Geotechnical Reliability Analysis: Theories, Methods and Algorithms 9811962537, 9789811962530

Citation preview

Jie Zhang · Te Xiao · Jian Ji · Peng Zeng · Zijun Cao

Geotechnical Reliability Analysis Theories, Methods and Algorithms

Geotechnical Reliability Analysis

Jie Zhang · Te Xiao · Jian Ji · Peng Zeng · Zijun Cao

Geotechnical Reliability Analysis Theories, Methods and Algorithms

Jie Zhang Tongji University Shanghai, China Jian Ji Hohai University Nanjing, China

Te Xiao The Hong Kong University of Science and Technology Hong Kong SAR, China Peng Zeng Chengdu University of Technology Chengdu, China

Zijun Cao Southwest Jiaotong University Chengdu, China

ISBN 978-981-19-6253-0 ISBN 978-981-19-6254-7 (eBook) https://doi.org/10.1007/978-981-19-6254-7 Jointly published with Tongji University Press, Shanghai, China The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Tongji University Press © Tongji University Press Co., Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

Mechanics for structural analysis consists of a multiple of disciplines including Theoretical mechanics, Elasticity, Mechanics of materials, Structural mechanics, Plasticity, etc. This advanced knowledge has made us evaluate the actions and resistance of a structure with high accuracy. However, when applying the analytical results to engineering practice, an empirical coefficient called factor of safety is applied which, to a large extent, absorbs the efforts we have made in developing such an elegant framework of the solid mechanics. In this sense, we may say that the discipline of Structural reliability analysis represents our efforts of transferring the empirical approach of factor of safety to a new area in Mechanics of structure that evaluates the uncertainties involved in the analysis on a probabilistic basis with scientific evaluations on the safety margin applied to the design. There are several well-known textbooks for engineering reliability analysis, such as Probability Concepts in Engineering Planning and Design, Vol I: Basic Principles and Probability Concepts in Engineering Planning and Design, Vol II: Decision, Risk and Reliability by Alfredo H-S Ang and Wilson H Tang. This book, Geotechnical Reliability Analysis: Theories, Methods, and Algorithms, by Jie Zhang, Te Xiao, Jian Ji, Peng Zeng, and Zijun Cao, is a welcome addition that specially focuses on reliability analysis in geotechnical engineering, as the characteristics of uncertainties and the limit state functions in geotechnical engineering are often quite different from those in other disciplines of civil engineering. Starting from the basics of probability theory, this book systematically introduces the first order reliability methods, Monte Carlo simulation, response surface methods, reliability-based design, and Bayesian methods within the context of geotechnical engineering. In the textbook, the theories and methods are clearly illustrated with typical geotechnical examples, such as bearing capacity of shallow foundations and stability of slopes. Concise computer codes are provided to help students have indepth understanding of the algorithms and recover the computational details of the examples. Such computer codes can also be easily adapted for other applications. At the end of each chapter, the recent developments in related topics are also described. In the past decades, the geotechnical reliability analysis field has experienced significant advancements. This book provides an excellent balance between the explanation of v

vi

Foreword

topics of practical importance and the state of the art in the geotechnical reliability field. The authors have made a significant contribution in synthesizing the knowledge in geotechnical reliability and translating such knowledge into an easy-to-access textbook. I believe this textbook will be an important, effective and timely tool for students and researchers to learn the theories, methods, and algorithms for geotechnical reliability analysis, and help facilitate the practical use of these methods in geotechnical engineering. Zuyu Chen Chinese Academy of Sciences Beijing, China

Preface

Uncertainties are pervasive in geotechnical engineering. Reliability analysis provides a scientific way to characterize, model, and assess the impact of uncertainties in geotechnical engineering. Due to the increase in processing power of personal computers, the field of geotechnical reliability has experienced significant advancements since 1990s. Learning geotechnical reliability, however, still seems challenging for many beginners. There are two possible reasons for such a phenomenon. Firstly, the concepts and theories involved are abstract and could be hard to follow. Secondly, the application of geotechnical reliability in practice typically involves considerable programming work, which often makes it difficult for beginners to get immediate hands-on experience on geotechnical reliability theory. The purpose of this textbook is to provide a convenient tool for graduate students and professionals to learn geotechnical reliability theories, methods, and algorithms. The unique features of the textbook include: (1) the reliability theories and methods are introduced and illustrated using geotechnical examples as many as possible; (2) simple computer codes are provided to illustrate the important algorithms such that the readers can adapt these codes to their own applications conveniently; and (3) the review on recent developments is provided to guide the readers to explore more on the topics they are interested in. We hope this textbook can facilitate the teaching and the practical application of geotechnical reliability theories, methods, and algorithms. There are seven chapters in this textbook. In Chap. 1, the relevant basics of the probability theory are explained in the context of geotechnical engineering. Readers who are familiar with probability theory can skip this chapter. In Chap. 2, the first-order reliability methods, including the mean value first-order reliability method, the advanced first-order reliability method, and system reliability methods based on first-order reliability analyses, are introduced. As first-order reliability methods are efficient and often quite accurate, knowledge from this chapter can be used to solve many geotechnical reliability problems. In Chap. 3, simulation-based methods, including Monte Carlo simulation, Latin hypercube sampling, importance sampling, and subset simulation, are described.

