x R , z R ) = 0.5B + x R − e − z R tan δ

(8.20)

where

Thereinto,

x N Aq (x R , z R , R0 ) =

q(x − x R )dx

(8.21)

0

cs0 Bz R tan δ Ac (x R , z R , R0 ) = Fϕ0 xi   csi

 + x − x R + h  (h − z R ) dx F ϕi i xi−1

(8.22)

76.478

40

0.628

0.682

0.693

0.814

0.893

1.090

1.253

1.585

15

20

25

30

35

40

1.502

1.244

1.055

0.909

0.801

0.718

0.662

0.680

1.267

1.042

0.910

0.713

0.625

0.557

0.434

Z m /B

10

0.1

53.891

25.607

13.497

7.654

4.591

2.868

1.787

Z m /B

Y m /B

76.61

36.07

18.69

10.44

6.196

3.859

2.501

0.0

5

ϕ(◦ )

tan δ

35.124

35

(b)

9.968

17.994

5.928

20

30

3.700

15

25

2.415

10

1.669

Pm

P0

Pm

1.634

0.1

0.0

tan δ

5

ϕ(◦ )

(a)

1.287

1.061

0.891

0.756

0.644

0.542

0.420

Y m /B

54.81

26.57

14.08

7.990

4.772

2.942

1.796

P0

1.058

0.864

0.692

0.607

0.463

0.344

Z m /B

0.2

36.701

18.073

9.719

5.579

3.326

1.950

Pm

0.2

1.078

0.880

0.725

0.593

0.469

0.318

Y m /B

38.04

18.90

10.18

5.802

3.408

1.945

P0

0.906

0.696

0.526

0.416

0.252

Z m /B

0.3

24.511

12.310

6.708

3.803

2.096

Pm

0.3

0.883

0.707

0.560

0.423

0.262

Y m /B

25.74

13.02

7.041

3.937

2.117

P0

0.659

0.523

0.377

0.252

Z m /B

0.4

15.890

8.121

4.364

2.279

Pm

0.4

0.705

0.543

0.396

0.232

Y m /B

17.05

8.674

4.604

2.350

P0

Table 8.1 a Comparison of total limit load (q/(γ B) = 0.1, c/(γ B) = 0.2). b Comparison of slip surface depth (q/(γ B) = 0.1, c/(γ B) = 0.2)

258 8 Limit Load on Ground with Heterogeneous Soil

8.2 Helicoid Calculation Mode in Surface Failure Mode



259

xi

Aγ (x R , z R , R0 ) =

wi (x − x R )dx

(8.23)

i x i−1

The slip surface is Eq. (8.2). (x R , z R , R0 ) is the parameter that makes the minimum total limit load be taken. According to Eq. (8.21), the calculation equation may be used for calculation of the non-uniformly distributed edge load. Similar to homogeneous soil, the eccentricity in Eq. (8.16) is introduced only for approximate treatment due to the calculation demands of the total limit load (it will be discussed later). It is easy to calculate the integral in the equation. For instance, according to the linear distribution of the soil weight within the soil strip, Eq. (8.23) may be calculated as below:   1 1 wi−1 xi−1 − x R + (xi − xi−1 ) Aγ (x R , z R , R1 ) = 2 3 i   1 + wi xi − x R − (xi − xi−1 ) (xi − xi−1 ) 3 wi is the soil strip weight at x = xi . 

xi

i x i−1

 1 csi

 csi

x − x R + h  (h − z R ) dx = (xi − x R )2 + (h i − z R )2 Fϕi 2 F ϕi i  − (xi−1 − x R )2 − (h i−1 − z R )2

Maximum depth of the slip surface: As for the slip surface {h i } that makes the minimum total limit load be taken, z m = {h i }max is true.

8.2.4 Layered Soil For the ground with layered soil, at present, it is commonly calculated according to the weighted average index of the soil layer thickness. The helicoid calculation mode is applicable to heterogeneous soil. The calculated results of two layers of soil grounds are as below:   Pm = Pz / γ B 2 . Case 1: Topsoil, 0.5B thick, ϕ1 = 25◦ and c1 /(γ B) = 0.2; subsoil, γ1 = γ , c2 /(γ B) = 0.2 and different ϕ is taken for calculation; edge load, q = 0.1/(γ B).

260

8 Limit Load on Ground with Heterogeneous Soil

Comparison of the total limit load Pm and slip surface depth Z m /B that are calculated respectively according to heterogeneous soil and the weighted average index is detailed in Table 8.2a, b. Case 2: The topsoil strength increases, and other calculation conditions are the same as those in Case 1. Comparison of the total limit load Pm and slip surface depth Z m /B that are calculated according to heterogeneous soil and the weighted average index, respectively, is detailed in Table 8.3a, b. Case 3: The topsoil strength decreases (ϕ1 = 20◦ , c1 /(γ B) = 0.1), and other calculation conditions are the same as those in Case 1. Comparison of the total limit load Pm and slip surface depth Z m /B that are calculated respectively according to heterogeneous soil and the weighted average index is detailed in Table 8.4a, b. It needs to be stated that, presently, the weighted average index of the soil characteristics on each soil layer within the maximum depth of slip surface in case of homogeneous soil is generally adopted for calculation of the limit load on the ground with heterogeneous soil, for instance, Tianjin University’s equation (“Code for Foundation in Port Engineering”) adopted in the above calculation. Even though the corresponding calculation equation is derived according to the field failure mode (e.g., the characteristic line method), such calculation process neither entirely belongs to the surface failure mode nor the field failure mode, because the same treatment and calculation for the strength index has not carried out for several slip surfaces (crossing several characteristic lines in the foundation bottom); however, the above logarithmic helicoid calculation mode entirely belongs to the surface failure mode. The calculated results clearly indicate that when the topsoil strength is visibly larger than the subsoil strength, the slip surface depth calculated according to heterogeneous soil is visibly larger than that calculated according to the average index and the corresponding limit load calculated according to heterogeneous soil is visibly smaller than that calculated according to the average index, which is owed to calculation according to the average strength index. Moreover, the slip surface depth is relatively small and will make the average strength index relatively large; thus, the calculated total limit load will be relatively large. When the topsoil strength is smaller than the subsoil strength, the slip surface depth calculated according to heterogeneous soil is visibly smaller than that calculated according to the average index. When Z max ≤ 0.5 is true, it indicates that the slip surface only passes through the subsoil. When the topsoil strength is relatively larger than the subsoil strength, the total limit load calculated according to heterogeneous soil is apparently different from that calculated according to the weighted average index of the soil layer thickness. If the strength difference is very large, for example, when the topsoil strength ϕ1 = 25◦ , c1 /(γ B) = 0.2; subsoil strength ϕ2 = 40◦ , c2 /(γ B) = 0.2, and tan δ = 0.1 is true; the total limit load (Pm ) calculated respectively according to the weighted average index of the soil layer thickness and heterogeneous soil is 20.348 and 10.388; when the ϕ1 = 20◦ , c1 /(γ B) = 0.1 and ϕ2 = 40◦ are true for the topsoil strength and the subsoil strength respectively, c2 /(γ B) = 0.2 and tan δ = 0.0 are true; the total limit load (Pm ) calculated respectively according to the weighted average index of the soil layer thickness and heterogeneous soil is

7.011

9.968

14.624

22.353

35.598

20.0

25.0

30.0

35.0

40.0

8.121

10.339

13.956

20.305

32.826

30.0

35.0

40.0

6.656

15.0

25.0

5.639

20.0

4.886

1.369

1.187

1.063

0.973

0.906

0.855

0.814

0.779

20.348

13.597

10.046

7.987

6.693

5.855

5.197

4.836

Pm

10.0

0.1

10.388

10.388

10.388

7.654

5.684

4.269

3.238

Pm

Z m /B

0.959

0.962

0.883

0.893

0.912

0.888

0.976

0.0

5.0

ϕ(◦ )

tan δ

5.062

15.0

(b)

3.708

10.0

2.460

Pm

Z m /B

1.016

Pm

2.722

0.1

0.0

tan δ

5.0

ϕ(◦ )

(a)

Z m /B

1.076

0.945

0.855

0.791

0.743

0.708

0.677

0.659

Z m /B

0.498

0.498

0.498

0.713

0.805

0.808

0.832

0.933

10.538

7.908

6.575

5.834

5.372

5.026

4.783

4.631

Pm

0.2

5.768

5.768

5.768

5.579

4.470

3.549

2.801

2.198

Pm

0.2 Z m /B

0.770

0.689

0.639

0.607

0.585

0.567

0.554

0.545

Z m /B

0.486

0.486

0.486

0.607

0.667

0.738

0.756

0.834

3.952

3.952

3.952

3.952

3.952

3.952

3.952

3.952

Pm

0.3

3.803

3.803

3.803

3.803

3.487

2.913

2.390

1.948

Pm

0.3 Z m /B

0.421

0.421

0.421

0.421

0.421

0.421

0.421

0.421

Z m /B

0.416

0.416

0.416

0.416

0.611

0.634

0.713

0.777

2.327

2.327

2.327

2.327

2.327

2.327

2.327

2.327

Pm

0.4

2.279

2.279

2.279

2.279

2.279

2.279

2.043

1.703

Pm

0.4

0.225

0.225

0.225

0.225

0.225

0.225

0.225

0.225

Z m /B

0.252

0.252

0.252

0.252

0.252

0.252

0.661

0.744

Z m /B

Table 8.2 a Calculation according to heterogeneous soil (Case 1). b Calculation according to weighted average index of thickness of soil layer [3] (Case 1)

8.2 Helicoid Calculation Mode in Surface Failure Mode 261

7.498

11.010

16.898

27.297

47.252

20.0

25.0

30.0

35.0

40.0

9.364

12.180

16.825

25.021

41.229

30.0

35.0

40.0

7.538

15.0

25.0

6.234

20.0

5.355

1.459

1.258

1.121

1.020

0.946

0.889

0.842

0.806

26.840

17.411

12.480

9.627

7.826

6.623

5.867

5.199

Pm

10.0

0.1

26.311

18.866

12.432

8.576

6.111

4.484

3.312

Pm

Z m /B

1.158

1.068

1.023

0.960

0.973

0.961

1.019

0.0

5.0

ϕ(◦ )

tan δ

5.279

15.0

(b)

3.780

10.0

2.493

Pm

Z m /B

0.979

Pm

2.771

0.1

0.0

tan δ

5.0

ϕ(◦ )

(a)

Z m /B

1.173

1.023

0.921

0.847

0.792

0.750

0.720

0.691

Z m /B

0.497

0.845

0.837

0.837

0.803

0.859

0.913

0.960

15.548

11.136

8.706

7.293

6.443

5.827

5.442

5.073

Pm

0.2

8.580

8.580

8.580

6.444

4.868

3.725

2.887

2.238

Pm

0.2 Z m /B

0.889

0.792

0.725

0.680

0.649

0.624

0.608

0.592

Z m /B

0.499

0.499

0.499

0.691

0.723

0.758

0.795

0.872

6.914

6.020

5.570

5.346

5.159

5.122

4.999

4.880

Pm

0.3

5.037

5.037

5.037

4.732

3.860

3.131

2.513

1.996

Pm

0.3 Z m /B

0.588

0.555

0.537

0.527

0.519

0.518

0.512

0.506

Z m /B

0.480

0.480

0.480

0.554

0.650

0.710

0.749

0.874

3.457

3.457

3.457

3.457

3.457

3.457

3.457

3.457

Pm

0.4

3.199

3.199

3.199

3.199

3.092

2.613

2.179

1.798

Pm

0.4

0.368

0.368

0.368

0.368

0.368

0.368

0.368

0.368

Z m /B

0.353

0.353

0.353

0.353

0.598

0.621

0.683

0.826

Z m /B

Table 8.3 a Calculation according to heterogeneous soil (Case 2). b Calculation according to weighted average index of thickness of soil layer [3] (Case 2)

262 8 Limit Load on Ground with Heterogeneous Soil

8.2 Helicoid Calculation Mode in Surface Failure Mode

263

Table 8.4 a Calculation according to heterogeneous soil (Case 3). b Calculation according to weighted average index of thickness of soil layer [3] (Case 3) (a) ϕ(◦ )

tan δ 0.0

0.1

Pm

Z m /B

5.0

2.157

10.0

2.892

15.0

0.2

Pm

Z m /B

Pm

0.919

1.908

0.767

0.804

2.451

0.716

3.862

0.764

3.086

20.0

5.183

0.659

25.0

5.460

0.499

30.0

5.460

0.499

0.3 Z m /B

Pm

Z m /B

1.663

0.707

1.412

0.213

2.031

0.603

1.412

0.213

0.607

2.301

0.406

1.412

0.213

3.352

0.486

2.301

0.406

1.412

0.213

3.352

0.486

2.301

0.406

1.412

0.213

3.352

0.486

2.301

0.406

1.412

0.213

(b) φ(◦ )

tan δ 0.0 Pm

0.1 Z m /B

Pm

0.2

0.3

Z m /B

Pm

Z m /B

Pm

Z m /B

5.0

3.188

0.695

2.944

0.564

2.416

0.425

1.469

0.229

10.0

3.612

0.720

3.113

0.576

2.416

0.425

1.469

0.229

15.0

4.223

0.753

3.338

0.591

2.416

0.425

1.469

0.229

20.0

5.103

0.795

3.688

0.613

2.416

0.425

1.469

0.229

25.0

6.436

0.849

4.235

0.644

2.416

0.425

1.469

0.229

30.0

8.680

0.925

5.123

0.688

2.416

0.425

1.469

0.229

35.0

12.730

1.033

6.823

0.757

2.416

0.425

1.469

0.229

40.0

20.896

1.194

10.329

0.867

2.416

0.425

1.469

0.229

20.896 and 5.460. It is obvious that the total limit load calculated according to the weighted average index of the soil layer thickness is completely distorted. It is well known that when the difference between the topsoil strength and the subsoil strength is not large, the total limit load calculated according to heterogeneous soil is basically equivalent to that calculated according to the weighted average index of the soil layer thickness. In addition, it is specified that the Tianjin University’s equation adopted in the “Code for Foundation in Port Engineering” is only applicable to the case with small strength difference of the topsoil.

8.3 Helicoid Calculation Mode in Field Failure Mode In the solutions of the former chapters, for the common helicoid applied for each slip surface in the slip surface family, either (x R , z R ) is limited to be a fixed point or to be on a plane [the interface of (A) and (B)]. As for the slip surface through any

264

8 Limit Load on Ground with Heterogeneous Soil

point on the foundation, (x R , z R ) may be different; thus, such slip surface family is more likely to be close to the real slip surface family.

8.3.1 Calculation Equation for Limit Load It shall be first noticed that Eq. (8.1) is correct for any width b, namely: 0

0 (x − x R + z R tan δ) pz dx = −

−b

−b

cs0 z R tan δdx Fϕ0

    − wγ i + qi (x − x R ) xi

i x i−1

 + csi z − z R − (x − x R )h  dx

(8.24)

Here, if x ≤ 0, the edge load qi does not exist. The foundation width is divided into M intervals in [−bk , −bk−1 ] (Fig. 8.3); thereinto, b M = B, b0 = 0. If the limit load in [−bk , 0] is obtained, the limit load in [−bk+1 , −bk ] will be as below: −bk

0 (x − x R + z R tan δ) pz dx = −

−bk+1

(x − x R + z R tan δ) pz dx

−bk

0 − −bk+1

cs0 z R tan δdx Fϕ0

( xR , z R) θ0

θN

q − Be

− bk +1

− bk

z

Fig. 8.3 Schematic diagram for helicoid calculation mode

x

8.3 Helicoid Calculation Mode in Field Failure Mode

265

    wγ i + qi (x − x R ) + csi − xi



i x i−1

 z − z R − (x − x R )h  dx

(8.25)

It is noticed that Eq. (8.25) is derived under boundary condition  −bk  bk −bk+1 px dx = tan δ −bk+1 ( pz + cs0 /Fϕ0 )dx of the foundation bottom. If the interval [−bk+1 , −bk ] is very small, the uniformly distributed limit load may be taken within the interval; therefore, it will be: −bk pz dx = −bk+1

⎧ ⎨ ⎩

0 −

(x − x R + z R tan δ) pz dx

−bk

0 − −bk+1

 cs0 z R tan δdx − Fϕ0 i

xi

  wγ i + qi (x − x R )

xi−1

   1 +csi z − z R − (x − x R )h  dx / − (bk + bk+1 ) 2 −x R + z R tan δ) 

(8.26)

The slip surface in Eq. (8.26) is Eq. (8.2), but for each k, the slip surface z = h and parameters x R , z R , R 0 are different. In addition, the minimum through (−bk+1 , 0) is the limit load in the interval [−bk+1 , −bk ]. Total limit load: 0 pz dx = −B

−bk−1   k

pz dx

(8.27)

−bk

Eccentricity of the limit load: ⎡ ⎤ 0 −b  k−1  B ⎣1  ⎦ e= + pz dx / pdx (−bk−1 − bk ) 2 2 k −bk

(8.28)

−B

In the interval [−bk+1 , −bk ], the limit load of linear distribution may also be taken; hence, pz =

 1

pz,k (x + bk+1 ) − pz,k+1 (x + bk ) b

(8.29)

266

8 Limit Load on Ground with Heterogeneous Soil

It is substituted in Eq. (8.24) to derive the calculation equations for pz,k . pz,0 at the front toe (x = 0) may also be determined according to the theoretical solution. According to the calculation results, the limit loads obtained according to the two calculation methods are basically consistent (in some cases, distribution of the calculated limit load is easy to be instable by the latter). Similarly, according to the principle that the total limit load, in general, is slightly smaller than the calculated value by the characteristic line method and after calculation and comparison, the following equation is taken: Fsc = 1.09 + 0.06 tan δ, Fsϕ = 1.05 + 0.06 tan δ

(8.30)

The calculated total limit load, slip surface depth, and eccentricity are quite consistent with those by the characteristic line method. How many subareas (M) of the foundation bottom are suitable? Through calculation and analysis by a great many calculation examples, generally, when M is 20, it is basically stable. As for different values of c, q, the calculated results are all well consistent with those by the characteristic line method. The followings are the calculated results of 20 slip surfaces.

8.3.2 Calculated Result of Ground with Homogeneous Soil (1) When q/(γ B) = 0.1 and c/(γ B) = 0.2 are true, comparison of the total limit load, slip surface depth, eccentricity with those by the characteristic line method is detailed in Table 8.5a, b, and c, respectively. (2) When q/(γ B) = 0.2 and c/(γ B) = 0.4 are true, comparison of the total limit load, slip surface depth, eccentricity with those by the characteristic line method is detailed in Table 8.6a, b, and c, respectively. The total limit load, slip surface depth, and eccentricity of the limit load are all well close to those by the characteristic line method. Thereinto, comparison of the total limit load Pm with P0 (by the characteristic line method) is as below: When q/(γ B) 8.42%. When q/(γ B) 8.12%. When q/(γ B) 6.14%. When q/(γ B) 4.37%. When q/(γ B) 3.21%. When q/(γ B) 2.62%.

= 0.0 and c/(γ B) = 0.1 are true, the maximum relative error is = 0.1 and c/(γ B) = 0.0 are true, the maximum relative error is = 0.1 and c/(γ B) = 0.2 are true, the maximum relative error is = 0.2 and c/(γ B) = 0.4 are true, the maximum relative error is = 0.4 and c/(γ B) = 0.6 are true, the maximum relative error is = 0.6 and c/(γ B) = 0.8 are true, the maximum relative error is

5.955

10.052

18.140

35.467

76.733

20

25

30

35

40

1.669

0.617

0.640

0.681

0.770

0.879

15.0

20.0

25.0

0.909

0.801

0.718

0.662

0.680

0.713

0.604

0.490

0.406

Z m /B

10.0

0.1

54.127

25.830

13.609

7.699

4.613

2.869

1.785

Z m /B

Y m /B

76.61

36.07

18.69

10.44

6.196

3.859

2.501

0.0

5.0

ϕ(◦ )

tan δ

3.706

15

(b)

1.615

2.409

Pm

10

0.1

Pm

P0

0.0

tan δ

5

ϕ(◦ )

(a)

0.756

0.644

0.542

0.420

Y m /B

54.81

26.57

14.08

7.990

4.772

2.942

1.796

P0

0.556

0.447

0.327

Z m /B

0.2

37.013

18.216

9.806

5.611

3.337

1.960

Pm

0.2

0.593

0.469

0.318

Y m /B

38.04

18.90

10.18

5.802

3.408

1.945

P0

0.417

0.242

Z m /B

0.3

24.625

12.417

6.757

3.835

2.123

Pm

0.3

0.423

0.262

Y m /B

25.74

13.02

7.041

3.937

2.117

P0

0.255

Z m /B

0.4

16.001

8.187

4.398

2.315

Pm

0.4

(continued)

0.232

Y m /B

17.05

8.674

4.604

2.350

P0

Table 8.5 a Total limit load Pm (P0 is the calculated value by the characteristic line method). b Slip surface depth (Y m is the calculated value by the characteristic line method). c Eccentricity of limit load (e0 is the calculated value by the characteristic line method)

8.3 Helicoid Calculation Mode in Field Failure Mode 267

1.525

40.0

e0 /B

−0.020

−0.036

−0.050

−0.064

−0.076

−0.088

−0.100

−0.111

e/B

−0.018

−0.032

−0.046

−0.059

−0.071

−0.083

−0.095

−0.106

10.0

15.0

20.0

25.0

30.0

35.0

40.0

0.0

1.502

1.244

1.055

5.0

ϕ(◦ )

tan δ

1.232

(c)

1.023

−0.101

−0.089

−0.077

−0.065

−0.053

−0.040

−0.026

e/B

0.1

1.312

1.038

0.828

Z m /B

35.0

0.1

Z m /B

Y m /B

0.0

tan δ

30.0

ϕ(◦ )

(b)

Table 8.5 (continued)

−0.104

−0.095

−0.083

−0.070

−0.057

−0.044

−0.029

e0 /B

1.287

1.061

0.891

Y m /B

−0.095

−0.083

−0.071

−0.058

−0.046

−0.031

e/B

0.2

1.063

0.829

0.719

Z m /B

0.2

−0.100

−0.086

−0.076

−0.064

−0.050

−0.036

e0 /B

1.078

0.880

0.725

Y m /B

−0.088

−0.075

−0.063

−0.050

−0.035

e/B

0.3

0.838

0.650

0.525

Z m /B

0.3

−0.094

−0.081

−0.066

−0.055

−0.040

e0 /B

0.883

0.707

0.560

Y m /B

−0.080

−0.067

−0.053

−0.038

e/B

0.4

0.647

0.476

0.354

Z m /B

0.4

−0.085

−0.073

−0.060

−0.045

e0 /B

0.705

0.543

0.396

Y m /B

268 8 Limit Load on Ground with Heterogeneous Soil

6.372

9.760

15.736

27.087

50.380

103.67

15.0

20.0

25.0

30.0

35.0

40.0

3.137

0.658

0.703

0.770

0.838

0.983

15

20

25

1.005

0.889

0.802

0.737

0.700

0.818

0.674

0.588

0.455

Z m /B

10

0.1

74.811

37.619

20.794

12.335

7.740

5.033

3.279

Z m /B

Y m /B

102.7

50.68

27.55

16.14

10.05

6.565

4.465

0.0

5

ϕ(◦ )

tan δ

4.336

(b)

3.050

Pm

10.0

0.1

Pm

P0

0.0

tan δ

5.0

ϕ(◦ )

(a)

0.835

0.716

0.603

0.464

Y m /B

75.04

38.16

21.22

12.63

7.907

5.112

3.277

P0

0.647

0.493

0.349

Z m /B

0.2

52.523

27.224

15.381

9.230

5.746

3.537

Pm

0.2

0.656

0.522

0.353

Y m /B

53.37

27.84

15.74

9.408

5.796

3.474

P0

0.436

0.281

Z m /B

0.3

36.022

19.119

10.917

6.493

3.776

Pm

0.3

0.469

0.291

Y m /B

37.11

19.72

11.20

6.575

3.719

P0

0.255

Z m /B

0.4

24.210

13.029

7.372

4.055

Pm

0.4

(continued)

0.258

Y m /B

25.32

13.55

7.563

4.068

P0

Table 8.6 a Total limit load Pm (P0 is the calculated value by the characteristic line method). b Slip surface depth (Y m is the calculated value by the characteristic line method). c Eccentricity of limit load (e0 is the calculated value by the characteristic line method)

8.3 Helicoid Calculation Mode in Field Failure Mode 269

1.610

40

e0 /B

−0.012

−0.022

−0.033

−0.044

−0.055

−0.066

−0.077

−0.089

e/B

−0.010

−0.020

−0.030

−0.040

−0.050

−0.062

−0.073

−0.086

10

15

20

25

30

35

40

0.0

1.617

1.351

1.154

5

ϕ(◦ )

tan δ

1.322

(c)

1.114

−0.080

−0.068

−0.056

−0.045

−0.035

−0.025

−0.015

e/B

0.1

1.406

1.109

0.970

Z m /B

35

0.1

Z m /B

Y m /B

0.0

tan δ

30

ϕ(◦ )

(b)

Table 8.6 (continued)

−0.084

−0.072

−0.060

−0.049

−0.038

−0.028

−0.017

e0 /B

1.388

1.155

0.977

Y m /B

−0.073

−0.061

−0.050

−0.039

−0.029

−0.019

e/B

0.2

1.149

0.935

0.740

Z m /B

0.2

−0.078

−0.066

−0.054

−0.043

−0.032

−0.022

e0 /B

1.168

0.961

0.797

Y m /B

−0.066

−0.054

−0.042

−0.031

−0.020

e/B

0.3

0.906

0.735

0.577

Z m /B

0.3

−0.071

−0.059

−0.047

−0.036

−0.025

e0 /B

0.960

0.774

0.617

Y m /B

−0.058

−0.046

−0.035

−0.022

e/B

0.4

0.705

0.537

0.427

Z m /B

0.4

−0.064

−0.051

−0.040

−0.028

e0 /B

0.769

0.597

0.438

Y m /B

270 8 Limit Load on Ground with Heterogeneous Soil

8.3 Helicoid Calculation Mode in Field Failure Mode

271

The above maximum relative errors are relatively small, and the relatively large ones are less than 2.5%. In addition, as for the total limit load and slip surface depth, the calculated results in surface failure mode and field failure mode are quite consistent with each other. Other Fsc and Fsϕ may also be selected. The calculated results are much closer to those by the characteristic line method. In fact, if Fsc = 1.06 + 0.05 tan δ, Fsϕ = 1.02 + 0.05 tan δ are true, for the above calculation, the total limit loads in other cases are larger than that by the characteristic line method except the total limit load P0 < 1.0, which indicates that the error of the strength index is less than 3%.

8.3.3 Calculated Results of Two Layers of Soil Grounds Subsoil: γ2 = γ , c2 /(γ B) = 0.2, and different ϕ is taken for calculation; edge load: q = 0.1/(γ B). Case 1: topsoil, 0.5B thick, ϕ1 = 25◦ , and c1 /(γ B) = 0.2. The calculation results of the total limit load, slip surface depth, and eccentricity are detailed in Table 8.7a and b, respectively. Case 2: topsoil, 0.5B thick, ϕ1 = 30◦ and c1 /(γ B) = 0.1. The calculation results of the total limit load, slip surface depth, and eccentricity are detailed in Table 8.8a and b, respectively. Case 3: topsoil, 0.5B thick, ϕ1 = 20◦ , and c1 /(γ B) = 0.1. The calculation results of the total limit load, slip surface depth, and eccentricity are detailed in Table 8.9a and b, respectively. When the topsoil strength is much larger than the subsoil strength, it is evident that the maximum depth of the slip surface is relatively large. Even though the slip surface is made to pass through the subsoil as much as possible by expanding hunting zone of the s parameters x R , z R of the slip surface, moreover, the slip surface near the foundation front toe only passes through soil or the topsoil arc, it is longer than that near the foundation rear toe, so the calculated value of the limit load nearby the front toe is relatively large, and thus, the limit load is caused to be subject to forward eccentricity. For example, the forward eccentricity in Table 8.8b is more than 0.1B. When the strength topsoil strength is relatively smaller than the subsoil strength, the maximum depth of the slip surface is obvious to be relatively small and even only passes through the topsoil, which is certain to be related to the relative thickness of topsoil. If the relative thickness of topsoil is relatively small, the slip surface may also pass through the subsoil, and in this case, the limit load nearby the rear toe is relatively large, thus leading to a quite large backward eccentricity, for instance, −0.177B eccentricity in Table 8.8b. If the topsoil strength is much smaller than the subsoil strength, the limit load nearby the rear toe will be relatively much larger. When the difference between the topsoil strength and subsoil strength is very large, if the calculation is carried out according to the existing weighted average index of the soil layer within the thickness of the supporting layer, the larger the internal friction angle of subsoil is, the larger the slip surface depth is, and the smaller the internal

4.469

5.958

7.815

10.052

12.180

12.406

12.406

15.0

20.0

25.0

30.0

35.0

40.0

tan δ

0.0189 −0.0230 −0.0648 −0.0875 −0.0875 −0.0875

0.0133

−0.0712

−0.1184

−0.1248

−0.1248

25.0

30.0

35.0

40.0

0.0586

0.1031

0.1

0.500

0.500

0.500

0.713

0.810

0.881

−0.0283

8.281

8.281

8.281

7.699

6.294

4.978

0.981

1.036

20.0

0.0553

2.940

3.875

Z m /B

15.0

0.1019

0.500

0.500

0.750

0.879

0.952

0.987

10.0

0.0

1.193

1.037

5.0

ϕ(◦ )

(b)

3.293

Pm

10.0

0.1

Pm

Z m /B

0.0

tan δ

5.0

ϕ(◦ )

(a)

−0.0604

−0.0604

−0.0604

−0.0578

−0.0210

0.0187

0.0602

0.1002

0.2

5.647

5.647

5.647

5.611

4.900

4.066

3.275

2.586

Pm

0.2

0.495

0.495

0.495

0.556

0.670

0.770

0.852

1.021

Z m /B

−0.0497

−0.0497

−0.0497

−0.0497

−0.0332

0.0066

0.0456

0.0873

0.3

3.835

3.835

3.835

3.835

3.673

3.211

2.717

2.229

Pm

0.3

Table 8.7 a Total limit load and slip surface depth (Case 1). b Eccentricity e/B of limit load (Case 1)

0.417

0.417

0.417

0.417

0.608

0.665

0.793

0.920

Z m /B

−0.0380

−0.0380

−0.0380

−0.0380

−0.0380

−0.0380

−0.0072

0.0416

0.4

2.315

2.315

2.315

2.315

2.315

2.315

2.143

1.843

Pm

0.4

0.255

0.255

0.255

0.255

0.255

0.255

0.699

0.799

Z m /B

272 8 Limit Load on Ground with Heterogeneous Soil

6.392

8.676

11.730

15.560

19.522

21.037

15.0

20.0

25.0

30.0

35.0

40.0

tan δ

0.0215 −0.0183 −0.0605 −0.1052 −0.1396 −0.1396

0.0190

−0.0215

−0.0636

−0.1080

−0.1564

−0.177

20.0

25.0

30.0

35.0

40.0

0.0648

0.0602

15.0

0.1078

0.1

0.500

0.500

0.778

0.864

0.893

0.998

1.011

10.0

12.787

12.787

11.396

9.092

7.018

5.370

4.043

0.1083

0.0

0.500

0.849

0.937

0.968

1.042

1.034

1.157

Z m /B 1.112

5.0

ϕ(◦ )

(b)

4.677

10.0

3.036

Pm

Z m /B

1.158

Pm

3.433

0.1

0.0

tan δ

5.0

ϕ(◦ )

−0.0955

−0.0955

−0.0952

−0.0587

−0.0172

0.0244

0.0612

0.1040

0.2

7.798

7.798

7.791

6.777

5.517

4.394

3.472

2.675

Pm

0.2 Z m /B

0.499

0.499

0.500

0.667

0.753

0.838

0.890

0.981

−0.0770

−0.0770

−0.0770

−0.0623

−0.0235

0.0140

0.0540

0.0915

0.3

5.101

5.101

5.101

4.885

4.229

3.555

2.908

2.346

Pm

0.3

Table 8.8 a Total limit load and slip surface depth (Case 2). b Eccentricity e/B of limit load (Case 2)

Z m /B

0.477

0.477

0.477

0.590

0.659

0.736

0.794

0.931

−0.0674

−0.0674

−0.0674

−0.0674

−0.0553

−0.0158

0.0234

0.0622

0.4

3.253

3.253

3.253

3.253

3.157

2.798

2.406

2.019

Pm

0.4 Z m /B

0.341

0.341

0.341

0.341

0.576

0.656

0.745

0.831

8.3 Helicoid Calculation Mode in Field Failure Mode 273

274

8 Limit Load on Ground with Heterogeneous Soil

Table 8.9 a Total limit load and slip surface depth (Case 3). b Eccentricity e/B of limit load (Case 3) ϕ(◦ )

tan δ 0.0

0.1

0.2

0.3

Pm

Z m /B

Pm

Z m /B

Pm

Z m /B

Pm

Z m /B

5.0

2.451

0.900

2.139

0.880

1.808

0.751

1.429

0.671

10.0

3.167

0.842

2.655

0.738

2.126

0.632

1.446

0.238

15.0

3.940

0.779

3.128

0.628

2.327

0.383

1.446

0.238

20.0

4.582

0.500

3.296

0.493

2.327

0.383

1.446

0.238

25.0

4.582

0.500

3.296

0.493

2.327

0.383

1.446

0.238

(b) ϕ(◦ )

tan δ 0

0.1

0.2

0.3

0.0332

0.0298

0.0142

−0.0433

10.0

−0.0110

−0.0141

−0.0287

−0.0483

15.0

−0.0533

−0.0534

−0.0599

−0.0483

20.0

−0.0941

−0.0702

−0.0599

−0.0483

25.0

−0.0941

−0.0702

−0.0599

−0.0483

5.0

friction angle of subsoil is, the smaller the slip surface depth is. However, the fact is that the slip surface shall pass through the soil layer with relatively strength as much as possible. In addition, the above calculated result reflects the actual situation of the ground with layered soil; namely, the larger the internal friction angle of subsoil is, the smaller the slip surface depth is, and the smaller the internal friction angle of subsoil, the larger the slip surface depth is.

8.3.4 Distribution of Limit Load on Ground When the strength difference of soil layers is small, distribution of the calculated limit load is similar to that by the characteristic line method under a very good distribution rule; namely, it is distributed in backward eccentricity form with small limit load nearby the front toe and large one nearby the rear toe. The calculated results of distribution pk = pz /(γ B) of the limit load on two layers of soil ground are as below: Case 1: For distribution of the limit load with large topsoil strength and small subsoil strength [topsoil, 0.5B thick, ϕ1 = 25◦ , and c1 /(γ B) = 0.4; subsoil, γ2 = γ , ϕ2 = 10◦ , and c2 /(γ B) = 0.2. q = 0.1/(γ B)], the calculated result is detailed in Table 8.10. Under such condition with large topsoil strength and small subsoil strength, due to control from the minimum limit load, the slip surface near the front toe only passes

8.3 Helicoid Calculation Mode in Field Failure Mode

275

Table 8.10 Distribution of limit load with large topsoil strength and small subsoil strength (Case 1) b/B

tan δ 0.0

0.2

0.4

pk

Z m /B

pk

Z m /B

pk

Z m /B

0.05

10.291

0.095

6.576

0.059

3.249

0.026

0.15

11.218

0.405

7.067

0.131

3.373

0.055

0.25

3.763

0.721

7.028

0.441

3.485

0.085

0.35

4.076

0.791

3.021

0.713

3.598

0.113

0.45

4.040

0.858

2.951

0.764

3.715

0.135

0.55

4.072

0.906

2.941

0.815

3.840

0.430

0.65

4.134

0.984

2.900

0.850

1.814

0.731

0.75

4.196

1.073

2.841

0.900

1.774

0.761

0.85

4.272

1.138

2.814

0.951

1.790

0.763

0.95

4.258

1.204

2.839

0.964

1.594

0.779

Total limit load

5.432

4.098

2.823

Eccentricity

0.0924

0.1034

0.0702

through the topsoil and the corresponding slip surface depth is very small; while the slip surface passes through the subsoil, the corresponding load is much smaller, the slip surface family will have a sudden change, and it will immediately crosses from the smaller subsoil strength to the subsoil. When the slip surface corresponding to the limit load within [−bk+1 , −bk ] only passes through the topsoil, the calculated value of the limit load is relatively large; once the slip surface passes through the subsoil, the limit load will decrease rapidly. However, as the horizontal force (tan δ) increases, after the slip surface passes through the subsoil, the limit load tends to slowly decrease. As for the above example, after the slip surface passes through the subsoil, the limit load is approximately distributed uniformly. If the subsoil strength increases (still smaller than the topsoil strength), the limit load will increase with b (distance from the front toe) after its distribution is reduced. In addition, after the slip surface passes through the subsoil, distribution of the calculated limit load is slightly instable. On the one hand, it is owed to the calculation error, if the calculation accuracy is improved (the soil strip is divided more densely and the grid for searching the parameters x R , z R of the slip surface is denser; with denser division of the soil strip, the calculated total limit load will slightly increase), the slightly instable distribution of the limit load will be improved; on the other hand, the quite large limit load near the front toe expands the error of parameters x R , z R , since no such instability can be seen when the strength differences of soil layers are not very large.

276

8 Limit Load on Ground with Heterogeneous Soil

Table 8.11 Distribution of limit load with small topsoil strength and large subsoil strength (Case 2) b/B

tan δ 0.0 pk

0.1 Z m /B

pk

0.2 Z m /B

pk

Z m /B

0.05

3.992

0.081

3.272

0.066

2.524

0.048

0.15

4.487

0.172

3.612

0.140

2.735

0.105

0.25

5.051

0.247

3.948

0.200

2.932

0.150

0.35

6.543

0.247

4.362

0.246

3.131

0.201

0.45

8.704

0.247

5.318

0.249

3.320

0.239

0.55

11.584

0.248

6.366

0.248

3.683

0.244

0.65

11.340

0.483

7.835

0.250

4.100

0.247

0.75

12.165

0.568

9.563

0.249

4.677

0.241

0.85

13.004

0.633

10.916

0.472

5.244

0.240

0.95

13.977

0.718

10.240

0.533

6.003

0.245

Total limit load Eccentricity

9.085

6.543

3.835

−0.112

−0.117

−0.080

Case 2: for distribution with small topsoil strength and large subsoil strength [topsoil, 0.25B thick, ϕ1 = 20◦ and c1 /(γ B) = 0.2; subsoil, γ2 = γ , ϕ2 = 25◦ , and c2 /(γ B) = 0.3, q = 0.1/(γ B)], the calculated result is detailed in Table 8.11. In case of topsoil strength and large subsoil strength, due to control from the minimum limit load, the slip surface depth is apparently small. The slip surface corresponding to the limit load within [−bk+1 , −bk ] is quite likely to pass the topsoil, and only when the calculated value of the limit load is still larger than that which makes the slip surface pass the subsoil after the slip surface is excessively flat, the slip surface is able to pass the subsoil. For the above example, if the topsoil thickness or the subsoil strength increases, less slip surfaces (in the slip surface family) will pass the subsoil. For instance, if the topsoil thickness is 0.25B and ϕ2 = 30◦ , the slip surface only passes through the topsoil. Even though the slip surface only passes through the topsoil, due to great restriction of the subsoil strength, the slip surface corresponding to the limit load within [−bk+1 , −bk ] nearby the rear toe is apparently flat and the limit load increases, e.g., in case of tan δ = 0.2. Only when the slip surface passes through the topsoil, the corresponding limit load is free from any restriction of the subsoil strength; thus, it will be the same as that with regard to homogeneous soil, e.g., in case of tan δ = 0.4. Unless the limit load is free from any restriction of the subsoil strength, it is inevitable that the backward eccentricity of the limit load is relatively large. It is worthy to notice that when the topsoil strength is far larger than the subsoil strength, the range [−bk , 0] of the limit load nearby the front toe is relatively large, which is bound to affect the limit load within [−bk+1 , −bk ]. With respect to this problem, it will be discussed later.

8.4 Analysis and Comparison of Surface Failure Mode and Field …

277

8.4 Analysis and Comparison of Surface Failure Mode and Field Failure Mode As for the ground with homogeneous soil, both of the two failure modes are based on the results by the characteristic line method, and the calculated total limit load and slip surface depth are basically consistent. As for heterogeneous soil, the total limit loads calculated in the two failure modes are apparently different; therefore, it is important for practical application to analyze advantages and disadvantages of the two failure modes.

8.4.1 Analysis of Surface Failure Mode The advantage of the surface failure mode lies in that only the total horizontal load is required to be considered, so the mentioned distribution of the horizontal force on the foundation bottom no longer exists. However, the following aspects need to be further discussed. Firstly, the method for determination of the eccentricity of the above limit load actually is only applicable to an approximation method with regard to homogeneous soil; for the ground with heterogeneous soil, non-uniformly distributed edge load, and non-horizontal soil mass surface, strictly speaking, it does not apply. Just for the calculation of two layers of soil, when the topsoil strength is large and the subsoil strength small, the calculated eccentricity (it is referred to the backward eccentricity; e ≤ 0, and it is relatively small) will be relatively large, thus resulting in relatively small total limit load; when the topsoil strength is small and the subsoil strength large, the calculated eccentricity (it is referred to the backward eccentricity; e ≤ 0, and it is relatively large) will be relatively small, thus resulting in relatively large total limit load. Therefore, how to determine the eccentricity of the limit load will become a problem when the surface failure mode is applied to calculate the limit load of the ground with heterogeneous soil. Secondly, it needs to be discussed that the value of the surface failure mode Fsϕ , Fsc is determined according to the calculated result by the characteristic line method, because it cannot be proven that the total limit loads in the two failure modes are the same with regard to heterogeneous soil. Therefore, how to determine Fsϕ , Fsc to be more suitable for characteristics of the surface failure mode remains to be further discussed. Additionally, for the surface failure mode, distribution of the limit load cannot be obtained and the partial failure of the ground cannot be discussed.

278

8 Limit Load on Ground with Heterogeneous Soil

8.4.2 Analysis on Field Failure Mode For the advantages of the field failure mode, distribution of the limit load may be obtained and rationality of calculation for the eccentricity is out of the question; moreover, it is very convenient for application. However, the followings need to be further discussed. Firstly, since the load boundary condition of the foundation bottom, the mentioned distribution of horizontal force on the foundation bottom, is required to be given for the field failure mode, it needs to be further discussed in the field failure mode. Under the load boundary condition px = tan δ( pz + cs0 /Fϕ0 ) of the foundation bottom, the obtained ground soil strength and distribution of the limit load are different. As for heterogeneous soil, there may exist great difference between the distribution form of the limit load with small topsoil strength and large subsoil strength and that of the limit load with large topsoil strength and small subsoil strength. This means that there exists great difference between the distribution of the calculated horizontal limit load along the foundation bottom and the actual distribution. Here, only for larger horizontal force, reducing the total limit load (homogeneous soil compared with the characteristic line method) with a smaller amplitude is nothing but a kind of rough consideration for the total limit load and it cannot be used to solve the consistence between the distribution of the calculated horizontal limit load and actual distribution of the horizontal force, which still needs to be further studied. Secondly, on condition that the topsoil strength is far larger than the subsoil strength (including the condition of larger edge load nearby the front toe), when the limit load nearby the front toe is calculated, whether because the slip surface only passes through the topsoil or because the slip surface passes through the subsoil with evidently increased slip surface depth, the calculated limit load appears apparently large. Whereas, during the calculation, the limit load at (−bk+1 , −bk ) is calculated according to that at (−bk , 0) and as bk increases, the restriction from the topsoil to minimization of the limit load reduces; moreover, since the limit load at (−bk , 0) is quite large, the calculated value of the limit load reduces evidently after the slip surface passes through the subsoil. In practical problems, if the load nearby the front toe is far smaller than the calculated limit load, the quite large limit load at (−bk , 0) nearby the front toe not only is of no meaning for the engineering but also results in smaller calculated value of the limit load at (−bk+1 , −bk ); thus, on the one hand, the forward eccentricity of the limit load may take place; on the other hand, the above problem may be more apparent when the strengths of rubble mound and soil nearby the front toe are very small or when the topsoil strength is far and away larger than the subsoil strength. If the quite large limit load at (−bk , 0) nearby the front toe is sure to be of no signification for the engineering and the calculated value of the limit load at [−B, −bk ) is unreasonable, without any violation of the yield criterion, the limit load at (−bk , 0) can properly reduce (into allowable load) and then the limit load at [−B, −bk ). And this is a better method to solve such a problem and will be discussed later.

8.4 Analysis and Comparison of Surface Failure Mode and Field …

279

Such similar problem will also take place when the subsoil strength is larger than the topsoil strength (including the condition of the ground with homogeneous soil). Because the limit load is eccentrical backward and when ϕ is relatively large, the limit load nearby the rear toe will be quite large. In practical problems, if the load nearby the rear toe is far smaller than the calculated limit load, for instance, under the condition of forward eccentricity of the design load, the quite large limit load nearby the rear toe is of no great signification for the practical engineering. In other words, even though the calculation of the limit load is solved, distribution of the limit load is generally impossible to be the same as that of the design load in engineering. Therefore, how to determine the ground bearing capacity also needs to be further studied. As for the ground with layered soil, the Tianjin University’s method (or other methods for the ground with homogeneous soil) may also be applied to calculate the limit load in the field failure mode. And it only requires that the total limit load at (−bk , 0) is calculated according to the mean parameter of the soil characteristics within the depth of each slip surface passing the foundation bottom, the total limit load at (−bk , 0) is subtracted, and then, the limit load at (−bk+1 , −bk ) may be obtained.

8.4.3 Comparisons of Two Failure Modes When the topsoil strength is large and the subsoil strength small, the slip surface depth will increase obviously compared with that of homogeneous soil, which is because under control action of the minimum limit load, the slip surface passes through the subsoil as much as possible. Correspondingly, the limit load in the surface failure mode is smaller than that in the field failure mode with the reason that for the field failure mode, the nearer the starting point of the slip surface is to the front toe, the longer the arc length over topsoil is compared with the total arc length of this slip surface. When the topsoil strength is low and the subsoil strength is high, the slip surface depth will reduce obviously compared with that of homogeneous soil and the limit load in the surface failure mode is larger than that in the field failure mode. For this, on the one hand, the nearer the starting point of the slip surface in the field failure mode is to the front toe, the longer the arc length over topsoil is compared with the total arc length of the slip surface; on the other hand, the slip surface is also subject to control of the minimum limit load. Whereas, in some cases, the control action of the minimum limit load is very evident, for example, under the condition of two layers of soil for the subsoil ϕ = 40◦ in Tables 8.8a and 8.3a with the same calculation condition and the slip surfaces in the field failure mode and surface failure mode can only be in the topsoil, but there exists great difference in the total limit load (Pm are 12.787 and 26.311, respectively). Obviously, under control action of the minimum limit load, the slip surface (family) can only in the topsoil. Generally, with large topsoil strength and small subsoil strength, the total limit load in the surface failure mode is relatively small; with small topsoil strength and

280

8 Limit Load on Ground with Heterogeneous Soil

large subsoil strength, the total limit load in the surface failure mode is relatively large. Discussion on the calculation of the limit load is aimed at solving calculation of the ground bearing capacity. Whether according to the surface failure mode or the field failure mode, the discussion on the limit load must be considered in combination with the design load in engineering when the ground bearing capacity is determined. In contrast, the limit load calculated in the field failure mode is more reasonable and is more convenient for discussion on the ground bearing capacity. However, it may be more convenient to discuss the ground bearing capacity by methods similar to the slope stability analysis than by those in the surface failure mode, which also has been applied overseas; for example, Bishop’s arc slipping analysis method is adopted in the “Technical Standard of Port Facilities” in Japan. Therefore, the following discussion on the limit load is only limited to the field failure mode.

8.5 General Calculation Mode by Generalized Limit Equilibrium Method As it has been stated before, by the generalized limit equilibrium method, the following two equations may be applied for solution: d [(h − z R )E(h s , h) − (x − x R )T (h s , h) − M(h s , h)] dx   

= −cs h − z R − h  (x − x R ) − (x − x R ) w(h s , h) + ( pz ) s  

   + (h s − z R )( px ) s + σ h  − Fϕ (h − z R ) + 1 + h  Fϕ (x − x R )   cs dh  2Fϕ h  − Fϕ dσ σ + − − γ =0 dx 1 + h 2 Fϕ dx 1 + Fϕ2

(8.31) (8.32)

According to Eq. (8.31), it will be: 0 −b



 −(x − x R ) pz + (h s − z R ) px dx =

xb



 cs h − z R − h  (x − x R )

−b



 + (x − x R )[w(h s , h) + q] − σ h  − Fϕ    (h − z R ) + 1 + h  Fϕ (x − x R ) dx (8.33)

where if x ≤ 0, q = 0 is true. For Eq. (10.32), the general form of solution is:

8.5 General Calculation Mode by Generalized Limit Equilibrium …

σ+

   cs cs = exp 2Fϕ arctan h  C + Fϕ Fϕ ⎤  xb     γ h − Fϕ − exp −2Fϕ arctan h  dx ⎦ 1 + Fϕ2

281

(8.34)

x

where C is a constant to be determined. C is the same on the slip surface passing the same soil layer. When the slip surface passes through two layers of soil, C will be different due to different strength indexes and unit weights of soil, so treatment is required to be carried out when the slip surface passes through the interface of the two layers of soil. According to the slice method, within [xi−1 , xi ], it will be: σ+

   csi csi = exp 2Fϕi arctan h  Ci + + gi (x) Fϕi Fϕi

(8.35)

where xi gi (x) = − x

    γ h  − Fϕ exp −2Fϕ arctan h  dx 1 + Fϕ2

(8.36)

If the ground soil is in layers horizontally and the two adjacent soil strips [xi−1 , xi ] and [xi , xi+1 ] are in different soil layers, the vertical stress at the division xi of the two soil strips shall be continuous, namely (σz )i− = (σz )i+ (if the two adjacent layers of soil is not in layers horizontally, (Z )i− = (Z )i+ shall be adopted). Since σz +

csi Fϕi

=

(1+Fϕi2 )+(1+Fϕi h  )2 (σ 1+h 2

(σz )i− +

(σz )i+ +

csi+1 = Fϕi+1

csi = Fϕi

+

csi Fϕi

), it will be:

  2  2  1 + Fϕi2 h i− + 1 + Fϕi h i−

2 1 + h i−     csi  exp 2Fϕi arctan h i− Ci + Fϕi   2  2 2  1 + Fϕi+1 h i+ + 1 + Fϕi+1 h i+

2 1 + h i+    csi+1  exp 2Fϕi+1 arctan h i+ Ci+1 + + gi+1 (xi ) Fϕi+1

The following equation may be derived: Ci =

Bi Ci+1 + Di Ai

(8.37)

282

8 Limit Load on Ground with Heterogeneous Soil

where Ai =

Bi =

  2  2  1 + Fϕi2 h i− + 1 + Fϕi h i− 2 h i−

1+    2 2 2  h i+ 1 + Fϕi+1 + 1 + Fϕi+1 h i+ 1+ Di =

2 h i+

   exp 2Fϕi arctan h i−

(8.38)

   exp 2Fϕi+1 arctan h i+

(8.39)

Bi 1 csi+1 1 csi gi+1 (xi ) + + (1 − Ai ) (Bi − 1) Ai Ai Fϕi Ai Fϕi+1

(8.40)

2h 

 If ϕi = 0, (1 − Ai ) F1ϕi = −( 1+hi−2 + 2 arctan h i− ), Ai = 1 is true. i−

2h 

1  If ϕi+1 = 0, (Bi − 1) Fϕi+1 = 1+hi+2 + 2 arctan h i+ , Bi = 1 is true. i+ For the slip surface passing the same soil layer, if ϕi = ϕi+1 and ci = ci+1 are true, Bi = Ai and Di = gi+1 (xi ) are true. Supposing v N = 1.0, s N = 0.0, D N = 0.0, Eq. (8.37) may be written in the following recursion form:

⎫ Ci = vi C N + si ⎬ i = N , N − 1, N − 2, . . . Bi Bi vi−1 = vi , si−1 = si + Di ⎭ Ai Ai

(8.41)

According to the boundary conditions (σz = q(x), τx z = 0) of the soil mass surface, it is easy to obtain:    1   q exp −2Fϕ N arctan h N 1 + Fϕ N h N    cs N

  1 − exp −2Fϕ N arctan h N − cs N h N − Fϕ N

CN =

(8.42)

 where h N = Fϕ N − 1 + Fϕ2N . Thus, Ci is determined. Equation (8.41) is substituted in Eq. (8.35) to obtain: σ+

   csi csi = exp 2Fϕi arctan h  vi C N + si + + gi (x) Fϕi Fϕi

(8.43)

The moment equation is substituted. Similar to the above discussion, the foundation width is divided into M intervals in [−bk , −bk−1 ] (Fig. 8.3); thereinto, b M = B, b0 = 0. If the limit load in [−bk , 0] is obtained, the limit load in [−bk+1 , −bk ] is as below:

8.5 General Calculation Mode by Generalized Limit Equilibrium …

−bk pz dx =

Ab −bk+1

 

283

  wγ i + qi xi∗ − x R

i

      + csi h i∗ − z R − h i xi∗ − x R + 1 − exp 2Fϕi arctan h i     csi − exp 2Fϕi arctan h i vi C N + si + gi+1 xi∗ Fϕi     

  h i − Fϕi h i∗ − z R + 1 + h i Fϕi xi∗ − x R xi 0 +

0 [(x − x R + z R tan δ ] pz dx −

−bk

(−z R ) tan δ

−bk+1

cs0 dx (8.44) Fϕ0

where Ab = 0.5(bk+1 + bk ) + x R − z R tan δ

(8.45)

xi∗ , h i∗ are the center point coordinates on the slip surface of the ith soil strip. The above calculation equation for limit load may be used for trial calculation of any slip surface (family). When different slip surfaces are adopted, the calculation mode will be different. In addition, the moment point (x R , z R ) may also be taken for the two optional parameters and they may not be those in the slip surface. Since the equation is derived by applying the extremum condition of the yield function, if the selected slip surface (family) is close to the real slip surface (family), the calculated limit load will also be close to the genuine solution.

8.6 Helicoid–Helicoid–Plane Calculation Mode 8.6.1 Slip Surface Family Supposing the slip surface as the helicoid–helicoid–plane (Fig. 8.4), (A): (−b ≤ x ≤ xa ): x − x R = R A exp(Fϕ θ ) cos θ , h − z R = R A exp(Fϕ θ ) sin θ , θ0 ≥ θ ≥ θa .   R A = exp Fϕ θ 0 (b + x R )2 + z 2R , z R = x R tan θa , z R = (b + x R ) tan θ0 (8.46) (B): (xa ≤ x ≤ xc ): x = R B exp(Fϕ θ ) cos θ , h = R B exp(Fϕ θ ) sin θ , θa ≥ θ ≥ θc .   R B = exp Fϕ θ a xa2 + h a2

(8.47)

284

8 Limit Load on Ground with Heterogeneous Soil

Fig. 8.4 Schematic diagram for helicoid–helicoid–plane calculation mode

(C): (xc ≤ x ≤ xb ): h = h c + (x − xc )h c , h c = Fϕ − θc =



1 + Fϕ2

  π − ϕc + arctan h c , ϕc = arctan Fϕ 2

(8.48)

It shall be noticed that the slip surface is continuous, for example, Ri+1 = Ri exp[(Fϕi+1 − Fϕi )θ i ] is true. The minimum under the condition of passing (−bi+1 , 0) is the limit load on the interval [−bk+1 , −bk ]. It is not difficult to prepare the calculation procedure for calculation according to the above-mentioned calculation equation. Similarly, under the condition of homogeneous soil, F sc , F sϕ are expected to be determined properly to make the calculated total limit load, eccentricity of the limit load, and slip surface depth be consistent with those by the characteristic line method; thus, they are used for the general condition with regard to heterogeneous soil. After calculation, analysis, and comparison, if the total limit load is also required to be slightly less than the calculated value by the characteristic line method, it will be: Fsc = Fsϕ = 1.01 + 0.03(tan ϕ − tan δ)

(8.49)

The calculated total limit load, eccentricity of the limit load, and slip surface depth all can be consistent with those by the characteristic line method. The following are the calculated results of 20 slip surfaces.

8.6 Helicoid–Helicoid–Plane Calculation Mode

285

8.6.2 Calculated Result of the Ground with Homogeneous Soil (1) When q/(γ B) = 0.1, c/(γ B) = 0.2, the comparison of the total limit load, slip surface depth, eccentricity with those by the characteristic line method is detailed in Table 8.12a, b, and c, respectively. (2) When q/(γ B) = 0.2, c/(γ B) = 0.4, the comparison of the total limit load, slip surface depth, eccentricity with those by the characteristic line method is detailed in Table 8.13a, b, and c, respectively. Thereinto, if the condition that the total limit load P0 is less than 1.0 is not considered, comparison of the total limit load Pm with P0 is as below: When q/(γ B) 4.26%. When q/(γ B) 5.34%. When q/(γ B) 4.60%. When q/(γ B) 3.64%. When q/(γ B) 4.04%. When q/(γ B) 4.10%.

= 0.0 and c/(γ B) = 0.1 are true, the maximum relative error is = 0.1 and c/(γ B) = 0.0 are true, the maximum relative error is = 0.1 and c/(γ B) = 0.2 are true, the maximum relative error is = 0.2 and c/(γ B) = 0.4 are true, the maximum relative error is = 0.4 and c/(γ B) = 0.6 are true, the maximum relative error is = 0.6 and c/(γ B) = 0.8 are true, the maximum relative error is

The above-mentioned maximum relative errors are all relatively small, while the relatively large maximum relative errors are all less than 0.6%. It is remarkable that the slip surface depth and eccentricity of the limit load are all very consistent with those by the characteristic line method. The calculation mode is applicable to the general condition with regard to the heterogeneous soil and non-uniformly distributed edge load. Compared with the helicoid calculation mode, this calculation mode is more reasonable because the slip surface is closer to the real slip surface under the condition of the homogeneous soil (F sc and F sϕ are closer to 1.0 than those in the helicoid mode).

8.6.3 Distribution of Limit Load As for the ground with homogeneous soil, the distribution pk = pz /(γ B) of the limit load along the foundation bottom is in the backward eccentricity form with the small front toe and large rear toe, and the distribution form nearby the front toe is slightly steeper than that nearby the rear toe and is entirely consistent with that by the characteristic line method. When the topsoil strength is small and the subsoil

35.65

75.64

35

40

1.669

0.663

0.697

0.801

0.893

1.059

10.0

15.0

20.0

25.0

30.0

1.055

0.909

0.801

0.718

0.662

0.680

0.859

0.752

0.640

0.527

0.405

Z m /B

0.640

Z m /B

54.40

26.37

13.97

7.917

4.713

2.891

1.749

0.1

Y m /B

76.61

36.07

18.69

10.44

6.196

3.859

2.501

0.0

5.0

ϕ(◦ )

tan δ

18.50

30

(b)

6.140

10.34

25

3.824

15

20

1.663

2.478

Pm

10

0.1

Pm

P0

0.0

tan δ

5

ϕ(◦ )

0.891

0.756

0.644

0.542

0.420

Y m /B

54.81

26.57

14.08

7.990

4.772

2.942

1.796

P0

0.689

0.577

0.445

0.296

Z m /B

0.2

37.71

18.70

10.04

5.696

3.325

1.880

Pm

0.2

0.725

0.593

0.469

0.318

Y m /B

38.04

18.90

10.18

5.802

3.408

1.945

P0

0.543

0.421

0.235

Z m /B

0.3

25.32

12.76

6.866

3.818

2.026

Pm

0.3

0.560

0.423

0.262

Y m /B

25.74

13.02

7.041

3.937

2.117

P0

0.384

0.217

Z m /B

0.4

16.58

8.396

4.427

2.242

Pm

0.4

(continued)

0.396

0.232

Y m /B

17.05

8.674

4.604

2.350

P0

Table 8.12 a Total limit load Pm (P0 is the calculated value of the characteristic line method). b Slip surface depth (Y m is the calculated value of the characteristic line method). c Eccentricity of limit load (e0 is the calculated value of the characteristic line method)

286 8 Limit Load on Ground with Heterogeneous Soil

e0 /B

−0.020

−0.036

−0.050

−0.064

−0.076

−0.088

−0.100

−0.111

−0.020

−0.036

−0.050

−0.064

−0.076

−0.089

−0.100

−0.111

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

0.0

1.502

1.244

e/B

ϕ(◦ )

tan δ

1.516

(c)

1.231

−0.107

−0.095

−0.084

−0.071

−0.058

−0.044

−0.028

e/B

0.1

1.304

1.070

Z m /B

40.0

0.1

Z m /B

Y m /B

0.0

tan δ

35.0

ϕ(◦ )

(b)

Table 8.12 (continued)

−0.104

−0.095

−0.083

−0.070

−0.057

−0.044

−0.029

e0 /B

1.287

1.061

Y m /B

−0.102

−0.090

−0.077

−0.064

−0.051

−0.039

e/B

0.2

1.095

0.856

Z m /B

0.2

−0.100

−0.086

−0.076

−0.064

−0.050

−0.036

e0 /B

1.078

0.880

Y m /B

−0.096

−0.083

−0.070

−0.057

−0.043

e/B

0.3

0.855

0.671

Z m /B

0.3

−0.094

−0.081

−0.066

−0.055

−0.040

e0 /B

0.883

0.707

Y m /B

−0.088

−0.075

−0.061

−0.048

e/B

0.4

0.687

0.518

Z m /B

0.4

−0.085

−0.073

−0.060

−0.045

e0 /B

0.705

0.543

Y m /B

8.6 Helicoid–Helicoid–Plane Calculation Mode 287

6.499

9.939

15.95

27.14

49.77

100.2

15.0

20.0

25.0

30.0

35.0

40.0

3.137

0.710

0.712

0.774

0.861

1.006

15

20

25

1.005

0.889

0.802

0.737

0.700

0.821

0.686

0.596

0.429

Z m /B

10

0.1

73.65

37.58

20.92

12.45

7.784

5.018

3.206

Z m /B

Y m /B

102.7

50.68

27.55

16.14

10.05

6.565

4.465

0.0

5

ϕ(◦ )

tan δ

4.421

(b)

3.125

Pm

10.0

0.1

Pm

P0

0.0

tan δ

5.0

ϕ(◦ )

(a)

0.835

0.716

0.603

0.464

Y m /B

75.04

38.16

21.22

12.63

7.907

5.112

3.277

P0

0.632

0.494

0.367

Z m /B

0.2

52.24

27.31

15.42

9.197

5.655

3.372

Pm

0.2

0.656

0.522

0.353

Y m /B

53.37

27.84

15.74

9.408

5.796

3.474

P0

0.464

0.276

Z m /B

0.3

36.07

19.18

10.88

6.387

3.593

Pm

0.3

0.469

0.291

Y m /B

37.11

19.72

11.20

6.575

3.719

P0

0.242

Z m /B

0.4

24.42

13.10

7.305

3.920

Pm

0.4

(continued)

0.258

Y m /B

25.32

13.55

7.563

4.068

P0

Table 8.13 a Total limit load Pm (P0 is the calculated value of the characteristic line method). b Slip surface depth (Y m is the calculated value of the characteristic line method). c Eccentricity of limit load (e0 is the calculated value of the characteristic line method)

288 8 Limit Load on Ground with Heterogeneous Soil

1.635

40

e0 /B

−0.012

−0.022

−0.033

−0.044

−0.055

−0.066

−0.077

−0.089

e/B

−0.012

−0.023

−0.034

−0.045

−0.056

−0.068

−0.080

−0.092

10

15

20

25

30

35

40

0.0

1.617

1.351

1.154

5

ϕ(◦ )

tan δ

1.344

(c)

1.147

−0.087

−0.075

−0.063

−0.051

−0.040

−0.029

−0.018

e/B

0.1

1.396

1.167

0.947

Z m /B

35

0.1

Z m /B

Y m /B

0.0

tan δ

30

ϕ(◦ )

(b)

Table 8.13 (continued)

−0.084

−0.072

−0.060

−0.049

−0.038

−0.028

−0.017

e0 /B

1.388

1.155

0.977

Y m /B

−0.081

−0.069

−0.057

−0.045

−0.035

−0.025

e/B

0.2

1.154

0.956

0.773

Z m /B

0.2

−0.078

−0.066

−0.054

−0.043

−0.032

−0.022

e0 /B

1.168

0.961

0.797

Y m /B

−0.074

−0.061

−0.049

−0.039

−0.029

e/B

0.3

0.921

0.758

0.579

Z m /B

0.3

−0.071

−0.059

−0.047

−0.036

−0.025

e0 /B

0.960

0.774

0.617

Y m /B

−0.066

−0.054

−0.042

−0.033

e/B

0.4

0.765

0.587

0.418

Z m /B

0.4

−0.064

−0.051

−0.040

−0.028

e0 /B

0.769

0.597

0.438

Y m /B

8.6 Helicoid–Helicoid–Plane Calculation Mode 289

290

8 Limit Load on Ground with Heterogeneous Soil

strength large, the distribution rule of the limit load is consistent with that in the helicoid calculation mode. Condition of large topsoil strength and small subsoil strength: As it is indicated previously, when the topsoil strength is far larger than the subsoil strength, the quite large limit load nearby the front toe is of no meaning for the engineering; thus, the calculated value of the subsequent limit load is unreasonable. Because the slip surface is restricted more (compared with the common helicoid) in this calculation mode, the area with quite large limit load nearby the front toe enlarges; thus, the calculated value of the subsequent limit load is unreasonable. For distribution of the limit load on ground with large topsoil strength and small subsoil strength [topsoil, 0.5B thick, ϕ1 = 25◦ , c1 /(γ B) = 0.4; subsoil, γ2 = γ , ϕ2 = 10◦ , c2 /(γ B) = 0.2, q = 0.1/(γ B)], the calculated result is detailed in Table 8.14. When the topsoil strength is far larger than the subsoil strength, the hunting zone of x R , z R shall be increased during the calculation so as to make the slip surface longer in the subsoil. When the slip surface only passes the topsoil (Z m ≤ 0.5B), the limit load is very large; this is useful to the partial failure nearby the front toe. As for the overall failure, once the slip surface passes the subsoil, the limit load reduces rapidly, if the corresponding load scope is relatively large while the slip surface only passes the topsoil, the subsequent limit load (e.g., when tan δ = 0, 0.325 ≤ bk /B ≤ 0.425) is obviously unreasonable. The similar problem also exists in the previous helicoid calculation mode, but it is just not so obvious. This shows that distribution of the limit load with actual engineering meaning may be unavailable if the soil masses in the slip mass are all required to be in the limit state. As it has been stated before, the field failure mode is just an “ideal” Table 8.14 Distribution of the limit load on ground ( pk = pz /(γ B)) b/B

tan δ 0.0 pk

0.2 Z m /B

pk

0.4 Z m /B

pk

Z m /B

0.05

9.656

0.123

6.115

0.078

2.829

0.027

0.15

11.094

0.235

6.722

0.151

2.966

0.054

0.25

12.330

0.344

7.290

0.220

3.103

0.082

0.35

0.064

0.584

5.521

0.592

3.242

0.109

0.45

1.765

0.686

2.110

0.694

3.366

0.126

0.55

4.396

0.794

2.612

0.704

3.485

0.151

0.65

4.229

0.872

3.006

0.693

3.672

0.160

0.75

4.435

0.963

3.106

0.703

1.858

0.681

0.85

4.764

1.067

2.808

0.760

1.349

0.683

0.95

4.508

1.117

2.956

0.764

1.511

0.687

Total limit load

5.724

4.224

2.738

Eccentricity

0.099

0.100

0.052

8.6 Helicoid–Helicoid–Plane Calculation Mode

291

Table 8.15 Distribution of limit load on ground ( pk = pz /(γ B)) b/B

tan δ 0.0

0.2

0.4

pk

Z m /B

pk

Z m /B

0.05

5.492

0.123

5.519

0.078

pk 2.079

0.027

Z m /B

0.15

6.310

0.235

6.066

0.151

2.181

0.054

0.25

7.013

0.344

6.579

0.220

2.281

0.082

0.35

6.110

0.596

5.905

0.592

2.383

0.109

0.45

5.465

0.686

3.047

0.694

2.475

0.126

0.55

4.413

0.785

2.618

0.704

2.561

0.151

0.65

4.313

0.872

3.036

0.693

2.699

0.160

0.75

4.494

0.963

3.114

0.703

2.857

0.681

0.85

4.613

1.026

2.814

0.760

2.841

0.683

0.95

4.673

1.117

2.981

0.764

3.305

0.687

Total limit load

5.290

4.168

2.566

Eccentricity

0.036

0.086

−0.038

failure mode; actually, as for great strength difference in the soil layers, it will be more consistent with the actual situation considering that the soil layer with small strength is in the limit state and the soil layer with large strength is in the stable state. Therefore, a coefficient k p is suggested; if the limit load in [−bk , −bk−1 ] is k p times that in [−bk+1 , −bk ], it is slightly smaller than the limit load in [−bk , 0]; then the limit load in [−bk+1 , −bk ] is recalculated until the limit load in [−bk , −bk−1 ] is not the k p times larger than that in [−bk+1 , −bk ]. And the reduced value of the limit load in [−bk , 0] is used as the calculated result. Distribution of the limit load on ground under the condition of the topsoil with large strength and the subsoil with small strength (topsoil, 0.5B thick, ϕ1 = 25◦ , c1 /(γ B) = 0.4; subsoil, γ2 = γ , ϕ2 = 10◦ , c2 /(γ B) = 0.2, q = 0.1/(γ B), k p = 1.5 is detailed in Table 8.15. Obviously, when the limit load in [−bk , 0] reduces properly, the limit load in [−bk+1 , −bk ] tends to be reasonable. Only from the calculation of the limit load, it seems not necessary to discuss which reasonable k p shall be taken. Because the calculation of the limit load is aimed at calculating the ground bearing capacity; moreover, the design load in the actual engineering is known, so it will be of more actual significance to correct the partially large limit load without any actual meaning according to the design load in the engineering. Such a treatment method is only proposed here and is verified to be feasible.

292

8 Limit Load on Ground with Heterogeneous Soil

8.6.4 Calculated Result of Two Layers of Soil Grounds Case 1: topsoil, 0.5B thick, ϕ1 = 25◦ , c1 /(γ B) = 0.2; subsoil, γ2 = γ , c2 /(γ B) = 0.2, different values are taken as ϕ2 for calculation; edge load, q = 0.1/(γ B). The calculated results of the total limit load, slip surface depth, and eccentricity (it is admissible load k p = 1.5 in [−bk , 0]) are detailed in Table 8.16a and b. Case 2: Topsoil strength is ϕ1 = 30◦ , c1 /(γ B) = 0.1, and the subsoil and edge load are the same as those in Case 1. The calculated results of the total limit load, slip surface depth, and eccentricity (it is admissible load k p = 1.5 in [−bk , 0]) are detailed in Table 8.17a and b. Except when the topsoil strength is far higher than the subsoil strength, the total limit load is slightly lower than that in the helicoid calculation mode, the other calculated results are basically consistent with those in the logarithmic helicoid calculation mode. When the strength between the soil layers does not differ greatly, distribution of the calculated limit load is similar to that by the characteristic line method; namely, it is distributed in the backward eccentricity form with small limit load nearby the front toe and large limit load nearby rear toe. As for the condition that the topsoil strength is far greater than the subsoil strength, then the distribution of the calculated limit load may be the forward eccentricity distribution with the large value nearby the front toe and small value nearby the rear toe. It is similar to that in the common helicoid calculation mode. Similarly, the distribution of the calculated limit load is slightly instable; if the calculation accuracy is improved (the soil strip is divided more densely, the grid for searching parameters x R , zR of the slip surface is denser; because the classification of the soil strip is finer, the calculated total limit load will slightly increase), the instable distribution of the limit load will be improved. Compared with the helicoid calculation mode, the regularity of the calculated result in the helicoid–helicoid–plane calculation mode is basically consistent. But F sc , F sϕ are obviously less than that in the helicoid calculation mode. If the problem that when the topsoil strength is far greater than the subsoil strength, the subsequent calculated value of the limit load is not so reasonable because of the large calculated value nearby, the front toe can be solved in further step, and it shall be a kind of good calculation mode. In addition, when the slip surface is the plane–helicoid–helicoid–plane, it is ground with homogeneous soil; if: Fsc = Fsϕ = 1.009 + 0.019(λ − tan δ), comparison of the total limit load Pm with P0 by the characteristic line method is as below: When q/(γ B) = 0.0 and c/(γ B) = 0.1 are true, the maximum relative error is 2.76%. When q/(γ B) = 0.1 and c/(γ B) = 0.0 are true, the maximum relative error is 3.10%. When q/(γ B) = 0.1 and c/(γ B) = 0.2 are true, the maximum relative error is 2.81%.

4.290

5.746

7.820

10.343

12.699

13.125

13.125

15.0

20.0

25.0

30.0

35.0

40.0

tan δ

0.018 −0.026 −0.071 −0.098 −0.098 −0.098

0.001

−0.076

−0.124

−0.135

−0.135

25.0

30.0

35.0

40.0

0.036

0.056

0.1

0.500

0.500

0.500

0.752

0.783

0.771

−0.035

8.664

8.664

8.664

7.917

6.205

4.801

0.870

0.947

20.0

0.027

2.930

3.732

Z m /B

15.0

0.068

0.499

0.499

0.775

0.893

0.958

0.954

10.0

0.0

0.997

1.026

5.0

ϕ(◦ )

(b)

3.312

Pm

10.0

0.1

Pm

Z m /B

0.0

tan δ

5.0

ϕ(◦ )

(a)

5.767

5.767

5.767

5.696

4.772

3.867

3.160

2.642

Pm

0.2

−0.069

−0.069

−0.069

−0.064

−0.021

0.023

0.047

0.077

0.2

0.500

0.500

0.500

0.577

0.685

0.703

0.727

0.782

Z m /B

−0.057

−0.057

−0.057

−0.057

−0.048

−0.048

−0.048

−0.048

−0.048

−0.048 −0.033

−0.078

−0.048

0.4

2.242

2.242

2.242

2.242

2.242

2.242

2.051

1.730

Pm

0.4

−0.054

0.421

0.421

0.421

0.421

0.587

0.657

0.689

0.712

Z m /B

0.029

0.069

0.3

3.818

3.818

3.818

3.818

3.578

2.843

2.591

2.213

Pm

0.3

Table 8.16 a Total limit load and slip surface depth (Case 1). b Eccentricity of limit load e/B (Case 1)

0.217

0.217

0.217

0.217

0.217

0.217

0.649

0.706

Z m /B

8.6 Helicoid–Helicoid–Plane Calculation Mode 293

4.273

5.863

8.198

11.721

15.925

20.461

22.471

15.0

20.0

25.0

30.0

35.0

40.0

tan δ

0.019 −0.027 −0.063 −0.108 −0.144 −0.144

0.012

−0.068

−0.112

−0.161

−0.186

25.0

30.0

35.0

40.0

0.041

0.067

0.1

−0.030

13.369

13.369

11.732

9.026

6.516

4.872

20.0

0.023

2.945

3.734

15.0

0.048

0.500

0.833

0.959

1.012

1.002

1.000

10.0

0.0

0.980

1.004

5.0

ϕ(◦ )

(b)

3.229

Pm

10.0

0.1

Pm

Z m /B

0.0

tan δ

5.0

ϕ(◦ )

(a)

0.499

0.499

0.818

0.846

0.883

0.844

0.860

0.953

Z m /B

−0.107

−0.107

−0.104

−0.059

−0.018

0.024

0.000

0.041

0.2

8.234

8.234

8.169

6.705

5.128

4.019

2.990

2.488

Pm

0.2

0.500

0.500

0.629

0.724

0.749

0.733

0.750

0.815

Z m /B

−0.084

−0.084

−0.084

−0.059

−0.015

0.012

0.005

0.033

0.3

5.260

5.260

5.260

4.837

3.990

3.233

2.563

2.108

Pm

0.3

Table 8.17 a Total limit load and slip surface depth (Case 2). b Eccentricity of the limit load e/B (Case 2)

0.492

0.492

0.492

0.617

0.648

0.682

0.695

0.723

Z m /B

−0.075

−0.075

−0.075

−0.075

−0.088

−0.006

−0.018

0.044

0.4

3.311

3.311

3.311

3.311

2.945

2.662

2.151

1.896

Pm

0.4

0.339

0.339

0.339

0.339

0.596

0.647

0.696

0.714

Z m /B

294 8 Limit Load on Ground with Heterogeneous Soil

8.6 Helicoid–Helicoid–Plane Calculation Mode

295

When q/(γ B) = 0.2 and c/(γ B) = 0.4 are true, the maximum relative error is 2.80%. When q/(γ B) = 0.4 and c/(γ B) = 0.6 are true, the maximum relative error is 2.87%. When q/(γ B) = 0.6 and c/(γ B) = 0.8 are true, the maximum relative error is 3.29%. The above-mentioned maximum relative errors are all relatively small, while the relatively large maximum relative errors are all less than 0.31%. Some more complex slip surface may be selected so that the calculated total limit load may be consistent with that by the characteristic line method when Fsc = Fsϕ = 1.0 is true with regard to homogeneous soil. However, as for the condition that the strength between the soil layers differs greatly, the numerical stability of the limit load distributed along the foundation bottom is often slightly poor.

8.7 Calculation of Limit Load According to Load Inclination 8.7.1 Treatment for Boundary Condition of Foundation Bottom During the previous discussion on the calculation of the limit load, for the convenience of the comparison with the characteristic line method, the treatment for the boundary condition of foundation bottom is the same as the characteristic line method, that is to say that the boundary condition of foundation bottom is taken as: τx z0 = tan δ(σz0 +cs0 /Fϕ0 ), so tan δ is determined according to the design load based on the boundary condition of foundation bottom in application, namely:     cs0 cs0 = Px / Pz + B tan δ = Ph / Pv + B Fϕ0 Fϕ0

(8.50)

where Ph , Pv are total horizontal and vertical design loads, respectively; Px , Pz are total horizontal and vertical limit loads, respectively. When c = 0 is true for the first layer of soil under the foundation bottom (in case of sand cushion), tan δ = Ph /Pv is no doubt true for Eq. (8.50). If c = 0 in the first layer of soil under the foundation bottom, there are two problems in the calculation mode corresponding to the boundary condition in application: Firstly, tan δ is not the load inclination with real meaning; the ratios of horizontal and vertical limit loads to design load are different: Px /Ph = [Pz +(Bcs0 /Fϕ0 )]/[Pv + (Bcs0 /Fϕ0 )] = Pz /Pv . Generally, the ratio of vertical ground bearing capacity to

296

8 Limit Load on Ground with Heterogeneous Soil

design load is defined as the load-carrying safety factor; however, the ratio of horizontal ground bearing capacity to design load is different from the safety factor; this may bring certain trouble for the safety factor to be used to measure the ground bearing capacity. Secondly, when ϕ = 0 in the first layer of soil under the foundation bottom, the calculation equation cannot apply. Actually, if the boundary condition of foundation bottom is determined as load inclination: tan δ  = Ph /Pv = Px /Pz

(8.51)

The above calculation mode is also applicable, and after the boundary condition of the foundation bottom is changed into Eq. (8.51), Eq. (8.44) shall be changed into: −bk pz dx =

Ab −bk+1

    wγ i + qi xi∗ − x R + csi i



      h i∗ − z R − h i xi∗ − x R + 1 − exp 2Fϕi arctan h i     csi − exp 2Fϕi arctan h i ki C N + si + gi xi∗ Fϕi     

  h i − Fϕi h i∗ − z R + 1 + h i Fϕi xi∗ − x R xi 0 +



 x − x R + z R tan δ  pz dx

(8.52)

−bk

where Ab = 0.5(bk+1 + bk ) + x R − z R tan δ  , and the meanings of other signs are the same as the previous ones. Similarly, all the potential slip surface families may be selected for calculation in this equation. In case of the helicoid slip surface, the equation is simplified as: −bk pz dx =

Ab −bk+1



    wγ i + qi xi∗ − x R + csi i

h i∗

  − z R − h i xi∗ − x R xi

0 +



 x − x R − (h s − z R ) tan δ  pz dx

(8.53)

−bk

Obviously, the two above-mentioned problems do not exist in the treatment method of the boundary condition of foundation bottom.

8.7 Calculation of Limit Load According to Load Inclination

297

Because values of tan δ corresponding to the two types of boundary conditions are different, the effect of the two types of boundary conditions on the limit load will be discussed below.

8.7.2 Comparison of Calculation Results Under Two Types of Boundary Conditions (1) As for the limit load, if tan δ  is given, the total limit load Pz and Px = Pz tan δ  may be obtained; it will be tan δ ≈ Px /[Pz + (Bc0 / tan ϕ0 )]; thus, the limit load corresponding to tan δ may be calculated. Because Px ≤ Pz tan ϕ0 + Bc0 shall be in accordance with along the foundation bottom, with the same calculation according to tan δ, tan δ  will be of meaning under the condition of tan δ < tan ϕ0 . The calculated results under two types of boundary conditions in the helicoid calculation mode are as follows: The comparison of total limit load on the ground with homogeneous soil (q/(γ B) = 0.1, c/(γ B) = 0.2) and the comparison of eccentricity and maximum depth of the slip surface are detailed in Table 8.18a and b, respectively. Table 8.18 a Comparison of total limit load on the ground with homogeneous soil. b Comparison of eccentricity of the limit load on the ground with homogeneous soil and maximum depth of the slip surface (a) ϕ(◦ )

Pm

tan δ 

Pm

tan δ

Pm

tan δ 

Pm

tan δ

15.0

2.394

0.200

2.398

0.152

20.0

3.652

0.200

3.650

0.174

25.0

5.857

0.200

5.847

0.186

30.0

9.965

0.200

9.950

0.193

4.921

0.400

4.901

0.374

35.0

18.314

0.200

18.290

0.197

8.565

0.400

8.532

0.387

40.0

36.935

0.200

36.903

0.199

16.189

0.400

16.185

0.394

3.018

0.400

3.014

0.350

(b) ϕ(◦ )

tan δ  = 0.2

tan δ  = 0.4

e/B

Z m /B

e/B

Z m /B

15.0

−0.033

0.434

−0.036

0.434

20.0

−0.045

0.483

−0.048

0.483

25.0

−0.058

0.569

−0.059

30.0

−0.070

0.698

−0.071

35.0

−0.082

0.818

40.0

−0.094

1.033

tan δ

tan δ

e/B

Z m /B

e/B

Z m /B

0.569

−0.039

0.299

−0.044

0.293

0.698

−0.052

0.398

−0.056

0.399

−0.083

0.818

−0.065

0.529

−0.068

0.529

−0.095

1.033

−0.079

0.653

−0.081

0.661

298

8 Limit Load on Ground with Heterogeneous Soil

Table 8.19 a Comparison of total limit load on the ground with heterogeneous soil. b Comparison of eccentricity of the limit load on the ground with heterogeneous soil and maximum depth of the slip surface (a) ϕ(◦ )

Pm

tan δ 

Pm

tan δ

Pm

tan δ 

Pm

tan δ

5.0

2.653

0.200

2.673

0.172

2.041

0.400

2.081

0.331

10.0

3.365

0.200

3.385

0.177

2.443

0.400

2.476

0.340

15.0

4.200

0.200

4.215

0.181

2.800

0.400

2.818

0.347

20.0

5.074

0.200

5.080

0.184

3.018

0.400

3.014

0.350

25.0

5.857

0.200

5.847

0.186

3.018

0.400

3.014

0.350

30.0

5.936

0.200

5.926

0.187

3.018

0.400

3.014

0.350

35.0

5.936

0.200

5.926

0.187

3.018

0.400

3.014

0.350

(b) ϕ(◦ )

tan δ  = 0.2 e/B

tan δ  = 0.4

tan δ Z m /B

e/B

Z m /B

5.0

0.106

1.037

0.110

0.975

10.0

0.059

0.883

0.061

15.0

0.018

0.778

0.018

20.0

−0.021

0.688

25.0

−0.058

30.0

−0.063

35.0

−0.063

e/B

tan δ Z m /B

e/B 0.082

Z m /B

0.074

0.853

0.846

0.884

0.030

0.720

0.033

0.774

0.779

−0.010

0.655

−0.012

0.631

−0.021

0.688

−0.039

0.299

−0.044

0.293

0.569

−0.059

0.569

−0.039

0.299

−0.044

0.293

0.499

−0.064

0.499

−0.039

0.299

−0.044

0.293

0.499

−0.064

0.499

−0.039

0.299

−0.044

0.293

Comparison of total limit load on the ground with heterogeneous soil, topsoil, ϕ = 25◦ , c/(γ B) = 0.2, the subsoil: c/(γ B) = 0.2, different values are taken for ϕ; q/(γ B) = 0.1 and comparison of eccentricity and maximum depth of the slip surface are detailed in Table 8.19 a and b, respectively. The total limit load, eccentricity of the limit load, and maximum depth of the slip surface under two types of boundary conditions are basically consistent. Because of the different treatment on the boundary condition, it is normal that the calculated results are slightly different. (2) In practical problems, tan δ and tan δ  are determined according to the design load; as for the total limit load calculated according to tan δ  corresponding to Eq. (8.51), Pz /Pv = K is supposed, and tan δ ≈ (Px /K )/[(Pz /K ) + (Bc0 / tan ϕ0 )] shall be true.

8.7 Calculation of Limit Load According to Load Inclination

299

Table 8.20 a Comparison of total limit load on the ground with homogeneous soil. b Comparison of eccentricity of the limit load on the ground with homogeneous soil and maximum depth of the slip surface (a) ϕ(◦ )

Pm

tan δ 

Pm

tan δ

Pm

15.0

2.394

0.200

2.664

0.123

20.0

3.652

0.200

3.908

0.154

25.0

5.857

0.200

6.087

0.174

30.0

9.965

0.200

10.170

0.187

35.0

18.314

0.200

18.497

0.194

40.0

36.935

0.200

37.090

0.197

16.189

3.018

tan δ 

Pm

tan δ

0.400

3.612

0.311

4.921

0.400

5.421

0.351

8.565

0.400

8.980

0.375

0.400

16.582

0.389

(b) ϕ(◦ )

tan δ  = 0.2

tan δ  = 0.4

tan δ

e/B

Z m /B

e/B

Z m /B

15.0

−0.033

0.434

−0.038

0.458

20.0

−0.045

0.483

−0.049

0.498

25.0

−0.058

0.569

−0.060

30.0

−0.070

0.698

−0.071

35.0

−0.082

0.818

40.0

−0.094

1.033

tan δ

e/B

Z m /B

e/B

Z m /B

0.609

−0.039

0.299

−0.048

0.373

0.699

−0.052

0.398

−0.058

0.426

−0.083

0.819

−0.065

0.529

−0.068

0.530

−0.095

1.033

−0.079

0.653

−0.081

0.697

If K = 2 is taken, the calculated results under the two types of boundary conditions in the helicoid calculation mode are as follows: The comparison of total limit load on the ground with homogeneous soil (q/(γ B) = 0.1, c/(γ B) = 0.2) and comparison of eccentricity and maximum depth of the slip surface are detailed in Table 8.20a and b, respectively. Comparison of total limit load on the ground with heterogeneous soil, topsoil, ϕ = 25◦ , c/(γ B) = 0.2, the subsoil: c/(γ B) = 0.2, different values are taken for ϕ; q/(γ B) = 0.1), and comparison of eccentricity and maximum depth of the slip surface are detailed in Table 8.21a and b, respectively. If tan δ and tan δ  differ relatively greatly, the difference of the total limit load under two types of boundary conditions is relatively obvious. Because while calculated according to tan δ  , Px /Ph = Pz /Pv = K ; Px /Ph = [Pz + (Bc0 / tan ϕ0 )]/[Pv + (Bc0 / tan ϕ0 )] ≤ K according to tan δ. That is, while calculated according to tan δ, the vertical limit load is relatively large caused by the relatively small horizontal limit load.