vii

viii

Preface

These methods can be used when first-order reliability methods are not applicable, but are generally computationally more demanding. In Chap. 4, the response surface method, which is a powerful tool to solve geotechnical reliability problems when the deterministic geotechnical model must be solved numerically, is explained. As numerical models are increasingly used in routine geotechnical design, the response surface method is expected to play an increasingly important role in geotechnical engineering. In Chap. 5, how to model the spatial variability of soils with the random field theory is introduced. Compared with many other fields, the effect of spatial variability of materials seems more obvious in geotechnical engineering. How to consider the effect of spatial variability of soils is an important and unique topic in geotechnical reliability analyses. In Chap. 6, how to perform reliability-based design in geotechnical engineering is briefly discussed. Currently, many geotechnical codes are experiencing the transition from the traditional working stress design to the reliability-based design. However, such a transition is not smooth. The challenges involved in reliability-based design are also discussed, and three interesting developments in this field are introduced. In Chap. 7, Bayesian methods are introduced for geotechnical applications. In geotechnical engineering, it is common practice to combine information from different sources to help decision making. Bayesian methods are particularly useful for information combination and are increasingly used in geotechnical engineering. Bayesian methods also provide a formal tool to implement the concept of observational method in geotechnical engineering. The authors are indebted to the late Prof. Wilson H. Tang, who was one of the pioneers in geotechnical reliability and the Ph.D. supervisor of the first author. The authors are thankful to Profs. Limin Zhang, Hongwei Huang, C. Hsein Juang, Dianqing Li, Bak Kong Low, Rafael Jiménez, and Yu Wang for their great mentoring and collaboration. We are also thankful to our graduate students who have helped a lot during the preparation of the manuscript, in particular, to Jinzheng Hu, Tianpeng Wang, Chengguang Wu, Shihao Xiao, Meng Lu, Yuan Sun, Ruisong Cheng, Shuangyi Wu, Hongzhi Cui, Yining Hu, Lepei Wang, Tianlong Zhang, Guohui Gao, and Qiang Zhou. Last but not least, we want to thank Ms. Jialin Yuan for her excellent editorial and production efforts. This textbook is published under the support from the Graduate School of Tongji University. The opinions expressed in the book are those of the authors and do not necessarily reflect the views of the Graduate School of Tongji University. Shanghai, China Hong Kong SAR, China Nanjing, China Chengdu, China Chengdu, China

Jie Zhang Te Xiao Jian Ji Peng Zeng Zijun Cao

About This Book

Uncertainties are pervasive in geotechnical engineering. It is important that such uncertainties and their effects on geotechnical design and decision making can be modeled and quantified. The purpose of this textbook is to systematically introduce the reliability theories, methods, and algorithms to characterize, model, and assess the impact of uncertainties in geotechnical engineering. The unique features of the textbook include: (1) reliability theories and methods are introduced and illustrated in detail using geotechnical examples as many as possible; (2) simple computer codes are provided to illustrate the important algorithms such that the readers can adapt these codes to their own applications conveniently; and (3) comprehensive review on recent developments is provided in each chapter to guide the readers to explore more on the topics they are interested in. This textbook provides a convenient and easyto-access tool to teach and learn the reliability theory in the context of geotechnical engineering.

ix

Contents

1 Basics of Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Elements of Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 De Morgan’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Axioms of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Conditional Probability and Multiplication Rule . . . . . . . . . . 1.3 Total Probability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Bernoulli Sequence and Binomial Distribution . . . . . . . . . . . 1.4.2 The Poisson Process and Poisson Distribution . . . . . . . . . . . . 1.5 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Multivariate Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Covariance and Correlation Coefficient . . . . . . . . . . . . . . . . . . 1.6.2 Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Multivariate Lognormal Distribution . . . . . . . . . . . . . . . . . . . . 1.7 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 6 7 7 9 11 12 15 16 17 19 21 23 23 26 28 31 32