300

8 Limit Load on Ground with Heterogeneous Soil

Table 8.21 a Comparison of total limit load on the ground with heterogeneous soil. b Comparison of eccentricity of the limit load on the ground with heterogeneous soil and maximum depth of the slip surface ϕ(◦ )

Pm

tan δ 

Pm

tan δ

Pm

tan δ 

Pm

tan δ

5.0

2.653

0.200

2.758

0.151

2.041

0.400

2.263

0.282

10.0

3.365

0.200

3.494

0.159

2.443

0.400

2.721

0.296

15.0

4.200

0.200

4.354

0.166

2.800

0.400

3.144

0.306

20.0

5.074

0.200

5.265

0.171

3.018

0.400

3.506

0.311

25.0

5.857

0.200

6.087

0.174

3.018

0.400

3.612

0.311

30.0

5.936

0.200

6.199

0.175

3.018

0.400

3.612

0.311

35.0

5.936

0.200

6.199

0.175

3.018

0.400

3.612

0.311

(b) ϕ(◦ )

tan δ  = 0.2 e/B

tan δ  = 0.4

tan δ Z m /B

e/B

Z m /B

e/B

tan δ Z m /B

e/B

Z m /B

5.0

0.106

1.037

0.111

0.989

0.074

0.853

0.094

0.929

10.0

0.059

0.883

0.062

0.900

0.030

0.720

0.046

0.775

15.0

0.018

0.778

0.018

0.817

−0.010

0.655

0.004

0.670

20.0

−0.021

0.688

−0.021

0.723

−0.039

0.299

−0.036

0.594

25.0

−0.058

0.569

−0.060

0.609

−0.039

0.299

−0.048

0.373

30.0

−0.063

0.499

−0.067

0.499

−0.039

0.299

−0.048

0.373

35.0

−0.063

0.499

−0.067

0.499

−0.039

0.299

−0.048

0.373

In addition, if the relationship between tan δ and tan δ  is not considered, when tan δ = tan δ  , it shows that the ratio of horizontal design load to the vertical design load that are corresponding to tan δ is larger than tan δ  , and it is normal that the limit load calculated according to tan δ is smaller than that calculated according to tan δ  . (3) As for the helicoid–helicoid–plane calculation mode, Eq. (8.52) is the calculation equation under the boundary condition of foundation bottom which is load inclination. The calculated result under two types of boundary conditions is as follows: The comparison of total limit load on the ground with homogeneous soil (q/(γ B) = 0.1, c/(γ B) = 0.2) and comparison of eccentricity and maximum depth of the slip surface are detailed in Table 8.22 a and b, respectively. Similarly, the total limit load under two types of boundary conditions, the eccentricity of the limit load and maximum depth of the slip surface, is basically consistent. The rule of calculated results under other conditions is basically consistent with that of the helicoid calculation mode.

8.8 Calculation of Limit Load with Undrained Shear Strength Index

301

Table 8.22 a Comparison of total limit load on the ground with homogeneous soil. b Comparison of eccentricity of the limit load on the ground with homogeneous soil and maximum depth of the slip surface (a) ϕ(◦ )

Pm

tan δ 

Pm

tan δ

Pm

tan δ 

Pm

tan δ

15.0

2.389

0.200

2.371

0.152

20.0

3.681

0.200

3.661

0.174

25.0

5.958

0.200

5.937

0.186

3.033

0.400

2.987

0.350

30.0

10.220

0.200

10.182

0.193

4.994

0.400

4.948

0.374

35.0

18.726

0.200

18.691

0.197

8.767

0.400

8.710

0.387

40.0

37.440

0.200

37.390

0.199

16.658

0.400

16.584

0.394

(b) ϕ(◦ )

tan δ  = 0.2

tan δ  = 0.4

tan δ

e/B

Z m /B

e/B

Z m /B

15.0

−0.037

0.444

−0.041

0.441

20.0

−0.050

0.488

−0.053

0.489

25.0

−0.063

0.590

−0.065

30.0

−0.077

0.728

−0.078

35.0

−0.089

0.872

40.0

−0.101

1.060

tan δ

e/B

Z m /B

e/B

Z m /B

0.597

−0.045

0.340

−0.051

0.321

0.699

−0.058

0.399

−0.063

0.395

−0.090

0.855

−0.072

0.530

−0.075

0.548

−0.102

1.060

−0.086

0.692

−0.088

0.691

8.8 Calculation of Limit Load with Undrained Shear Strength Index If there is saturated soft clay in the ground soil, undrained shear strength index should be used to calculate the ground bearing capacity during the construction period [3].

8.8.1 Sand Cushion Under Foundation Bottom As for the condition that ϕ0 = 0, the previous calculation method (Eq. 8.52 or 8.53) is still applicable. If the helicoid calculation mode is adopted, the calculated results are as follows: Calculation Case 1: Sand cushion, 0.2B thick, bottom width B, top width 1.4B, ϕ = 30◦ ; ground soil, γ = γ sand , different values are taken for c1 /(γ B); edge load, q = 0.2/(γ B). The total limit load, slip surface depth, and eccentricity calculated according to heterogeneous soil are detailed in Table 8.23a and b, respectively. Calculation Case 2: sand cushion, 0.2B thick, bottom width B, top width 1.4B, ϕ = 30◦ ; two layers for ground soil: the upper layer thickness is 0.3B (sand cushion is excluded) γ 1 = γ sand , c1 /(γ B) = 0.6; the subsoil, γ 2 = γ sand , different values are

2.740

3.939

5.123

6.292

0.6

0.8

1.0

0.723

0.783

0.783

0.783

0.783

5.537

4.515

3.487

2.441

1.362

0.649

0.649

0.649

0.727

0.727

Z m /B

0.0643

0.0540

0.0469

0.0409

0.0358

0.4

0.6

0.8

1.0

0.0

tan δ

0.2

c1 /(γ B)

0.0330

0.0376

0.0439

0.0519

0.0654

0.1

b Eccentricity calculated according to heterogeneous soil e/B (Case 1)

1.523

Pm

0.4

0.1

Pm

Z m /B

0.0

tan δ

0.2

c1 /(γ B)

4.770

3.895

3.011

2.120

1.188

Pm

0.2

0.0296

0.0351

0.0420

0.0529

0.0698

0.2

0.507

0.507

0.507

0.507

0.495

Z m /B

0.0304

0.0365

0.0449

0.0580

0.0807

0.3

3.981

3.249

2.513

1.768

0.996

Pm

0.3

0.357

0.357

0.357

0.357

0.333

Z m /B

0.0356

0.0423

0.0523

0.0683

0.0977

0.4

3.198

2.607

2.014

1.415

0.799

Pm

0.4

0.248

0.246

0.246

0.246

0.243

Z m /B

Table 8.23 a Total limit load and slip surface depth calculated according to heterogeneous soil (Case 1). b Eccentricity calculated according to heterogeneous soil e/B (Case 1)

302 8 Limit Load on Ground with Heterogeneous Soil

8.8 Calculation of Limit Load with Undrained Shear Strength Index

303

taken for c2 /(γ B); edge load, q = 0.2/(γ B). The total limit load, slip surface depth, and eccentricity calculated according to heterogeneous soil are detailed in Table 8.24 a and b, respectively. Similarly, when the topsoil strength is large and the subsoil strength is small, the slip surface depth is obviously large; when the topsoil strength is small and the subsoil strength is large, the slip surface depth is obviously small. It shows the characteristics that the dangerous slip surface passes the soil layer with small strength as much as possible. In order to investigate the action of the sand cushion, small thickness (0.05B) is taken, and it is only confined to the foundation bottom with sand cushion; thus, it is taken to compare with the result of the theoretical solution, which is the one under the condition of homogeneous soil (ϕ = 0) discussed in Chap. 4. If Ph = Pz tan δ is taken, the following equation is derived according to Chap. 4: Pc =

Pz c q = Nc + γ B2 γB γB

(8.54)

h

where Nc = 2(arctan h a + 1+ha 2 ) + (π + 2)/2 a  ph /c h a = 1− . If p /c ≥ 1.0, it is calculated according to h a = 0. h 1+ ph /c Sand cushion with small thickness (0.05B) and the total limit load of theoretical solution are detailed in Table 8.25. As for the calculated results in Tables 8.25 and 8.23a, its comprehensive analysis is that except for the condition of sand cushion with small thickness and tan δ = 0.4 [here, Pc tan δ ≥ c/(γ B)], when there is sand cushion, the limit load is larger than that without sand cushion; when the thickness of the sand cushion is small, the limit load in both conditions are basically consistent (see Table 8.26); when the thickness of the sand cushion increases, the limit load increases, and this shows the action of sand cushion. When tan δ = 0.4, it is easy to verify Pc tan δ  ≥ c/(γ B); it shows that the theoretical solution is obtained under the condition of ph /c ≥ 1.0, which is obviously wrong.

8.8.2 ϕ0 = 0 for Soil on Lower Foundation Bottom The boundary condition Ph /Pv = Px /Pz = tan δ  shall be used to discuss the condition of ϕ = 0 for ground soil; the calculated result in the helicoid calculation mode is detailed in Table 8.26. When tan δ  ≤ 0.2, the calculated total limit load is consistent with the theoretical solution. When tan δ  = 0.3, the calculated total limit load is slightly less than the theoretical solution; when tan δ  = 0.4, the calculated total limit load is obviously less than the theoretical solution; the following is the discussion on the problem. As for the theoretical solution, the stress field is derived according to Chap. 2:

3.190

4.096

4.841

5.404

0.6

0.8

1.0

0.0112 0.0021

0.0543

0.0200

−0.0117

0.6

0.8

1.0

0.0480

0.0896

0.1402

0.0926

0.1

0.298

0.586

0.586

0.762

0.890

0.1449

4.164

4.078

3.533

2.789

1.881

0.4

0.0

tan δ

0.586

0.586

0.715

0.902

0.922

Z m /B

0.2

c2 /(γ B)

(b)

2.097

Pm

0.4

0.1

Pm

Z m /B

0.0

tan δ

0.2

c2 /(γ B)

3.173

3.173

3.018

2.443

1.676

Pm

0.2

0.0287

0.0287

0.0439

0.0835

0.1377

0.2

0.300

0.300

0.516

0.644

0.853

Z m /B

0.0434

0.0434

0.0447

0.0803

0.1350

0.3

2.520

2.520

2.513

2.128

1.498

Pm

0.3

0.298

0.298

0.337

0.520

0.754

Z m /B

0.0536

0.0536

0.0536

0.0774

0.1314

0.4

2.009

2.009

2.009

1.834

1.337

Pm

0.4

0.242

0.242

0.242

0.435

0.709

Z m /B

Table 8.24 a Total limit load and slip surface depth calculated according to heterogeneous soil (Case 2). b Eccentricity calculated according to heterogeneous soil e/B (Case 2)

304 8 Limit Load on Ground with Heterogeneous Soil

3.412

4.477

5.542

0.6

0.8

1.0

5.342

4.313

3.285

2.257

4.974

4.018

3.062

2.106

Note If Pc tan δ ≥ c/(γ B), it indicates ph /c ≥ 1.0

2.347

0.4

1.148

Pm

Pc

1.228

Pm

1.281

0.1

0.0

tan δ

0.2

c/(γ B)

Table 8.25 Comparison of total limit load (q/(γ B) = 0.2)

Pc

4.730

3.817

2.906

1.994

1.082

4.305

3.475

2.645

1.816

0.984

Pm

0.2 Pc

4.007

3.230

2.455

1.676

0.895

3.469

2.787

2.105

1.422

0.738

Pm

0.3 Pc

3.239

2.603

1.964

1.323

0.714

2.637

2.117

1.597

1.077

0.557

Pm

0.4 Pc

2.771

2.257

1.742

1.228

0.714

8.8 Calculation of Limit Load with Undrained Shear Strength Index 305

1.198

2.201

3.204

4.207

5.210

0.6

0.8

1.0

5.342

4.313

3.285

2.257

1.228

4.675

3.774

2.872

1.970

1.068

Pm

0.4

0.1

Pm

Pc

0.0

tan δ 

0.2

c1 /(γ B)

4.730

3.817

2.906

1.994

1.082

Pc

Table 8.26 Total limit load calculated according to homogeneous soil

4.030

3.248

2.467

1.686

0.899

Pm

0.2

4.007

3.230

2.455

1.676

0.895

Pc

3.099

2.481

1.864

1.246

0.629

Pm

0.3

3.239

2.603

1.964

1.323

0.714

Pc

2.364

1.893

1.422

0.951

0.480

Pm

0.4

2.771

2.257

1.742

1.228

0.714

Pc

306 8 Limit Load on Ground with Heterogeneous Soil

8.8 Calculation of Limit Load with Undrained Shear Strength Index

  ⎫ c h ⎪ ⎪ + σx = γ z + 2c arctan h  − + 2) + q (π ⎪ ⎪ ⎪ 1 + h 2 2 ⎪ ⎪   ⎬  c h  + + 2) + q σz = γ z + 2c arctan h + (π ⎪ 1 + h 2 2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ 1−h ⎪ ⎭ c τx z = 1 + h 2

307

(8.55)

If the boundary condition tan δ  = Ph /Pv = Px /Pz = Bpx /(Bpz ) = τx z0 /σz0 is applied, the following shall be derived: tan δ  =

   c 1 − h a2 h a  + c/ 2c arctan h + + 2) + q (π a 1 + h a2 1 + h a2 2

(8.56)

where h a is the first derivative of the slip surface on the foundation bottom. It is easy to learn that when tan δ  is greater than a certain value, Eq. (8.56) has no solution, namely h a in accordance with Eq. (8.56) does not exist. Then, the range of tan δ  is restricted, and tan δ  ≤ 1/[(π + 2)/2 + q/c] shall be derived at least. It is the reason why the total limit load becomes relatively small when tan δ  is greater than a certain value; when tan δ  = 0.4, the corresponding theoretical solution Pc tan δ  ≥ c/(γ B) is derived. This shows that when tan δ  is relatively large, theoretical solution of the limit load which is in accordance with the boundary condition tan δ  = Ph /Pv = Px /Pz cannot be obtained. This is also the reason why the boundary condition of the horizontal design load is adopted for the discussion of the homogeneous soil in Chap. 4. Actually, as for tan δ  which is relatively large, the lateral resistance of the soil on lower foundation bottom carrying horizontal load shall be considered first. Namely, Hk ≤ Bc or ph ≤ c shall be derived; the stress field of Eq. (8.54) also shows that px ≤ c shall be derived. In the helicoid (which has already retrograded into the arc surface) calculation mode, as for tan δ  adopted in calculation, Pm tan δ  ≤ c/(γ B) are derived; namely, the horizontal load is in accordance with px ≤ c. However, just because of the restriction of Pm tan δ  ≤ c/(γ B), the calculated value of the corresponding limit load is smaller than the theoretical solution. Therefore, the result calculated by the helicoid calculation mode is correct. If there are two layers of the ground soil: topsoil, 0.3B thick, c1 /(γ B) = 0.6; the subsoil, γ2 = γ1 , different values are taken for c2 /(γ B); edge load, q = 0.2/(γ B). The total limit load, slip surface depth, and eccentricity calculated according to heterogeneous soil are detailed in Table 8.27a and b, respectively. Similarly, the calculated results show the characteristics that the dangerous slip surface passes through the soil layer with small strength as much as possible. As for the condition of ground soil ϕ = 0, helicoid–helicoid–plane calculation mode shall be used; the calculated total limit load is very close to that by the helicoid calculation mode.

3.204

3.599

3.599

0.6

0.8

1.0

0.0 −0.0130 −0.0130

0.0

−0.0346

−0.0346

0.8

1.0

0.0380

0.0362

0.6

0.0910

0.1

0.300

0.300

0.476

0.681

0.4

2.981

2.981

2.872

2.394

0.0902

0.0

tan δ 

0.300

0.300

0.636

0.841

Z m /B 1.028

0.2

c2 /(γ B)

(b)

2.606

0.4

1.728

Pm

Z m /B

1.104

Pm

1.828

0.1

0.0

tan δ 

0.2

c2 /(γ B)

2.471

2.471

2.467

2.152

1.602

Pm

0.2

0.300

0.300

0.333

0.556

−0.0007

−0.0007

0.0

0.0354

0.0893

0.2

Z m /B 0.787

1.864

1.864

1.864

1.832

1.439

Pm

0.3

0.3

0.0

0.0

0.0

0.0077

0.0723

Z m /B

0.076

0.076

0.076

0.424

0.747

0.0

0.0

0.0

0.0

0.0417

0.4

1.422

1.422

1.422

1.422

1.274

Pm

0.4

0.076

0.076

0.076

0.076

0.662

Z m /B

Table 8.27 a Total limit load and slip surface depth calculated according to heterogeneous soil. b Eccentricity calculated according to heterogeneous soil e/B

308 8 Limit Load on Ground with Heterogeneous Soil

8.8 Calculation of Limit Load with Undrained Shear Strength Index

309

The discussion above shows that two coefficients Fsϕ ≥ 1.0 and Fsc ≥ 1.0 are determined properly, and the reduction (csi = ci /Fsc and Fϕi = tan ϕi /Fsϕ ) of the strength index is used to calculate the limit load; simple and common slip surface may be used. Thereinto, the helicoid calculation mode is simple; it is applicable to various conditions of the ground soil, but Fsϕ and Fsc are slightly large. While the reduction of the strength index of the helicoid–helicoid–plane calculation mode is less than 4%, and the calculation accuracy is higher, it is just a little complicated. As is well known, the study on the limit load on ground has been a long history in the soil mechanics community. However, as for the general condition in the engineering, such as the calculation of the limit load on the ground with heterogeneous soil, non-uniformly distributed edge load and non-horizontal soil mass surface, no better calculation methods are available at present yet. The previous calculations under many conditions are aimed at obtaining a better understanding of the basic rule of the ground with heterogeneous soil limit load. Certainly, many problems need to be further discussed, and certain errors in the calculated results of values will also take place. In addition, the calculation of the limit load is discussed to calculate the ground bearing capacity and the above-mentioned problems will be further considered during discussion on the calculation of the ground bearing capacity.

References 1. Luan MT, Nian TK, Zhao SF (2003) Summary of research process of earth structure and side slope. In: 9th conference proceedings on soil mechanics and geotechnical engineering. Tsinghua University Press, Beijing, pp 37–55 2. Code for Design of Building Foundation (GB 5007-2002) (2002) China Architecture and Building Press, Beijing 3. Code for Foundation in Port Engineering (JTJ 250-98) (1998) China Communications Press, Beijing 4. Luan MT, Jin CQ, Lin G (1988) Ultimate bearing capacity of shallow footings on nonhomogeneous soil foundations. Chin J Geotech Eng 10(4):14–27 5. Huang CZ, Zhang J, Sun WH (2000) Calculation method of foundation bearing capacity for heterogeneous soil. China Harbour Eng (5):38–43

Chapter 9

Ground Bearing Capacity

9.1 Design Load and Ground Failure Mode Previous content discusses calculation of the ground limit load and has stated that the limit load is the maximum load for the ground to bear and distribution forms of the limit load in field failure mode are determined. Actually, major factors for determining the ground limit load are two: One is load boundary condition of the foundation bottom (e.g., tan δ) as well as surface shape of and load (hereinafter referred to as edge load) on the ground on both sides of the foundation; the other is relevant characteristics of the ground soil such as distribution of soil layer and the strength index. The load that ground of actual engineering bears is the actual load (design load), and its distribution form is determined. Generally, distribution form of the limit load and the design load are different. Ground bearing capacity shall be the load the ground may bear under the design load condition. Thus, discussion on calculation of the ground bearing capacity shall also include the possible failure mode of the ground under the design load condition and developing corresponding calculation mode on the basis of failure mode.

9.1.1 Design Load of Actual Engineering Action width and distribution characteristics of the design load are the main contributors to affect the ground limit load and ground bearing capacity, so the discussion on it is necessary. Usually, if the action point of vertical resultant is at the foundation center (ev = 0), it will be referred to as central load; if it is at the rear side (ev ≤ 0) of foundation center, the load will be referred to as eccentric backward; if it is at the front side (ev ≥ 0) of foundation center, the load will be referred to as eccentric forward. © Springer Nature Singapore Pte Ltd. and Zhejiang University Press, Hangzhou, China 2020 C. Huang, Limit Analysis Theory of the Soil Mass and Its Application, https://doi.org/10.1007/978-981-15-1572-9_9

311

312

9 Ground Bearing Capacity p1 p0

p2

a c

p2

p0

p1

Front toe

a

b

c

d

c'

(a) Backward eccentricity of design load;

c'

Front toe b d

(b) Forward eccentricity of design load

Fig. 9.1 Action width of design load

Rubble foundation bed or cushion is provided for most gravity-type structure engineering. If the top surface of rubble foundation bed is regarded as load action surface of calculated limit load, it will be no doubt that action width and distribution form of the design load are determined according to the common method at present. In the recent calculation of ground bearing capacity, bottom surface of the rubble foundation bed is mostly regarded as the load action surface and the action width and distribution (Fig. 9.1) of design load is determined according to the following methods. If action width of load “B = ab” for the top surface of foundation bed, vertical stress on the rear and front toe will be p1 , p2 , respectively. Then, the actual width of load action for the bottom surface of foundation bed is: B1 = cd = B + 2h d

(9.1)

The vertical stress on the rear toe (c) of bottom surface for the foundation bed is: pv1 =

B p1 + γ h d B1

(9.2)

The vertical stress on the front toe (d) of bottom surface for the foundation bed is: pv2 =

B p2 + γ h d B1

(9.3)

where h d , γ are thickness and weight of the rubble foundation bed, respectively. Such determination of load action surface width and load distribution may be applicable under the condition that difference between edge loads on both sides of bottom surface of the structure is small or far less than the design load within the bottom surface width of the structure. However, as for engineering such as the gravity wharf, the load action width and load distribution determined by the load condition completely ignoring structure behind wharf are very unreasonable. For example, if difference between the design load distribution near the rear toe and

9.1 Design Load and Ground Failure Mode

313

edge load distribution behind the bottom surface for the wharf structure is small, the top surface stress of the foundation bed will diffuse backward and simultaneously the edge load will also diffuse forward in the foundation bed. At the bottom surface of foundation bed vertical to the rear toe of the wharf structure, generally, both stresses diffused to the surface are basically equivalent. The load action surface extended toward the backside of wharf structure will certainly increase the width of load action surface and make the distribution of design load unreasonable. In addition, the load action surface is extended toward the backside of wharf structure; however, the edge load (the load between the horizontal distance of c, c and above the top surface of foundation bed in Fig. 9.1) behind the wharf structure in the extension section is not considered, thus making the total design load for the actual width of bottom surface of foundation bed relatively small. At present, effective width is often adopted to calculate the ground bearing capacity. But, the effective width is just the treatment measure adopted to consider the eccentricity of design load on the basis of the actual width. It is totally different from the actual width and cannot explain the rationality of actual width and load distribution. It is no exaggeration to say that if the foundation bed is relatively thick, the load action width and distribution status determined according to the above method will be enough to make the calculated ground bearing capacity obviously larger. As for the gravity wharf, two approaches for solving the reasonable determination of the action width and distribution status for the design load are available. One is to discuss the reasonable method for the load action width and distribution status of the bottom surface of foundation bed. If the rear toe load for the top surface of foundation bed and the edge load behind the wharf are basically equivalent or for the sake of safety before more reasonable method is proposed, backward extension of the load action width may not be considered for the bottom surface of foundation bed. Namely, the actual width of the load action determined according to the same horizontal position of rear toe (c ) for the bottom surface of foundation bed and the rear toe (a) for the top surface of foundation bed is: B1 = c d = B + h d

(9.4)

As for the vertical stress on the front (d) and rear toe of the bottom surface for the foundation bed, the following may be considered: If the top surface of foundation bed is provided with forward eccentricity ( p1 ≤ p2 ) and hBd is relatively small (e.g., hBd ≤ 0.5), the vertical stress on the rear toe (c ) of bottom surface for the foundation bed is: pv1 = p1 + γ h d According to the total load on foundation bottom: B( p1 + p2 )/2 + γ h d B1 = B1 ( pv1 + pv2 )/2.

(9.5a)

314

9 Ground Bearing Capacity

The vertical stress on the front toe (d) of bottom surface for the foundation bed will be: pv2 =

B hd p2 − p1 + γ h d B1 B1

(9.5b)

Generally, according to the treatment method for the backward extension condition of load action width, the vertical stress on the front toe (d) of bottom surface for B B the foundation bed shall be provided with “ B+2h p2 +γ h d ≤ pv2 ≤ B+h p2 +γ h d ”, d d B and that on the rear toe (c ) shall be provided with “ pv1 ≥ B1 p1 + γ h d ”. So, 1 p2 + γ h d ” is taken as the vertical stress on the front toe (d) of bottom “ pv2 = BB1 1+α   α p2 + γ h d ” is that of rear surface for the foundation bed, and “ pv1 = BB1 p1 + 1+α toe (c ). It is easy to derive: In the case of “0 ≤ α ≤ hBd1 ”, the vertical stress of front and rear toe is in accordance with the above in equation, respectively. When “α = 0.5 hBd1 ,” hence:   B 0.5h d p1 + p2 + γ h d (9.6a) pv1 = B1 B1 + 0.5h d   0.5h d B pv2 = 1− p2 + γ h d (9.6b) B1 B1 + 0.5h d The other is to take the top surface of foundation bed as the load action surface, and the calculation is carried out through the rubble foundation bed according to the slip surface. Another thing for the design load is the treatment to the variable load of top surface for the structure. If the vertical variable load of top surface for the structure is counted, the total vertical design load will increase, forward eccentricity will decrease, and tan δ will decrease, which may lead to the increase of calculated limit load; if the vertical variable load of top surface for the structure is not counted, the total vertical design load may relatively be small, forward eccentricity may increase, and tan δ may relatively be large, thus decreasing the calculated limit load. Because the variable load of top surface for the structure sometimes exists and sometimes not, in case of the adverse condition, both conditions shall be checked. And the smaller one of calculated safety factor is taken as the calculated result. The horizontal load of the design load usually provides only the total load. Generally, its distribution is difficult to determine. How to reasonably consider the ground bearing capacity? It is a difficulty to solve, and the study on this respect is still rare at present. Of course, if the surface failure mode (the slip surface is just through the rear toe of the calculated surface) is adopted, it will not exist.

9.1 Design Load and Ground Failure Mode

315

9.1.2 Failure Mode of Ground Bearing Capacity Distribution of the limit load and the design load shall be considered simultaneously for the failure mode of ground bearing capacity. Due to the diversity of ground condition and design load, possible failure mode for the ground is also diverse. (1) Compression failure, retroversion failure, and anteversion failure [1]: Usually, according to different positions of resultant action points for the design load, the possible failure modes for the ground are: The corresponding failure modes are compression failure, retroversion failure, and anteversion failure (Fig. 9.2). The possible failure modes determined for the ground with homogeneous soil, uniformly distributed edge load and horizontal ground surface by the different positions of resultant action points for the design load are correct. But for the ground with heterogeneous soil, non-uniform distributed edge load, and non-horizontal ground surface, the modes may not be correct. In fact, the failure of ground soil is related to not only the design load, but also its strength and edge load. (2) One-way failure and two-way failure: As for no action of horizontal force and symmetric condition of same edge load and surface shape on both sides of the foundation, if the ground undergoes failure, it must be the soil mass on both sides of the foundation simultaneously undergoes failure. And consequence of such failure may be that the ground is presented with greater abnormal sedimentation. Generally, according to differences of large and small for horizontal force, large and small for edge load and ground surface shape on both sides of the foundation,

(a)

(b)

(c)

Fig. 9.2 Possible failure mode for the ground. a Compression failure; b Retroversion failure; c Anteversion failure

316

9 Ground Bearing Capacity

the ground may suffer slip failure to just one side. It may be also that the ground suffers slip failure to one side, and simultaneously the soil mass on the other side is also with failure. This is the two-way failure mode. Unless enough reasons can be used to determine that it is not possible for the soil mass on the other side to suffer failure (such as the edge load on the other side of the foundation for the gravity wharf is very large), the possibility of failure to both sides simultaneously may also exist (the failure area of the soil mass on the other side is smaller). For example, because obvious ridge soil mass (the south and north of the caisson are provided with mud mounds of 1.5 m height and width respectively which are not normal deformation) are provided with on both sides of the breakwater in some engineering [2], the failure mode is two-way failure for the engineering. And because the wave force acted on both sides is different, the failure mode shall be dissymmetric (Fig. 9.3). As for dissymmetric two-way failure mode, calculation mode of corresponding limit load has not been found yet at present. However, it can be sure that the total limit load of two-way failure mode is obviously less than that of one-way failure mode. According to present one-way failure mode, the ground bearing capacity will be overestimated. (3) Surface failure and field failure: These are two most basic types of failure mode in the previous discussion. Just as the ground bearing capacity, the following cases (Fig. 9.4) may exist: Case 1: The ratio of ground bearing capacity to total design load of the foundation width [−B, 0] calculated by the slip surface through foundation rear toe (−B) is relatively small. Such condition may occur when the subsoil is relatively soft. Case 2: The ratio of ground bearing capacity to general design load of the foundation width [−B, 0] calculated by the slip surfaces (slip surface family or slip line family) all in the limit state through every point of the foundation width is relatively small. Case 3: The ratio of the total limit load within [−b, 0] to the general design load of corresponding width calculated by the slip surface through some point (−b) of the

Limit load of One-way failure

Limit load of One-way failure

Fig. 9.3 Limit load of two-way failure

9.1 Design Load and Ground Failure Mode

−B

317

x −b

(C )

z

Fig. 9.4 Schematic diagram for failure mode

foundation width is relatively small. Such condition may occur when the topsoil is relatively soft. Case 4: The ratio of the total limit load within [−b, 0] to the general design load of corresponding width calculated by the slip surfaces through every point of the width [−b, 0] is relatively small. The Cases 1 and 3 are the surface failure mode, and the Cases 2 and 4 are the field failure mode. At present, both failure modes for the calculation of ground bearing capacity are applied. All the bearing capacity determined by the calculation equation for the limit load which is derived under condition of applying surface failure mode is ground bearing capacity of surface failure mode, just like the upper bound solutions of Hansen equation, Vesic equation, Meyerhof equation as well as Chen W.-F. Calculation equation of Tianjin University’s methods (“Code for Foundation in Port Engineering” (JTJ 250-98) [3]) is derived based on the characteristic line method. When being used for ground with homogeneous soil, it is the ground bearing capacity of field failure mode. However, when being used for the ground with heterogeneous soil, it cannot be regarded as the ground bearing capacity of field failure mode because its weighted average index is determined by the maximum depth of forced layer; namely, the slip surface is the maximum slip surface through the rear toe of the calculated surface. (4) Overall failure and partial failure: The Cases 1 and 2 above are the overall failure modes, and Cases 3 and 4 above are the partial failure modes. Adequate attention shall be paid to the partial failure because it may make part of the ground with relatively large and abnormal deformation which result in the incline of engineering structure to affect the normal use and even result in the overall failure [2]. “Code for Design of Building Foundation” (GB 5007-2002) [4] of our nation specifies: When the eccentric load acts, the maximum pressure of edge of the foundation bottom shall not be greater than 1.2 times characteristic value of the ground bearing capacity. The specification is just used to avoid the possible partial failure which may affect the normal engineering. The relationship between the partial failure and edge load is particularly close. Actually, the status for the distribution of edge load makes a great impact on the distribution of the limit load (Fig. 9.5). It is especially important for judging whether

318

9 Ground Bearing Capacity

Limit load according to the rubble-mould Limit load according to the uniformly-distributed edge load Design load

Fig. 9.5 Limit load for different edge loads

the partial failure may occur for the ground. Thus, the design of height and width for the rubble mound and the analysis (especially the analysis for partial failure) of the ground bearing capacity shall be one of the major bases. For a different engineering, it is essential to study the applicable calculation mode for the ground bearing capacity according to different failure modes. It is no doubt that the gravity wharf engineering is considered according to oneway failure mode, overall failure mode, and partial failure modes. For the vertical face breakwater engineering, because the load which may lead to failure of the engineering is the dynamic load, the failure mode will be specially discussed later. However, it is also no doubt that the engineering is considered according to two-way failure mode, overall failure mode, and partial failure mode.

9.2 Ground Bearing Capacity and Allowable Bearing Capacity 9.2.1 Limit Load and Ground Bearing Capacity For the design load bore in the actual engineering ground, generally, some conditions of design load have been known such as the ratio tan δ of the total horizontal load to the total vertical load and the eccentricity ev of load (or the distribution of load on bottom surface of the foundation bed). As it has been stated previously, the ground bearing capacity of engineering shall be the maximum load the ground may bear under the design load (distribution) condition. If the distribution of limit load and that of design load are the same, the limit load (ultimate bearing capacity) will be the ground bearing capacity. However, the limit load (including the joint action of vertical force and horizontal force; namely, tan δ is considered) under the field failure mode condition is unique.

9.2 Ground Bearing Capacity and Allowable Bearing Capacity

319

The total limit load of the foundation bottom is unique, so is its distribution form (including eccentricity e). But, distribution of this limit load (or eccentricity e) is generally impossible to be the same as that of design load (or eccentricity ev ). As a matter of fact, eccentricity of the design load is just considered generally when the ground bearing capacity of engineering is determined by the ground limit load. For the stress field method and generalized limit equilibrium method of the field failure mode, parameters to be determined are still included in its calculation equation under the condition that the equation is the same as tan δ of design load. So, in theory, undetermined parameters in the equation may be appropriately determined to make the calculated eccentricity the same as that of design load. However, it is tend to be difficult achieve. For example, the eccentric condition that the load of tan δ is relatively small while ev is relatively large. Because distribution characteristics of the design load shall be considered for the ground bearing capacity and it shall be reasonably derived under the condition that the soil mass is in accordance with the yield criterion, generally, the ground bearing capacity must be less than the limit load. An easy example is taken: As for two-way failure without regard to the horizontal force, according to present study, the limit load shall be non-uniformly distributed; if the limit design load is uniformly distributed and the soil mass is required to be in accordance with the yield criterion, the calculated total limit load of the latter must be less than that of the former. For the limit equilibrium method and generalized limit equilibrium method of the surface failure mode, because tan δ and ev (supposing “e = ev ”) of the design load may be directly substituted in the equation for calculation, the limit load (namely the ultimate bearing capacity) in accordance with the design load tan δ and ev may be obtained. However, the calculated value is also obviously greater than the limit load of the characteristic line method. And it is still possible to cause great difference between the calculated horizontal force distributed along the foundation bottom and the actual situation (Chap. 8). In addition, for the surface failure mode, although only the total limit load is calculated to avoid the distribution of the limit load for the foundation bottom, as it has been stated previously, eccentricity included in the calculation equation is still the eccentricity of calculated value of the limit load. When it is used to determine the ground bearing capacity, the eccentricity of design load and that included in the calculated result of limit load shall be also simultaneously considered just as the field failure mode does. In fact, ground bearing capacity is limited by two factors, characteristics of design load and ground soil (including the edge load). On the one hand, it is connected with the design load. On the other hand, it is independently influenced by the characteristic of the ground soil. From the previous discussion for the limit load, if the edge load is uniformly distributed, the limit load will be eccentric forward only under the condition of the topsoil strength being greater than subsoil strength of the ground. However, relatively small horizontal force may just make the design load eccentric forward without any relationship to the ground soil. That is, it is impossible to make conditions for tan δ and e of the limit load and tan δ and ev of the design load the same under the field failure mode. In other words, for the design load in actual engineering,

320

9 Ground Bearing Capacity

the limit load is the same as its tan δ and ev under the field failure mode in accordance with limit equilibrium may not exist at all. Some treatment process may be certainly adopted to make the calculated eccentricity the same as that of design load; for example, distribution (when it is eccentric forward, horizontal force near the rear toe increases and that near the front toe decreases) of the horizontal force may be properly selected. However, the calculation proves that under more conditions, it cannot be achieved by such treatment process and whether distribution of horizontal force obtained in this way is consistent with the actual situation is also a problem. In conclusion, just simple conditions such as the ground with homogeneous soil and uniformly distributed edge load, the same calculation method just like the method of seeking tan δ and e of the limit load and tan δ and ev of the design load for the ground bearing capacity is also not directly obtained. As for general conditions such as heterogeneous soil and non-uniform distributed edge load, distribution forms of the limit load are more complicated. Frequently, difference between the distribution forms of the limit load and design load is very great. Therefore, calculation of ground bearing capacity is carried out according to two steps: The first step is to calculate the limit load under the condition of meeting the relationship (tan δ) of vertical force and horizontal force; the second step is to determine the ground bearing capacity according to the distribution characteristics of design load and calculated value of the limit load. Under the condition of the calculated value of limit load being obtained, the ground bearing capacity is calculated with effective width rather than the actual width, and it is one method considering the eccentricity of the design load (but the eccentricity of the calculated value for limit load is not included).

9.2.2 Allowable Load on Ground and Allowable Bearing Capacity In order to solve the calculation of ground bearing capacity, in the calculation of limit load for the ground with heterogeneous soil, the coefficient Fsϕ , Fsc is introduced to further consider the uncertainty of strength index and the safety degree of engineering. Suitable value is taken for Fsϕ , Fsc , and actual meaning is provided for discussion of allowable load. From the discussion in Chap. 7, critical load ( p1/4 equation) is an allowable load actually. Supposing “Fsc = F0 (1.09 + 0.06 tan δ), Fsϕ = F0 (1.05 + 0.06 tan δ),” if “F0 > 1.0,” what calculated by corresponding equation will be just an allowable load. If the homogeneous soil has no horizontal force, the distribution of allowable load can be written as:

9.2 Ground Bearing Capacity and Allowable Bearing Capacity

321

pz = γ bNγ + ck Nc + q Nq (ck = c/Fsc )

(9.7)

The calculated allowable load coefficient (in accordance with helicoid calculation mode in Chap. 8) is detailed in Table 9.1. The coefficients above are close to the calculated value of characteristic value calculation equation for the ground bearing capacity in “Code for Design of Building Foundation” (GB 5007-2002). That is, the calculated value of p1/4 equation for the critical load is generally equivalent to the limit load after the strength index c, tan ϕ is divided by 1.6–1.7. Based on the limit load, the uncertainty of strength index is considered and allowable load is derived by properly selecting coefficient F0 . According to the distribution characteristics (eccentricity) of design load and eccentricity of the calculated value for the allowable load, to determine ground allowable bearing capacity is another way to calculate the ground bearing capacity. Table 9.1 Allowable load coefficient ϕ(◦ )

q γB

= 0,

Nγ

c γB

= 0.1, F0 = 1.6

q γB

Nc

Nγ

Nq

= 0.1,

c γB

= 0.2, F0 = 1.66

Nc

Nq

2

0.045

3.354

1.071

0.047

3.288

1.094

4

0.088

3.476

1.158

0.096

3.431

1.198

6

0.133

3.628

1.262

0.147

3.597

1.319

8

0.183

3.810

1.380

0.204

3.783

1.453

10

0.241

4.000

1.520

0.273

3.967

1.601

12

0.306

4.222

1.682

0.351

4.180

1.773

14

0.380

4.480

1.871

0.436

4.432

1.971

16

0.466

4.781

2.093

0.537

4.700

2.197

18

0.570

5.103

2.353

0.655

5.004

2.458

20

0.698

5.414

2.637

0.801

5.313

2.746

22

0.845

5.839

2.996

0.964

5.701

3.097

24

1.028

6.262

3.393

1.159

6.147

3.511

26

1.251

6.760

3.867

1.412

6.581

3.965

28

1.522

7.379

4.465

1.705

7.134

4.531

30

1.871

7.966

5.121

2.078

7.721

5.174

32

2.302

8.730

5.942

2.539

8.412

5.949

34

2.855

9.585

6.925

3.110

9.248

6.912

36

3.554

10.733

8.225

3.851

10.172

8.042

38

4.485

11.933

9.726

4.800

11.295

9.450

40

5.717

13.430

11.641

6.031

12.681

11.246

322

9 Ground Bearing Capacity

9.3 Measurement for Stability of Ground Bearing Capacity 9.3.1 Measurement of Ground Stability with Ultimate Bearing Capacity Generally, safety factor of bearing capacity (partial safety factor of resistance) is relatively and universally applied by the engineering circle to measure the stability of ground bearing capacity. Under the condition of considering the distribution characteristics (eccentricity), the ground bearing capacity is determined according to the limit load and the safety factor K of ground bearing capacity is defined as: the ratio of total vertical limit load to total vertical design load (partial safety factor of resistance is defined as the ratio of characteristic value for the ground bearing capacity to design value for the vertical resultant). Because the action of horizontal load has been considered according to boundary condition of foundation bottom in calculation of the limit load, the ratio of total horizontal limit load to total horizontal design load is also K. Such safety factor is just a kind of measurement for overall stability of ground and is used to judge whether the possibility of overall failure is effective. However, we shall have a distinct knowledge of the safety factor. First of all, the safety factor cannot reflect the possible partial failure. As stated before, the field failure mode requires all the soil mass in the slip mass must be in the limit state, so the corresponding limit load is the maximum load for ground to bear when the strength of ground soil is extremum. Distribution forms of the limit load are determined. Generally, the forms of the limit load are impossible to be the same as that of the design load of actual engineering. In general, distribution forms of the limit load under the condition of homogeneous soil, uniformly distributed edge load, and load boundary of foundation bottom must be eccentric backward (including relatively large horizontal force), while the design load under the condition of relatively large horizontal force is eccentric forward. If a comparison of distribution forms is made between the limit load and design load, the possible partial failure of the ground soil may be reflected. However, the comparison of total limit load and total design load can just reflect the overall failure. Although safety factor is relatively large, it may be in the area near the front toe and the design load is greater than the limit load (Fig. 9.6). This is the partial failure that the safety factor cannot reflect but may occur. “Code for Design of Building Foundation” (GB 5007-2002) of our nation specifies: When the eccentric load acts, the maximum pressure for the edge of the foundation bottom shall not be larger than 1.2 times characteristic value of the ground bearing capacity. It is obviously reasonable and worth using for reference. Secondly, the soil mass state corresponding to the safety factor is not definite. When the ground bearing capacity is determined with the limit load, usually, only the distribution (eccentricity) of design load is considered but not the distribution of the limit load. However, the limit load is also the eccentric load and shall be considered. For example, when distribution forms of the design load and the limit

9.3 Measurement for Stability of Ground Bearing Capacity

323

Limit load

Design load

−B

− B'

0

Failure area

Fig. 9.6 Partial failure of ground

load are just the same, the ground bearing capacity shall be the total limit load. Actual width B of the load is converted into effective width B  and is regarded as the ground bearing capacity according to the total limit load within the effective width. However, the ground bearing capacity is obviously underestimated. When the design load is eccentric forward, the limit load is eccentric backward, and distribution of the limit load is not considered, the ground bearing capacity may be overestimated. In fact, the distribution forms of the limit load are corresponding to the limit state of ground soil if distribution forms of the limit load are not considered for the determination of the ground bearing capacity just like the actual state of ground soil. For example, when the eccentricity of the design load is zero (eccentricity of the design load is zero after being converted into the effective width), that the ground bearing capacity determined under distribution condition of the limit load is the limit load is not considered; if the safety factor is 1.0, under the action of design load, the soil mass in the slip surface through a point [−B1 , 0] of the foundation bottom is in the failure state (Fig. 9.7) and B1 ≈ 0.5B  . That is, the safety factor of 1.0 actually reflects the ground condition that partial soil mass is in the failure state and partial soil mass in the stable state. The soil mass state corresponding to such safety factor is undetermined. Limit load

− B'

− B1'

Design load

0

Failure area

Fig. 9.7 Failure area of ground with safety factor 1.0

324

9 Ground Bearing Capacity

The safety factor is analyzed by the above two points from different angles, but same in the essence. Namely, the coefficient cannot reflect the partial failure. In addition, because the safety factor cannot reflect the partial failure, it leads to the uncertainty of major factors for the engineering failure. Under the condition of relatively large horizontal design load, appropriate increase of the vertical design load benefits the stability. This fact is well known for the stability of the horizontal slip, but is often neglected in the ground bearing capacity. According to the calculated result of the characteristic line method, the total vertical design load is supposed as Pv and the total limit load is Pz when the edge load “q/(γ B) = 0.1”, strength index “c/(γ B) = 0.2”, “tan δ = 0.4, 0.3, and 0.2”, respectively, and Pz /(γ B 2 ) is 2.35, 3.94, and 5.80 (the calculation is accuracy enough). That is to say, in the case of the constant horizontal load, when the total vertical load is Pv , 1.5Pv , and 2Pv , respectively, the safety factor will be 2.35, 2.63, and 2.9. Obviously, when the vertical load increases, the safety factor also increases. If the condition that after the vertical load increases, the eccentricity of design load will generally reduce is considered, the effective width B will increase, and the safety factor will increase further. One engineering [2] is taken as example to state this problem. Partial safety factor of resistance of the bearing capacity is calculated in original design of engineering: “service life ≥3.02” and “construction period ≥2.55” which completely comply with corresponding specification requirements. However, part of the engineering is presented with severe subsidence failure: “The caisson is with sudden and severe sedimentation with distance of over 1 m downward, and its both sides are provided with obvious ridge soil mass. It is proved that the ground bearing capacity is insufficient and the overall stability of the ground has suffered failure.” In fact, vertical load this engineering ground bears is just the self-weight of the building and the variability is relatively small; the variability of ground soil strength is relatively large (with great influence on the limit load), and for the horizontal load (wave force), the variability is not only large but also dynamic. From angle of probabilistic method, horizontal load and ground soil strength are more possible major factors to influence the ground bearing capacity. It is impossible for the ground to bear the actual vertical load which reaches 2.55 times of the design load, or the probability for the actual vertical load to reach 2.55 times of the design load is zero. Objectively, it causes false impression that the engineering has been provided with enough safety degree. In some degree, major factor leading to the failure of the engineering is not the increase of vertical load. On the contrary, because “the caisson with backfill sand is a severe sand loss after the wind,” reduction of vertical load is caused, and this is one of the factors for the engineering failure. Generally, it is basically impossible for the vertical load (above the foundation bottom) of port engineering to reach 2 times (the lower limit of partial safety factor of resistance in corresponding specifications) of the design load. Although the hor-

9.3 Measurement for Stability of Ground Bearing Capacity

325

izontal load (wave force) is the possible major failure load for the breakwater, the more important factor may be the reduction [2] of the soil mass strength under the condition of dynamic load. Therefore, whether it is scientific that the stability of the ground bearing capacity is measured by a single safety factor of bearing capacity needs to be studied again.

9.3.2 Evaluation of Ground Stability in Accordance with Allowable Bearing Capacity Under the condition of considering the distribution characteristics (eccentricity), the allowable bearing capacity of ground is determined according to the allowable load and the safety factor of ground bearing capacity is defined as: After the strength index is divided by the safety factor, the ground will be in the limit state (the total vertical limit load is the same as the total vertical design load). This safety factor is just F0 mentioned above (hereinafter referred to as strength safety factor). If the load distribution of foundation bottom is known (including the horizontal load), definition of such strength safety factor and the safety factor of general slope stability will be the same. Application [5] for the condition that surface failure mode of slip surface through the rear toe of the foundation bottom is only calculated and the normal slope stability analysis method is used to calculate the ground bearing capacity has existed. Of course, due to the different processes of calculation for the ground bearing capacity and slope stability, the calculated safety factor will be different. If a different F0 is defined for a different slip surface (through a different horizontal position of the foundation bottom), F0 will be available to measure the local stability of ground. Certainly, the strength safety factor may be also properly determined so as to make the calculated allowable load less than the limit load and greater than the design load. In this way, the safety factor of bearing capacity for the allowable load may be also defined similar to the limit load, and at the same time, the safety factor of bearing capacity and strength safety factor are used to measure the load bearing capacity of ground. From the aspect of probabilistic method based on the reliability theory, both uncertainty of soil mass strength and actual load are the major factors to affect the reliable probability of the engineering. As for reliability theory-based partial safety factor method, if the load bearing capacity of ground is jointly measured by the partial safety factor of strength and partial load coefficient, the strength and uncertainty of the load may be reflected simultaneously. There is no doubt that a further step is obtained, compared with application of the single partial safety factor.

326

9 Ground Bearing Capacity

9.4 Determination of Ground Bearing Capacity with Limit Load It is noticed that if the boundary condition of foundation bottom is “τx z0 = tan δ(σz0 + c/ tan ϕ)”, tan δ will be determined under the boundary condition of foundation bottom according to the design load, namely: tan δ = Ph /(Pv + Bc/ tan ϕ)

(9.8a)

If the boundary condition of foundation bottom is “τx z0 = tan δσz0 ”, tan δ shall be: tan δ = Ph /Pv

(9.8b)

where Ph , Pv are the horizontal and vertical design loads within the width of calculated surface, respectively. If the bottom surface of foundation bed is regarded as the load action surface, B shall be changed to B1 . Corresponding calculation equations shall be, respectively, adopted for the calculated limit load. As it has been stated previously, if the ground bearing capacity is determined by applying the limit load, distribution characteristics (eccentricity) of the design load and distribution characteristics of the limit load shall be simultaneously considered.

9.4.1 Determination of Ground Bearing Capacity Without Prejudice to Yield Criterion Generally, practical problems may provide the stress “ pv1 = pv (−B) and pv2 = pv (0)” of the design load corresponding to edge points on the load action surface pv1 ” will be determined according to and the corresponding eccentricity “ev = B6 ppv2v2 − + pv1 consideration of the linear distribution. If the general design load and eccentricity are provided with, the stress of edge points on the load action surface may be also obtained. The previous content (Chap. 8) has pointed out that the calculated partial limit load under certain conditions is very large. In this part, the probability for design load to reach the limit load is almost zero, so it is free from actual meaning. Appropriate correction shall be provided for determining the ground bearing capacity. Suppose “K ∗ = Pz /Pv ” and Pz , Pv are the total limit load and general design load, respectively; distributions of the design load and limit load along the foundation bottom, respectively, are pv (x), pz (x), (−B ≤ x ≤ 0), and “ pv∗ (x) = K ∗ pv (x)”. When the total load of the two is the same, some partial soil mass must be in the stable state and other partial soil mass must be in the failure state (Fig. 9.8) according to the design load. Obviously, it is improper to take K ∗ as the safety factor.

9.4 Determination of Ground Bearing Capacity with Limit Load

327

pv*1

p z1

pv*2

pz 1

pz 2

pv*1 pz 2 − Be

−b

pv*2

0

− Be

(a) Forward eccentricity of design load

−b

0

(b) Backward eccentricity of design load

Fig. 9.8 Schematic diagram for calculation of ground bearing capacity I

Thus, when the limit load is used to determine the ground bearing capacity, the design load of actual engineering and distribution forms of the limit load shall be simultaneously considered and necessary correction shall be corrected out for the limit load. The ground bearing capacity is calculated by the following equation: 0 Pu =

min{ pz (x), pv∗ (x)}dx

(9.9)

−B

Overall safety factor of bearing capacity is: K = Pu /Pv

(9.10)

Actual meaning of Eq. (9.9) is to determine the ground bearing capacity in accordance with the soil mass without prejudice to the yield criterion: If K ∗ times design load of one point is still less than the limit load, the part of the limit load greater than K ∗ times design load will be regarded with no actual meaning. Then, the limit load corrected as K ∗ times design load will be regarded as the ground bearing capacity (distribution). If the limit load of the point is less than K ∗ times design load, it will just be the ground bearing capacity. So, in area for K ∗ times design load less than the limit load under the condition of safety factor K, the soil mass on the corresponding slip surface family is in the stable state; as is shown in Fig. 9.8a, slip surface of any point in area [−Be , −b) passes through the foundation bottom and in Fig. 9.8b slip surface of any point in area (−b, 0) passes through the foundation bottom. However, in the area for K ∗ times design load still greater than the limit load under the condition of safety factor K, the soil mass on the corresponding slip surface family is either in the limit state (as shown in Fig. 9.8a), slip surface of any point in area [−b, 0) through the foundation bottom) or in the stable state (as shown in Fig. 9.8b), slip surface of any point in area [−Be , −b] through the foundation bottom).

328

9 Ground Bearing Capacity p v*1

p z1

p v*2

p z1

pv*1

pz 2

pz2

−B

− B/2

0

(a) Forward eccentricity of design load

pv*2

0 − B /2 −B (b) Backward eccentricity of design load

Fig. 9.9 Schematic diagram for calculation of ground bearing capacity II

The above methods are applicable to the conditions of ground with heterogeneous soil, non-uniform distributed edge load, and non-horizontal ground surface. As for the conditions of ground with homogeneous soil, uniformly distributed edge load, and horizontal ground surface, the limit load is considered according to the linear distribution and the error will not be large. Suppose the calculated limit load is pz1 = pz (−B), pz2 = pz (0); if the design ∗ ∗ ∗ ∗ = K ∗ pv1 , pv2 = K ∗ pv2 , and B2 ( pv1 + pv2 ), the calculated total load increases pv1 limit load will be equivalent and the eccentricity is still the eccentricity of design load. That is, if the general design load increases to the calculated total limit load under the condition that eccentricity is constant, its distribution within the 1/2 foundation width will be greater than the calculated limit load (see Fig. 9.9). Thus, distribution forms of the ground bearing capacity with consideration for distribution forms of the design load are determined according to the following three points (Fig. 9.9): ∗ }, p(−B/2) = p(0) = min{ pz1 , pv1

1 ∗ ( pz1 + pz2 ), p(−B) = min{ pz2 , pv2 }. 2

The ground bearing capacity is: Pu = Pz −

  B B  ∗  = ( pz1 + pz2 ) 1 − pz2 − pv2 4 2

  ∗ ∗  − pv1 1  pz2 − pz1 pv2  − ∗ ∗  4  pz2 + pz1 pv2 + pv1

namely:   ev  3  e Pu = Pz 1 −  −  2 B B

(9.11)

This is an exceptional case for Eq. (9.9), because the design load and limit load are both under the condition of the linear distribution, it is easy to prove:

9.4 Determination of Ground Bearing Capacity with Limit Load

0 Pu =

 min{ pz (x), pv∗ (x)}dx = Pz 1 −

−B

329

 3  e ev   −  2 B B

The actual meaning of ground bearing capacity is determined according to Eq. (9.11): Under the condition of “K ∗ = 1.0”, if the eccentricity of the design load is greater than that of the limit load, on the slip surface through the foundation bottom [−B/2, 0], the soil mass will be in the limit state; on the slip surface through the foundation bottom [−B, −B/2), the soil mass will be in the stable state (Fig. 9.9a). If the eccentricity of the design load is smaller than that of the limit load, on the slip surface through the foundation bottom [−B/2, 0], the soil mass will be in the stable state; because the design load of (−B/2, 0] is smaller than the limit load, on the slip surface through the foundation bottom [−B, −B/2], the soil mass will be in the stable or limit state (Fig. 9.9b).

9.4.2 Determination of Ground Bearing Capacity with Requirements of Basic Effective Width Usually, treatment methods of calculation are carried out with basic effective width replacing the actual width. If distribution of the design load and the limit load is simultaneously considered, the limit load is required to meet the eccentricity of the 0 design load: −B pz (x)(x + B/2 − ev )dx = 0. Because this condition cannot be met generally, the action width of the load is reduced to meet. If “ev ≥ e ” (the design load is eccentric forward, ev ≥ 0), it will be derived that within the [−B, −B  ], “ pz (x) = 0” (Fig. 9.10a) and 0 “ −B  pz (x)(x + B/2 − ev )dx = 0.” And the calculated value of the limit load shall 0 meet: −B  pz (x)(x + B  /2 − e )dx = 0. So, the effective width is: B  = B − 2(ev − e ) pz

(9.12) pz p v*

p v*

−B

− B'

0

(a) Forward eccentricity of design load

−B

− b' 0

(b) Backward eccentricity of design load

Fig. 9.10 Schematic diagram for calculation of ground bearing capacity with effective width

330

9 Ground Bearing Capacity

the total limit load is: 0 Pu =

pz (x)dx

(9.13)

−B 

If “ev ≤ e ” (the design load is eccentric backward, ev ≤ 0), it will be derived that within the [−b , 0], “ pz (x) = 0” (Fig. 9.10b) and  −b “ −B pz (x)(x + B/2 − ev )dx = 0.” And the calculated value of the limit load shall  −b meet: −B pz (x)[x + (B − b )/2 − e ]dx = 0. So, b = −2(ev − e ); namely, the effective width is: B  = B + 2(ev − e )

(9.14)

the total limit load is: −b Pu =

pz (x)dx

(9.15)

−B

This method is also the correction for the limit load, and the limit load after the correction shall be regarded as the ground bearing capacity. However, it is slightly simple to correct the limit load as zero in area of [−B, −B  ) or (−b , 0].

9.4.3 Partial Bearing Capacity of Partial Failure The total limit load and design load within the partial width [−bm , 0](m = 1, 2, . . .) are calculated as the following equations: 0 Pz,m =

pz (−b)db

(9.16)

pv (−b)db

(9.17)

−bm

0 Pv,m = −bm

Partial failure of the soil mass having relatively small influence on the engineering failure is considered, so the safety factor of partial bearing capacity may be determined with the total limit load and design load within the partial width: K min = min{Pz,m /Pv,m }

(9.18)

9.4 Determination of Ground Bearing Capacity with Limit Load

331

9.4.4 Several Discussions The above method that the ground ultimate bearing capacity is determined by the limit load is also applicable to the allowable bearing capacity determined by the allowable load. The eccentricity included in the calculated value of the limit load is not considered generally, and the effective width is taken as: “B  = B − 2|ev |”. It is thought to be that the limit load within the effective width is uniformly distributed: 0  −B  pz (x)(x + B /2)dx = 0; this is not reasonable. In some conditions, the calculated value of the ground bearing capacity will be relatively small; for example, when the eccentricity of the design load is just the same as that of the calculated value for the limit load, the eccentricity of the design load is also used to correct the calculated value of the limit load, and the ground bearing capacity is artificially reduced. In other conditions, the calculated value of ground bearing capacity will be relatively large; for example, when the eccentricity of design load is zero, ground bearing capacity determined is actually the maximum value of the limit load when the eccentricity is the calculated value. If the safety factor is 1.0, the soil mass in partial area will be in the failure state. Obviously, the ground bearing capacity is overestimated. According to the treatment method of using basic effective width to replace actual width, because e is needed to be obtained in the calculation process, the bearing capacity only can be obtained through the trial calculation (iteration). The calculation process is relatively complicated. If the basic effective width is determined with ev − e , when the design load is eccentric forward, the ground bearing capacity will be reduced. An extreme case is considered: When “q = 0, c = 0 and ev = B/6,” if the general boundary condition of foundation bottom is adopted, “e ≈ −B/6” shall be provided with; the effective width is “B  = B − 2|ev − e| = B/3,” and it means only about 1/3 width of the bottom is bearing the load and will make the ground bearing capacity about 1/9 of the calculated value for the total limit load. However, according to Eq. (9.9), the ground bearing capacity is about 1/2 of the calculated value for the total limit load. Distribution of the limit load under the condition of ground with heterogeneous soil, non-uniform distributed edge load, and non-horizontal ground surface is relatively complicated, such as the topsoil strength is far larger than that of the subsoil, and rubble foundation bed and rubble mound near the front toe are provided with. The calculated partial (near the front toe) limit load is very large without the actual meaning. However, the calculated value of the limit load reduces in the rear area. If the correction is just carried out when the ground bearing capacity is determined, the calculated value of ground bearing capacity may be relatively small. If the correction is carried out when the limit load is calculated, it may meet actual engineering conditions. However, the specific correction method is needed to be discussed. Because distribution of the limit load cannot be calculated under the surface failure mode, eccentricity of conditions such as ground with heterogeneous soil,

332

9 Ground Bearing Capacity

non-uniform distributed edge load, and non-horizontal ground surface is unable to be determined. Therefore, for the surface failure mode, in theory, the above method that the ground bearing capacity is determined by the limit load is just applicable to the conditions such as ground with homogeneous soil, uniformly distributed edge load, and horizontal ground surface. If the ground bearing capacity is determined according to the calculation mode of calculating the eccentricity of limit load in Chap. 8, larger error may exist.

9.5 Ground Bearing Capacity of Gravity Wharf 9.5.1 Calculation Method The ground of the gravity wharf shall be in the one-way failure mode. In the above discussion, multiple calculation methods are given; for example, in different failure modes, with different load action surfaces, the ground bearing capacity is determined without prejudice to the yield criterion or according to the effective width. At present, many calculation methods are available for the ground bearing capacity. However, these methods are mostly only applicable to the ground with homogeneous soil and apply only when the difference between strength indexes of the soil in the layers is not large [3]. Here, these methods are applied in the above case just for the comparison with the calculation method according to the heterogeneous soil; the selected calculation methods are as follows: (1) Tianjin University’s method [3]: The load action surface is the bottom surface of the rubble foundation bed, and its width is determined according to Eq. (9.1) and calculated according to effective width; the ground soil index is calculated according to the weighted average of the soil layer thickness (in the supporting layer) as the following equation. Pz = B 



1 γk B  Nγ + qk Nq + ck Nc 2

 (9.19)

where B  is the effective width “B  = B − 2|ev |” of the foundation bottom, γk is the mean weight of the soil below the foundation bottom, qk is the main of the edge load above the foundation bottom, ck is the mean cohesion, and ϕk is the mean internal friction angle. Nγ , Nq , Nc are the bearing capacity coefficients for the ground soil in the limit state. They may be obtained through fitting of the calculated results by the characteristic line method:

9.5 Ground Bearing Capacity of Gravity Wharf

333

 

 π ϕk 1 + sin ϕk sin(2α − ϕk ) + 2α − ϕk tan ϕk tan2 45◦ + Nc = exp − 1 / tan ϕk 2 2 1 + sin ϕk

(9.20)

Nq = Nc tan ϕk + 1

(9.21)

Nγ = f (η, tan ϕk , tan δ) ≈ 1.25{(Nq + 0.28 + tan δ) tan[ϕk − 0.72δ(0.9455 + 0.55 tan δ]}   1 1+ √ 1 + 0.8[tanϕk − 0.7(1 − tan δ)] + (tan ϕk − tan δ)η

(9.22)

α is determined according to the following equation: 

ϕk tan α − = 2



1 − (tan δ/ tan ϕk )2 − tan δ tan δ 1 + sin ϕk

(9.23)

Thickness of the supporting layer Z max is calculated according to the following equation: Z max = B  exp(ε tan ϕk ) sin ε exp[−0.87η0.75 /(4.8 + η0.75 )]

(9.24)

where η = γk B  /(ck + qk tan ϕk ) ε=

  sin δ ϕk δ 1 π + − − sin−1 4 2 2 2 sin ϕk

(2) Hansen’s method [6]: The load action surface and its width as well as the thickness of the supporting layer are the same as those in (1). Calculation equation for the ground bearing capacity is as below: Pz = B 



   1 ck ck Nq dq i q − γk B  N γ i γ + q k + 2 tan ϕk tan ϕk

(9.25)

where  Nq = exp(π tan ϕk ) tan2 45◦ + Nγ = 1.5(Nq − 1) tan ϕk dq = 1 + 2(1 − sin ϕk )2 tan ϕk where D is the embedded depth of foundation;

ϕk 2



D B

(9.26) (9.27)

334

9 Ground Bearing Capacity

5 ⎫   ck ⎪ ⎪ ⎪ i γ = 1 − 0.7Ph / Pv + B ⎬ tan ϕk 5 ⎪   ck ⎪ ⎪ ⎭ i q = 1 − 0.5Ph / Pv + B tan ϕk

(9.28)

(3) The ground soil index is determined according to the field failure mode—calculation mode for the weighted average of the soil layer thickness in the field failure mode; the load action surface and its width are the same as those in (1); the soil index is determined according to those stated in (1). However, the slip surface depth is determined according to the minimum total limit load. (4) Field failure mode—calculation mode for the heterogeneous soil; the load action surface is the top surface of the rubble foundation bed, and its width is the actual width. (5) Field failure mode—calculation mode for the heterogeneous soil (foundation bed bottom a); the load action surface is the bottom surface of the rubble foundation bed, and its width is determined and calculated [same as those in (1)] according to Eq. (9.1) and effective width, respectively. (6) Field failure mode—calculation mode for the heterogeneous soil (foundation bed bottom b); the load action surface is the bottom surface of the rubble foundation bed, and its width, the actual width, is determined according to Eq. (9.4). When the load action surface is the bottom surface of the rubble foundation bed, during the calculation, the active soil pressure (rear toe) in the thickness of the foundation bed is counted, but the passive soil pressure (front toe) in the thickness of the foundation bed is not. Logarithmic helicoid calculation mode is adopted as the calculation mode for the limit load on the ground with heterogeneous soil. The boundary condition of foundation bottom is “τx z0 = tan δσz0 ”, where “tan δ = Ph /Pv ”.

9.5.2 Calculated Results of Bearing Capacity The calculated results—safety factors for the bearing capacity: K is the distribution considering the limit load and design load, and corresponding safety factor of the ground bearing capacity is determined without prejudice to the yield criterion; K ∗ is the distribution only considering the design load, and the total limit load is regarded as the corresponding safety factor of the ground bearing capacity. K m is the safety factor of the partial bearing capacity. The strength safety factor F0 is of the field failure mode meaning; for the calculation according to heterogeneous soil, when “K = 1”, “Fsc = F0 (1.09 + 0.06 tan δ) and Fsϕ = F0 (1.05 + 0.06 tan δ)” will be true. Calculated results of the bearing capacity safety factors K, K ∗ , and K m as well as the strength safety factor F0 are detailed in Table 9.2. Calculated results of the ground soil and the slip surface depth are detailed in Table 9.3.

4. Caisson wharf Forward eccentricity of design load

3. Block wharf Slightly forward eccentricity of design load

2. Block wharf Backward eccentricity of design load

2.214 0.588

3.238

Km

3.266

2.897

3.204

K∗

K

1.225

F0

1.312

1.383

2.351 2.518

3.075

Km

3.115

K∗

3.472

1.423

1.167

K

F0

2.017 2.119

4.786

Km

3.788 2.152

4.186

K∗

K

1.160

1.160

1.974

F0

2.465

Foundation bed top Heterogeneous soil (4)

1.136

2.470

Mean index (3)

2.336

2.744

Hansen’s method (2)

Km

K

1. Caisson wharf Forward eccentricity of design load

Tianjin University’s method (1)

K∗

Safety factor

Engineering case

Table 9.2 Bearing capacity safety factors

3.083

3.247

3.042

1.343

2.504

3.517

3.243

1.362

1.715

3.716

3.158

1.205

1.846

2.615

2.373

Foundation bed bottom a Heterogeneous soil (5)

2.220

2.802

2.428

1.329

2.206

2.722

2.589

1.320

2.784

3.251

3.122

1.178

1.362

2.382

2.003

(continued)

Foundation bed bottom b Heterogeneous soil (6)

9.5 Ground Bearing Capacity of Gravity Wharf 335

8. Buttressed wharf Forward eccentricity of design load

7. Buttressed wharf Forward eccentricity of design load

6. Block wharf Forward eccentricity of design load

5. Block wharf Slightly backward eccentricity of design load

Engineering case

Table 9.2 (continued)

1.118

F0

1.191

1.142

1.595 2.036

3.806

Km

3.753

K∗

3.993

1.214

1.056

K

F0

1.295 0.867

3.903

Km

3.945 1.643

4.480

K∗

K

1.159 1.283

F0

1.392

Km

3.049 3.453

4.208

K∗

4.373

1.384

4.625

1.658

F0

K

2.895

2.217

1.150

Foundation bed top Heterogeneous soil (4)

Km

3.908

Mean index (3)

2.895

3.766

Hansen’s method (2)

K∗

4.204

1.376

F0

K

Tianjin University’s method (1)

Safety factor

1.172

1.502

2.363

2.054

1.148

1.364

2.544

2.060

1.477

3.909

4.912

4.671

1.597

3.629

3.785

3.474

1.357

Foundation bed bottom a Heterogeneous soil (5)

1.168

1.099

2.427

1.849

1.124

1.015

2.320

1.722

1.485

2.929

4.172

3.831

1.639

3.020

3.020

2.686

1.335

(continued)

Foundation bed bottom b Heterogeneous soil (6)

336 9 Ground Bearing Capacity

11. Caisson wharf Slightly backward eccentricity of design load

10. Buttressed wharf Forward eccentricity of design load

1.414

1.367

F0

2.149 2.880

2.705

Km

2.834 2.880

2.824

K∗

K

0.940 1.118

1.264

1.730

F0

2.965

Km

3.002 2.430

3.212

K∗

K

1.071

1.185

1.412

F0

2.681

Foundation bed top Heterogeneous soil (4)

0.672

2.736

Mean index (3)

1.832

2.877

Hansen’s method (2)

Km

K

9. Buttressed wharf Forward eccentricity of design load

Tianjin University’s method (1)

K∗

Safety factor

Engineering case

Table 9.2 (continued)

1.512

2.632

3.204

3.080

1.218

1.195

2.991

2.564

1.164

1.358

2.672

2.225

Foundation bed bottom a Heterogeneous soil (5)

1.491

2.472

2.472

2.420

1.209

0.915

2.988

2.321

1.142

0.997

2.547

1.870

(continued)

Foundation bed bottom b Heterogeneous soil (6)

9.5 Ground Bearing Capacity of Gravity Wharf 337

4.750

4.916

2.835 1.929 1.266

1.446

3.767

Km

F0

K Mean

3.518

5.725

K∗

K

1.398

2.106

5.242

4.327

1.579

3.053

3.053

2.701

Foundation bed bottom a Heterogeneous soil (5)

2.523

1.400

1.685

4.695

3.673

1.607

3.050

3.050

2.282

Foundation bed bottom b Heterogeneous soil (6)

Note The engineering information on the calculation example in the specification is extracted from references [7], and other engineering information is provided by the drafting group of “Code for Foundation in Port Engineering”

13. Caisson wharf Slightly forward eccentricity of design load

1.361

1.524

2.250

F0

3.938

Foundation bed top Heterogeneous soil (4)

2.887

3.527

Mean index (3)

3.224

3.428

Hansen’s method (2)

Km

K

12. Calculation example in the specification (block) Slightly backward eccentricity of design load Wave force is counted in

Tianjin University’s method (1)

K∗

Safety factor

Engineering case

Table 9.2 (continued)

338 9 Ground Bearing Capacity

2.0 Foundation bed bottom Front

7. Buttressed wharf

1.9

Non-layered soil

6. Block wharf

27.3

4.0

3.0 Foundation bed bottom Front

21.6

19

1.0

4. Caisson wharf

5. Block wharf

20.4

0

Homogeneous soil

3. Block wharf

17.5

32.6

30

18.3

8.04

0

32

0

27.0

17

48.0

32

5.95

35

25.26

0

14.6

25.1

40.8

32 30

0

0

3.3

2. Block wharf

26.5

Friction angle (°)

12.1

Cohesion (kPa)

3.65

Thickness of soil layer (m)

Ground soil condition

1. Caisson wharf

Engineering case

Table 9.3 Ground soil condition and slip surface depth

2.5

3.45

6.5

5.4

1.9

2.5

3.0

Thickness of rubble foundation bed (m)

8.8

12.5

6.0

7.80

7.15

3.88

8.25

9.68

4.95

5.26

6.58

4.63

Foundation bed bottom a Heterogeneous soil

Slip surface depth (m) Tianjin University’s method

3.88

6.89

9.69

4.60

4.61

5.95

4.61

(continued)

Foundation bed bottom b Heterogeneous soil

9.5 Ground Bearing Capacity of Gravity Wharf 339

30 30.6

0

24.4

0.6

30

23.5

21

38.0

Non-layered soil

36.0

3.5

13. Caisson wharf

16

45.0

3.6

12. Block wharf

0

30–32

26

17.5

28

32.6

37.0

Homogeneous soil

12.8

11. Caisson wharf

32

24.0

28

0

37.0

2.0 Foundation bed bottom Front

12.8

24.0

10. Buttressed wharf

32

0

2.9 Foundation bed bottom Front

18.6

0

38.8

0

1.20

Friction angle (°)

Cohesion (kPa)

1.27

Thickness of soil layer (m)

Ground soil condition

9. Buttressed wharf

8. Buttressed wharf

Engineering case

Table 9.3 (continued)

4.0

2.0

6.5

2.1

2.4

2.5

Thickness of rubble foundation bed (m)

15.46

5.30

7.90

6.30

7.97

13.39

7.16

9.62

7.34

5.88

2.46

Foundation bed bottom a Heterogeneous soil

Slip surface depth (m) Tianjin University’s method

11.51

6.98

8.26

7.33

4.65

2.46

Foundation bed bottom b Heterogeneous soil

340 9 Ground Bearing Capacity

9.5 Ground Bearing Capacity of Gravity Wharf

341

9.5.3 Analysis of Calculated Results (1) Calculation method comparison of homogeneous soil: By (1), (2), and (3) calculation methods in accordance with the mean index, the calculated safety factors K ∗ are very close to each other; due to different treatments for the distribution characteristics of the design load and different calculations of the slip surface depth, it is normal to have relatively small difference, for example, when the design load is eccentric backward, that by Method (3) is slightly greater than that by the Method (1) (Cases 2 and 12), and when the design load is eccentric forward, that by Method (3) is slightly less than that by the Method (1). (2) Calculation method comparison of homogeneous soil and heterogeneous soil: For the calculation method of the heterogeneous soil, in Method (5), other calculation conditions are close to the mean index; by comprising calculated results by the Methods (5) and (1), respectively, on the whole, except in Cases 7 and 8, those of other engineering cases are relatively close to each other, which shows that the Tianjin University’s method applies to the majority of the engineering. Because Method (1) only can be applied for the calculation with the edge load determined with by the ground soil in front of the rubble foundation bed; by Method (5), the edge load may be determined according to the partial block stone and partial ground soil of the foundation bed. Due to different treatment for the edge load, that by Method (5) is slightly greater than that by Method (1), for example, Cases 3, 6 and 11 of ground with homogeneous soil. Because strength indexes of soil in layers of the ground are different, that by Method (5) will be generally less than that by Method (1). If the difference between the strength indexes is small, the calculated safety factors K ∗ will be relatively close to each other; if the difference is relatively large, the difference between the calculated safety factors K ∗ will be relatively large, too; especially in Cases 7 and 8 with low topsoil strength and high subsoil strength, the calculated safety factors for the bearing capacity will have very large difference, which shows relatively large difference between the soil layer strength is the main reason for its large difference. In Cases 7 and 8, the slip surface depth calculated by Method (5) is obviously less than that calculated by calculated value; for Case 12 with the topsoil strength greater than the subsoil strength, the slip surface depth calculated by Method (5) is obviously greater than that calculated by Method (1). The safety factor of the bearing capacity in Case 7 has the maximum difference; those by Methods (1) and (5) are 4.48 and 2.54, respectively, and the differences between the slip surface depths are also very large, 8.8 m and 3.89 m, respectively after further analysis. The engineering calculation section is shown in Fig. 9.11, and the strength indexes of the soil layer are detailed in Table 9.3. When the mean index is determined by Method (1), 2-m-thick sand cushion below the bottom surface of the rubble foundation bed or soft soil in the front cannot be considered simultaneously and sand cushion is adopted for the calculated value.

342

9 Ground Bearing Capacity

Backfill sand

Rubber foundation bed Sand cushion Soil layer 1 ϕ = 30 Soil layer 2

Soft soil

ϕ = 38.8

Fig. 9.11 Buttressed wharf section in Case 7

However, in Method (5), the strength of the slip surface (family) is adopted. Moreover, ϕ of the soil layer 2 is relatively large, the larger the ϕ of the soil is, the larger the calculated slip surface depth by Method (1) will be, which is because the slip surface depth by this method is relatively large and the average weighted ϕ is 35.57°. It is easy to see that the calculated slip surface depth by Method (5) is exactly on the bottom surface of the soil layer 1 with relatively small strength. In Case 7, the distribution (with the structure bottom front toe as the horizontal coordinate origin) of the limit load, calculated according to Method (5), along the foundation bottom is detailed in (Table 9.4). Because strength of the soil layer 2 is relatively large, the slip surface only passes through the soil layer 1 (the slip surface depth is less than 3.9 m); according to the calculated results, it fully shows that the slip surface passes through the soil layer with relatively small strength. From the strength index of the soil layer and the slip surface analysis, the calculation method for the heterogeneous soil is reasonable. Obviously, when the strength indexes of layers of such ground soil have relatively large difference, the partial safety factor of the ground bearing capacity, approximately calculated according to the homogeneous soil, cannot truthfully reflect actual load bearing capacity of the engineering ground any longer. (3) Comparison considering and without considering distribution of the limit load: During the determination of the ground bearing capacity, if the distributions of the limit load and design load are considered simultaneously, the safety factor of the bearing capacity K is generally less than K ∗ only considering the distribution of the design load. For Method (5), the design load is simplified to the uniformly distributed load according to the effective width; therefore, the closer to the uniformly distributed load the limit load is, the closer to K ∗ the K will be; in Method (6), the closer to the distribution form of the design load that of the limit load is, the closer to K ∗ the K will be. (4) Calculation method comparison of heterogeneous soil:

9.5 Ground Bearing Capacity of Gravity Wharf

343

Table 9.4 Distribution of the limit load along foundation bottom x (m)

Slip surface depth (m)

2.870

0.622

2.210

1.961

1.549

Limit load (kPa)

Design load (kPa)

Safety factor of partial failure

461.3

269.5

1.711

288.0

269.5

1.390

1.884

353.9

269.5

1.364

0.889

1.956

405.2

269.5

1.399

0.229

2.544

411.6

269.5

1.425

−0.432

2.814

410.1

269.5

1.441

−1.092

3.165

446.3

269.5

1.471

−1.752

3.344

463.1

269.5

1.502

−2.413

3.570

514.3

269.5

1.547

−3.073

3.883

563.3

269.5

1.602

−3.733

3.848

617.9

269.5

1.664

−4.394

3.708

778.8

269.5

1.767

−5.054

3.816

700.2

269.5

1.830

−5.714

3.829

756.5

269.5

1.900

−6.375

3.803

865.3

269.5

1.988

−7.035

3.841

897.9

269.5

2.072

−7.695

3.871

977.0

269.5

2.163

−8.356

3.851

1182.3

269.5

2.286

−9.016

3.773

1320.0

269.5

2.424

−9.676

3.875

1299.5

269.5

2.544

For the three Methods (4), (5), and (6) calculated according to heterogeneous soil, safety factor K for top surface of the rubble foundation bed (load action surface) calculated by Method (4) is minimum, especially for safety factor K m of partial bearing capacity for some engineering. As the edge load is relatively small, no cohesion (rubble) is provided with for the lower part near the calculated surface front toe, and the limit load near the front toe is frequently small; if the design load is eccentric forward, the safety factor K m of partial bearing capacity will certainly be relatively small. Actually, the load action surface at this time is the bottom surface of the engineering structure whose integral action is obvious; however, K m cannot reflect such integral action. For ground bearing capacity currently, it is universal for the load action surface to be adopted with bottom surface of rubble foundation bed, and the safety factors K and K ∗ calculated by Method (5) correspondingly are relatively large and by Method (6) are relatively small, because width of the surface calculated by Method (6) is generally less than that by Method (5). Simultaneously, the safety factor K determined according to the design load and the limit load is generally less than K ∗ determined only by considering the design load distribution. In contrast, Method (6)

344

9 Ground Bearing Capacity

is also more reasonable for determining the ground bearing capacity due to the width of the load action surface and its consideration for distribution of the design load and the limit load, and the safety factor K rule calculated by it is fine. (5) Analysis of strength safety factor: After the strength indexes of different soil layers are divided by safety factor, their difference decreases, and thickness of the supporting layer reduces as well; therefore, the strength safety factors calculated by different methods are with small difference. Please note that the better the cohesion of the ground soil, the larger the strength safety factor (relative to the safety factor of the bearing capacity); for example, for Cases 6, 11, and 12, their strength safety factors calculated by Method (6) are relatively large due to this obvious condition. And it is possible for this reason, the partial safety factor of adhesion is 2–2.5, and that of tan ϕ for internal friction angle is 1.2–1.5, in Canadian geotechnical engineering design handbook. The safety factor of bearing capacity calculated by methods for homogeneous soil and heterogeneous soil is relatively large apparently; the strength safety factor is not. Comparing Methods (1) and (6), strength safety factors of most engineering are close to each other, while the safety factors of bearing capacity differ a lot. Obviously, stability for the strength safety factor is better. The above analysis of the calculated result indicates that the calculated result rule of the strength safety factor is basically consistent with the safety factor of bearing capacity, thus mutually verifying the feasibility of the calculation method for bearing capacity of ground with heterogeneous soil; the analysis of the safety factor of bearing capacity and slip surface depth indicates that calculation method for bearing capacity of ground with heterogeneous soil can better reflect the stability of the ground bearing capacity.

9.5.4 Calculation for Allowable Bearing Capacity Strength safety factor “F0 > 1.0” is taken, after the soil strength index is divided by it, the ground ultimate bearing capacity and safety factor of bearing capacity are calculated, and it is the safety factor of bearing capacity under allowable bearing capacity. In practical application, the safety factor of bearing capacity and strength safety factor are considered simultaneously. On the basis of the reliability theory, it may be further transformed to partial safety factor of load (or action) (partial safety factor of bearing capacity) and partial safety factor of strength (partial safety factor of resistance); thus, the stability of the ground bearing capacity can be measured simultaneously with the partial safety factors of bearing capacity and strength index.

9.5 Ground Bearing Capacity of Gravity Wharf

345

9.5.5 Reliability of Ground Bearing Capacity JC method [7, 8] is adopted for the calculation of reliability, and the correlation [3] between c and tan ϕ is considered. In calculation, the first derivative for each variable is adopted with numerical calculation. The calculated results are detailed in Table 9.5, where K R is the partial safety factor of bearing capacity, β is the reliability index, and F0 is the stable partial safety factor. During calculation of the reliability, frequency value and quasi-permanent value [9] are applied for changeable action of load on the wharf surface, so that the corresponding partial safety factor is larger than the safety factor calculated with the standard value. Calculated results of reliability index are similar to the safety factors in Table 9.2. Mostly, the index is the minimum when it is calculated for the top surface of foundation bed by Method (4). However, for the partial safety factor (ratio of reliability index to partial safety factor), it is maximum under most conditions when calculated by Method (4), due to the relatively small variation coefficient for strength of rubble.

9.5.6 Calculation for Bearing Capacity Stability of Ground by Overall Stability Analysis Method Currently, method for slope stability has been widely used for bearing capacity calculation, and some countries even list it into “specification” [5]. Surface failure calculation mode is adopted as the method for slope stability is calculated directly with the design load. The corresponding safety factor (or partial safety factor) is the stability safety factor (or partial safety factor). In calculation, the load action surface is treated according to top surface of foundation bed. For vertical load, when partial stability is calculated, load distribution under the same condition is calculated according to ground bearing capacity. And when overall stability is calculated, if load is eccentric backward, the calculation will be in accordance with the ground bearing capacity; if the load is eccentric forward, it will be in accordance with uniform distribution, and simultaneously, width of the acting surface will be taken as the effective width (B1 = B − 2ev ). The vertical load is treated according to the above-mentioned methods for the following factor: When the total load remains unchanged within width of the calculated surface, forward eccentricity of design load increases, together with the safety factor calculated according to slope stability. For example, for gravity wharf of ground with homogeneous soil, under constant total load, distribution forms of load on the foundation bottom are backward eccentricity, uniform distribution, and forward eccentricity, and the calculated results of safety factor Fs are detailed in Table 9.6. In order to calculate partial stability, multiple slip surfaces (Fig. 9.12) through calculated surface point (−bm , 0) are taken, and the load distribution within the width

4.458

3.944

3.630

4.375

5.050

5.273

4.753

3.383

3.635

2.898

6.146

3.916

2 Block wharf

3 Block wharf

4 Caisson wharf

5 Block wharf

6 Block wharf

7 Buttressed wharf

8 Buttressed wharf

9 Buttressed wharf

10 Buttressed wharf

11 Caisson wharf

12 Caisson wharf

β Mean

β

4.162

2.525

3.560

3.432

4.607

5.203

2.786

5.351

3.942

2.954

5.579

2.308

F0

1.476

1.377

1.304

1.226

1.249

1.250

1.431

1.671

1.430

1.369

1.458

1.291

Note Variability information for a lack of soil in Case 12

4.388

KR

98 specifications (1)

1 Caisson wharf

Engineering case

Table 9.5 Calculated results of reliability

3.008

2.160

1.953

1.616

1.864

1.446

3.067

2.257

2.465

2.580

2.214

2.625

KR

3.416

2.089

2.702

2.054

2.548

1.675

2.903

2.308

2.714

2.341

2.979

2.150

β

Foundation bed top (4) F0

1.289

1.420

1.148

1.103

1.163

1.085

1.306

1.412

1.184

1.266

1.192

1.265

4.665

3.081

2.917

2.580

2.513

2.415

5.027

3.578

3.413

3.649

3.368

3.486

KR

3.816

3.013

3.406

2.813

2.782

3.380

3.292

3.444

2.536

2.986

4.088

2.420

β

1.430

1.519

1.259

1.204

1.234

1.183

1.521

1.612

1.407

1.402

1.389

1.340

F0

Foundation bed bottom a (5)

2.836

3.954

2.447

2.623

2.170

2.241

1.977

4.123

2.704

2.718

2.913

3.330

2.848

KR

3.426

2.473

3.324

2.563

2.619

2.813

2.982

2.910

2.293

2.541

3.888

2.161

β

1.433

1.503

1.248

1.188

1.228

1.163

1.535

1.682

1.402

1.395

1.351

1.325

F0

Foundation bed bottom b (6)

346 9 Ground Bearing Capacity

9.5 Ground Bearing Capacity of Gravity Wharf

347

Table 9.6 Safety factor Fs under backward eccentricity, uniform distribution of load and forward eccentricity Load condition

Simple slice method

Simplified Bishop method

Logarithmic helicoid method

Backward eccentricity ( p1 = 3 p2 )

0.867

1.321

1.290

Uniformly distributed

0.980

1.453

1.393

Forward eccentricity (3 p1 = p2 )

1.120

1.519

1.500

Note Horizontal load is not considered in calculation, and the slip surface is only adopted with the surface failure mode through foundation rear toe. The calculation equation for stability analysis method is detailed in Chap. 10 Vertical load is calculated according to uniform distribution

( xR , z R ) θ0

Vertical load distribution

θN

q − Be

− bm

− bm −1

x

z

Fig. 9.12 Schematic diagram for calculating ground bearing capacity stability by overall stability analysis method

of calculated surface requires to be known. For vertical load, if the design load is eccentric backward, it may be calculated according to the design load distribution; if the load is eccentric forward, the vertical load within the width [−bm , 0] will be calculated according to uniformly distributed load. Generally, the horizontal load within width of calculated surface is unknown, and its specific treatment method requires studying. For example, the following treatment may be adopted: For engineering with soil pressure as the main horizontal load, distribution of horizontal force on the foundation bottom is calculated according to the uniform distribution; for engineering with wave force as the main horizontal load, the distribution of horizontal force on the foundation bottom is considered to be proportional (tan δ) to the vertical load during calculation, for safety. Safety factor of slip surface through calculated surface rear toe (−Be , 0) is regarded as the overall stability safety factor and the minimum safety factor of multiple slip surfaces through point (−bm , 0) the partial stability safety factor. The calculated results are detailed in Table 9.7.