2 First Order Reliability Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Concept of Geotechnical Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mean Value First Order Second Moment Method (MVFOSM) . . . . 2.3 Advanced First Order Reliability Method (AFORM) . . . . . . . . . . . . . 2.3.1 Hasofer-Lind Reliability Index for Uncorrelated Normal Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 AFORM for Uncorrelated Non-normal Variables . . . . . . . . . 2.3.3 AFORM for Correlated Normal Variables . . . . . . . . . . . . . . . 2.3.4 AFORM for Correlated Non-normal Variables . . . . . . . . . . . 2.3.5 EXCEL-Based AFORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 38 41 41 47 49 52 54

xi

xii

Contents

2.3.6 AFORM for Implicit Performance Function . . . . . . . . . . . . . . 2.4 System Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Ditlevsen’s Bounds for System Reliability Analyses . . . . . . . 2.4.2 Linearization Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 67 67 72 78 82

3 Simulation-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Random Sampling for Univariate Variable . . . . . . . . . . . . . . . . . . . . . 3.1.1 Inverse Transformation Method . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Acceptance-Rejection Method . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Markov Chain Monte Carlo Simulation . . . . . . . . . . . . . . . . . 3.2 Random Sampling for Multivariate Variables . . . . . . . . . . . . . . . . . . . 3.2.1 Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Correlated Normal Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Correlated Non-normal Variables . . . . . . . . . . . . . . . . . . . . . . . 3.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Latin Hypercube Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Subset Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 88 90 92 94 94 96 98 100 106 109 112 120 124

4 Response Surface Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Classical Response Surface Method (RSM) . . . . . . . . . . . . . . . . . . . . 4.1.1 Calibration of a Second-Order Polynomial Function . . . . . . . 4.1.2 Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Iterative RSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Kriging-Based RSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Kriging Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Determination of Experimental Points . . . . . . . . . . . . . . . . . . . 4.2.3 Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Active-Learning Kriging Model . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Support Vector Machine (SVM)-Based RSM . . . . . . . . . . . . . . . . . . . 4.3.1 SVM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Calibration of SVM and Reliability Analysis . . . . . . . . . . . . . 4.3.3 Active-Learning SVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Application in Slope Reliability Analysis . . . . . . . . . . . . . . . . 4.4 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 128 128 131 134 137 137 140 142 144 153 154 156 160 165 167 170

5 Spatial Variability of Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Modeling of Spatial Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Random Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Spatial Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Characterization of Spatial Variability . . . . . . . . . . . . . . . . . . . . . . . . .

173 173 173 177 181

Contents

xiii

5.2.1 Mean-Crossings Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . 5.3 Simulation of Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Covariance Matrix Decomposition . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Karhunen-Loève Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Expansion Optimal Linear Estimation . . . . . . . . . . . . . . . . . . . 5.3.4 Sequential Gaussian Simulation . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Multidimensional and Multivariate Random Field . . . . . . . . . . . . . . . 5.4.1 Spatial Correlation Modeling with Separable Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Simulation of Multidimensional Random Field . . . . . . . . . . . 5.4.3 Simulation of Multivariate Random Field . . . . . . . . . . . . . . . . 5.5 Effects of Spatial Variability on Geotechnical Reliability . . . . . . . . . 5.6 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

182 183 185 188 189 192 197 200 203

6 Reliability-Based Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Calibration of a Single Resistance Factor . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Assessment of Reliability Level of an Existing Design . . . . . 6.1.2 Calibration of Resistance Factor . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Calibration of Multiple Resistance Factors . . . . . . . . . . . . . . . . . . . . . 6.2.1 Design Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Challenges in Implementation of Load and Resistance Factor Design (LRFD) in Geotechnical Engineering . . . . . . . . . . . . . . . . . . . 6.3.1 Methods for Applying Partial Factors . . . . . . . . . . . . . . . . . . . 6.3.2 Robustness of the Resistance Factors . . . . . . . . . . . . . . . . . . . . 6.3.3 Difficulties in Specifying the Characteristic Values . . . . . . . . 6.3.4 Selection of Target Reliability Index . . . . . . . . . . . . . . . . . . . . 6.4 Full Probabilistic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 General Design Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Direct MCS-Based Reliability-Based Design . . . . . . . . . . . . . 6.4.3 Reliability-Based Design Using Subset Simulation . . . . . . . . 6.5 Robust Geotechnical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Concept of Robust Geotechnical Design . . . . . . . . . . . . . . . . . 6.5.2 Measures of Design Robustness . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Procedure for Implementing Robust Geotechnical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Method of Ratio of Safety Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227 228 229 231 233 233