348

9 Ground Bearing Capacity

Table 9.7 Bearing capacity calculation Fs by stability analysis method Engineering case

Simple slice method Partial/Integral (1)

Simplified Bishop method Partial/Integral (2)

Logarithmic helicoid Partial/Integral (3)

1 Caisson wharf

1.024/1.135

1.343/1.338

1.354/1.365

2 Block wharf

0.861/0.961

1.265/1.276

1.264/1.285

3 Block wharf

0.866/1.114

1.337/1.469

1.267/1.502

4 Caisson wharf

0.754/1.044

1.164/1.492

1.167/1.517

5 Block wharf

0.981/1.108

1.510/1.880

1.456/1.761

6 Block wharf

0.867/1.277

1.351/1.706

1.272/1.662

7 Buttressed wharf

0.805/1.055

1.162/1.138

1.134/1.177

8 Buttressed wharf

0.855/1.141

1.300/1.238

1.232/1.268

9 Buttressed wharf

0.786/1.088

1.183/1.193

1.153/1.215

10 Buttressed wharf

0.743/1.114

1.169/1.383

1.114/1.395

11 Caisson wharf

0.994/1.087

1.565/1.923

1.492/1.755

12 Block wharf

0.968/1.292

1.174/1.870

1.226/1.737

13 Caisson wharf

0.865/1.180

1.345/1.505

1.334/1.540

It shall be stated that, compared with the common analysis method for slope stability, the load treatment is different when the bearing capacity is calculated by method for slope stability, and their safety factors are different, either. When slip circle of slip surface through “h  < 0” is relatively long, the non-slip moment generated by corresponding slip circle will be relatively small (it will be discussed in Chap. 10); therefore, for such horizontal slope surface bearing load, its safety factor calculated by simple slice method is relatively small. The safety factors calculated by the simplified Bishop method and the logarithmic helicoid method are close to each other. When bearing capacity stability of ground is calculated by overall stability analysis method, one important advantage is that the distribution of horizontal force on the calculated surface may be calculated with different distribution forms. For calculation of ground bearing capacity with distribution of horizontal force inconvenient to determine, it may be with more actual meaning.

9.6 Ground Bearing Capacity of Breakwater 9.6.1 Possible Failure Mode of Breakwater Ground Main load on the breakwater ground is the self-weight (vertical load) of the breakwater and the wave force (horizontal load). The wave force is dynamic load, and it acts

9.6 Ground Bearing Capacity of Breakwater

349

pz px

( A)

1 2

Fig. 9.13 Direction of horizontal force from inner breakwater to outer breakwater

pz px

(B )

Fig. 9.14 Direction of horizontal force from inner breakwater to outer breakwater

from inner breakwater to outer breakwater within this period (Fig. 9.13, hereinafter referred to as forward) and reverse for another period (Fig. 9.14, hereinafter referred to as backward). If the wave force is forward, distribution of the vertical load is necessarily eccentric forward. Though the total design load is less than the total limit load, calculated results of limit load in field failure mode for ground with homogeneous soil or ground with small topsoil strength and large subsoil strength indicate that the limit load that the ground may bear is eccentric backward generally. As the design load near the front toe is relatively large, while the limit load is relatively small, the soil mass (action of dumped riprap before breakwater is considered) between slip surfaces 1 and 2 has been yielded, and partial soil mass has been squeezed out, which result in the reduction of soil mass strength within slip surface 2. And for ground with large topsoil strength and small subsoil strength, slip surface 1 through subsoil also exists to achieve the above-mentioned phenomenon. When the next wave arrives, partial load will be inevitably added on the adjacent soil mass beyond slip surface 2 as the soil mass within slip surface cannot bear corresponding load, which will result in expansion of the yielded soil mass and increase of squeezed soil mass. If the wave force is backward, the vertical load is necessarily eccentric backward, which is similar to that of the forward wave force, with the difference lying at the wave force strength and damaged soil mass area. If the design load is static, relatively small partial failure may only result in increase of abnormal sedimentation for ground, and it is uneasy to cause overall

350

9 Ground Bearing Capacity

failure. However, under the action of relatively large dynamic wave force, self-weight of the breakwater is dynamic load, too. Under the action of dynamic load, on the one hand, for the reoccurring of unstable pore water pressure (covering Mandel effect [10, 11]), soft soil with relatively high moisture content will be softer and easy for flowing; on the other hand, partial soil mass yields due to insufficiency of its strength to bear the actual shear stress (as a result of the Mandel effect, maximum shear stress under the foundation edge may reach about 1.8 times of its initial value [10]) thus result in the squeezing out of soil mass and reduction of its strength. After several wave periods, the soil mass increases along with the increasing of damaged soil mass area. Taking certain engineering [2] as an example, about 1.5 m high and 1.5 m wide mud mound is on the south and north sides of the caisson, which is abnormal obviously; simultaneously, the caisson will be with relatively large abnormal sedimentation necessarily. Due to the difference of forward and backward wave force strength, the caisson will also move horizontally.

9.6.2 Calculation Mode for Ground Bearing Capacity of Breakwater According to the above discussion, possible failure mode of ground soil shall be twoway failure mode. Currently, calculation methods for ground bearing capacity are considered according to one-way failure mode. For such asymmetrically two-way failure mode, no appropriate limit load calculation mode is available. In principle, the foundation bottom can be divided into two parts under calculation of two-way failure mode, it is calculated with one-way failure at different directions, and the division point is determined to achieve equal limit load at this point. In theory, it is not very difficult. However, for boundary condition of foundation bottom as has been stated before, only the total horizontal force can be generally provided within practical problems, while calculation of limit load (and field failure mode) requires load boundary condition of foundation bottom, general boundary condition is assumed as: px = tan δ( pz + c/ tan ϕ), where pz is the limit load (vertical distribution load) to be solved and px is the distribution of the horizontal force along the foundation bottom. The limit load corresponding to ground with homogeneous soil is eccentric backward, which means that horizontal limit load is in distribution form with small front toe and large rear toe. If the resultant action point of wave force is relatively far to the foundation bottom, the actually delivered wave force may be in distribution form with large front toe and small rear toe (Fig. 9.15) along the foundation bottom. Obviously, when the horizontal force is distributed with small front toe and large rear toe, vertical limit load near the front toe is greater than that when the horizontal force is with limit load in distribution with large front toe and small rear toe. Without doubt, under relatively large horizontal force, the different distributions of the horizontal force along the foundation bottom greatly affect the distribution

9.6 Ground Bearing Capacity of Breakwater

351

Actual horizontal load distribution

Assumed horizontal load distribution

Fig. 9.15 Schematic diagram for distribution of horizontal force

status of limit load; the reasonable distribution form of horizontal design load along the foundation bottom cannot be avoided in the study of whether the ground soil will suffer partial failure. Especially, the different dynamic load conditions at the action direction of the horizontal force within a period and another period are required to be considered simultaneously, but it is a pity that such study is still rare at present. Considered from the overall stability covering breakwater structure and ground, overall failure may be impossible due to integrity of breakwater structure, though it is partial failure of relatively small ground soil. If it is calculated according to the most possible failure direction, and the ground soil is free from partial failure, it will be impossible for the other side to suffer from failure. Therefore, bearing capacity stability of breakwater ground may be similar to the gravity wharf, and it may be simplified to be calculated according to the oneway failure mode, namely the most possible failure direction. And moreover, the followings shall be noted. Width of the load action surface when rubble foundation bed is provided with: When the horizontal load is relatively large, the width shall be determined considering the vertical and horizontal design loads simultaneously; after the forward horizontal load acting on the top surface of foundation bed is delivered to the bottom surface of foundation bed, backward diffusion should not be considered. Therefore, for the sake of safety, rear toe of the load action surface should not be expanded backward (bottom surface of foundation bed) from rear toe (horizontal position) for top surface of the foundation bed. Overall safety factor of bearing capacity: For two-way failure mode with wave force as the main failure load, if soil mass of slip surface through calculated surface midpoint (−Be /2, 0) suffers from failure, the engineering may suffer from overall failure. Therefore, the minimum safety factor of multiple slip surfaces through calculated surface (−Be , −Be /2) shall be regarded as the integral safety factor of bearing capacity. However, calculated result indicates that, with large topsoil strength and relatively thick soil layer, the dangerous slip surface may cross deep soil, the safety factor corresponding to the slip surface through calculated surface rear toe may be smaller, and in addition, when the minimum safety factor of multiple slip surfaces through calculated surface (−Be /2, 0) is regarded as the safety factor of the partial

352

9 Ground Bearing Capacity

bearing capacity, the failure of slip surface soil mass through calculated surface midpoint (−Be /2, 0) is included. Therefore, the calculation of overall and partial safety factors is the same as the ground bearing capacity of gravity wharf.

9.6.3 Calculated Result of Bearing Capacity Calculated result: Safety factors of bearing capacity K, K ∗ , and K m as well as the strength safety factor F0 are detailed in Table 9.8; ground soil conditions and calculated results of slip surface depth are detailed in Table 9.9; safety factors Fs of ground bearing capacity calculating by stability analysis method are detailed in Table 9.10.

9.6.4 Analysis of Calculated Result (1) Tianjin University’s method and Hansen method: For semi-circular caisson Case 1, the safety factor of bearing capacity calculated by the Hansen method is apparently greater than that by the Tianjin University’s method, which is different from other engineering. It is because boundary conditions of them are different: boundary condition of foundation bottom by the Hansen method is “tan δ = Ph /(Pv + Bc/ tan ϕ)”; for limit load calculation, boundary condition of Tianjin University’s method is the same as the Hansen method; however, in calculation, “tan δ = Ph /Pv ” in “Code for Foundation in Port Engineering” is applied for calculating the ground bearing capacity. Because Pv of the engineering is relatively small, the above-mentioned result is obtained. (2) Action of edge load: Rubble foundation bed of semi-circular caisson engineering is open, as the Tianjin University’s method (including Hansen method) can only calculate the uniformly distributed edge load, edge load formed with partially dumped riprap before the structure can only be treated approximately. Considering that width of partial dumped riprap is smaller than that of the foundation bottom, and the method (including the Hansen method) in the table is calculated regardless of the edge load, its safety factor of bearing capacity is relatively small. The method for heterogeneous soil is calculated according to the edge load formed with the partial dumped riprap before the structure entirely. (3) The overall safety factor calculated by Method (6) is greater than that by Method (5), which is different from the condition of gravity wharf. As for the gravity wharf, the effective width of Method (5) is generally greater than the actual width of Method (6); it is just the way for the above 4 breakwater engineering. (4) Semi-circular caisson (Case 2) is analyzed as key point below, and the engineering structure section is detailed in Fig. 9.16.

9.6 Ground Bearing Capacity of Breakwater

353

Table 9.8 Safety factor of bearing capacity Engineering case

Safety factor

Tianjin University’s method (1)

Hansen method (2)

Mean index (3)

Foundation bed bottom a Heterogeneous soil (5)

Foundation bed bottom b Heterogeneous soil (6)

0 Semi-circular caisson (consolidated quick shear)

K

2.761

3.496

2.295

2.098

2.293

2.293

2.551

F0

1.722

Semi-circular caisson (undrained triaxial shear)

K

2.562

F0

1.683

1 Semi-circular caisson (consolidated quick shear)

K

2.244

F0

1.628

Semi-circular caisson (undrained triaxial shear)

K

1.926

F0

1.599

2 Vertical caisson

K

3.602

3 Vertical caisson

K∗ Km 3.365

2.138

2.293

2.375

1.670

1.852

1.474

1.723

K∗

1.664

1.914

Km

1.664

1.859

1.132

1.293

1.601

1.685

K∗

1.746

1.881

Km

1.729

1.764

1.303

1.389

1.118

1.232

1.260

1.388

2.251

2.112

1.872

1.678

K∗ Km

2.999

2.868

3.301

3.636

Km

2.143

1.633

F0

1.758 2.963

K∗ Km F0

1.313

2.539

3.514

1.340 1.145

K∗

K

3.253

1.260 1.071

2.898

1.588

1.681

2.570

2.240

2.865

3.160

1.765

1.125

1.290

1.277

Note The engineering information is provided by the drafting group of “Code for Foundation in Port Engineering”

The result calculated according to heterogeneous soil and that calculated approximately according to the weighted average index for thickness of soil layer within the supporting layer differ a lot, and the calculated safety factor of ground bearing capacity is also relatively large though the action of edge load is not counted. It may be obtained from the maximum depth of the slip surface that the former one is apparently greater than the latter one, as topsoil strength of the ground is large, while its subsoil strength is small. The strength index calculated according to the weighted average for thickness of soil layer is also relatively large, which is the main reason

12.5

8.5

13.5

4.0

0.6

Homogeneous soil

4 Vertical caisson

14.7

4.0

0

33

32 19.68

29.94

35.2

35.2

0

22.5

3.2

2.0

3 Vertical caisson

Semi-circular caisson (undrained triaxial shear)

9.0

3.2

2.0

0

14.7

2 Semi-circular caisson (consolidated quick shear)

34

0

1.7

7.0

Semi-circular caisson (undrained triaxial shear)

34

11.6

Friction angle (°)

0

Cohesion (kPa)

1.7

Thickness of soil layer (m)

Ground soil condition

1 Semi-circular caisson (consolidated quick shear)

Engineering case

Table 9.9 Ground soil condition and slip surface depth

2.0

4.0

1.45

1.0

Thickness of rubble foundation bed (m)

12.9

3.90

4.60

2.80

3.20

3.69

10.96

5.98

5.92

7.21

5.87

Foundation bed bottom a Heterogeneous soil

Slip surface depth (m) Tianjin University’s method

3.69

11.15

5.97

5.96

7.0

5.17

Foundation bed bottom b Heterogeneous soil

354 9 Ground Bearing Capacity

9.6 Ground Bearing Capacity of Breakwater

355

Table 9.10 Bearing capacity Fs calculation by stability analysis method Engineering case

Simple slice method Partial/Integral (1)

Simplified Bishop method Partial/Integral (2)

Logarithmic helicoid Partial/Integral (3)

1 Semi-circular caisson (consolidated quick shear)

1.552/1.555

1.742/1.677

1.808/1.729

(Undrained triaxial shear)

1.390/1.331

1.532/1.466

1.612/1.471

2 Semi-circular caisson (consolidated quick shear)

0.954/1.017

1.156/1.201

1.203/1.234

(Undrained triaxial shear)

0.879/0.854

1.102/1.038

1.158/1.063

3 Vertical caisson

0.856/1.130

1.238/1.459

1.251/1.551

4 Vertical caisson

0.817/0.971

0.961/1.112

1.038/1.131

Fig. 9.16 Sectional drawing for semi-circular caisson structure

for the relatively large ground bearing capacity calculated by corresponding method. If it is calculated according to heterogeneous soil, it will be relatively reasonable for the calculated ground bearing capacity to be relatively small as the slip surface crosses the soft soil at sublayer. Partial failure of engineering occurs in construction period; if undrained triaxial shear strength index and calculation method for bearing force for ground with heterogeneous soil are adopted for the soft soil layer, the engineering will not be presented with failure though the calculated safety factor of bearing capacity is not large. Another basis of this saying is the calculated stability safety factor and strength safety factor; in the Japanese “Technical Standard of Port Facilities [5],” “arc slipping analysis method of Bishop is the standard calculation method for the bearing capacity of shallow foundation under the action of heart and inclined load,” and it is “the calculation method has been verified, and with enough bearing capacity.” For breakwater with wave force counted in, the standard safety factor is greater than 1.0.

356

9 Ground Bearing Capacity

The safety factor calculated by this method and given in Table 9.10 is 1.038, and it is satisfactory; though its meaning is not the same as the strength safety factor 1.071 under identical calculation condition, their safety degree is consistent. If soft soil layer is adopted with dynamic undrained triaxial shear strength index [2] (5.32 kPa) of simulated wave load action, the calculated safety factors K ∗ of bearing capacity will be 1.458 [Method (1)], 1.480 [Method (2)], 0.667 [Method (5)], and 0.737 [Method (6)], respectively. Though the calculated safety factor of bearing capacity is relatively small for that action of edge load is not considered in Methods (1) and (2), it cannot give reasonable explanation for the partial failure occurred, while the safety factors calculated by the other two methods have reflected the fact of partial failure that the engineering suffers. Actually, reasonable determination and calculation method of soil index are the two main problems in engineering analysis calculation. Calculation and analysis of this engineering indicate that contributors of partial failure in engineering cover not only the decrease of strength index under the action of dynamic load, but also that the ground bearing capacity calculated approximately according to the weighted average index for thickness of soil layer within the supporting layer cannot reflect the calculation method for the actual load bearing capacity of engineering ground.

9.7 Ground Bearing Capacity by Generalized Limit Equilibrium Method For the ground bearing capacity calculated above, limit equilibrium method (helicoid calculation mode) is adopted for the calculation of its limit load. The generalized limit equilibrium method (helicoid–helicoid–plane calculation mode) in Chap. 8 is adopted below to calculate the ground limit load, and moreover, the ground bearing capacity is determined by the method as same as the helicoid mode; thereinto. If the limit load near the front toe of the calculated surface is very large as the topsoil strength is far larger than that of the subsoil or relatively large edge load is provided with near the front toe of the calculated surface, the following treatment may be carried out according to the conditions of design load and limit load: If the upper limit load within [−bk , −bk−1 ] is 1.25 times greater than the upper limit load within [−bk+1 , −bk ], and the safety factor of partial bearing capacity within [−bk , 0] is greater than 1.25, it will be slightly less than the limit load within [−bk , 0]; the limit load within [−bk+1 , −bk ] will be recalculated, until the limit load within [−bk+1 , −bk ] meets the above-mentioned conditions. Thus, the ground bearing capacity is more reasonable with enough safety factor of partial bearing capacity guaranteed. For equation “Fsc = Fsϕ = 1.01 + 0.03(tan ϕ − tan δ)” in the calculation mode, when “Fsc = Fsϕ < 1.0”, it is calculated according to “Fsc = Fsϕ = 1.0”. Tables 9.11 and 9.12 show calculated results of Tianjin University’s method, Hansen method, helicoid calculation mode, and helicoid–helicoid–plane calculation mode.

6 Block wharf Forward eccentricity of design load

5 Block wharf Slightly backward eccentricity of design load

4 Caisson wharf Forward eccentricity of design load

3 Block wharf Slightly forward eccentricity of design load

2 Block wharf Backward eccentricity of design load

K∗

K

4.625

4.373

4.912

4.671

3.629

3.474 3.785

3.766

Km

4.204

K∗

K

3.083

3.042 3.247

3.266

Km

3.204

K∗

K

2.504

3.243

Km

3.115 3.517

3.472

K∗

K

1.715

Km

1.158 3.716

3.788

K∗

K

4.186

2.373 1.846

2.470

4.854

4.493

3.259

3.732

3.427

2.693

3.112

2.910

2.273

3.543

3.223

1.587

3.878

3.221

1.749

2.686

2.356

4.172

3.831

3.020

3.020

2.686

2.220

2.802

2.428

2.206

2.722

2.589

2.784

3.251

3.122

1.362

2.382

2.003

4.113

3.682

2.919

2.919

2.685

1.955

2.754

2.308

2.001

2.728

2.556

2.582

3.394

3.197

1.278

2.453

1.998

(continued)

Helicoid–helicoid–plane

Foundation bed bottom b Helicoid

Helicoid

Helicoid–helicoid–plane

Foundation bed bottom a

Km

2.744

Hansen method

2.615

K

1 Caisson wharf Forward eccentricity of design load

Tianjin University’s method

K∗

Safety factor

Engineering case

Table 9.11 Safety factor comparison for ground bearing capacity of gravity wharf

9.7 Ground Bearing Capacity by Generalized Limit Equilibrium … 357

10 Buttressed wharf Forward eccentricity of design load

9 Buttressed wharf Forward eccentricity of design load

8 Buttressed wharf Forward eccentricity of design load

7 Buttressed wharf Forward eccentricity of design load

Engineering case

1.195

2.564

Km

3.002 2.991

3.212

K∗

K

1.358

2.225 2.672

2.736

Km

2.877

K∗

K

1.502

Km

2.054 2.363

3.753

K∗

K

1.364

3.993

3.909 2.060

3.636

1.113

3.288

2.787

1.282

2.969

2.435

1.393

2.524

2.117

1.279

2.850

1.193

2.929

0.915

2.988

2.321

0.997

2.547

1.870

1.099

2.427

1.849

1.015

2.320

1.722

0.856

3.285

2.529

0.931

2.847

2.053

1.015

2.576

1.884

1.011

2.597

1.827

2.690

(continued)

Helicoid–helicoid–plane

Foundation bed bottom b Helicoid

Helicoid

Helicoid–helicoid–plane

Foundation bed bottom a

Km

3.945

Hansen method

2.544

4.480

Tianjin University’s method

K∗

K

Km

Safety factor

Table 9.11 (continued)

358 9 Ground Bearing Capacity

13 Caisson wharf Slightly forward eccentricity of design load

12 Block wharf Slightly backward eccentricity of design load Wave force is counted in

2.106

4.327

Km

4.750 5.242

5.725

K∗

K

3.053

Km

2.701 3.053

3.527

K∗

K

3.428

3.080 2.632

2.834

1.930

5.511

4.469

2.812

2.812

2.607

2.543

3.215

3.050

1.685

4.695

3.673

3.050

3.050

2.282

2.472

2.472

2.420

1.552

4.874

3.744

2.801

2.801

2.173

2.480

2.480

2.456

Helicoid–helicoid–plane

Foundation bed bottom b Helicoid

Helicoid

Helicoid–helicoid–plane

Foundation bed bottom a

Km

2.824

Hansen method

3.204

K

11 Caisson wharf Slightly backward eccentricity of design load

Tianjin University’s method

K∗

Safety factor

Engineering case

Table 9.11 (continued)

9.7 Ground Bearing Capacity by Generalized Limit Equilibrium … 359

4 Vertical caisson

1.765

Km

2.570 2.865

2.539

K∗

2.963

2.143

Km

K

3.301

2.999

K∗

3.253

K

3 Vertical caisson

3.602

1.260

1.118 1.260

2.112

Km

1.926

K∗

K

1.729

1.601 1.746

2.251

Km

2.244

K∗

K

1.664

1.474 1.664

3.365

Km

2.562

K∗

K

Semi-circular caisson (undrained triaxial shear)

2 Semi-circular caisson (consolidated quick shear)

Semi-circular caisson (undrained triaxial shear)

2.098 2.293

3.496

1.637

2.946

2.580

1.824

3.413

3.063

1.245

1.245

1.121

1.524

1.679

1.547

1.690

1.690

1.515

2.007

2.123

1.954

1.125

3.160

2.240

1.633

3.636

2.868

1.340

1.388

1.232

1.764

1.881

1.685

1.859

1.914

1.723

2.375

2.551

2.293

1.069

3.269

2.258

1.402

3.765

2.926

1.291

1.347

1.213

1.470

1.812

1.593

1.799

1.905

1.709

1.874

2.389

2.085

Helicoid–helicoid–plane

Foundation bed bottom b Helicoid

Helicoid

Helicoid–helicoid–plane

Foundation bed bottom a

Km

2.761

Hansen method

2.293

K

1 Semi-circular caisson (consolidated quick shear)

Tianjin University’s method

K∗

Safety factor

Engineering case

Table 9.12 Safety factor comparison for ground bearing capacity of breakwater

360 9 Ground Bearing Capacity

9.7 Ground Bearing Capacity by Generalized Limit Equilibrium …

361

Safety factor of bearing capacity under helicoid–helicoid–plane calculation mode is close to that under the helicoid calculation mode, and they mutually verify the reliability of bearing capacity method for ground with heterogeneous soil and nonuniformly distributed edge load. Certainly, for safety factor of partial bearing capacity, the calculated value for minor engineering under helicoid–helicoid–plane calculation mode is apparently less than that under the helicoid calculation mode due to the above-mentioned treatment [e.g., breakwater Case 1 (consolidated quick shear)], and it is normal. Combining the above discussion, the safety factor of bearing capacity (or partial safety factor) calculated by the calculation method for bearing capacity of ground with heterogeneous soil and non-uniformly distributed edge load can basically reflect the actual load bearing capacity of engineering ground. Thereinto, the load action surface is bottom surface of the rubble foundation bed, width of the load action surface is the actual width, and simultaneously, the distribution of design load and limit load is considered; besides, it is recommended that the ground bearing capacity be determined by calculation method without violating the yield criterion. Additionally, further studies on main fields are in need, for example, safety factor of bearing capacity and strength safety factor, which one is more scientific to measure stability of ground bearing capacity. Considered from the probabilistic method, together with the uncertainty of load and strength, it is more scientific to measure the stability jointly with the partial load coefficient (or partial factor of action) and the partial safety factor of strength (or partial safety factor of resistance); however, the relationship between the partial load coefficient and the partial safety factor of strength shall be solved reasonably.

References 1. Shen ZJ (2000) Theoretical soil mechanics. China Water Power Press, Beijing 2. Fan QJ, Li NY (2004) Reasons and countermeasures for North Bank’s part failure in the second phase regulation project of Yangtze estuary. China Harbour Eng 2:1–8 3. JTJ 250-98 Code for Foundation in Port Engineering 4. GB 5007-2002 Code for Design of Building Foundation 5. Japanese Port and Harbour Association (1993) Explained by technical standard of port facilities (revision, vol 1). Translated by Committee of Water Borne Transportation, CECS. References by Committee of Water Borne Transportation, CECS 6. The Ministry of Communications of the People’s Republic of China (1988) Technical standard for port engineering (1987). China Communications Press, Beijing 7. Wu SW (1990) Analysis of structure reliability. China Communications Press, Beijing 8. Drafting group of “Unified standard of reliability of structural design for harbour engineering” (1992) Reliability of harbor engineering structures. China Communications Press, Beijing 9. JTJ 215-98 Load Code for Harbour Engineering 10. Wei RL (1987) Strength and deformation of soft clay. China Communications Press, Beijing 11. Huang WX (1983) Engineering properties of soil. China Water & Power Press, Beijing

Chapter 10

Slope Stability

10.1 Introduction The limit equilibrium method, main theoretical foundation of the methods for studying the slope stability analysis in a long time [1, 2], develops with the equilibrium analysis of force and moment by dividing several isolators as the starting point. Its study has been active all the time, and the slice methods are numerous to enumerate. However, as what it summarizes is static indetermination, for planes, except the specific plane and logarithmic helicoid, selected as the slip surface, the corresponding calculation equation can be derived generally after the force on the soil mass is assumed or simplified. Theoretically, these methods are not distinguishable essentially, except for the assumption made for the inter-slice force and the interaction force on the slip surface to eliminate the static indetermination and the methods for calculating the safety factor [3]. The research work has been relatively sufficient and gained extensive application. The starting point of the study is somehow limited; for example, it is not easy to observe that the horizontal force equilibrium equation and moment equilibrium equation of the soil strip are not mutually independent. And individual analysis methods, derived by the simultaneous solution for the vertical force, horizontal force, moment equilibrium equation, and yield condition of the soil strip, are not correct. It is generally believed that pursuing the slice method is unlikely to achieve essential progress. In this chapter, force and moment equation on slip surface are established at first with the fundamental equation, and then, limit analysis is carried out according to the force and the moment equilibrium. With this study clue, some common analysis methods on engineering at present may be derived easily, for instance, logarithmic helicoid method, simple slice method, simplified Bishop method, unbalanced thrust transmission method, and Morgenstern–Price method. Obviously, in order to achieve essential progress in the study of slope stability analysis, the problem that the quantity of the equations is less than that of the © Springer Nature Singapore Pte Ltd. and Zhejiang University Press, Hangzhou, China 2020 C. Huang, Limit Analysis Theory of the Soil Mass and Its Application, https://doi.org/10.1007/978-981-15-1572-9_10

363

364

10 Slope Stability

unknowns must be solved, so that the slope stability becomes statically determinate. The derivation of the extremum condition for the yield function is one way to solve it. How to establish the slope stability analysis method with the extremum condition of the yield function is worth studying. Assumption or simplification of the force on the soil mass is made in this chapter by reference to the extremum condition of the yield function, and it will be more reasonable than the artificial assumption or simplification. On this basis, compound slip surface method is derived. In addition, variational principle of the surface failure mode is established in Chap. 7, which lays a solid theoretical foundation for the establishment of the generalized limit equilibrium method. For general homogeneous soil slope, analysis method with no force assumption may be derived by the generalized limit equilibrium method. Therefore, the study topic of this chapter is to establish such analysis method for general slope.

10.2 Analysis Method for Homogeneous Soil Slope The slope stability analysis needs to give solution to the analysis method and slip surface. Chaps. 5 and 7 discuss the stability analysis method for homogeneous soil slope; thereinto, the slip surface may be helicoid and plane–helicoid, or arc surface and plane–arc surface. At present, many common stability analysis methods for soil slope on engineering are available, for instance, the simplified Bishop method and the simple slice method for arc slip surface. In order to discuss the effect of different analysis methods for different slip surfaces on the calculated result, the helicoid and arc surface methods in Chaps. 5 and 7 are calculated similarly according to the slice method (the calculated result is slightly different with that given in Chaps. 5 and 7), so that the calculation process is consistent with that of the simplified Bishop method and the simple slice method. The following is a comparison of the calculated results. (1) Comparison for the safety factor of the homogeneous soil slope (c/(γ H ) = 0.1) is detailed in Table 10.1a, b, c and d. (2) The slope crest is with load (when “x/H ≤ −1.0,” it is uniform load: q/γ H = 1.0; when “−1.0 ≤ x/H ≤ 0,” it is linear distribution), the comparison of safety factor for homogeneous soil slope is detailed in Table 10.2a, b, c and d. For homogeneous soil slope, the slip surface is the logarithmic helicoid, and the safety factors calculated by the analysis method and simplified Bishop method of the arc surface are very close to each other, and their difference is generally less than or equal to the permissible error in engineering. Generally, the safety factor calculated by simple slice method of the arc surface is relatively small, which is long known to people. As is well known, when it is arc slip surface, specific calculation equation can be derived after necessary simplification is carried out for force on the slip mass according to limit equilibrium method; though the virtual work equation-based generalized limit equilibrium method is adopted, for heterogeneous soil, the derivation of calculation equation for the safety factor when no other simplification condition is added

10.2 Analysis Method for Homogeneous Soil Slope

365

Table 10.1 a Safety factor of slope (slip surface: logarithmic helicoid). b Safety factor of slope (slip surface: arc surface). c Safety factor of slope (simple slice method: arc slip surface). d Safety factor of slope (simplified bishop method: arc slip surface) β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

60.0

0.667

0.788

0.913

1.020

1.144

1.267

1.383

1.533

55.0

0.704

0.845

0.971

1.100

1.239

1.359

1.510

1.677

50.0

0.744

0.901

1.039

1.178

1.331

1.478

1.636

1.809

45.0

0.785

0.960

1.117

1.276

1.433

1.601

1.778

2.002

40.0

0.823

1.015

1.193

1.372

1.568

1.754

1.958

2.202

35.0

0.875

1.093

1.296

1.505

1.717

1.934

2.175

2.451

30.0

0.932

1.185

1.425

1.661

1.907

2.171

2.458

2.865

25.0

0.999

1.300

1.586

1.879

2.179

2.492

2.880

20.0

1.091

1.470

1.835

2.207

2.590

3.044

15.0

1.255

1.780

2.296

2.823

60.0

0.660

0.779

0.894

1.006

1.122

1.238

1.365

1.503

55.0

0.697

0.831

0.956

1.082

1.210

1.344

1.488

1.642

50.0

0.735

0.883

1.026

1.168

1.308

1.458

1.614

1.790

45.0

0.775

0.942

1.101

1.260

1.419

1.588

1.767

1.964

40.0

0.820

1.009

1.189

1.368

1.550

1.739

1.942

2.168

35.0

0.870

1.086

1.291

1.496

1.705

1.921

2.157

2.419

30.0

0.929

1.180

1.419

1.654

1.900

2.153

2.429

2.742

25.0

0.996

1.298

1.582

1.867

2.162

2.475

2.814

20.0

1.089

1.468

1.831

2.198

2.578

2.989

15.0

1.255

1.777

2.289

2.816

60.0

0.657

0.781

0.890

0.999

1.115

1.231

1.351

1.481

55.0

0.690

0.822

0.945

1.065

1.190

1.319

1.453

1.598

50.0

0.731

0.866

1.004

1.136

1.274

1.418

1.563

1.725

45.0

0.761

0.925

1.072

1.219

1.369

1.531

1.700

1.882

40.0

0.804

0.980

1.147

1.317

1.493

1.671

1.861

2.073

35.0

0.845

1.048

1.240

1.430

1.623

1.829

2.044

2.288

30.0

0.890

1.128

1.350

1.569

1.799

2.036

2.298

2.587

25.0

0.945

1.225

1.492

1.755

2.027

2.314

2.627

20.0

1.021

1.363

1.692

2.023

2.363

2.732

15.0

1.151

1.608

2.048

2.501

(a)

(b)

(c)

(continued)

366

10 Slope Stability

Table 10.1 (continued) β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

60.0

0.653

0.773

0.884

1.000

1.121

1.237

1.366

1.504

55.0

0.692

0.827

0.956

1.082

1.210

1.345

1.487

1.639

50.0

0.728

0.876

1.025

1.170

1.317

1.460

1.618

1.795

45.0

0.770

0.940

1.102

1.261

1.425

1.595

1.769

1.966

40.0

0.814

1.005

1.190

1.375

1.551

1.740

1.949

2.172

35.0

0.865

1.084

1.291

1.496

1.707

1.931

2.166

2.429

30.0

0.921

1.177

1.420

1.658

1.896

2.155

2.441

2.751

25.0

0.988

1.291

1.576

1.863

2.163

2.475

2.813

20.0

1.079

1.463

1.825

2.194

2.577

2.990

15.0

1.248

1.769

2.279

2.809

(d)

Table 10.2 a Safety factor of slope with load at crest (slip surface: logarithmic helicoid). b Safety factor of slope with load at crest (slip surface: arc surface). c Safety factor of slope with load at crest (simple slice method: arc slip surface). d Safety factor of slope with load at crest (simplified Bishop method: arc slip surface) β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

0.473

0.658

0.837

1.016

1.182

1.377

1.556

1.774

40.0

0.497

0.696

0.893

1.089

1.289

1.489

1.722

1.963

35.0

0.528

0.751

0.967

1.187

1.412

1.647

1.914

2.194

30.0

0.565

0.813

1.055

1.309

1.572

1.853

2.152

2.490

25.0

0.613

0.900

1.185

1.479

1.783

2.107

2.465

2.883

20.0

0.690

1.033

1.356

1.686

2.037

2.392

2.803

15.0

0.808

1.179

1.550

1.931

2.332

2.766

45.0

0.476

0.657

0.832

1.007

1.179

1.358

1.549

1.758

40.0

0.504

0.702

0.896

1.091

1.288

1.495

1.712

1.949

35.0

0.537

0.758

0.974

1.195

1.421

1.660

1.914

2.194

30.0

0.578

0.829

1.077

1.329

1.593

1.875

2.183

2.518

25.0

0.635

0.927

1.219

1.519

1.832

2.169

2.531

2.937

20.0

0.720

1.075

1.407

1.739

2.087

2.457

2.861

15.0

0.847

1.216

1.574

1.937

2.321

2.730

0.443

0.606

0.764

0.925

1.094

1.263

(a)

(b)

(c) 45.0

1.444

1.638 (continued)

10.2 Analysis Method for Homogeneous Soil Slope

367

Table 10.2 (continued) β(◦ )

ϕ(◦ ) 5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

40.0

0.465

0.638

0.807

0.981

1.165

1.356

1.562

1.787

35.0

0.487

0.675

0.863

1.051

1.249

1.462

1.695

1.951

30.0

0.514

0.721

0.927

1.139

1.359

1.596

1.856

2.150

25.0

0.552

0.784

1.021

1.261

1.513

1.784

2.082

2.419

20.0

0.612

0.881

1.156

1.434

1.723

2.033

2.373

2.752

15.0

0.716

1.019

1.314

1.615

1.931

2.267

2.635

3.052

45.0

0.472

0.654

0.830

1.010

1.183

1.363

1.557

1.760

40.0

0.497

0.698

0.892

1.085

1.281

1.491

1.708

1.945

35.0

0.527

0.747

0.961

1.179

1.410

1.641

1.890

2.164

30.0

0.563

0.810

1.052

1.299

1.561

1.840

2.139

2.473

25.0

0.612

0.896

1.181

1.470

1.774

2.099

2.454

2.850

20.0

0.690

1.029

1.356

1.681

2.014

2.373

2.771

15.0

0.810

1.170

1.517

1.870

2.241

2.643

(d)

for discontinuous vx , vz is also difficult, because it is difficult to obtain continuous vx , vz . When the slip surface is logarithmic helicoid, though it is heterogeneous soil, the continuous vx , vz is easy to obtain; therefore, the calculation equation of safety factor may be obtained when no other simplification condition is added. In conclusion, except the arc slip surface, it is reasonable to select the logarithmic helicoid as the approximate slip surface. However, through the discussion in Chaps. 5 and 7, the plane–helicoid and the plane–arc surface are more close to the real slip surface, compared with the helicoid and the arc surface.

10.3 Analysis Method for Helicoid Slip Surface At present, the common slope stability analysis methods on engineering are mostly derived on the basis of the limit equilibrium method for force and moment equilibrium. For logarithm helical slip surface, the analysis method of slope stability may be derived without making assumptions or simplifications for the force on the soil mass. If virtual work equation is used to derive the upper bound solution method [4, 5], Luan Maotian et al. will use Lagrange multiplier and improved limit equilibrium method of variational principle [6]. Actually, for logarithm helical slip surface, the limit equilibrium method and the virtual work equation-based generalized limit equilibrium method are the same. For more convenient discussion on slope stability and derivation of calculation equation, limit equilibrium method is adopted for discussion.

368

10 Slope Stability

10.3.1 Basic Solving Idea If “z = h B , z = h A ” are respectively the soil mass surface and the slip surface,  B = s : h B = h s , A =  : h A = h. The fundamental equations of limit equilibrium method given in Chap. 5 are: d E(h s , h) − px dx

(σ h  − τ ) =

(σ + τ h  ) = wγ (h s , h) + pz − h  E(h s , h) − T (h s , h) =

d T (h s , h) dx

d M(h s , h) − (h − h s ) px dx

(10.1) (10.2) (10.3)

where h E(h s , h) =

σx dz

(10.4)

τx z dz

(10.5)

(h − z)σx dz

(10.6)

hs

h T (h s , h) = hs

h M(h s , h) = hs

h wγ (h s , h) =

γ dz

(10.7)

hs

px = (τ − σ h  )s , pz = (σ + τ h  )s

(10.8)

In slope stability analysis of saturated soil, the action of pore water pressure u is often considered, and the corresponding yield condition is: f =τ−

1 [(σ − u) tan ϕ + c] = 0 Fs

(10.9)

Horizontal force and vertical force equilibrium Eqs. (10.1) and (10.2), moment equilibrium Eq. (10.3), and yield condition (10.9) are equations often used in the limit equilibrium method. As it has been pointed out in Chap. 5 that the horizontal force equilibrium equation and the moment equilibrium equation are not mutually independent, they cannot be solved simultaneously. In theory, the horizontal force and vertical force equilibrium equations shall be satisfactory; therefore, only the

10.3 Analysis Method for Helicoid Slip Surface

369

integral moment equilibrium equation for the whole slip surface can be used in the moment equilibrium equation to derive the slope stability analysis method. τ, σ are solved by using the horizontal force and vertical force equilibrium equations, substituted in the yield condition; hence, (1 + h  λ F )

     dE dT + (h  − λ F ) = (h  − λ F wγ (h s , h) + pz − 1 + h 2 (c F − uλ F ) dx dx   + 1 + h  λ F px

(10.10)

According to the moment equilibrium Eq. (10.3), for any point (x R , z R ): d dE dT − (x − x R ) − (h − h s ) px [(h − z R )E − (x − x R )T − M] = (h − z R ) dx dx dx (10.11) Equation (10.10) is substituted in, hence:

 d  h − zR (h − z R )E − (x − x R )T − M = (wγ + pz )(h  − λ F ) − (c F − uλ F )(1 + h 2 ) dx 1 + λF h   dT 1 (h − z R )(h  − λ F ) + (1 + λ F h  )(x − x R ) + (h s − z R ) px − 1 + λF h dx

(10.12)

where λ F = tan ϕ/Fs , c F = c/Fs For the integral of Eq. (10.12) along the slip surface, it is the integral moment equilibrium equation for the whole slip surface. If “(h − z R )(h  − λ F ) + (1 + λ F h  )(x − x R ) = 0,” namely when it is logarithm helical slip surface, the calculation equation of safety factor may be obtained.

10.3.2 Analysis Method for Simple Slope with Two Types of Soils One simple slope (Fig. 10.1) with two types of soils is considered firstly, it may be divided into two sections to give moment equation similar to Eq. (10.12):  d  h − zR  (h − z R )E − (x − x R )T − M = (wri + pz )(h  − λ Fi ) dx 1 + λ Fi h    dT h − λ Fi − (c Fi − uλ Fi )(1 + h 2 ) + (h s − z R ) px − (h − z R ) + (x − x R ) 1 + λ Fi h  dx

(10.13)

x ∈ (xi−1 , xi ), i = 1, 2 Equation (10.13) is integrated into (x0 , x1 ) and (x1 , x2 ), respectively, and then added together; hence,

370

10 Slope Stability

Fig. 10.1 Simple slope with two types of soils x 2 i

i=1xi−1

=

h − zR w + pz )(h  − λ Fi ) − (c Fi − uλ Fi )(1 + h 2 ) dx + 1 + λ Fi h  ri

x2 (h s − z R ) px dx x0

x 2 i  

dT h − λ Fi (h − z R ) + (x − x R ) dx  1 + λ Fi h dx

(10.14)

i=1xi−1

“E = 0, T = 0, and M = 0” at the slope entry point x 0 and the slope exit point x 2 is used to obtain the above equation. The slip surface is taken to satisfy the equation: (h − z R )(h  − λ Fi ) + (1 + λ Fi h  )(x − x R ) = 0

(10.15)

It may be derived that the slip surface is two sections of logarithmic helixes: ⎫ x − x R = Ri exp(−λ Fi θ ) cos θ ⎬ h − z R = Ri exp(−λ Fi θ ) sin θi ⎭ x ∈ (xi−1 , xi ) i = 1, 2

(10.16)

In order to achieve a continuous slip surface, it shall be R2 = R1 exp[(λ Fi+1 − λ Fi )θ1 ]

(10.17)

where R1 is the constant to be determined. Hence, the calculation equation of safety factor can be obtained: Fs = M R /M0 2 

   (wri + pz − u) tan ϕi + ci h − z R − h  (x − x R ) dx MR =

(10.18)

xi

i=1 x

i−1

(10.19)

10.3 Analysis Method for Helicoid Slip Surface 2 

  (wri + pz )h  (h − z R ) + (h s − z R ) px dx M0 =

371

xi

i=1 x

(10.20)

i−1

Compared with the mean slope condition, the calculation equation is only different in the slip surface; namely, the smooth logarithm helically curved surface is changed as a continuous curved surface connected by two sections of smooth logarithm helically curved surfaces. When surface (Fig. 10.1) of the slope is relatively simple, integrals of Eqs. (10.19) and (10.20) may be worked out. For example, when the slope surface is: ⎧ x λ F − 1 + λ2F ” for the arc surface, the plane section adjacent to the slope base will not exist.

10.6.4 Calculation Equation for Plane–Helicoid–Plane Slip Surface Similarly, if slip surface is selected as the plane–helicoid–plane, it may be more close to the real slip surface than the helicoid. Slip surface (Fig. 10.5) is taken: When “x ≤ xk ,” it is plane slip surface and

390

10 Slope Stability

( xR , z R )

x0

xN

xk

xl Fig. 10.5 Schematic diagram for plane–helicoid–plane slip surface

* “h  = λ F + 1 + λ2F .” When “xk ≤ x ≤ xl ,” it is logarithm helical slip surface, and (h − z R )(h  − λ F ) + (x − x R )(1 + h  λ F ) = 0. *

When “xl ≤ x,” it is plane slip surface and “h  = λ F − 1 + λ2F .” xk , z k , xl , and zl shall be determined by smoothness of the slip surface at the point. Equation (10.72) may be simplified as: x N MR =

  (wγ + pz − u) tan ϕ + c A x dx

(10.77)

x0

⎧ x ≤ xk ⎨ 2(h − z R )  where A x = h − z R − h (x − x R ) xk ≤ x ≤ xl ⎩ 2(h − z R ) xl ≤ x * 2  Similarly, if “h < λ F + 1 + λ F ” for the helicoid, the plane section adjacent to * the slope crest will not exist, and if “h  > λ F − 1 + λ2F ” for the helicoid, the plane section adjacent to the slope base will not exist. The above-mentioned calculation equations are all applicable to general heterogeneous soil slope; thereinto, “ϕ = ϕ(x, h),” “c = c(x, h),” and “wγ = wγ (x, h)”; namely, ϕ and c are indexes of any point on the slip surface, and wγ is the soil weight per unit width. In actual calculation, integral in the equation can be written as cumulative sum for calculation according to the slice method. Considering that the analysis method of Eqs. (10.72) and (10.73) is applicable to slip surfaces of any form, and the recommended slip surfaces cover plane, helicoid, or that composited by plane, helicoid or plane and arc surface, the method is referred to as composite slip surface method for convenience. Calculation methods for four kinds of slip surfaces, namely logarithmic helicoid, arc surface, plane–logarithmic helicoid–plane, and plane–arc surface–plane, are covered in it, and for slope with weak interlayer, other slip surfaces may be selected.

10.7 Calculated Results and Their Discussion

391

10.7 Calculated Results and Their Discussion 10.7.1 Treatment in Particular Case In actual calculation with seepage flow, when it is above the low water level while below the zero-pressure line, the slip moment M0 is calculated with saturated unit weight; and when it is below the zero-pressure line, the slippage resistance moment M R is calculated with the buoyant unit weight [9]. When it is necessary to count in the horizontal force on the straight wall surface (such as vertical face breakwater), the one concerning horizontal force in the slip moment shall be: x N

h 2 (h s − z R ) px dx =

x0

(z − z R ) px (z)dz = (h x − z R )Px

(10.78)

h1

where Px is the total horizontal force on the straight wall surface, and h x is its action point.

10.7.2 Calculated Results For composite slip surface method, four types of slip surfaces, arc slip surface, helicoid, plane–arc surface–plane and plane–helicoid–plane, are taken respectively for calculation; for the convenience of comparison, both the simple slice method and the simplified Bishop method of the arc slip surface are adopted in the calculation. The calculated results are detailed in Table 10.5.

10.7.3 Comparison of Calculation Methods When the arc slip surfaces are the same, the safety factor calculated by the simple slice method is generally less than those by the simplified Bishop method and the composite slip surface method. When the indexes are calculated according to the consolidated quick shear, the results calculated by the simplified Bishop method do not differ greatly from that by the composite slip surface method, and it is consistent with that calculated according to the arc slip surface. However, when the ground soil is calculated according to the cross plate strength or direct and quick shear strength, the safety factors calculated by the simplified Bishop method differ obviously from the composite slip surface method which differ obviously in general. The following engineerings differ greatly: breakwater engineering

Simple slice method (1)

1.252

1.301

1.315

1.381

1.714

1.148

1.179

1.274

1.382

1.461

1.539

2 Block wharf

3 Block wharf

4 Caisson wharf

5 Block wharf

6 Block wharf

7 Buttressed wharf

8 Buttressed wharf

9 Buttressed wharf

10 Buttressed wharf

11 Caisson wharf

12 Block wharf

0.683

1.233

1.443

13 Slope embankment (cross plate)

14 Specification 98 calculation examples (slope embankment)

15 Specification 98 calculation examples (vertical breakwater)

Breakwater engineering (5 items)

1.285

1 Caisson wharf

Gravity wharf engineering (12 items)

Engineering case

1.504

1.575

1.453

1.911

1.852

1.601

1.546

1.458

1.337

2.029

1.721

1.733

1.555

1.573

1.605

Simplified Bishop method (2)

Table 10.5 Calculated results of safety factor for slope stability

1.444

1.515

0.966

1.851

1.807

1.420

1.463

1.401

1.363

1.916

1.688

1.623

1.524

1.419

1.513

Minimum (3)

1.553

1.533

0.998

1.906

1.850

1.662

1.586

1.511

1.380

2.053

1.708

1.759

1.567

1.577

1.638

Arc surface (4)

Composite slip surface method

1.466

1.579

0.966

1.933

1.860

1.448

1.501

1.430

1.400

1.916

1.761

1.658

1.548

1.489

1.537

Helicoid (5)

1.510

1.515

1.056

1.851

1.812

1.568

1.515

1.449

1.363

2.008

1.688

1.667

1.524

1.531

1.578

Plane–arc surface (6)

1.444

1.549

1.035

1.870

1.807

1.420

1.463

1.401

1.381

1.917

1.733

1.623

1.537

1.419

1.513

(continued)

Plane–helicoid (7)

392 10 Slope Stability

1.347

1.362

1.129

16 Semi-circular caisson

17 Semi-circular caisson (partial failure)

(Indexes of undrained triaxial shear)

1.128

1.113

1.253

1.180

1.035

0.920

1.510

1.152

1.090

0.823

19 Wharf slope 2

20 Wharf slope 3

21 Wharf slope 4

22 Wharf slope 5

23 Wharf slope 6

24 Wharf slope 7

25 Wharf slope 8

26 Wharf slope 9

27 Wharf slope 10

28 Wharf slope 11

29 Coastal revetment

1.749

Revetment engineering (4 items)

0.938

18 Wharf slope 1

High pile wharf engineering (11 items)

Simple slice method (1)

Engineering case

Table 10.5 (continued)

1.988

0.982

1.362

1.368

1.636

0.990

1.192

1.497

1.329

1.237

1.253

1.029

1.257

1.470

1.438

Simplified Bishop method (2)

1.816

0.932

1.352

1.343

1.622

0.972

1.186

1.449

1.353

1.229

1.216

1.007

1.168

1.401

1.430

Minimum (3)

1.970

0.932

1.382

1.361

1.632

0.989

1.186

1.478

1.353

1.238

1.233

1.011

1.175

1.442

1.430

Arc surface (4)

Composite slip surface method

1.816

0.948

1.396

1.378

1.635

0.989

1.213

1.455

1.403

1.277

1.234

1.031

1.213

1.441

1.466

Helicoid (5)

1.949

0.949

1.352

1.349

1.629

0.972

1.193

1.469

1.419

1.229

1.216

1.007

1.168

1.401

1.443

Plane–arc surface (6)

1.826

0.974

1.375

1.343

1.622

0.979

1.199

1.449

1.401

1.275

1.226

1.022

1.228

1.458

1.496

(continued)

Plane–helicoid (7)

10.7 Calculated Results and Their Discussion 393

0.783

0.844

0.927

30 Coastal revetment (cross plate)

31 Coastal revetment (cross plate)

32 Coastal revetment (cross plate)

1.391

1.193

1.395

Simplified Bishop method (2)

1.005

1.027

1.198

Minimum (3)

1.234

1.144

1.198

Arc surface (4)

Composite slip surface method

1.005

1.054

1.386

Helicoid (5)

1.197

1.110

1.271

Plane–arc surface (6)

Notes (1) Consolidated quick shear strength is the index when no strength index is noted, and the pore water pressure is zero in calculation (2) For high pile wharf, slippage resistance of the pile is not considered (3) Slip circle of “1 + h  tan ϕ/Fs ≤ 0.1” is removed by the simplified Bishop method (4) Wave force is counted in vertical breakwaters 15, 16, and 17 (5) The engineering information is mainly collected by “Code for Foundation in Port Engineering” or taken for references [10]

Simple slice method (1)

Engineering case

Table 10.5 (continued)

1.028

1.027

1.291

Plane–helicoid (7)

394 10 Slope Stability

10.7 Calculated Results and Their Discussion

395

13, revetment engineering 30, 31 and 32. Their breakwater or revetment is provided with relatively wide dumped ripraps (Fig. 10.3). Width of the dumped riprap for breakwater engineering 13 and revetment engineering 30 is 10 m that of the revetment engineering 31 is 5 m and of revetment engineering 32 is 6.6 m. As ϕ of block stone is relatively large, “h  < 0” near slope exit point of the slip surface, the safety factor of the ground soil calculated according the strength of the cross plate is relatively small and “1 + h  tan ϕ/Fs ” is minimum, the result calculated by the simplified Bishop method is relatively large apparently; thus, the method is no longer applicable. If the safety factor is calculated according to the consolidated quick shear, it will be relatively large, which is an improvement. For arc slip surface, if pore water pressure and horizontal force are not considered, according to the moment Eq. (10.53), the overall moment equilibrium equation on the slip surface will be: 

  (x − x R )(wγ + pz ) + (σ λ F + c F )(1 + h 2 )(h − z R ) dx = 0

(10.79)

In limit state, strength “σ λ F + c F ” of the slip surface reflects the differences of methods. According to the equilibrium equation: and σ (1 + h  λ F ) + h  c F = wγ + pz − dT hence: σ (h  − λ F ) − c F = dE dx dx  dE Simple slice method: It is assumed that “ dT = h ”; therefore, dx dx σ λF + cF =

 1  (wγ + pz )λ F + (1 + h 2 )c F 1 + h 2

(10.80)

Simplified Bishop method: It is assumed that “ dT = 0”; therefore, dx σ λF + cF =

  1 (wγ + pz )λ F + c F 1 + hλF

Composite slip surface method:  1+λ2F −(h  −λ F )2  dE λ + h  c F ”; therefore, 1+λ2 +(h  −λ )2 dx F F

It

is

assumed

(10.81) that

“ dT dx

=

F

σ λF + cF =

 1 + λ2F + (h  − λ F )2  (wγ + pz )λ F + c F 2 2 (1 + λ F )(1 + h )

(10.82)

Obviously, if “ϕ = 0,” the three methods will be the same. At the point “h  = λ F ” on the slip surface, strength of the simple slice method is the maximum. Within the area “0 < h  ≤ λ F ” on the slip surface, strength by the simple slice method is greater than that by the simplified Bishop method; within the area “h  < 0” on the slip surface, strength by the simple slice method is the minimum and it is the main reason that the calculated safety factor is relatively small. In Chap. 8, the ground bearing capacity is calculated by the analysis method for slope stability. As the slope surface

396

10 Slope Stability

is horizontal and its section “h  < 0” is relatively long, the calculated safety factor is relatively small. Example: Slope exit point of the slip surface for the breakwater engineering 12 is located at the breakwater. 1+h 2 Compared with the simplified Bishop method, 1+λ  , for the slippage resistance Fh 

−λ F ) moment, is changed as “1 + (h1+λ ,” and the inapplicability due to the small or even 2 F  negative value of “1 + λ F h ” is eliminated for the composite slip surface method, when it is only judged from the calculation equation. Among the common analysis methods at present, methods [11] such as Morgenstern–Price method are also equipped with the above-mentioned problem, which shows the irrationality of their assumptions. However, the assumption of the composite slip surface is made by reference to the extremum condition of the yield function, so that is relatively rational. The composite slip surface method is apparently superior to the simplified Bishop method in both the theory established by the method and the actual calculated result. 2

10.7.4 Comparison of Different Slip Surfaces For different slip surfaces, the calculated safety factors are also different. Among the four kinds of slip surfaces in the composite slip surface method, most of the calculated safety factors for the arc surface and the helicoid are consistent. The difference is relatively large for revetment engineering 29 and 31. When consolidated quick shear strength is adopted, the safety factors for the slip surfaces of plane–arc surface–plane and plane–helicoid–plane are generally smaller than those of the arc surface and the helicoid; when cross-plate strength or direct and quick shear strength is adopted, the safety factors for the four kinds of slip surfaces are different, especially between the maximum one and the minimum one, which indicates that calculation and analysis only with the arc slip surface may be insufficient. As stated earlier, if the slope crest is horizontal, and the load on it is uniformly distributed, the slip surface shall be plane from the slop crest to a certain depth; and if the slope base is horizontal, similar conclusion may be derived. The previous calculated result for the homogeneous soil slope without force assumption, safety factor of the plane–arc surface is less than that of the arc surface, and safety factor of the plane–helicoid is less than the helicoid, demonstrates the correctness of the conclusion. Therefore, slip surfaces of the plane–arc surface–plane and the plane– helicoid–plane are more close to the real slip surface compared with the other two, and moreover, their calculated safety factors are relatively smaller in most cases. In conclusion, the minimum calculated value of various slip surfaces by the composite slip surface method shall be regarded as the safety factor. It shall be pointed out that engineering 17 is presented with partial failure; however, when triaxial quick shear index is applied for calculation, the safety factors by the simple slice method, composite slip surface method, and simplified Bishop

10.7 Calculated Results and Their Discussion

397

method are greater than 1.1 for that the failure is regarded as integral considering that the caisson structure cannot be sheared. Therefore, calculation of ground bearing capacity shall also be carried out for such gravity-type structure engineering.

10.7.5 Reliability Analysis for Slope Stability JC method is adopted for the calculation of reliability [12, 13], considering the relativity [9] of “c, tan ϕ.” When composite slip surface method is adopted, the first derivatives of various variables may be obtained through numerical calculation. The calculated results of the reliability index “β” and partial safety factor of the resistance “Fs ” are detailed in Table 10.6. During calculation of the reliability, frequency value and quasi-permanent value [14] are applied for changeable action of load on the wharf surface, so that the corresponding partial safety factor is larger than the safety factor calculated with the standard value. Basic conditions of the calculated reliability index are similar to the safety factor. As it is calculated according to the consolidated quick shear index, the reliability indexes calculated by the simplified Bishop method and the composite slip surface method are relatively close; especially when it is arc slip surface, engineering 12 is excluded.

10.8 Analysis Method for Slope with Weak Interlayer Integral stability analysis is always carried out for slop with weak interlayer. At present, many analysis and calculation methods fan any slip surfaces are available, and they are all applicable for the integral stability analysis of such slop. However, in actual calculation, different slip surfaces shall be assumed artificially for calculation, which depends on the experience of the calculator (the slip surface selected is close to the most dangerous slip surface). For special engineering with most dangerous slip surface easy to be judged, corresponding slip surface can be selected for the calculation, and for general engineering with the most dangerous slip surface difficult to be judged, analysis method for obtaining the minimum safety factor rapidly and accurately is sought, which is significant for actual engineering. As has been stated earlier, the composite slip surface method is applicable to any form of slip surfaces. Likewise, forms of arc surface–bottom surface with weak interlayer–arc surface, helicoid–bottom surface with weak interlayer–helicoid, plane–arc surface–bottom surface with weak interlayer–arc surface–plane and plane–helicoid– bottom surface with weak interlayer–helicoid–plane may be taken as slip surfaces, and then integral stability analysis and calculation is carried out for slop with weak interlayer.

Gravity wharf mean

11 Caisson wharf (new)

10 Buttressed wharf

9 Buttressed wharf

8 Buttressed wharf

7 Buttressed wharf

6 Block wharf

5 Block wharf

4 Caisson wharf

3 Block wharf

2 Block wharf

1 Caisson wharf

Gravity wharf (11 items)

Engineering case

4.294

β

Fs

1.369

2.612

β

Fs

1.419

3.919

β

Fs

1.313

4.221

β

Fs

1.284

1.609

β

Fs

1.165

4.349

β

Fs

1.757

1.932

β

Fs

1.429

1.973

β

Fs

1.381

1.897

β

Fs

1.330

3.952

β

Fs

1.289

2.185

Fs

1.350

Fs

β

Simple slide method (1)

1.686

6.138

1.697

3.355

1.653

4.230

1.600

6.450

1.564

3.410

1.371

5.484

2.086

2.734

1.794

3.671

1.812

2.884

1.652

5.676

1.620

3.141

1.696

Simplified Bishop method (2)

1.604

5.376

1.628

3.254

1.465

4.053

1.530

4.878

1.485

3.478

1.370

5.253

1.978

2.564

1.763

3.296

1.720

2.771

1.628

5.294

1.468

2.819

1.603

Maximum (3)

6.331

1.727

3.331

1.700

4.112

1.667

6.196

1.589

3.593

1.418

5.490

2.106

2.659

1.785

3.443

1.838

2.870

1.670

5.487

1.642

3.506

1.749

Arc surface (4)

Composite slip surface method

Table 10.6 Reliability of calculated results (β—reliability index; Fs —partial safety factor of resistance)

5.705

1.679

3.313

1.515

4.301

1.530

5.210

1.497

4.034

1.402

6.016

1.980

2.987

1.825

3.808

1.732

2.968

1.684

5.580

1.534

3.024

1.637

Helicoid (5)

6.010

1.628

3.254

1.616

4.117

1.585

6.119

1.522

3.479

1.392

5.253

2.062

2.564

1.763

3.296

1.744

2.771

1.628

5.294

1.535

3.003

1.637

Plane–arc surface (6)

5.376

1.640

3.318

1.465

4.053

1.537

4.878

1.485

3.478

1.370

5.393

1.978

2.662

1.776

3.339

1.720

2.900

1.667

5.330

1.468

2.819

1.603

(continued)

Plane–helicoid (7)

398 10 Slope Stability

β

1.327

3.387

Fs

β

Simple slide method (1)

2.820

β

2.252 1.423

0.991

β

1.438

2.935

1.432

4.233

1.713

2.507

1.267

2.921

1.578

2.486

1.363

3.104

1.320

2.124

1.277

Fs

1.156

1.677

β

Fs

1.209

3.743

β

Fs

1.569

1.267

β

Fs

1.110

1.707

β

Fs

1.257

2.082

β

Fs

1.297

2.066

β

Fs

1.191

1.316

Fs

1.155

Fs

β

6.009

1.675

4.288

Simplified Bishop method (2)

2.749

1.407

2.201

1.428

2.641

1.393

4.359

1.698

2.645

1.258

2.725

1.543

2.506

1.388

3.013

1.313

1.899

1.238

4.726

1.545

3.912

Maximum (3)

2.239

1.449

2.789

1.422

4.359

1.709

2.657

1.258

2.764

1.556

2.506

1.388

3.085

1.325

2.026

1.256

4.726

1.545

Arc surface (4)

Composite slip surface method

Note It is calculated only for the consolidated quick shear index and when the safety factor is greater than 1.0

High pile wharf mean

20 Wharf slope 10

19 Wharf slope 9

18 Wharf slope 8

17 Wharf slope 6

16 Wharf slope 5

15 Wharf slope 4

14 Wharf slope 3

13 Wharf slope 2

High pile wharf engineering (8 items)

12 Vertical breakwater (new)

Breakwater engineering (1 item)

Engineering case

Table 10.6 (continued)

2.433

1.476

2.895

1.416

4.575

1.713

2.958

1.298

2.972

1.543

2.740

1.445

3.375

1.357

2.012

1.259

6.744

1.753

Helicoid (5)

2.201

1.428

2.749

1.409

4.489

1.698

2.645

1.264

2.725

1.551

2.773

1.459

3.013

1.313

1.899

1.238

4.892

1.556

Plane–arc surface (6)

2.287

1.449

2.641

1.393

4.495

1.703

2.886

1.310

2.733

1.540

2.671

1.442

3.192

1.345

1.959

1.243

6.572

1.737

Plane–helicoid (7)

10.8 Analysis Method for Slope with Weak Interlayer 399

400

10 Slope Stability

Then, the determination process for the slip surface is described with the plane–helicoid–bottom surface with weak interlayer–helicoid–plane as an example (Fig. 10.6). Supposing, the internal friction angle of the weak interlayer soil is ϕa2 , a set of “(x R , z R )” and a point a2 on the bottom surface with weak interlayer, then the whole slip surface is determined. The specific process is as follows: Slip surface of section a2 ∼ a1 : ⎫ x − x R = Ri ex p(−λ Fi θ )cos θ ⎪ ⎪ ⎬ i = N2 , N2 − 1 . . . h − z R = Ri ex p(−λ Fi θ )sin θ ⎪ Ri−1 =,Ri ex p[(λ Fi − λ Fi−1 )θi ] ⎪ ⎭ R N2 = (xa2 − x R )2 + (h a2 − x R )2 ex p(θa2 λ Fa2 )

(10.83)

* It is calculated from a2 to a1 , if “h  ≥ Fϕ + 1 + Fϕ2 ,” section a2 ∼ a1 will be obtained.  * For slip surface of section a1 ∼ a, it will be obtained by taking “h = Fϕ +

1 + Fϕ2 and h = h a1 + h  (x − xa1 ).” For bottom surface with weak interface a2 −a3 , it is calculated from a2 to a3 along −z R .” the bottom surface with weak interlayer; it is supposed as “θa3 = arctan hxa3a3−x R , 2 2 If (xa3 − x R ) + (h a3 − x R ) ≥ R N2 exp(−θa3 Fϕa2 ), section a2 ∼ a3 will be obtained. For slip surface of section a3 ∼ a4 : ⎫ x − x R = Ri exp(−λ Fi θ ) cos θ ⎪ ⎪ ⎬ x ∈ (xi−1 ,xi ) i = N4 , N4 + 1, · · · h − z R = Ri exp(−λ Fi θ ) sin θ (10.84) ⎪ Ri+1 = Ri exp[(λ Fi+1 − λ Fi )θi ] ⎪ ⎭ RN 4 = RN 2 It is calculated from a3 to a4 . If “h  ≤ λ F − obtained.

*

1 + λ2F ,” section a3 ∼ a4 will be

Fig. 10.6 Schematic diagram for slip surface of slope with weak interlayer

10.8 Analysis Method for Slope with Weak Interlayer

401

* For slip surface of section a4 ∼ b: If “h  = λ F − 1 + λ2F ” and “h = h a4 + h  (x − xa4 ),” section a4 ∼ b will be obtained. The safety factor corresponding to the slip surface will be obtained by substituting the determined slip surface into Eqs. (10.72) and (10.73). Different minimum safety factors will be obtained for different “(x R , z R )” and different a2 on the bottom surface with weak interlayer. It is easy to determine the minimum safety factor and its corresponding slip surface automatically by the above-mentioned method, and that is its biggest advantage. Additionally, the section of the slip surface passing the weak interlayer may be , selected randomly; namely, the determination of point a3 is not stricted by (xa3 − x R )2 + (h a3 − x R )2 ≥ R N2 exp(−θa3 λ Fa2 ); in this way, the safety factor calculated is relatively small. For general soil layer (relatively thick), such slip surface will be plane–helicoid– plane–helicoid–plane. If the plane on the bottom of the slip surface can be any plane in the soil layer, and determination of point a3 is not stricted by the above-mentioned conditions, the plane–helicoid–plane will be part of such slip surface only, and the calculated safety factor will not surely be greater than that calculated according to the plane–helicoid–plane, and it will be relatively small in most cases. The same result may be obtained for the arc surface if it is extended to the arc surface–plane–arc surface. Though safety factor of such slip surface is relatively small, the slip surface is generally not adopted unless thickness of the weak interlayer is very small, as the slip surface in the same soil layer is not smooth.

10.9 Generalized Limit Equilibrium Method-Based Analysis Method The basic idea of slope stability analysis method established for homogeneous soil is discussed by the generalized limit equilibrium method in Chap. 7, and now, it is extended to general heterogeneous soil slope, and analysis method with no force assumed is established.

10.9.1 Basic Solving Idea The yield condition Eq. (10.9) is substituted in the horizontal and vertical force equilibrium equation; hence, σ (h  − λ F ) − (c F − uλ F ) =

d E(h s , h) − px dx

σ (1 + h  λ F ) + h  (c F − uλ F ) = wγ + pz −

d T (h s , h) dx

(10.85) (10.86)

402

10 Slope Stability

The moment equilibrium Eq. (10.60) can be rewritten as:  d  (h − z R )E(h s , h) − (x − x R )T (h s , h) − M(h s , h) dx    = (wγ + pz )h  (h − z R ) + (h s − z R ) px − (wγ + pz − u)λ F + c F h − z R − h  (x − x R )    + (σ − wγ − pz ) (h − λ F )(h − z R ) + (1 + h  λ F )(x − x R )

  According to the discussion in Chap. 2, “ ∂∂zf namely: (h  − λ F )

z=h

(10.87)

= 0” along the slip surface,

dτx z dσx ∂u + (1 + h  λ F ) = γ (h  − λ F ) + λ F (1 + h 2 ) dx dx ∂z

(10.88)

Total derivative is calculated for Eq. (10.65) along the slip surface and then substituted in Eq. (10.88), and through rearrangement, it can be derived along the slip surface that:  2λ F 1 dh  ∂u ∂u dσu    γ (h − = − (h σ − λ ) − (1 + h λ ) − λ ) u F F F dx 1 + h 2 dx ∂x ∂z 1 + λ2F (10.89) where σu = σ − u + λc FF It is the same as the stress equation along the slip surface derived in Chap. 2, except that the pore water pressure is considered in it. According to the vertical force equilibrium equation and moment equilibrium equation, the following equations are derived:  b   cF cF + u − wγ − pz dx = 0 [(1 + h  λ F ) − σ −u+ λF λF

(10.90)

a

b a

   (wγ + pz )h  (h − z R ) + (h s − z R ) px − (wγ + pz − u)λ F + c F h − z R − h  (x − x R )

  + (σ − wγ − pz ) (h  − λ F )(h − z R ) + (1 + h  λ F )(x − x R ) dx = 0

(10.91)

For Eq. (10.89), the general form of solution is:  c c σ − u + F = exp(2λ F arctan h  ) C + F λF λF ⎫  x ⎬ 1  − λ ) − (1 + h  λ ) ∂u − (h  − λ ) ∂u exp(−2λ arctan h  )dx γ (h + F F F F 2 ⎭ ∂x ∂z 1 + λF a

where C is the constant to be determined.

(10.92)

10.9 Generalized Limit Equilibrium Method-Based Analysis Method

403

Two equations for two unknowns C and Fs may be obtained by substituting Eq. (10.92) into Eqs. (10.90) and (10.91), and then the safety factor Fs may be obtained through simultaneous solution.

10.9.2 Calculation Equation of Safety Factor For heterogeneous soil slope, when the slip surface crosses the interface of two types of soils, its normal stress is generally not continuous, which means that for different soil, the constant C to be determined may be different. According to the slice method: Within [xi−1 , xi ]: σ −u+ x gi (x) = xi−1

1 1 + λ2Fi

 c Fi c Fi = exp(2λ Fi arctan h  ) Ci + + gi (x) λ Fi λ Fi 

γ (h  − λ Fi ) − (1 + h  λ Fi )

∂u ∂u − (h  − λ Fi ) exp(−2λ Fi arctan h  )dx ∂x ∂z

(10.93) (10.94)

At xi , the vertical effective stress “(σz − u)i− = (σz − u)i+ ” shall be continuous, as:  2 + (1 + λ h  )2 (1 + λ2Fi )h i− c Fi i−  ) C + c Fi + g (x ) (σz − u)i− + Fi = exp(2λ Fi arctan h i− i i i 2 λ Fi λ 1 + h i− Fi   2 + (1 + λ  2 (1 + λ2Fi+1 )h i+ c Fi+1 c Fi+1 Fi+1 h i+ )  ) C (σz − u)i+ + = exp(2λ Fi+1 arctan h i+ i+1 + 2 λ Fi+1 λ 1 + h i+ Fi+1

Therefore: Ci+1 =

Bi Ci + Di Ai

(10.95)

where 2  2 + (1 + λ Fi h i− ) (1 + λ2Fi )h i−  exp(2λ Fi arctan h i− ) 2 1 + h i−

(10.96)

2  2 + (1 + λ Fi+1 h i+ ) (1 + λ2Fi+1 )h i+  exp(2λ Fi+1 arctan h i+ ) 2 1 + h i+

(10.97)

Bi 1 c Fi 1 c Fi+1 gi (xi ) + (Bi − 1) + (1 − Ai ) Ai Ai λ Fi Ai λ Fi+1

(10.98)

Bi = Ai =

Di =

If “ϕi = 0” and “(Bi − 1) λ1Fi = “Bi = 1.”

 2h i− 2 1+h i−

 + 2 arctan h i− ,” it will be obtained that

404

10 Slope Stability

   2h 1  ,” it will be obtained If “ϕi+1 = 0” and “(1− Ai ) λ Fi+1 = − 1+hi+2 + 2 arctan h i+ i+ that “Ai = 1.” If two adjacent soil strips “[xi−1 , xi ]” and “[xi , xi+1 ]” are within the same soil   = h i+ ” at xi , it will layer (ϕi = ϕi+1 , ci = ci+1 ), and the slip surface is smooth “h i− be obtained that “Ai = Bi ” and “Di = gi (xi ) .” Supposing “k1 = 1.0 and s1 = 0.0,” hence: Ci = ki C1 + si ki+1 = ABii ki , si+1 =

Bi s Ai i

+ Di

i = 1, 2, 3, . . .

(10.99)

Equation (10.99) is substituted in Eq. (10.93) and then into Eqs. (10.90) and (10.91), and moreover, it is written as the form by slice method:

  exp(2λ Fi arctan h i ) ki C1 + si + gi (xi∗ ) (1 + h i λ Fi )

i

   c Fi + (1 + h i λ Fi ) exp(2λ Fi arctan h i ) − 1 − (wγ i + pzi − u i ) xi = 0 λ Fi (10.100)







(wγ i + pzi )h i (h i∗ − z R ) + (h s − z R ) pxi − [(wγ i + pzi − u i )λ Fi + c Fi ] [h i∗ − z R − h i (xi∗ − x R )]

i  c − [wγ i + pzi − u i + (1 − exp(2λ Fi arctan h i )) Fi λ Fi

    − exp(2λ Fi arctan h i )(ki C1 + si + gi (xi∗ ))][(h i − λ Fi )(h i∗ − z R ) + (1 + h i λ Fi )(xi∗ − x R )] xi = 0

(10.101) where xi∗ , h i∗ is the point coordinate in the slip surface of the ith soil strip, and “xi = xi − xi−1 ” is the width of the soil strip. Then, the two equations of unknowns C1 and Fs are obtained. From Eq. (10.100), it may be obtained that: C1 =

fc gc

(10.102)

where ⎧ ⎫   ⎬

⎨ − exp(2λ Fi arctan h i )(si + gi (xi ))(1 + h i λ Fi )  c Fi  fc = xi   ⎩ + 1 − (1 + h i λ Fi ) exp(2λ Fi arctan h i ) + wγ i + pzi − u i ⎭ i λ Fi (10.103)

exp(2λ Fi arctan h i )(1 + h i λ Fi )ki xi (10.104) gc = i

Equation (10.102) is substituted in Eq. (10.101), and then, the safety factor may be solved:

10.9 Generalized Limit Equilibrium Method-Based Analysis Method

Fs = M R /M0

405

(10.105)

where M0 = MR =



i

i

 (wγ i + pzi )h i (h i∗ − z R ) + (h s − z R ) pxi xi

(10.106)

  [(wγ i + pzi − u i ) tan ϕi + ci ] h i∗ − z R − h i (xi∗ − x R )



c Fi + Fs wγ i + pzi − u i + (1 − exp(2λ Fi arctan h i )) λ Fi  − exp(2λ Fi arctan h i )(ki C1 + si + gi (xi∗ )) (h i − λ Fi )(h i∗ − z R )  + (1 + h i λ Fi )(xi∗ − x R ) xi (10.107) Equations (10.105)–(10.107) are the slope stability analysis methods based on the generalized limit equilibrium method. Compared with the limit equilibrium method, the generalized limit equilibrium method applies the stress Eq. (10.89) along the slip surface; thus, it is not necessary to assume the relationship between E, T for deriving the analysis method for slope stability, which is different from the limit equilibrium method. The stress equation along the slip surface is derived according to the yield condition “f =0,” and the ∂f ∂f extremum condition of the yield function “ ∂h  and ∂h ,” while the analysis method is derived with the equilibrium equation, yield condition, and stress equation along the slip surface, which means that the real slip surface shall also meet the extremum ∂f ∂f condition of the yield function ∂h  or ∂h ; however, such real slip surface is difficult to obtain. In nature, the above-mentioned analysis method is still expressing the safety factor as functional with unknown as slip surface, though it is more complex than the functional derived under certain assumptions (limit equilibrium method). For general heterogeneous soil slope, it may be more difficult to calculate the minimum safety factor and its corresponding real slip surface by such functional than by other analysis methods. Therefore, the safety factor may only be calculated when slip surface is assumed; in other words, when the safety factor of the slope stability is calculated by the generalized limit equilibrium method, different slip surfaces shall be selected, and the minimum calculated value of the slip surface is regarded as the safety factor. Obviously, for logarithm helical slip surface, calculation equations of the safety factor by the generalized limit equilibrium method and the limit equilibrium method are identical. For arc slip surface, it is easy to derive that the calculation equation for the slippage resistance moment is: MR =

 i

     ki C1 + si + gi (xi∗ ) tan ϕi + ci exp(2λ Fi arctan h i ) h i∗ − z R − h i (xi∗ − x R ) xi

(10.108)

406

10 Slope Stability

It shall be stated that, when the slip surface crosses the interface of the two types of soils, it is only required that the vertical effective stresses on both sides of the interface are equal; actually, the vertical force and the horizontal force shall also be equal; hence, the corresponding calculation equation may be derived, and corresponding limit condition shall be provided for the slip surfaces on both sides of the interface. However, it is not very significant for method requiring the selection of different slip surfaces for calculation. The slope stability analysis method based on the generalized limit equilibrium method is also applicable to any forms of slip surfaces. Likewise, forms of arc surface–bottom surface with weak interlayer–arc surface and helicoids–bottom surface with weak interlayer–helicoid may be taken as slip surfaces, and then, integral stability analysis and calculation are carried out for slop with weak interlayer.

10.9.3 Calculated Results and Comparison Programming of calculation procedure according to the above-mentioned calculation equation is also not difficult; the calculation process is similar to the analysis method for helicoid slip surface. During actual calculation, the buoyant unit weight is used  = for calculating C1 when it is below the zero-pressure line, and moreover, “h i−   = 0.5(h i + h i+1 )” is taken for approximate calculation. The other calculation h i+ conditions are the same as the composite slip surface method of the limit equilibrium. For the convenience of comparison, arc slip surface, helicoid (the same as the limit equilibrium method of composite slip surface), plane–arc surface–plane and plane– helicoid–plane are still taken for the calculation. The comparison of calculated results by the simple slice method, the simplified Bishop method, the composite slip surface method, and the generalized limit equilibrium method is detailed in Table 10.7. If the minimum of the four slip surfaces is regarded as the safety factor, the safety factors by the generalized limit equilibrium method and the composite slip surface method of limit equilibrium are considerably consistent. Similar to the composite slip surface method, for consolidated quick shear index, the corresponding safety factors of different slip surfaces are much closer to each other, while those of several engineering with ground soil calculated with cross-plate strength or triaxial quick shear strength index are different apparently; the calculated safety factor of common arc slip surface in engineering is relatively large sometimes. Therefore, it may be insufficient to carry out calculation and analysis only with the arc slip surface. For the slope stability analysis method based on the generalized limit equilibrium method, the stress equation along the slip surface is used in replace of the assumptions or simplifications necessary for deriving the analysis method when the limit equilibrium method is adopted. As the stress equation along the slip surface is obtained by applying the extremum condition of the yield function, the extremum condition of the yield function is only the quantitative description for the yield criterion that the slop stability shall meet, and no new concept or content is introduced in it. Therefore, it is analysis method with no assumption and simplification.

Simple slice method (1)

1.301

1.315

1.381

1.714

1.148

1.179

1.274

1.382

1.461

1.539

2 Block wharf

3 Block wharf

4 Caisson wharf

5 Block wharf

6 Block wharf

7 Buttressed wharf

8 Buttressed wharf

9 Buttressed wharf

10 Buttressed wharf

11 Caisson wharf

12 Block mt

0.683

1.233

13 Slope embankment (cross plate)

14 Specification 98 calculation examples (slope embankment)

Breakwater engineering (5 items)

1.285

1.252

1 Caisson wharf

Gravity wharf engineering (12 items)

Engineering case

1.575

1.453

1.911

1.852

1.601

1.546

1.458

1.334

2.029

1.721

1.733

1.555

1.573

1.605

Simplified Bishop method (2)

Table 10.7 Comparison of safety factors for slope stability

1.515

0.966

1.851

1.807

1.420

1.463

1.401

1.363

1.916

1.688

1.623

1.524

1.419

1.513

Minimum by the composite slip surface method (3)

1.542

0.966

1.933

1.860

1.417

1.481

1.430

1.385

1.916

1.761

1.658

1.548

1.489

1.537

Maximum (4)

1.613

1.176

1.942

1.976

1.564

1.571

1.579

1.385

2.005

1.826

1.792

1.607

1.651

1.633

Arc surface (5)

1.649

1.328

2.043

2.039

1.572

1.572

1.610

1.419

2.114

1.855

1.853

1.654

1.653

1.650

Plane–arc surface (6)

Generalized limit equilibrium method

1.542

0.981

2.033

1.916

1.417

1.481

1.445

1.492

1.968

1.769

1.747

1.641

1.595

1.556

(continued)

Plane–helicoid (7)

10.9 Generalized Limit Equilibrium Method-Based Analysis Method 407

1.443

1.347

1.362

1.129

15 Specification 98 calculation examples (vertical breakwater)

16 Semi-circular caisson

17 Semi-circular caisson (partial failure)

(Indexes of undrained triaxial shear)

1.128

1.113

1.253

1.180

1.035

0.920

1.510

1.152

1.090

0.823

19 Wharf slope 2

20 Wharf slope 3

21 Wharf slope 4

22 Wharf slope 5

23 Wharf slope 6

24 Wharf slope 7

25 Wharf slope 8

26 Wharf slope 9

27 Wharf slope 10

28 Wharf slope 11

Revetment engineering (3 items)

0.938

18 Wharf slope 1

High pile wharf engineering (11 items)

Simple slice method (1)

Engineering case

Table 10.7 (continued)

0.982

1.362

1.368

1.636

0.990

1.192

1.497

1.329

1.237

1.253

1.029

1.257

1.470

1.438

1.504

Simplified Bishop method (2)

0.932

1.352

1.343

1.622

0.972

1.186

1.449

1.353

1.229

1.216

1.007

1.168

1.401

1.430

1.444

Minimum by the composite slip surface method (3)

0.931

1.396

1.366

1.623

0.980

1.204

1.450

1.363

1.247

1.234

1.021

1.213

1.441

1.466

1.420

Maximum (4)

0.974

1.396

1.366

1.639

0.981

1.205

1.467

1.363

1.247

1.262

1.029

1.308

1.529

1.524

1.420

Arc surface (5)

0.974

1.409

1.377

1.651

0.989

1.204

1.475

1.363

1.248

1.261

1.042

1.408

1.594

1.579

1.479

Plane–arc surface (6)

Generalized limit equilibrium method

0.931

1.409

1.367

1.623

0.980

1.219

1.450

1.404

1.277

1.234

1.021

1.257

1.469

1.500

1.459

(continued)

Plane–helicoid (7)

408 10 Slope Stability

0.844

0.927

30 Coastal revetment (cross plate)

31 Coastal revetment (cross plate)

32 Coastal revetment (cross plate)

1.391

1.193

1.396

1.988

Simplified Bishop method (2)

1.005

1.012

1.198

1.816

Minimum by the composite slip surface method (3)

1.005

0.993

1.169

1.816

Maximum (4)

1.208

1.168

1.169

2.046

Arc surface (5)

1.298

1.225

1.395

2.038

Plane–arc surface (6)

Generalized limit equilibrium method

Note Minimum by the generalized limit equilibrium method includes the calculated result of the helicoid slip surface

1.749

0.783

29 Coastal revetment

Simple slice method (1)

Engineering case

Table 10.7 (continued)

1.021

0.993

1.175

1.825

Plane–helicoid (7)

10.9 Generalized Limit Equilibrium Method-Based Analysis Method 409

410

10 Slope Stability

10.10 Slope Stability Analysis Method 10.10.1 Soil Mass Cannot Withstand Tension For composite slip surface method, according to Eq. (10.1):     cF cF d σ −u+ (h  − λ F ) − h  E(h s , h) − px −u = λF λF dx

(10.109)

Equation (10.69) (η = 0) is substituted in, and hence:   1 + λ2F + (h  − λ F )2 cF cF wγ + pz − u + = σ −u+ λF λF (1 + λ2F )(1 + h 2 )

(10.110)

Equation (10.65) is substituted in, then the stress field on the slip surface is obtained, and its horizontal stress is:   cF (1 + λ2F + (h  − λ F )2 )2 cF σx − u + wγ + pz − u + = λF λF (1 + λ2F )(1 + h 2 )2

(10.111)

In order to ensure that the soil mass does not generate tension, effective stress “σx = σx − u ≥ 0” shall be provided with, namely: gh = wγ + pz − u −

 (1 + λ2F )(1 + h 2 )2 cF − 1 ≥0 λ F (1 + λ2F + (h  − λ F )2 )2

(10.112)

Generally, if the cohesion c is relatively large, area near the slop crest may be not in accordance with Eq. (10.112). Along with the increase of the buried depth for the slip surface, gh is enlarged generally until it is satisfactory. Buried depth h c of “gh = 0” is generally referred to as height of tension joint. In order to guarantee that the soil mass is free from tension during the calculation of the safety factor, the slip surface in accordance with Eq. (10.112) is generally calculated; namely, the section of slip circle not in accordance with that equation is not calculated. As shown in Fig. 10.7, the slip surface of “xa ≤ x” is merely calculated. * Please note that, when “h  = λ F + 1 + λ2F ,” it can be derived that (1+λ2 )(1+h 2 )2 2 2 F) ) F

F “[ (1+λ2 +(h  −λ

− 1]max = 2h  λ F ”; hence,

(gh )min = wγ + pz − u − 2c F tan where “tan ϕ =

tan ϕ .” Fs

π 4

+ϕ



≥0

(10.113)

10.10 Slope Stability Analysis Method

411

x0 hc xa

xN

Fig. 10.7 Schematic diagram for calculation of soil mass cannot bear tension

If the height of tension joint is within the same soil layer “wγ = γ h c ,” and  moreover, “ pz − u = 0,” it will be obtained that “gh = 0” is “h c = γ2cFs tan π4 + ϕ ,” and it is the same as the height of the tension joint obtained by the field failure mode. For the generalized limit equilibrium method, according to Eq. (10.93) and within [xi−1 , xi ]:  1 + λ2F + (h  − λ F )2 c  ) k C + s + c F + g (x) σx − u + F = arctan h exp(2λ 1 F i i i λF λF (1 + λ2F )(1 + h 2 )

(10.114)

In order to guarantee that the soil mass is not bearing tension during the calculation of the safety factor, the following may be solved during the calculation: Step I: Initial value Fs(0) of the safety factor is taken, and whether the soil mass is bearing tension is not considered, then the first-order approximations C1(1) and Fs(1) are calculated; Step II: C1(1) is substituted in Eq. (10.107) instead of C1 , that the soil mass is free from tension is considered; namely, the second-order approximations C1(2) and Fs(2) are calculated under “σx = σx − u ≥ 0.” Then, the calculation of the k step is conducted: C1(k−1) is substituted in Eq. (10.107) instead of C1 , that the soil mass is free from tension is considered  to calculate the k-order approximations C1(k) and Fs(k) until “ Fs(k) − Fs(k−1)  < the calculated accuracy” is met; thereinto, Fs(k) is the safety factor. If the height of tension joint*is within the same soil layer, and “u = 0,” it will be derived that when “h  = λ F + 1 + λ2F ” and according to Eqs. (10.94) and (10.65):  cF 1 h − λF C + γ (h − h s ) σx + =   λF h (h − λ F ) h  cF h h − λF C + γ (h − h s ) σz + =  λF h − λF h

(10.115) (10.116)

  where “C = (1 + λ2F ) C1 + λc FF ” is the constant to be determined. According to the boundarycondition,  when “h = h s ,” it will be obtained that h  −λ F cF pz + λ F ”; therefore, “σz = pz ”; hence, “C = h 

412

10 Slope Stability

 cF 1 cF = 2 pz + + γ (h − h s ) λF h λF

σx + If “ pz = 0,” “h c =

2c γ Fs

tan

π 4

(10.117)

 + ϕ ” will also be obtained.

10.10.2 Slip Surface No matter what analysis method is adopted, it will be calculated until under assumed slip surface. The calculated result has proved that the most common arc slip surface in engineering currently is sometimes with relatively large safety factor. However, for various engineering, how to select approximate real slip surface is a confronted research topic. According to Eq. (10.65), if “1 + λ2F − (h  − λ F )2 < 0,” τx z < 0 must be provided with for*the shear stress, which indicates that the soil mass bears tension. Therefore, *

“λ F − 1 + λ2F ≤ h  ≤ λ F + 1 + λ2F ” shall be provided with. For slip surface selected by this way, any point on it is in accordance with “τx z ≥ 0.” Obviously, the above plane–arc surface–plane and plane–helicoid–plane are in accordance with the requirements of shear stress “τx z ≥ 0,” which is one of the reasons that the two types of slip surfaces are more approximate to the real slip surface.

10.10.3 Analysis Method It is well known that the safety factor calculated by the simple slice method is generally less than the ones obtained by other methods. However, it is not deemed to be less than the exact one. For the same slip surface, its safety factor calculated by the simplified Bishop method is close to that by the generalized limit equilibrium method, except for conditions under which the simplified Bishop method is not applicable. Actually, it is easy to extend the simplified Bishop method to be applicable to general slip surfaces [15], if different slip surfaces are taken for calculation, and the minimum value is regarded as the safety factor; the inapplicability of the method may be improved or even eliminated; thus, its application effect will be better. Morgenstern–Price method is applicable to any slip surface, but if the slip surface selected is not suitable, it will be presented with the same problem as above. Janbu method is derived based on simultaneous solution of dependent equations, and it is imperfect in theory. The composite slip surface method is equipped with two advantages: 1. Assumption in it is relatively reasonable, so that it is free from inapplicability; 2. for general slope, its slip surface is determined relatively reasonable. Of course, it may be further

10.10 Slope Stability Analysis Method

413

improved, for example, to introduce two undetermined constants in the assumption. b b According to Eqs. (10.69) and (10.70): “ a dE dx = 0 and a dT dx = 0,” the two dx dx undetermined constants may be determined, and then, the calculation equation of moment equilibrium is derived. However, such improvement still belongs to the solution system of the limit equilibrium method; it will make the calculation more complex and will not obviously affect the calculated result of the safety factor, as such calculated safety factor is presented with adequate accuracy compared with that calculated by the generalized limit equilibrium method. The generalized limit equilibrium method-based slope stability analysis method is exact, and up to now, it is the only one with no assumption and simplification among the methods derived by the Coulomb yield criterion (currently common yield criterion)-based theory of ultimate equilibrium. However, the slip surface shall be solved if this method is to be used to obtain the exact safety factor. Under assumed slip surface, only the approximate safety factor can be obtained, and moreover, the approximate solution derived is not necessarily to be less than the exact one (error in numerical calculation is excluded). If the slip surface selected is approximate to the real one, the approximate solution will be close to the exact one. In addition, if the pore water pressure is considered, the calculation of this method will be relatively complex. For composite slip surface method and generalized limit equilibrium methodbased slope stability analysis method, the calculation process of the safety factor is the same as that of the above-mentioned analysis method for helicoid slip surface, and the calculation equation for the safety factor may be written as “G(Fs ) = M0 −M R /Fs = 0.” However, G(Fs ) is a monotonic function, so its solution is unique. The solution of such monotonic function equation is very easy, so is the calculation of the minimum safety factor at a time computer is extensively popularized and the study of numerical calculation method is rather advanced.

References 1. Qian JH, Yin ZZ (1996) Principle and calculation of geotechnical. China Water Power Press, Beijing 2. Chen ZY (2003) Soil slope stability analysis—theory, methods and programs. China Water Power Press, Beijing 3. Luan MT, Nian TK, Zhao SF (2003) Summary of research process of earth structure and side slope. In: 9th conference proceedings on soil mechanics and geotechnical engineering. Tsinghua University Press, Beijing, pp 37–55 4. Chen HF (1995) Limit Analysis and Soil Plasticity (trans: Zhan S, Proofread by Han Dajian). China Communications Press, Beijing 5. Shen ZJ (2000) Theoretical soil mechanics. China Water Power Press, Beijing 6. Luan MT, Jin CQ, Lin G (1992) Improved limit equilibrium method and its applications to stability analysis of soil masses. Chinese J Geotech Eng 14(Supplement):20–29 7. Feng K et al (1978) Numerical calculation method. National Defense Industry Press, Beijing 8. Yin ZZ, Lv QF (2005) Finite element analysis of soil slope based on circular slip surface assumption. Rock Soil Mech 26(10):1525–1529

414

10 Slope Stability

9. JTJ 250-98 Code for Foundation in Port Engineering 10. CCCC First Harbor Engineering Investigation and Design Institute (1999) Calculation example for structure design of port engineering. China Communications Press, Beijing 11. Wei RL (1987) Strength and deformation of soft clay. China Communications Press, Beijing 12. Wu SW (1990) Analysis of structure reliability. China Communications Press, Beijing 13. Drafting group of unified standard of reliability of structural design for harbour engineering. Reliability of Harbor Engineering Structures. Beijing: China Communications Press, 1992 14. JTJ 215-98 Load Code for Harbour Engineering 15. Zhang LY, Zheng YR (2003) An extension of simplified bishop method and its application. In: 9th conference proceedings on soil mechanics and geotechnical engineering. Tsinghua University Press, Beijing pp 989–994

Chapter 11

Slope Stability During Construction and Pore Water Pressure

11.1 Slope Stability During Construction According to engineering practices, landslide mostly happens during the construction period or at the beginning of the engineering and several items of calculated stability results of landslide engineering are detailed in Table 11.1. In Case 1, it is calculated according to the section after the landslide, and the safety factor at the beginning of the landslide will be slightly larger. In Case 2, the engineering deformation is so large that it has affected normal use in the engineering, and corresponding safety factor is also relatively small. In Case 3, it is a test engineering on the slide resistance of the geotextile reinforced cushion; here, the safety factor is only given for the natural ground (without geotextile reinforced cushion). It should be notable that the safety factor by the composite slip surface method is slightly smaller than that by the simple slice method, because the embankment is the compacted cohesive soil with relatively large cohesion and plane– arc surface–plane, and plane–helicoid–plane slip surfaces may reduce the bearing tension of soil mass. In Case 4, when the landslide happens, detailed cause analysis [1] has been carried out. Here, the calculation condition and calculated safety factor are consistent with those at that time. In Case 5, with relatively long engineering construction duration, strength increase of the ground soil is considered in the calculation. Except in Case 2 (the strength index applied in the calculation is the reduction of the consolidated quick shear index), the information on the other engineering is relatively detailed and, the calculated safety factor shall accurately reflect the actual situation. In addition, the safety factor is generally much smaller than 1.0. In China, many ports are established on the soft soil ground; the slope stability is often the key of the engineering success [2]. It is known that when the ground soil is the saturated soil, strength of the saturated soil mass increases over the time with increasing load during the construction and increasing consolidation of the ground © Springer Nature Singapore Pte Ltd. and Zhejiang University Press, Hangzhou, China 2020 C. Huang, Limit Analysis Theory of the Soil Mass and Its Application, https://doi.org/10.1007/978-981-15-1572-9_11

415

Simple slice method (1)

0.765

0.690

0.871

0.783

0.701

Engineering case

1. Island ring road landslide (quick shear)

2. High pile wharf slope (very large deformation)

3. Embankment failure test (vane)

4. Artificial island revetment landslide (vane)

5. Seawall landslide (vane)

0.715

0.810

0.885

0.702

0.872

Simplified Bishop method (2)

Table 11.1 Safety factor for slope stability of landslide engineering

0.701

0.783

0.847

0.697

0.851

Minimum (3)

0.736

0.796

0.912

0.701

0.863

Arc surface (4)

Composite slip surface method

0.725

0.795

0.945

0.711

0.922

Helicoid (5)

0.701

0.791

0.847

0.733

0.851

Plane–arc surface (6)

0.702

0.783

0.865

0.697

0.855

Plane–helicoid (7)

416 11 Slope Stability During Construction and Pore Water Pressure

11.1 Slope Stability During Construction

417

soil. Especially after reinforcement treatment is carried out for the ground (such as arranging sand pile and plastic plate), the ground soil is rapidly consolidated, and the soil mass strength is significantly increased; therefore, strength increase of the soil mass shall be considered for the slope stability analysis. Practically, when the soil mass is not fully consolidated, the strength of the soil mass varies. The actual strength of the soil during the construction period or at the beginning of the engineering is generally less than that after the service for some time, and the soil is more easily subjected to instability. Therefore, in consideration of the soil mass consolidation, the stability analysis can be carried out until it has been served in engineering (service load is counted). In other words, the aforesaid construction period includes the service duration at the beginning of the engineering. For the calculation of the slope stability in engineering, effective stress method or total stress method is generally adopted. For the calculation of the slope stability during construction, the effective stress method is stated as follows: According to the effective shear strength index of the soil mass, the pore water pressure is calculated according to the consolidation theory, and it is substituted in the calculation equation for the slope stability analysis method, and then, safety factor for the slope stability during construction may be calculated. The total stress method is stated as follows: According to undrained shear strength index of the soil mass (such as the vane and quick shear strength indexes), strength increase of the ground soil is counted, for which the pore water pressure (or consolidation degree) needs to be calculated according to the consolidation theory. Therefore, determination of the ground soil strength and calculation of the pore water pressure are the main tasks of the stability analysis during the construction period.

11.2 Effective Stress Method When calculated according to the effective stress method, the strength of the saturated   soil is τ f = c + (σ − u) tan ϕ  , where c , ϕ are the effective shear strength indexes. As long as the pore water pressure is known, various stability analysis methods in Chap. 10 may be applied. For one slope on the ground with saturated soil shown in Fig. 11.1, for example, plane–arc surface–plane slip surface by the composite slip surface method, the slip moment M0 and non-slip moment M R are as below, respectively: M0 =

x N 

  (h − z R )h (wr + pz ) + (h s − z R ) px dx

(11.1)

  (h − z R ) (wγ + pz − u) tan ϕ + c A x dx

(11.2)

x0

x N MR = x0

418

11 Slope Stability During Construction and Pore Water Pressure

Fig. 11.1 Schematic diagram for plane–arc surface–plane slip surface

where ⎧ x ≤ xk ⎨2  A x = 1 + (h − λ F )2 /(1 + λ2F ) xk ≤ x ≤ xl ⎩ 2 xl ≤ x

(11.3)

Because the slip moment excludes the pore water pressure and strength index, it may be calculated according to Eq. (11.1). For the non-slip moment, when the slip surface passes through the filling (u = 0), hence: x j MRA =

 

(h − z R ) (wγ + pz ) tan ϕ + c A x dx

(11.4)

x0

When the slip surface passes through the ground soil: x N MRB =

     (h − z R ) (W A + pz )U tan ϕ + W B tan ϕ + c A x B dx

(11.5a)

xj

where W A is the weight of the filling in unit width of the slip mass and W B is the weight of the ground soil in the unit width of the slip mass; u = (W A + pz )(1 − U ) 

where U is the consolidation degree and A x B is A x with the strength index of tan ϕ . Equation (11.5a) is given when it is considered that the ground soil has been thoroughly consolidated under the self-weight condition; if the ground soil is not thoroughly consolidated under the self-weight condition, the strength of the ground   soil W B tan ϕ + c shall be properly reduced.

11.2 Effective Stress Method

419

For the calculation equation for the arc surface, helicoid and plane–helicoid–plane slip surface by the composite slip surface method and other methods (e.g., simplified Bishop method), equations quite similar to the above cases are also available. If the construction is divided into stage loading, the weight of the filling in the unit width of the slip mass at kth stage of loading is written as W Ak and the consolidation degree at kth stage of loading as Uk ; thus, the calculation equation for the non-slip moment when the slip surface passes through the ground soil is as below: x N MRB =



     (W Ak + pzk )Uk tan ϕ + W B tan ϕ + c A x B dx (h − z R )

xj

(11.5b) Safety factor: Fs = (MRA + MRB )/M0

(11.6)

11.3 Simple Slice Method According to the simple slice method, the undrained shear strength index of the soil mass is applied and counted in the strength increase of the ground soil for the calculation of the slope stability during construction, which is the common method in the engineering. The calculation equation for the simple slice method (Fig. 11.2) is as below: x N M0 =

(x R − x)(wr + pz )dx x0

Fig. 11.2 Schematic diagram for arc slip surface

(11.7)

420

11 Slope Stability During Construction and Pore Water Pressure

x N MR =

   (h − z R ) wr + pz − u(1 + h 2 ) tan ϕ + c(1 + h 2 ) dx

(11.8)

x0

Similarly, because the slip moment excludes the pore water pressure and strength index, it may be calculated according to Eq. (11.7). The non-slip moment shall be considered from two aspects. When the slip surface passes through the filling (u = 0), hence: x j MRA =

  (h − z R ) (wr + pz ) tan ϕ + c(1 + h 2 ) dx

(11.9)

x0

When the slip surface passes through the ground soil, the non-slip moment is also composed of two parts: one is the non-slip moment generated by the undrained shear strength of the ground soil; the other is the non-slip moment generated by consolidation of the upper filling; and the strength increment [3]  Su = σz U tan ϕcu generated by consolidation of the upper filling. Hence: x N MRB =

  (h − z R ) σz U tan ϕcu (1 + h 2 ) + W B tan ϕ B + c B (1 + h 2 ) dx

xj

(11.10a) where σz is the vertical additional stress of the upper filling on the ground, U is the consolidation degree, ϕcu is the internal friction angle of the consolidated quick shear, W B is the weight of the ground soil in unit width of the slip mass, and ϕ B and c B are undrained shear strength indexes of the ground soil. The safety factor equation is the same as Eq. (11.6). If the construction is divided into multistage loading, the additional stress of the filling in the unit width of the slip mass at kth stage of loading will be supposed as σzk ; thus, calculation equation for the non-slip moment when the slip surface passes through the ground soil is as below: x N MR B =



  2 2 σzk Uk tan ϕcu (1 + h ) + W B tan ϕ B + c B (1 + h ) A x B dx (h − z R )

xj

(11.10b) For Eqs. (11.10a) or (11.10b), multiple issues may be proposed for further discussion, such as rationality of the calculated strength increment of the additional stress according to the elasticity theory, consistency between definition of corresponding consolidation degree and Eq. (11.8). In fact, how to determine the soil mass strength of the ground during the construction period consistent with the actual situation

11.3 Simple Slice Method

421

remains to be a problem that has not been well solved. In a sense, it is reasonable to consider Eq. (111.10a, b ) as a semiempirical and semi-rational equation. According to the actual analysis and calculation, it is not rare that the engineering is stable, but the calculated safety factor is less than 1.0. According to undrained shear strength index of the soil mass, the strength increase of the ground soil is counted for the calculation of the slope stability during construction; other analysis methods shall be considered to be adopted. Because the simplified Bishop method is inapplicable to some cases, the composite slip surface method is discussed below.

11.4 Composite Slip Surface Method For the plane–arc surface–plane slip surface (Fig. 11.1), it is easy to obtain the calculation equation by the total stress method: The slip moment similarly may be calculated according to Eq. (11.1). For the non-slip moment, when the slip surface passes through the filling (u = 0), similarly it may be calculated according to Eq. (11.4): When the slip surface passes through the ground soil [4]: x N MRB =

  (h − z R ) (W A + pz )U tan ϕcu + W B tan ϕ B + c B A x B dx

(11.11)

xj

where W A is the weight of the filling in unit width of the slip mass, W B is the weight of the ground soil in the unit width of the slip mass; u = (W A + pz )(1 − U ), where U is the consolidation degree;A x B is A x with the strength index of tan ϕ B , ϕcu is the internal friction angle of the consolidated quick shear, and ϕ B and c B are undrained shear strength indexes of the ground soil. Obviously, in Eq. (11.11), strength increment generated by the consolidation of the ground soil in the upper filling (including external load) is as below: Su = (W A + pz )U tan ϕcu This is different from Su = σz U tan ϕcu applied in the current simple slice method. If the construction is divided into multistage loading, similarly corresponding calculation equation for the non-slip moment may be written. Additionally, it is noted that when the slip surface is plane section, the slip surface is also related to the strength index; if the plane section is within the filling, tan ϕ of the filling is adopted; if within the ground soil, tan ϕ B is adopted. Calculation equation for the plane–helicoid–plane slip surface is quite similar to the above cases. Only the slip surface is related to the strength index tan ϕ, and no disagreement is holded for the index adopted for the slip surface in the filling, and

422

11 Slope Stability During Construction and Pore Water Pressure

it is recommended that tan ϕ B be adopted for the slip surface in the ground soil. Actually, because the calculated safety factor is very close to the plane–arc surface– plane, adoption of tan ϕ B or tan ϕcu for the slip surface in the ground soil will not have large influence on the calculated safety factor. When the analysis method is the same as the composite slip surface method, the slip moment by the effective stress method and that by the total stress method are the same. For the non-slip moment, when the slip surface passes through the filling (u = 0), that by the effective stress method and that by the total stress method are also the same; when the slip surface passes through the ground soil, that by the effective stress method and that by the total stress method are Eqs. (11.5a, b) and (11.11), respectively. The difference lies in that of the strength indexes in the following two aspects. Strength generated by the filling on the consolidation of the ground soil: By the  effective stress method, it is (W A + pz )U tan ϕ ; by the total stress method, it is (W A + pz )U tan ϕcu . Because different strengths are adopted, this is logical.   Strength of ground soil: By the effective stress method, it is W B tan ϕ + c ; by the total stress method, it is W B tan ϕ B + c B . If the ground soil has been thoroughly consolidated under the self-weight condition, ϕ B and c B may also serve as the consolidated quick shear strength indexes of the ground soil. If not, no matter by the effective stress method or total stress method, strength consistent with the actual situation shall be taken as the strength of the ground soil or it shall be properly reduced [4]. At present, vane strength is universally adopted in the engineering and usually considered as the natural strength [2, 5] of the ground soil; meanwhile, it is the strength of the ground soil during the vane test; because the continuous consolidation of the ground soil under the self-weight condition is not counted in the ground soil during the construction period, the calculated safety factor is relatively safe. Certainly, if, during the construction period, the ground soil continues to generate very small consolidation under the self-weight condition, the calculated safety factor will be basically consistent with the actual situation.

11.5 Calculated Results and the Comparison Several items of calculated results are detailed in Table 11.2. The first four items of engineering are stable; thus, if the strength increase of the soil mass is not considered, the calculated safety factor cannot reflect the actual engineering conditions. If the strength increase of the soil mass is considered, the calculated safety factor by the simple slice method will be smaller. In practice, if vane strength index is adopted for the ground soil, when the slip surface passes through the ground soil, the non-slip moments of the same slip surfaces are the same. For the difference, when the slip surface passes through the embankment, the non-slip moment calculated by the simple slice method is apparently smaller. Therefore, when, according to undrained shear strength index of the soil mass, the strength

0.946

0.701

1.162

0.827

1.112

0.730

1.302

0.977

1.753

1.453

Simplified Bishop method (2)

0.933

0.681

1.112

0.832

1.090

0.722

1.171

0.886

1.230

0.966

Minimum (3)

0.933

0.683

1.137

0.832

1.097

0.722

1.224

0.906

1.317

0.998

Arc surface (4)

Composite slip surface method Helicoid (5)

1.079

0.805

1.186

0.919

1.199

0.888

1.187

0.886

1.329

0.966

Plane–arc surface (6)

0.949

0.681

1.112

0.832

1.090

0.736

1.222

0.894

1.230

1.056

Note The counted strength increase is calculated according to the requirements of “Code for Foundation in Port Engineering” (JTJ 250-98)

0.638

0.854

1.049

Strength increase is counted

Strength increase is counted

0.685

3. North embankment (vane) at certain port

5. Port dam (vane)

0.950

Strength increase is counted

0.705

0.731

2. East slope embankment (quick shear) at certain port

0.972

0.906

Strength increase is counted

Strength increase is counted

0.683

1. Port slope embankment (vane)

4. Port diking (vane)

Simple slice method (1)

Engineering case

Table 11.2 Calculated results of safety factor for slope stability Plane–helicoid (7)

0.949

0.689

1.123

0.835

1.111

0.726

1.171

0.886

1.370

1.035

11.5 Calculated Results and the Comparison 423

424

11 Slope Stability During Construction and Pore Water Pressure

increase of the ground soil is counted for the calculation of the slope stability during construction, other analysis methods shall be considered to be adopted. Because the simplified Bishop method is inapplicable to some cases while the composite slip surface method can always get reasonable and creditable calculated results, the composite slip surface method is worth being generalized and applied in the engineering. In Case 5, with ever partial failure, the calculated safety factor is also relatively small. Additionally, among the above engineering, measures such as laying geotextile reinforced cushion are not taken for some of them, and they are not considered in the calculation.

11.6 Pore Water Pressure in Terzaghi’s Consolidation Theory For the natural ground, in the current engineering, one-dimensional consolidation theory is mainly applied for the calculation of the consolidation degree. According to the engineering practice, the measured settling rate is obviously quicker than the calculated consolidation degree. The safety factor for the overall stability when the strength increase is considered according to one-dimensional consolidation degree is less than 1.0, and it is not rare that the engineering is stable. It is evident that no counting of the lateral consolidation contributes to smaller calculated value of the safety factor. Presently, analytic solutions in two-dimensional and three-dimensional Terzaghi’s consolidation theory and Biot’s consolidation theory has been obtained [6–9], which improves smaller consolidation degree calculated according to the one-dimensional consolidation theory. It is of practical meaning to calculate the strength of the ground soil with increasing loading in the two-dimensional consolidation theory and discuss actual analysis effect of the overall stability in two-dimensional consolidation theory and the engineering feasibility. The consolidation degree defined in the dissipation rate of the pore water pressure u is U = (u 0 − u)/u 0 , where u 0 is the initial pore water pressure. The calculation equation in two-dimensional consolidation theory and that for the pore water pressure u are as follows.

11.6.1 Terzaghi’s Consolidation Equation ∂u ∂ 2u ∂ 2u = Ch 2 + Cv 2 ∂t ∂x ∂z

(11.12)

11.6 Pore Water Pressure in Terzaghi’s Consolidation Theory

425

where C h , Cv are horizontal and vertical consolidation coefficients. Boundary condition: permeable boundary z = 0 or z = 2H : u = 0; = 0; Impermeable boundary z = H : ∂u ∂z Initial condition: the pore water pressure is generally as below when t = 0: u = u0 =

1 (σx + σz ) 2

where σx , σz is the normal stress; for the concentrated load p, u 0 = For the distributed load p(x), when A ≤ x ≤ B, 2 u0 = π

B p(ξ ) A

z dξ (x − ξ )2 + z 2

(11.13) 2p z π x 2 +z 2

(11.14)

Because σx , σz are normal stresses in the elasticity theory, which shall be the normal stresses of the ground soil in the stress field under normal use. The above initial condition is referred to as Class I initial condition. For the limit analysis, the stress field in the limit state is considered; for the composite slip surface method and simplified Bishop method, the  strength in their  equations are [(σ − u) tan ϕ + c] = (wγ + pz − u) tan ϕ + c A x (A x = 1/(1 +  h λ F ) by simplified Bishop method. The pore water pressure is associated with the weight of the filling u = (W A + pz )(1 − U ) in accordance with the definition U = (u 0 − u)/u 0 of the consolidation degree. In order to make the calculation of the pore water pressure consistent with the stress field in the limit state for the calculation of the slope stability, the initial condition shall be as below: u 0 = W A + pz = p(x)

(11.15)

Obviously, solution of the pore water pressure according to this initial condition is consistent with the calculation of the safety factor for the slope stability according to Eqs. (11.5a, b) and (11.11); such initial condition is referred to as Class II initial condition.

11.6.2 General Equation for the Pore Water Pressure If it is permeable when z = 0, but not when z = H as shown in Fig. 11.3, it will be referred as single-side drainage. If it is impermeable when z = 0 and z = 2H as shown in Fig. 11.4, it will be referred to as double-side drainage where H is the vertical drainage distance. The pore water pressure in accordance with the consolidation equation, boundary condition, and initial condition u 0 = u 0 (x, z) is as below: Single-side drainage:

426

11 Slope Stability During Construction and Pore Water Pressure

Fig. 11.3 Single-side drainage

Fig. 11.4 Double-side drainage

1 1 u=√ √ 4π ptC h t 4πCv t

∞ H −∞ 0

 (x − ξ )2 f 1 (z, η, t)dηptdξ u 0 (ξ, η) exp − 4C h t 

(11.16) where f 1 (z, η, t) =

   ∞ Cv t z  η 2 sin bn 4π Cv t exp −bn2 2 sin bn H H H H n=1

(11.17)

(2n − 1)π 2

(11.18)

bn = Double-side drainage:

u=√

1 1 √ 4π C h t 4π Cv t

∞ 2H −∞ 0

  (x − ξ )2 f 2 (z, η, t)dη dξ (11.19) u 0 (ξ, η) exp − 4C h t

where    ∞ z  η 1 2 Cv t f 2 (z, η, t) = sin an 4π Cv t exp −an 2 sin an H H H H n=1 an =

nπ 2

(11.20) (11.21)

11.6 Pore Water Pressure in Terzaghi’s Consolidation Theory

427

11.6.3 Pore Water Pressure Under Class I Initial Condition (1) Concentrated load (Fig. 11.5) Calculation equation for the pore water pressure in case of single-side drainage is as below: 

     ∞ p cos bn Hx ξ z 2 2 Cv t 2 Ch t 2 exp −bn 2 sin bn exp −b ξ u= n π H n=1 H H 1 + ξ2 H2 0   1 + cos(nπ )ξ exp(−bn ξ ) dξ (11.22) Calculation equation for the pore water pressure in case of double-side drainage is as below: 

     ∞ p cos an Hx ξ z 1 2 Cv t 2 Ch t 2 exp −an 2 sin an exp −an 2 ξ u= π H n=1 H H 1 + ξ2 H 0   1 − cos(nπ ) exp(−nπ ξ ) dξ (11.23) (2) Distribution load (Fig. 11.6) If the distribution load within A ≤ x ≤ B is p(x), only p cos(bn ξ x) in Eqs. (11.22) and (11.23) needs to be changed as below: B p(α) cos(bn ξ(x − α)/H )dα A

This is the equation for the distribution load. Fig. 11.5 Concentrated load

p

x

z

Fig. 11.6 Distribution load

x A

B z

428

11 Slope Stability During Construction and Pore Water Pressure

Because the distribution load in any form may be approximating with multisection linear distribution load, specific equation for the linear distribution load is 1 given. When x1 ≤ x ≤ x2 and p(x) = x2 −x [ p2 (x − x1 ) + p1 (x2 − x)], hence: 1 x2 p(α) cos[bn ξ(x − α)/H ]dα = x1

+

H { p1 sin[bn ξ(x − x1 )/H ] − p2 sin[bn ξ(x − x2 )/H ]} bn ξ

H p2 − p1 {cos[bn ξ(x − x2 )/H ] − cos[bn ξ(x − x1 )/H ]} (x2 − x1 )/H bn2 ξ 2

(11.24)

Single-side drainage: u=

      ∞ ξ 2 Ch t Cv t z  q(x/H, ξ ) bn exp −bn2 2 sin bn exp − 2 ξ 2 1 + cos(nπ) exp(−ξ ) dξ 2 2 π H H bn + ξ H bn n=1

0

(11.25) Double-side drainage: u=

     ∞  Cv t z  q(x/H, ξ ) 1 Ch t 2  1 − cos(nπ ) exp(−2ξ ) dξ an exp −an2 2 sin an exp − ξ 2 + ξ2 2 π H H H a n n=1 0

(11.26) where 1 { p1 sin[ξ(x − x1 )/H ] − p2 sin[ξ(x − x2 )/H ]} ξ p2 − p1 1 {cos[ξ(x − x2 )/H ] − cos[ξ(x − x1 )/H ]} + (x2 − x1 )/H ξ 2 (11.27)

q(x/H, ξ ) =

If p2 = p1 , x1 → −∞, and x2 → ∞ are true, Eqs. (11.25) and (11.26) will be the calculation equations for one-dimensional consolidation. For the multi-section linear distribution load, only calculated results of the loads on each section needs to be superposed. (3) Increasing load with time In general, increasing building self-weight in the process of construction may be approximately increased according to linear load relationship with time:  p(x, t) = Supposing,

t−t0 1 [ p2 (x − x1 ) + p1 (x2 − t f −t0 x2 −x1 1 (x − x1 ) + p1 (x2 − x)] p [ 2 x2 −x1

x)] t0 ≤ t ≤ t f tf ≤ t

(11.28)

11.6 Pore Water Pressure in Terzaghi’s Consolidation Theory

429

   1 exp −(Th − Th∗ )ξ 2 − bn2 (Tv − Tv∗ ) 2 (T f h − T0h + (T f v − T0v )bn   − exp −(Th − T0h )ξ 2 − bn2 (Tv − T0v ) (11.29)

s(t, ξ, bn ) =

Th =

)ξ 2

Ch t ∗ Ch t ∗ C h t0 Cv t ∗ Cv t ∗ C v t0 , T = , T = ; T = , T = , T0v = (11.30) 0h v h v H2 H2 H2 H2 H2 H2

where t∗ =



t t0 ≤ t ≤ t f tf t ≥ tf

(11.31)

Single-side drainage:    z  ∞ q(x/H, ξ ) ξ 2 bn sin bn s(t, ξ, bn ) 1 + cos(nπ ) exp(−ξ ) dξ u= π n=1 H bn2 + ξ 2 bn 0

(11.32) Double-side drainage:  z   q(x/H, ξ )   1 an sin an s(t, ξ, a ) 1 − cos(nπ ) exp(−2ξ ) dξ n π n=1 H an2 + ξ 2 ∞

u=

0

(11.33) Because the above calculation equations are similar to the common Terzaghi’s one-dimensional consolidation equation in the form, only the generalized integral needs to be calculated by the numerical integration method so that existing standard procedure has been directly available for the calculation of the generalized integral. During the numerical integration, it shall be noted that q(x/H, 0) = 21 ( p1 + p2 )(x2 − x1 )/H ; when the load doesn’t change over time:    s(t, ξ, bn )t f =t0 = exp −(Th − T0h )ξ 2 − bn2 (Tv − T0v )

11.6.4 Pore Water Pressure Under Class II Initial Condition (1) Distribution load: Because u 0 = p(x), the single-side and double-side drainage may be integrated into the following equation:

430

11 Slope Stability During Construction and Pore Water Pressure

u = u z (z, t)u x (x, t)

(11.34)

where u z (z, t) =

∞ n=1

    4 z 2 Cv t exp −bn 2 sin bn (2n − 1)π H H

1 u x (x, t) = √ 4π C h t

B

 (x − ξ )2 dξ p(ξ ) exp − 4C h t

(11.35)



(11.36)

A

If p(x) is the uniformly distributed load p, A → −∞, B → ∞, u x (x, t) = p will be true, and Eq. (11.34) will be the calculation equation in one-dimensional consolidation theory. B If p(ξ ) in the equation is changed as q(x/H, ξ ) = p(α) cos[ξ(x − α)/H ]dα, A

Eq. (11.6) also may be written as below: 1 u x (x, t) = π

∞

  Ch t q(x/H, ξ ) exp − 2 ξ 2 dξ H

(11.37)

0

Similarly, if the distribution load within A ≤ x ≤ B is p(x), multi-section linear distribution load may be used for approximating, and specific equation for the linear 1 distribution load may be given. Supposing p(x) = x2 −x [ p2 (x − x1 ) + p1 (x2 − x)] 1 is true when x1 ≤ x ≤ x2 , q(x/H, ξ ) will be the same as that in Eq. (11.27). (2) Increasing load with time If the load increases with time as shown in Eq. (11.28), hence: u=

∞ n=1

 ∞ 1 4 sin(bn z/H ) q(x/H, ξ )s(t, ξ, bn )dξ (2n − 1)π π

(11.38)

0

where q(x/H, ξ ), s(t, ξ, bn ), Th , Th∗ , T0h , Tv , Tv∗ , and T0v are the same as those in Eqs. (11.27), (11.29), and (11.30). (3) Multistage loading Supposing the construction is loaded in M stages and the load in each stage is pi (x) and Ai ≤ x ≤ Bi ; thereinto, ti,0 and ti, f are the start and finish time of loading at ith stage, respectively. Thus, the loading is as below:

11.6 Pore Water Pressure in Terzaghi’s Consolidation Theory

431

⎧ i−1  ⎪ ⎪ ⎪ p j (x) + t t−t−ti,0i,0 pi (x) ti,0 ≤ t ≤ ti, f ⎪ i, f ⎪ ⎨ j=1 i  p(x, t) = p j (x) ti, f ≤ t ≤ ti+1,0 ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎩ i = 1, 2, . . . M

(11.39)

The pore water pressure may be calculated stage by stage and obtained after the superposition.

11.6.5 Approximate Calculation of Pore Water Pressure In the pore water pressure under Class II initial condition: 1 u x (x, t) = √ 4π C h t

B

  (x − ξ )2 dξ p(ξ ) exp − 4C h t

(11.40)

A

If the distribution load p(x) within A ≤ x ≤ B may be approximating with multisection linear distribution load, specific equation for the linear distribution load can 1 be given. Supposing x1 ≤ x ≤ x2 and p(x) = x2 −x [ p2 (x − x1 ) + p1 (x2 − x)], 1 hence: p(ξ ) = p(x) + u x (x, t) = p(x)u 1 (x, t) −

p2 − p1

(ξ − x)

x2 −x1  √ p2 − p1 √C h t (x2 −x)2 exp − x2 −x1 4C h t π

  2 1 −x) − exp − (x4C ht

(11.41)

where 1 u 1 (x, t) = √ 4πC h t

x2 x1

  Bx 1 (x − ξ )2 dξ = √ exp − exp(−ξ 2 )dξ 4C h t π

(11.42)

Ax

x1 − x x2 − x , Bx = √ Ax = √ 4C h t 4C h t For Eq. (11.42), existing numerical integration equation may be applied. If the load increases with time, t ≥ t f may be approximately calculated according to the following equation:  u = uz

   t0 + t f t0 + t f u x x, t − z, t − 2 2

432

11 Slope Stability During Construction and Pore Water Pressure

During loading stage by stage, each loading may be calculated, respectively, and t ≥ tif may be approximately calculated according to the following equation:  u = uz

   ti0 + tif ti0 + tif z, t − u x x, t − 2 2

And then, it is obtained after superposition. Analytic solution of the Biot’s consolidation theory under Class I initial condition has been obtained by Huang Chuanzhi [8], but the calculation is relatively inconvenient and the calculated results of the pore water pressure and those in Terzaghi’s consolidation theory have a small difference, which will not be introduced here. The analytic solution under Class II initial condition remains to be studied.

11.7 Non-slip Role of Geosynthetic Reinforced Cushion 11.7.1 Basic Consideration for the Analysis Method Treatment on the non-slip role of the geotextile reinforced cushion by the current stability analysis method mainly includes two aspects. One is the common treatment method in the current engineering: Only non-slip moment generated by the tension of the reinforced cushion at the slip surface is counted; the other is the treatment method proposed in recent years: The effect of the tensile stress of the reinforced cushion at the soil mass interface on the stability: Many specific recommendation methods have been available [10–12] to consider force of the reinforced cushion at the soil mass interface and calculate the safety factor for the stability. At present, effect of the bearing tension of the reinforced cushion at the soil mass interface on the stability shall be considered, which has become a common understanding. In order to guarantee that the analysis method can be effectively applied in the engineering, not only the non-slip role of the reinforced cushion shall be considered, but also other aspects (including basic assumption, slip surface, and method theory) shall be consistent with common analysis methods in the engineering (non-slip role of the reinforced cushion is not considered) so that the calculated results can be guaranteed of direct comparability.

11.7.2 Non-slip Mechanism of Geosynthetic Reinforced Cushion An embankment is built on the soft soil ground, and geotextile reinforced cushion (Fig. 11.7) is laid on the embankment bottom.

11.7 Non-slip Role of Geosynthetic Reinforced Cushion

433

Fig. 11.7 Force diagram for reinforced cushion

For the embankment and ground within a ≤ x ≤ b, h s (x) ≤ z ≤ h(x), the area enclosed by the slip surface and the slope surface is divided into (A) and (B) for consideration. It is firstly assumed that the reinforced cushion is of adequate tensile strength and frictional resistance against the embankment bottom and the ground surface so that the reinforced cushion cannot be snapped or pulled out. In (A), besides the gravity, the force on the interface of (A) and (B) shall also include the horizontal force E k and vertical shear Tk generated by the embankment. In (B), besides the gravity, it shall also include the horizontal force E h and vertical shear Th generated by the embankment. Axial tensile stress τ R and normal stress σ R are borne by the reinforced cushion at each point (if it is considered that the reinforced cushion cannot bear any normal stress, σ R = 0 will be true). According to the equilibrium equation, at any point (x, h h+ ) along the bottom surface of the reinforced cushion, hence: −τx z | h+ =

dE h − τR dx

σz |h+ = wγ h (h s , h k ) + pz −

(11.43)

dTh + σR dx

(11.44)

h h where E h = h sk σx dz and Th = h sk τx z dz . h According to Eq. (11.43), for the horizontal stress dE generated by the embankdx ment (the soil pressure is distributed along the surface of the reinforced cushion), after one part passes through the reinforced cushion, τ R is borne by the reinforced cushion and the other part is transferred to the ground τx z |h+ . Supposing the bearing ratio of the reinforced cushion as ηx (0 ≤ ηx ≤ 1.0), hence: h τ R = ηx dE dx h −τx z h + = (1 − ηx ) dE dx

! (11.45)

If ηx = 1.0, all of the horizontal force generated by the embankment will be borne by the reinforced cushion. If ηx = 0, the reinforced cushion will not bear the horizontal force generated by the embankment but transfer it to the ground totally.

434

11 Slope Stability During Construction and Pore Water Pressure

According to Eq. (11.44), η y (0 ≤ η y ≤ 1.0) also may be obtained. σR = ηy

⎫ ⎪ ⎬

dTh dx

 dTh ⎪ ⎭ σz h + = wγ h + pz − (1 − η y ) dx

(11.46)

Moment equilibrium is considered below; according to Eqs. (11.43) and (11.44), a moment is taken for any point (x R , z R ), hence:   (σz − wγ h − pz )(x − x R ) − τx z (h k − z R ) h k+ + [τ R (h k − z R ) − σ R (x − x R )]h k   dE h dTh (h k − z R ) − (x − x R ) = dx dx h k− Total moment of the bottom reinforced cushion in the slip mass is as below: x B xk



 (σz − wγ h − pz )(x − x R ) − τx z (h k − z R ) dx +

  = − E k (h k − z R ) − Tk (xk − x R ) − Mxk

x B



 τ R (h k − z R ) − σ R (x − x R ) dx

xk

(11.47)

According to Eq. (11.47), the moment generated by the soil pressure at the interface of (A) and (B) may be break up into that generated by the force transferred to the ground and that generated by the force borne by the reinforced cushion; ηt (0 ≤ ηt ≤ 1.0) is necessarily obtained as below. x B [τ R (h k − z R ) − σ R (x − x R )]dx = −ηt [E k (h k − z R ) − Tk (xk − x R ) − Mxk ] xk

(11.48) x B



   (σz − wγ h − pz )(x − x R ) − τx z (h k − z R ) dx = −(1 − ηt ) E k (h k − z R ) − Tk (xk − x R ) − Mxk

(11.49)

xk

It may be seen that force and the moment borne by the reinforced cushion are just one part of the soil pressure and the moment generated by the embankment, which may be obtained through the limit analysis theory. Therefore, if the analysis method for the stability is required to be in accordance with the moment equilibrium equation, as along as ηt can be determined, the moment generated by the force borne by the reinforced cushion will be determined too. According to the above analysis, due to tension-bearing characteristic of the reinforced cushion, partial soil pressure generated by the embankment is borne by the reinforced cushion so that the soil pressure transferred to the ground is reduced, and thus, the ground stability is improved.

11.7 Non-slip Role of Geosynthetic Reinforced Cushion

435

11.7.3 Simple Slice Method For the slip area shown in Fig. 11.7, the slip mass is divided into several vertical soil strips [xi−1 , xi ] i = 1, 2 . . ., supposing the strength indexes for the bottom surface of each soil strip as ci , ϕi , when the slip surface is an arc surface, on any vertical surface in each soil strip, the following moment equation will be true:   d  (h − z R )E − (x − x R )T − M = (h − z R )[(wγ i + pzi )(h − λ Fi ) − c Fi (1 + h 2 )] dx       dE dT + (h k − z R )τ R − (x − x R )σ R − λ Fi (h − z R ) h − τR − − σR dx dx

where λ F = tan ϕ/Fs , c F = c/Fs . It is noted that τ R , σ R are inexistent in (A). Both sides of the equation are integrated from xi−1 to xi and added for all the soil strips; hence, within [x0 , xk ] in (A) − [(h k − z R )E k − (xk − x R )Tk − Mxk ] xi      (h − z R ) (wγ i + pzi )(h − λ Fi ) − c Fi (1 + h 2 ) dx = δa +

(11.50)

i x i−1

where 

xi

δa =

i x i−1



dE dT λ Fi (h − z R ) h − dx dx 

 dx

Within [xk , x N ] in (B), the following equation is derived according to Eq. (11.48): (1 − ηt )[(h k − z R )E k − (xk − x R )Tk − Mxk ] xi     (h − z R ) wγ i + pzi )(h − λ Fi ) − c Fi (1 + h 2 ) dx = δb + i x i−1

where 

xi

δb =

i x i−1

     dE dT  − τR − − σR dx λ Fi (h − z R ) h dx dx

[(h k − z R )E k − (xk − x R )Tk − Mxk ] is eliminated, hence: 

xi

(1 − ηt )

i x i−1

   (h − z R ) (wγ i + pzi )(h − λ Fi ) − c Fi (1 + h 2 ) dx

(11.51)

436

11 Slope Stability During Construction and Pore Water Pressure



xi

+

   (h − z R ) (wγ i + pzi )(h − λ Fi ) − c Fi (1 + h 2 dx = (1 − ηt )δa + δb

i x i−1

(11.52) Thus, the following equation is derived: Fs = M R /M0

(11.53)

   (wγ i + pzi ) tan ϕi + ci (1 + h 2 ) (h − z R )dx M R = (1 − ηt ) xi

i=1 x

i−1

xi    (wγ i + pzi ) tan ϕi + ci (1 + h 2 ) (h − z R )dx + δF +

(11.54)

i=k+1x i−1 M0 = (1 − ηt )

xi 

(wγ i + pzi )(h − z R )h  dx +

i=1xi−1



xi

(wγ i + pzi )(h − z R )h  dx

(11.55)

i=k+1xi−1

where δ F = (1 − ηt )δa + δb . After omitting δ F , it will be the calculation equation considering the non-slip role of the reinforced cushion. Compared with the calculation equation without considering the non-slip role of the reinforced cushion, when the slip circle is within [x0 , xk ] in the embankment, the non-slip moment and slip moment are multiplied by factors 1 − ηt considering non-slip role of the reinforced cushion. When ηt = 0, if without considering non-slip role of the reinforced cushion, it is completely the same as the calculation equation by the common simple slice method. When ηt = 1.0, all the internal force generated by the embankment is borne by the reinforced cushion; when 0 ≤ ηt ≤ 1.0, internal force generated by the embankment is borne by the reinforced cushion, and some are borne by the ground. In fact, the calculation equation is written as Eqs. (11.54) and (11.55) so as to be convenient for the calculation of the safety factor; they completely may be rewritten as below: MR =

xi  

 (wγ i + pzi ) tan ϕi + ci (1 + h 2 ) (h − z R )dx

i=1xi−1

+ ηt Fs

x  ! k i   tan ϕi c + i (1 + h 2 ) (h − z R )dx (wγ i + pzi )h − (wγ i + pzi ) Fs Fs

(11.56)

i=1xi−1



xi

M0 =

i=1 x

i−1



(wγ i + pzi )(h − z R )h dx

(11.57)

11.7 Non-slip Role of Geosynthetic Reinforced Cushion

437

k xi    (wγ i + pzi )h − clearly seen that i=1 xi−1   tan ϕi ci 2 (wγ i + pzi ) Fs + Fs (1 + h ) (h − z R )dx is the moment generated by the soil pressure on the embankment. According to the equilibrium of the horizontal force,

Thus,

it

may

be

 ! k xi  tan ϕi ci  2 (wγ i + pzi )h − (wγ i + pzi ) + (1 + h ) dx = E k . Fs Fs i=1 xi−1

This is the soil pressure on the embankment by the simple slice method. According to the above analysis on the non-slip role, some of common slope stability analysis methods also may be used for the consideration considering the nonslip role of the reinforced cushion, such as simplified Bishop method and composite slip surface method. Only non-slip moment and the slip moment generated on the slip surface above the reinforced cushion are needed to be multiplied by the coefficient (1 − ηt ).

11.7.4 Engineering Practice of Non-slip Mechanism and Model Test Verification (1) Ground topsoil of the breakwater at certain port is relatively thick soft clay, and geotextiles are laid in the engineering. Tianjin Harbor Engineering Research Institute has conducted field monitoring on the construction process of the breakwater and measured the distribution [13] of the geotextiles tension along the breakwater section after the dumping and filling (see Fig. 11.8 for the section). In addition, Department of Hydraulic Engineering of Tsinghua University and China Communication Planning and Design Institute for Water Transportation have conducted the centrifugal model test and numerical simulation calculation [14]. Measured geotextile tension and calculated soil pressure at corresponding position are detailed in Table 11.3.

Fig. 11.8 Schematic diagram for measuring point of geotextile tension on (1-1) Section

9.86

Mean

26.3

27.0

29.1

13.0

5.40

4.20

4.60 12.5

12.9

14.4

10.3

4.0

4.60

3.90*

5.10

5.60

3.90

Measured tension

4.72

4.99

5.90

3.41

4.57

Calculated value of E k

No. 5 measuring point

Notes a E k is calculated with Rankine’s active soil pressure equation, according to the mean at the high and low water levels b Values marked with * are obtained according to the interpolation of adjacent points

17.0

6.70

4–4

Numerical simulation

6.24

3–3

23.0

26.0 5.20

12.2

11.2

2–2

6.60a

15.3

Calculated value of E k

Measured tension

Measured tension

Calculated value of E k

No. 4 measuring point

No. 2 measuring point

1–1

Sections

Table 11.3 Measured geotextile tension and calculated soil pressure E k (kN/m)

2.0

3.60

3.50

3.50

3.80

3.60

Measured tension

2.85

3.06

3.77

1.85

2.72

Calculated value of E k

No. 6 measuring point

438 11 Slope Stability During Construction and Pore Water Pressure

11.7 Non-slip Role of Geosynthetic Reinforced Cushion

439

Fig. 11.9 Schematic diagram for measuring point of geotextile tension on east embankment at certain port

At No. 2 and No. 4 measuring points, measured geotextile tension of 1–1 and 2–2 sections is about 50% of the soil pressure. Measured geotextile tension of 3–3 and 4–4 sections is obviously smaller; because the geological conditions are almost the same and the embankment is slightly higher than 1–1 and 2–2 sections, it may be caused by relatively poor tension of the geotextile. At No. 5 measuring point, measured geotextile tension is close to the soil pressure. At No. 6 measuring point, the mean of the measured geotextile tension is slightly greater than the soil pressure, which shows that the geotextile also bears the horizontal force generated due to lateral displacement of the soil mass. (2) For the east embankment at certain port, when the embankment is dumped and filled to ∇2.5 (see Fig. 11.9 for the section), the distribution of the geotextile tension along the breakwater section is measured; the measured geotextile tension and calculated soil pressure at corresponding position are detailed in Table 11.4. At No. 2 and No. 4 measuring points, similar to 3–3 and 4–4 sections of the breakwater, the measured geotextile tension is obviously small; at No. 5 measuring point, the measured geotextile tension is slightly greater than 50% of the soil pressure; at No. 6 measuring point, the measured geotextile tension is close to the soil pressure. (3) Non-slip role analysis; the above measured geotextile tension is obviously smaller than the tensile strength (the design tensile strengths are 45 and 57.1kN/m, respectively). In fact, because the integral ground is stable, the measured geotextile tension must be relatively small. According to the centrifugal model test [10], it shows that “when the acceleration reaches the limit value (corresponding acceleration in case of failure), the measured geotextile tension at the symmetrical place of the dam axis will be 90 kN/m.” According to the stability analysis results, the intersection of the most dangerous slip circle and the reinforced cushion is located near No. 2 measuring point, and vertical position of the arc center is basically consistent with the dumping and filling height of the embankment. Considering that the stability analysis method is in accordance with the moment equilibrium and the arm from the geotextile tension to the arc center is the dumping and filling height and while the arm from E k to the arc center is about 2/3 of the dumping and filling height. The ratios of the moment from the geotextile tension/E k to the arc center are ηt = 0.56(mean)/ηt = 0.38, respectively, in two engineering; the mean of five sections in two engineering is

1–1

Section

11.0

43.6

9.50

30.7

Calculated value of E k (kN/m)

Measured tension (kN/m)

Measured tension (kN/m)

Calculated value of E k (kN/m)

No. 4 measuring point

No. 2 measuring point

Table 11.4 Measured geotextile tension and calculated soil pressure E k (kN/m)

9.20

Measured tension (kN/m) 16.6

Calculated value of E k (kN/m)

No. 5 measuring point

6.00

Measured tension (kN/m)

5.90

Calculated value of E k (kN/m)

No. 6 measuring point

440 11 Slope Stability During Construction and Pore Water Pressure

11.7 Non-slip Role of Geosynthetic Reinforced Cushion

441

ηt = 0.53. In addition, the geotextile tension of the breakwater at certain port subject to the numerical calculation is completely correspondent with the moment from the geotextile tension/soil pressure to the arc center, with the ratio of ηt = 0.97. Therefore, it is reasonable and creditable to properly select ηt and consider actual non-slip role of the reinforced cushion with ηt E k . (4) Determination of combined influence coefficient; actually, ηt is a combined influence coefficient. For its determination, the following factors shall be considered: Firstly, the reinforced cushion is not horizontal due to the ground deformation, and it is noted that non-slip role of the reinforced cushion is mainly subjected to axial tensile so that partial soil pressure shall be transferred to the ground soil; secondly, for the tension of the reinforced cushion and the geotextile pore (e.g., geotextile grid), when the reinforced cushion is not fully tensioned or seen with a pore, partial soil pressure will be transferred to the ground soil too. In addition, calculated safety factor shall be considered. According to the calculated results, when ηt is determined, the larger safety factor, the more significant non-slip role of the reinforced  x cushion, which is because total tension of the reinforced cushion is TR = xkB τ R = ηt E k and E k is the soil pressure when the strength indexes are c = c/Fs , tan ϕ = tan ϕ/Fs . When Fs is relatively small (≤1.0), c, ϕ are increased, and thus, E k and bearing tension of the reinforced cushion are reduced so that the non-slip role will be small. On the contrary, its non-slip role will be increased. However, in fact, when Fs is relatively large (≥1.0), the ground, in general, will have a relatively small deformation, its tension will necessarily be poor; relatively large soil pressure will be transferred to the ground soil and the corresponding non-slip role will be relatively low. Therefore, the above three influence factors are related to each other; large deformation of the ground must cause relatively good tension of the reinforced cushion and small safety factor. Calculated stability results of five engineering are detailed in Table 11.5. According to the analysis on calculated results of five engineering in Table 11.5, because the integral ground is stable, the safety factor shall not be less than 1.0, and correspondingly, ηt ≥ 0.5 shall be true. If ηt = 0.5 is taken, it is correspondent with the mean ηt = 0.53 of the non-slip role of measured geotextile tension of five sections in two engineering. Calculated results [15] of the test engineering (failure) for the embankment with soft soil ground at certain harbor are detailed in Table 11.6. In the calculation, vane strength indexes are adopted; filling: ϕ = 20◦ and c = 8 kPa; tensile strength of the geotextile: 25 kN/m. When the embankment height according to the method in “Technical Specifications for Application of Geotextiles in Water Transport Engineering” [15] is relatively small, it is apparently unreasonable that the non-slip role is greater than the embankment height.

Vane

Quick shear

Quick shear

Quick shear

Vane

Port breakwater

East breakwater at certain harbor

North breakwater at certain port

Port dam

Wharf revetment

56.7

50.0

57.1

57.1

45

Tensile strength of the cushion kN/m

0.988

0.911

0.973

0.950

0.906

Safety factor for the cushion is not counted

1.056

0.976

1.015

0.992

1.132

6.88

7.13

4.38

4.42

24.9

1.071

1.002

1.020

1.048

1.035

Safety factor

Safety factor

Increase, %

ηt = 0.5

Methods in current specification [15]

Note Strength increase of the soil mass with loading is considered in the calculation

Strength index

Engineering case

Table 11.5 Calculated stability results of five engineering

8.40

9.99

4.83

10.3

14.2

Increase, %

1.121

1.060

1.043

1.086

1.141

Safety factor

ηt = 0.75

13.5

16.4

7.19

14.3

25.9

Increase, %

442 11 Slope Stability During Construction and Pore Water Pressure

Safety factor for the reinforced cushion is not counted

0.871

0.811

1.032

Test section

Failure Sect. 11.1: natural ground (without reinforced cushion); it is seen with a failure when heightened to 4.04 m

Failure Sect. 11.2: reinforced cushion and sand cushion; it is seen with a failure when heightened to 4.35 m

Stable section: reinforced cushion, sand cushion, and inserted plastic plate; strength increase is counted. The embankment is 6.40 m high

Table 11.6 Calculated results of test engineering (failure)

1.057

0.844

2.42

4.19

1.083

0.826

Safety factor

Safety factor

Increase, %

ηt = 0.5

Method in the Specification [15]

4.94

1.85

Increase, %

1.110

0.832

Safety factor

ηt = 0.75

7.56

2.59

Increase, %

11.7 Non-slip Role of Geosynthetic Reinforced Cushion 443

444

11 Slope Stability During Construction and Pore Water Pressure

Calculated results by new method provided here reflect the actual engineering conditions. For the failure Sect. 11.2, because the embankment height is relatively small, the non-slip role of the reinforced cushion is tiny; for the stable section, because the embankment height is increased, the non-slip role of the reinforced cushion is also increased. In the new method, considering that the non-slip role is related to the dumping and filling height of the embankment, not only the calculated safety factor is relatively reasonable when the dumping and filling height of the embankment is relatively large, but also the disadvantages of relatively large safety factor calculated according to the method in the specification is avoided when the dumping and filling height of the embankment is relatively small. It is well known that loading rate usually needs to be controlled when a dam is built on the soft soil ground; multistage loading is often adopted to check the overall stability of the loading at each stage. Meanwhile, the overall failure tends to be caused by the first failure of the ground. When the dumping and filling heights of the embankment are different, actual (including potential) non-slip effects of the reinforced cushion will be different; if calculated according to the same design tensile strength, when the dumping and filling height of the embankment is relatively small, it is likely to cause the relatively large calculated value.

References 1. Gao ZY (2006) Analysis of causes of landslide of revetments for an artificial island and rehabilitation of revetments. China Harbour Eng 3:13–15 2. Chen H (1987) Slope stability calculation theory and engineering experience analysis. Tianjin Soft Soil Ground. Tianjin Science and Technology Press, Tianjin 3. JTJ 250-98 Code for Foundation in Port Engineering 4. Qian JH, Yin ZZ (1996) Principle and calculation of geotechnical. China Water Power Press, Beijing 5. Tianjin University (1980) Soil mechanics and foundation. China Communications Press, Beijing 6. Huang CZ (1991) Exact solution of multi-dimensional consolidation theory. Tianjin Harbor Engineering Research Institute 7. Huang CZ (1991) A solution to Terzaghi’s consolidation equation. Chinese Journal of Geotechnical Engineering 13(1):34–47 8. Huang CZ, Xiao Y (1996) Analytic solution of two-dimensional consolidation theory. Chinese J Geotech Eng 18(31):47–54 9. Shen ZJ (2000) Theoretical soil mechanics. China Water Power Press, Beijing 10. Liu JF, Gong XN, Wang SY (1996) Foundation treatment by a stability analysis method considering non-slip role of geotextile 7(2):1–5 11. Shen ZJ (1998) Limit analysis of soft ground reinforced by geosynthetics. Chinese J Geotech Eng 20(4):82–86 12. Huang CZ, Miao ZH, Yu ZQ et al (2002) Non-slip role of geotextile reinforced cushion and the slop stability analysis method. In: National reinforced earth engineering conference abstracts. Modern Knowledge Press, Hongkong, pp 152–165 13. Tianjin Harbor Engineering Research Institute. Field Monitorig Report on Breakwater in Huanghua Port. 2002, 8

References

445

14. Department of Hydraulic Engineering of Tsinghua University, China Communication Planning and Design Institute for Water Transportation. Study Report on Centrifugal Model Test for the Ground with Geotextile Reinforced Cushion. 2002, 11 15. JTJ/T 239-98 Technical Specifications for Application of Geotextiles in Water Transport Engineering

Chapter 12

Soil Pressure

Generally, during the study of soil pressure, the distribution of it along the wall surface has to be obtained; therefore, it is the study of field failure mode.

12.1 Stress Field Method 12.1.1 Active Soil Pressure with Plane–Helicoid Fracture Surface The stress field behind wall (active soil pressure) is considered at first, when the plane–logarithm helically curved surface shown in Fig. 12.1 is the fracture plane, the soil mass behind wall is divided into (A) and (B) in consideration. (1) Stress field For plane fracture surface (A), the stress component has been obtained in Chap. 4: ⎫ σz = q + γ z ⎬ τx z = 0 ⎭ σx = (q + γ z)s02 + 2cs0

(12.1)

Where s0 = λ −



1 + λ2

s = (x − x R )/(z − z R )

(12.2) (12.3)

The interface of areas (A) and (B): z − z R = (x − x R )/s0 In (B), the fracture surface (family) is logarithmic helicoid: © Springer Nature Singapore Pte Ltd. and Zhejiang University Press, Hangzhou, China 2020 C. Huang, Limit Analysis Theory of the Soil Mass and Its Application, https://doi.org/10.1007/978-981-15-1572-9_12

447

448

12 Soil Pressure

Fig. 12.1 Schematic diagram for calculation of active soil pressure

x − x R = R exp(−λθ ) cos θ z − z R = R exp(−λθ ) sin θ

 (12.4)

The stress component in accordance with the equilibrium equation and yield condition has been derived in Chap. 4:  

exp(−2λ arctan s) s+λ c + A0 1 + λ2 + (s − λ)2 + γ (z − z R ) 1 − 3λ 2 λ 1 + 9λ 1 + s2

exp( − 3λ arctan s) 1  + f u (s) + A1 (z − z R ) 1 + λ2 + 2(s − λ)2 z − zR 1 + s2

λ(1 + 3sλ) 2 )s − (s − λ)(1 + sλ) exp(−2λ arctan s) τx z = γ (z − z R ) + A (1 + λ 0 1 + 9λ2 1 + s2

exp( − 3λ arctan s) s  + f u (s) + A1 (z − z R ) (1 + λ2 )s − 2(s − λ)(1 + sλ) z − zR 1 + s2

exp(−2λ arctan s) 1 − sλ c σx = − + γ (z − z R ) + A0 (1 + λ2 )s 2 + (1 + sλ)2 λ 1 + 9λ2 1 + s2

exp( − 3λ arctan s) s2  + f u (s) + A1 (z − z R ) (1 + λ2 )s 2 + 2(1 + sλ)2 z − zR 1 + s2 σz = −

(12.5)

(12.6)

(12.7)

where f u (s) is the undetermined function that satisfies f u (s0 ) = 0. At the interface of areas (A) and (B), the horizontal force shall be equal to the vertical force, which requires: c 1 1 + s02 (q + + γ z R ) exp(2λ arctan s0 ) 2 λ 1 + λ2  2 γ λ(s0 + λ) 1 + s0 A1 = exp(3λ arctan s0 ) 1 + 9λ2 1 + λ2

A0 =

(12.8)

(12.9)

12.1 Stress Field Method

449

And moreover:

∂f 2 1 = (σz − σx )(1 + 2h  tan φ − h 2 ) − τx z (2h  − (1 − h 2 ) tan φ)  2 2 ∂h (1 + h ) 2   f u (s) 3λ arctan s) 2(1 + sλ)2 2λ + s + 3sλ2 2 ) exp( −  = (z − z ) + γ 2λ A (1 + λ − 1 R 1 + λ2 (z − z R )2 (1 + 9λ2 )(1 + s 2 ) 1 + s2

(12.10)

Equations (12.1), (12.5), and (12.7) give the analytical expressions of the stress component of soil mass behind wall; they are in accordance with the static equilibrium equation, the yield condition, and the boundary condition for soil mass surface. In (A), the fracture surface is the plane, and the soil mass satisfies the extremum condition of the yield function, and in (B), the fracture surface is the logarithmic helicoid, and it is a pity that the soil mass does not satisfy the extremum condition of the yield function. (2) Equation of soil pressure Actually, the soil pressure is the value of the stress component at the boundary of a straight wall. At any height h on the straight wall (x = 0), it shall be: −x R = s1 h − zR Therefore,

−z R h−z R

=

s1 s0

or z R =

−s1 h . s0 −s1

According to Eq. (12.10), supposing

∂f ∂h 

 z=h

= 0, namely:



2λ + s + 3sλ2 2 exp(−3λ arctan s) f u (s) = −(h − z R ) γ 2λ − A1 (1 + λ ) √ (1 + 9λ2 )(1 + s 2 ) 1 + s2 2

On the straight wall, the stress component will be in accordance with the extremum condition of the yield function. It is hoped that the fracture surface determined in such a way would be close to the most dangerous fracture surface; thus, the obtained calculation equation for the soil pressure is close to the genuine solution. It is easy to verify that f u determined according to the above equation is in accordance with “ f u (s0 ) = 0”. Then: ⎤ ⎡ 2 2 2 s 2γ λ(s1 + 2λ + 3s1 λ ) 1+λ − A1  f u (s1 ) = −z 2R 02 ⎣ exp(−3λ arctan s1 )⎦ s1 (1 + 9λ2 )(1 + s12 ) 1 + s2 1

(12.11) It is substituted in Eqs. (12.6) and (12.7). Through rearrangement, the horizontal and vertical components of the soil pressure on the straight wall are:

450

12 Soil Pressure

  c c  + q+ Aq + γ h Aγ (1 + λ2 )s12 + (1 + s1 λ)2 λ λ

   c Aq + γ h Aγ (1 + λ2 )s1 − (s1 − λ)(1 + s1 λ) = q+ λ

σx0 = −

(12.12)

τx z0

(12.13)

where Aγ =



1 − 3s1 λ s1 s0 + A − A q s0 − s1 (1 + 9λ2 )(1 + s12 ) s0

1 + s02 exp[2λ(arctan s0 − arctan s1 )] 2(1 + λ2 )(1 + s12 )  2λ(s1 + λ) 1 + s02 A= exp[3λ(arctan s0 − arctan s1 )] 2 2 (1 + 9λ )(1 + λ ) 1 + s12 Aq =

(10.14) (12.15)

(12.16)

If the shear stress at any point of wall soil is the shear strength reduction: τx z0 = tan δ(σx0 + c/λ)

(12.17)

According to Eqs. (12.12) and (12.13), the following will be derived:  λ(λ − tan δ) − (1 + λ2 )(λ2 − tan2 δ) s1 = λ(1 + λ tan δ) + (1 + λ2 ) tan δ

(12.18)

Soil pressure equation may be written as the common form: σx0 = q K xq − cK xc + γ h K xγ

(12.19)

  K xq = Aq (1 + λ2 )s12 + (1 + s1 λ)2

(12.20)

where

K xc =

1 (1 − K xq ) λ

K xγ = Aγ [(1 + λ2 )s12 + (1 + s1 λ)2 ]

(12.21) (12.22)

(3) Calculated result for coefficient of soil pressure The calculated results for coefficient of soil pressure are detailed in Table 12.1. When “δ,” the calculated results are the same as the Rankine soil pressure.

12.1 Stress Field Method

451

Table 12.1 Coefficient of active soil pressure (δ = 0.5ϕ) ϕ(◦ )

Ka

K xq

K xγ

K xc

1.0

0.958

0.959

0.959

2.334

2.0

0.919

0.920

0.920

2.280

3.0

0.881

0.883

0.882

2.228

4.0

0.845

0.848

0.846

2.177

5.0

0.810

0.814

0.812

2.128

6.0

0.777

0.781

0.780

2.080

7.0

0.746

0.750

0.748

2.034

8.0

0.716

0.721

0.719

1.989

9.0

0.687

0.692

0.690

1.945

10.0

0.660

0.665

0.663

1.902

11.0

0.633

0.638

0.636

1.860

12.0

0.608

0.613

0.611

1.819

13.0

0.584

0.589

0.587

1.780

14.0

0.561

0.566

0.564

1.741

15.0

0.539

0.544

0.542

1.703

16.0

0.517

0.522

0.520

1.666

17.0

0.497

0.502

0.500

1.630

18.0

0.477

0.482

0.480

1.594

19.0

0.458

0.463

0.461

1.560

20.0

0.440

0.445

0.443

1.526

21.0

0.422

0.427

0.425

1.493

22.0

0.406

0.410

0.408

1.460

23.0

0.389

0.394

0.392

1.429

24.0

0.374

0.378

0.376

1.397

25.0

0.359

0.363

0.361

1.367

26.0

0.344

0.348

0.346

1.337

27.0

0.330

0.334

0.332

1.307

28.0

0.317

0.320

0.319

1.278

29.0

0.304

0.307

0.306

1.250

30.0

0.291

0.294

0.293

1.222

31.0

0.279

0.282

0.281

1.195

32.0

0.267

0.270

0.269

1.168

33.0

0.256

0.259

0.258

1.141

34.0

0.245

0.248

0.247

1.115

35.0

0.235

0.237

0.236

1.089

36.0

0.225

0.227

0.226

1.064 (continued)

452

12 Soil Pressure

Table 12.1 (continued) ϕ(◦ )

Ka

K xq

K xγ

K xc

37.0

0.215

0.217

0.216

1.039

38.0

0.205

0.208

0.207

1.014

39.0

0.196

0.198

0.197

0.990

40.0

0.187

0.189

0.188

0.966

41.0

0.179

0.181

0.180

0.942

42.0

0.171

0.172

0.172

0.919

43.0

0.163

0.164

0.164

0.896

44.0

0.155

0.156

0.156

0.873

45.0

0.148

0.149

0.148

0.851

Note K a is Coulomb coefficient of soil pressure

12.1.2 Passive Soil Pressure of Plane–Helicoid Fracture Surface The passive soil pressure is the force (Fig. 12.2) on the wall surface when the soil mass before wall is in the limit state, like the discussion on the active one, it may be obtained: σx0 = pz G xq + cG xc + γ hG xγ

(12.23)

  G xq = Aq (1 + λ2 )s12 + (1 + s1 λ)2

(12.24)

Fig. 12.2 Schematic diagram for calculation of passive soil pressure

12.1 Stress Field Method

453

G xc = (G xq − 1)/λ

(12.25)

  G xγ = Aγ (1 + λ2 )s12 + (1 + s1 λ)2

(12.26)

where the expressions of G xq , G xr are the same as those of K xq , K xr under active condition are the same respectively; however, s0 , s1 shall be: s0 =



1 + λ2 + λ

 λ(λ − tan δ) + (1 + λ2 )(λ2 − tan2 δ) s1 = λ(1 + λ tan δ) + (1 + λ2 ) tan δ

(12.27)

(12.28)

The calculated results for coefficient of passive earth pressure are detailed in Table 12.2. Generally, the passive soil pressure is less than the Rankine soil pressure; when ϕ is relatively large, the difference is especially apparent.

12.2 Stress Equation-Based Stress Field Method The soil pressure is obtained according to the following equation: ⎫ 1 + λ2 + (h  − λ)2 c c ⎪ ⎪ ⎪ σx + = (σ + ) ⎪ 2 ⎪ λ 1+h λ ⎪ ⎪ ⎬ 2 2  2 (1 + λ )h + (1 + λh ) c c σz + = (σ + ) λ 1 + h 2 λ ⎪ ⎪ ⎪ ⎪ 2  2 ⎪ 1 + λ − (h − λ) c ⎪ ⎪ ⎭ ) τx z = λ (σ + 2 1+h λ dσ 2λ  h − λ c  dh  − = γ σ + dx 1 + h 2 λ dx 1 + λ2

12.2.1 Soil Pressure of Plane–Helicoid Fracture Surface (A): (Fig. 12.1): the stress field is as Eq. (12.1). (B): according to Eq. (12.30), it shall be:

(12.29)

(12.30)

454

12 Soil Pressure

Table 12.2 Coefficient of passive earth pressure (δ = 0.5ϕ) ϕ(◦ )

KP

G xq

G xγ

G xc

1.0

1.044

1.043

1.043

2.447

2.0

1.090

1.088

1.089

2.506

3.0

1.139

1.135

1.136

2.567

4.0

1.190

1.184

1.186

2.630

5.0

1.244

1.236

1.239

2.696

6.0

1.301

1.290

1.294

2.764

7.0

1.361

1.348

1.353

2.834

8.0

1.425

1.409

1.415

2.907

9.0

1.492

1.472

1.480

2.983

10.0

1.563

1.540

1.549

3.061

11.0

1.639

1.611

1.622

3.143

12.0

1.720

1.686

1.699

3.228

13.0

1.806

1.766

1.781

3.317

14.0

1.897

1.850

1.868

3.409

15.0

1.995

1.939

1.960

3.505

16.0

2.099

2.034

2.058

3.606

17.0

2.210

2.135

2.162

3.711

18.0

2.330

2.242

2.273

3.822

19.0

2.458

2.356

2.392

3.937

20.0

2.595

2.477

2.518

4.058

21.0

2.744

2.607

2.654

4.185

22.0

2.904

2.745

2.799

4.319

23.0

3.077

2.893

2.955

4.460

24.0

3.265

3.051

3.122

4.608

25.0

3.468

3.222

3.302

4.764

26.0

3.690

3.404

3.496

4.929

27.0

3.932

3.601

3.706

5.104

28.0

4.197

3.812

3.933

5.289

29.0

4.487

4.041

4.178

5.486

30.0

4.807

4.288

4.444

5.694

31.0

5.160

4.555

4.734

5.917

32.0

5.551

4.845

5.050

6.154

33.0

5.986

5.160

5.394

6.407

34.0

6.472

5.504

5.772

6.677

35.0

7.016

5.879

6.186

6.968

36.0

7.630

6.289

6.642

7.279 (continued)

12.2 Stress Equation-Based Stress Field Method

455

Table 12.2 (continued) ϕ(◦ )

KP

37.0

G xq

G xγ

G xc

8.324

6.738

7.144

7.615

38.0

9.114

7.232

7.699

7.977

39.0

10.019

7.777

8.315

8.368

40.0

11.062

8.378

9.001

8.793

41.0

12.272

9.044

9.765

9.254

42.0

13.688

9.784

10.622

9.756

43.0

15.359

10.609

11.584

10.305

44.0

17.351

11.532

12.669

10.906

45.0

19.752

12.567

13.898

11.567

Note K p is Coulomb coefficient of soil pressure



σ+

c λ

⎡ = exp(2λ arctan h  )⎣C +

x γ x1



⎤

h −λ exp(−2λ arctan h  )dx ⎦ (12.31) 1 + λ2

According to the force equilibrium condition at the interface of (A) and (B), it shall be: C=

q + γ z 1 + c/λ  exp(−2λ arctan h 0 )  1 + h0λ

(12.32)

where   ϕ π 3π − h 0 = tan θ0 − + ϕ = − tan θ0 and θ0 = 2 4 2

(12.33)

When it is helical slip surface, x

   (h − λ) exp(−2λ arctan h  ) dx

x1

=

π 

1 + λ2 exp 2λ − ϕ {exp(−2λθ)[z − z R − 3λ(x − x R )] − exp(−2λθ0 )[z 1 − z R − 3λ(x1 − x R )]} 2 2 1 + 9λ

where (x1 , z 1 ) is one point at the interface of (A) and (B). Therefore, on the straight wall:   c q + γ z 1 + c/λ = exp −2λ(θ0 − θ1 )  λ 1 + h0 λ      γ h − z R − 3λ(−x R ) − exp −2λ(θ0 − θ1 ) z 1 − z R − 3λ(x1 − x R ) + 1 + 9λ2

σ0 +

θ0 Because: z R = x R tan θ0 and −x R tan θ1 = h − z R , hence: z R = h tan θtan 0 −tan θ1

456

12 Soil Pressure

θ0 As: z 1 − z R = (x1 − x R ) tan θ0 and z 1 − z R = (h − z R ) sin exp[−λ(θ0 − θ1 )] sin θ1 Therefore,  c c Aq + γ h Aγ σ0 + = q + (12.34) λ λ

where Aq =

1 exp[−2λ(θ0 − θ1 )] 1 − λ tan θ0

(12.35)

 1 tan θ0 tan θ1 − 3λ + exp [−2λ(θ0 − θ1 )] Aγ = − 2 tan θ0 − tan θ1 1 + 9λ 1 − λ tan θ0  exp[−3λ(θ0 − θ1 )] cos θ0 (12.36) (1 + λ tan θ0 ) −4λ cos θ1 (1 + 9λ2 )(1 − λ tan θ0 ) Substituted in Eq. (12.29), the horizontal and vertical stress components of soil pressure on the wall surface are respectively: 

1 + λ2 + (h 1 − λ)2  c c Aq + γ h Aγ σx0 + = q+ 2 λ λ 1 + h1 

2 2  c 1 + λ − (h 1 − λ) Aq + γ h Aγ q+ τx z0 = λ 2 λ 1 + h1

(12.37) (12.38)

According to the boundary condition of the straight wall, it will be: h 1

   π λ − tan δ = tan θ1 − + ϕ = λ ± (1 + λ2 ) 2 λ + tan δ

(12.39)

where plus sign is taken as the active soil pressure and minus sign as the passive soil pressure. It may be verified that Eqs. (12.37), (12.38), (12.12), and (12.13) are the same.

12.2.2 Soil Pressure of Plane–Arc Fracture Surface (A): the stress field is still as Eq. (12.1). (B): it is arc fracture surface (Fig. 12.3), because:

σ+

  c γR (1 − 2λ2 ) sin θ − 3λ cos θ = C0 exp[2λ(θ − θ0 )] + 2 2 λ (1 + λ )(1 + 4λ )   (12.40) − (1 − 2λ2 ) sin θ0 − 3λ cos θ0 exp[2λ(θ − θ0 )]}

12.2 Stress Equation-Based Stress Field Method

457

Fig. 12.3 Schematic diagram for calculation of soil pressure for plane–arc surface

On the straight wall: σ0 +

   c γ q + γ z 1 + c/λ (1 − 2λ2 )(h − z R ) − 3λ(−x R ) = exp −2λ(θ0 − θ1 ) +  2 2 λ (1 + λ )(1 + 4λ ) 1 + h0 λ − exp[−2λ(θ0 − θ1 )][(1 − 2λ2 )(z 1 − z R ) − 3λ(x1 − x R )]

(12.41)

where  ϕ π 3π  , θ0 = + , x1 − x R = (z 1 − z R )/ tan θ0 h 0 = tan θ0 − 2 4 2 −x R = (h − z R )/ tan θ1 , and z 1 − z R = (h − z R )

sin θ0 sin θ1

Substituted in Eq. (12.41), and after rearrangement, Eq. (12.34) can be derived likewise; however Aq , Aγ in the equation shall be Eqs. (10.42) and (10.43), respectively: Aq = 

Aγ = A

1 − 2λ2 + 3λh 1

+ (1 + λ2 )(1 + 4λ2 )

A=

1 exp(−2λ(θ0 − θ1 ))  1 + h0λ ⎡

⎛



(12.42) ⎞⎤

4λ(h 0 − λ) sin θ0 exp[−2λ(θ0 − θ1 )] ⎣ ⎠⎦ 1 − A ⎝1 +  1 + 4λ2 sin θ1 1 + h0 λ

(12.43)

1    1 − h 0 h 1 + 2h 0 λ sin θ0 / sin θ1

(12.44)

Thus, the soil pressure equation which is still Eqs. (12.37) and (12.38) can be derived, and moreover, according to the boundary condition of the straight wall, it will be:

458

12 Soil Pressure

  π λ − tan δ = λ ± (1 + λ2 ) h 1 = tan θ1 − 2 λ + tan δ 

(12.45)

where plus sign is taken as the active soil pressure and minus sign the passive soil pressure. Compared with the plane–helicoid, the load and cohesion coefficients are totally the same, and the soil weight coefficients are very close to each other (see Tables 12.3 and 12.4). % & Table 12.3 Comparison for coefficients of active soil pressure K xγ ϕ(◦ )

10

δ(◦ )

Sokolovskii method

Limit analysis method

Improved limit equilibrium method

Stress field method Plane–helicoid

Plane–arc surface

5

0.670

0.664

0.665

0.663

0.663

10

0.650

0.642

0.644

0.622

0.622

10

0.450

0.448

0.449

0.443

0.443

20

0.440

0.434

0.439

0.394

0.394

30

15

0.300

0.302

0.303

0.293

0.293

30

0.310

0.303

0.304

0.250

0.252

40

20

0.200

0.200

0.200

0.188

0.189

40

0.220

0.214

0.215

0.155

0.156

20

& % Table 12.4 Comparison for coefficients of passive soil pressure G xγ ϕ(◦ )

δ(◦ )

Sokolovskii method

Limit analysis method

Plane–helicoid

1.560

10

1.660

1.680

1.673

1.684

1.688

20

10

2.550

2.580

2.528

2.518

2.520

20

3.040

3.170

3.129

3.069

3.115

15

4.620

4.710

4.649

4.444

4.455

30

6.550

7.100

6.929

6.384

6.746

20

9.690

10.10

9.815

9.001

9.076

40

18.200

20.90

20.01

1.549

Plane–arc surface

5

40

1.557

Stress field method

10

30

1.560

Improved limit equilibrium method

16.614

1.549

20.08

12.3 Comparison of Existing Methods

459

12.3 Comparison of Existing Methods Soil pressure is solved according to limit equilibrium; generally, there are the following methods. Limit equilibrium method: for plane fracture surface, Rankine and Coulomb equation is generally believed that the method is only applicable to non-cohesive soil, but some improves its equation to be applicable to cohesive soil for approximate calculation [1]. Improved limit equilibrium method: it is the calculation equation [2] for soil pressures of plane fracture surface and logarithmic helicoid. Upper bound method of limit analysis: the fracture surface is plane–logarithm helically (cylindrical) curved surface plane and the equation derived is as introduced in reference [3]. Characteristic line method: numerical solution of Sokolovskii. Tables 12.3 and 12.4 give the comparison for coefficients of active soil pressure and that for coefficients of passive soil pressure, respectively. The results calculated by the Sokolovskii method and limit analysis method are quoted from Ref. [3], and those (logarithmic helicoid) by the improved limit equilibrium method form Ref. [2]. When “δ = 0,” the values calculated by various methods are basically the same. Seen from the calculated results, the methods are approximately the same within a considerably large range. When ϕ is relatively large, for active soil pressure, the coefficients of active soil pressure calculated by slip line method, upper bound method, and improved limit equilibrium method increase as δ is relatively large and is increasing, which is abnormal. For passive soil pressure, the coefficient of passive soil pressure is the minimum when it is plane–helicoid fracture surface for the stress field method. “Code for Design and Construction for Quay Wall of Sheet Pile” [1] in China gives the calculation equation applicable to cohesive soil, and form of the equation is entirely similar to the stress field method, and the comparison between the calculated result for the coefficient of soil pressure and the stress field method is detailed in Tables 12.5 and 12.6. Load coefficient and soil weight coefficient active soil pressure as well as the sheet pile specification method and stress field method are considerably approximate; however, the cohesion coefficient differs greatly. When ϕ(≥ 30◦ ) is relatively large, the cohesion coefficient reduces along with the increase of δ, which signifies that along with the increase of δ, the active soil pressure increases, and it is abnormal. For load coefficient and soil weight coefficient of passive soil pressure, the value calculated by the sheet pile specification method is greater than that by the stress field method, especially when ϕ is relatively large. The three coefficients in the sheet pile specification method increase obviously along with the increase of δ, and they are far larger than those calculated by the stress field method. Sheet pile specification equation is derived based on the method the same as the Coulomb soil pressure equation; namely, it is plane fracture surface.  When “δ = 0,” √   δ 2 ” as “h 0 = λ ± 1 + λ ” on the soil mass surface, and “h 1 = λ ± (1 + λ2 ) λ−tan λ+tan δ

40

30

20

10

ϕ(◦ )

0.187

0.179

0.161

20.00

26.67

40.00

0.257

30.00

0.197

0.279

20.00

13.33

0.291

15.00

0.401

20.00

0.304

0.426

13.33

10.00

0.440

0.625

10.00

0.455

0.647

6.67

6.67

0.660

5.00

10.00

0.673

0.172

0.182

0.189

0.197

0.273

0.288

0.294

0.305

0.422

0.434

0.445

0.457

0.646

0.656

0.665

0.675

0.161

0.179

0.187

0.197

0.257

0.279

0.291

0.304

0.401

0.426

0.440

0.455

0.625

0.647

0.660

0.673

Sheet pile specifications

0.155

0.180

0.188

0.197

0.250

0.282

0.293

0.305

0.394

0.430

0.443

0.456

0.622

0.652

0.663

0.675

Stress field method Plane–helicoid

Soil weight coefficient K xγ

Sheet pile specifications

Stress field method

Load coefficient K xq

3.33

δ(◦ )

Table 12.5 Comparison for coefficients of active soil pressure

0.156

0.180

0.189

0.197

0.252

0.282

0.293

0.305

0.394

0.430

0.443

0.456

0.622

0.652

0.663

0.675

Stress field method Plane–arc surface

0.591

0.714

0.772

0.827

0.804

0.922

0.980

0.038

1.075

1.180

1.234

1.288

1.445

1.520

1.559

1.598

Sheet pile specifications

0.987

0.975

0.966

0.956

1.259

1.238

1.222

1.203

1.589

1.554

1.526

1.492

2.008

1.951

1.902

1.841

Stress field method

Cohesion coefficient K xc

460 12 Soil Pressure

40

30

20

10

ϕ(◦ )

16.73

21.59

70.92

26.67

30.00

40.00

8.734

30.00

11.06

6.982

22.50

20.00

5.737

20.00

3.312

20.00

4.807

2.926

15.00

15.00

2.811

1.704

10.00

2.595

1.634

7.50

13.33

1.610

6.67

10.00

1.563

10.92

10.09

9.573

8.378

5.026

4.776

4.633

4.288

2.699

2.625

2.582

2.477

1.597

1.578

1.567

1.540

70.92

21.59

16.73

11.06

8.734

6.982

5.737

4.807

3.312

2.926

2.811

2.595

1.704

1.634

1.610

1.563

Sheet pile specifications

16.61

11.93

10.91

9.001

6.384

5.214

4.953

4.444

3.069

2.739

2.666

2.518

1.684

1.603

1.585

1.549

Stress field method Plane–helicoid

Soil weight coefficient G xγ

Sheet pile specifications

Stress field method

Load coefficient G xq

5.00

δ(◦ )

Table 12.6 Comparison for coefficients of passive soil pressure

20.08

12.36

11.15

9.076

6.746

5.267

4.985

4.455

3.115

2.745

2.670

2.520

1.688

1.603

1.585

1.549

Stress field method Plane–arc surface

77.25

22.00

16.74

10.75

11.20

8.680

6.957

5.712

4.934

4.257

4.059

3.702

2.934

2.792

2.743

2.647

Sheet pile specifications

11.95

10.84

10.22

8.793

6.974

6.540

6.292

5.694

4.668

4.465

4.348

4.069

3.383

3.276

3.214

3.061

Stress field method

cohesion coefficient G xc

12.3 Comparison of Existing Methods 461

462

12 Soil Pressure

Fig. 12.4 Maximum fracture surface of plane–helicoid

on the wall surface, fracture surface is impossible to be plane. It may be the reason why the passive soil pressure is greatly different from the stress field method. For the two types of fracture surface of the stress field method, their active soil pressures are almost the same; however, the plane–helicoid calculation mode of the passive soil pressure is slightly smaller than the plane–arc surface. How to determine δ of passive soil pressure is another notable thing. It is easy  to obtain: if “tan δ ≥ λ/(1 + 2λ2 ),” it must be provided with that “h 1 ≥ 0” for the fracture surface of the wall surface, which means that the depth of the fracture surface is greater than the height of the straight wall (Fig. 12.4), though its rationality still requires discussion. If it is required that depth of the fracture surface is not greater than height of the straight wall, and determination of δ shall be in accordance with “tan δ ≤ λ/(1+2λ2 )” when the passive soil pressure is calculated according to the plane–helicoid fracture surface.

References 1. JTJ 292-98 Code for Design and Construction for Quay Wall of Sheet Pile 2. Luan MT, Jin CQ, Lin G (1992) Improved limit equilibrium method and its applications to stability analysis of soil masses. Chinese J Geotech Eng 14(Supplement):21–29 3. Chen HF (1995) Limit analysis and soil plasticity (trans: Zhan S, Proofread by Han D). China Communications Press