203 205 210 212 221 223

236 236 237 238 239 241 241 241 244 247 247 249 252 256 259 260

xiv

Contents

7 Bayesian Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Concept of Bayesian Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Bayes’ Theorem for a Continuous Random Variable . . . . . . 7.1.2 Bayes’ Theorem for Multiple Random Variables . . . . . . . . . . 7.2 Conjugate Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Conjugate Priors for Normal Distributions . . . . . . . . . . . . . . . 7.2.2 Misuse of Conjugate Distributions . . . . . . . . . . . . . . . . . . . . . . 7.3 Direct Integration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Maximum a Posterior Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Markov Chain Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . 7.6 System Identification Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 System Identification Method for Linear Models . . . . . . . . . . 7.6.2 System Identification Method for Non-linear Models . . . . . . 7.7 Summary and Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 265 267 269 270 270 277 277 283 286 293 294 297 303 304

Appendix: MATLAB Script of AFORM Analysis of Example 2.8 Based on HLRF-x Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

About the Authors

Jie Zhang is a Professor from the Department of Geotechnical Engineering, Tongji University, China. His research mainly focuses on georisk assessment and management. He is one of the founding managing editors of the journal of Underground Space, an editorial board member of the journal of Georisk, and the vice chair of TC304 (Engineering Practice of Risk Assessment and Management) in the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE). He is the recipient of the GEOSNet Young Researcher Award and the IASSAR Early Achievement Research Award, and the instructor of the ISSMGE online course Probability Analysis in Civil Engineering. Te Xiao is a Research Assistant Professor of Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology. He earned his Ph.D. from Wuhan University in 2018. His main research interests include geotechnical uncertainty and risk, probabilistic site characterization, landslide and flooding risk, and machine learning and digital modeling. He is a corresponding member of ISSMGE TC304 (Risk), TC309 (Machine Learning), and TC222 (Digital Twins), and the recipient of the ISSMGE Bright Spark Lecture Award. Jian Ji is a Jiangsu Specially-Appointed Professor of geotechnical engineering at Hohai University, Nanjing, China. He earned a Ph.D. from the Nanyang Technological University of Singapore in 2012. His research interests include numerical analysis of geotechnical problems with probabilistic considerations, slope stability and landslide mitigation, ground excavation and earth retaining systems, underground pipeline safety, etc. He received the 2018 Outstanding Paper Award from the journal Computers and Geotechnics. He is an editorial board member of Computers and Geotechnics, and member of American Society of Civil Engineers. Peng Zeng received his Ph.D. degree from the Technical University of Madrid in 2015. He is currently a professor at the State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology. His main research interests include geotechnical reliability analysis and design, landslide xv

xvi

About the Authors

risk assessment, tunnel squeezing, and rockburst hazard prediction. He is the chief scientist for the Everest Scientific Research Program of the Chengdu University of Technology. Zijun Cao is currently a professor at the MOE Key Laboratory of High-Speed Railway Engineering, Southwest Jiaotong University, China. He earned his Ph.D. in geotechnical engineering from City University of Hong Kong in 2012. He is the assistant editor and an editorial board member of the journal of Georisk. His main research areas include probabilistic site characterization with particular interests in the quantification of uncertainties in soil properties, efficient probabilistic analysis and risk assessment of slope stability, and practical reliability-based design of geotechnical structures.

Chapter 1

Basics of Probability Theory

Uncertainties are pervasive in geotechnical engineering, such as the inherent variability of the soil, the measurement error due to imperfect testing, the statistical uncertainties due to limited amount of testings, and uncertainties due to modeling assumptions. As a result, the performance of a geotechnical system can hardly be predicted deterministically, and an absolutely safe design may not be guaranteed. Traditionally, the factor of safety (FOS) is widely used to measure the safety of geotechnical systems. As the uncertainties involved in different problems are different, the same FOS may not imply the same level of safety. As a consequence, the target FOS is usually problem-specific. The FOS is not an objective measure of the safety of a geotechnical system. Since the last century, many pioneering studies have been conducted on how to develop methods for solving various types of geotechnical problems where the effect of uncertainties can be quantitatively characterized [1–6]. Among these studies, the probability-based methods have received most attention, although other types of methods such as the fuzzy sets have also been extensively studied [7]. The focus of this book is on probability-based methods, and the relevant studies will be called collectively as geotechnical reliability analyses. The review of development of geotechnical reliability methods in different periods can be found in Tang [8], Whitman [9], Christian [10], and Phoon [11]. The probability theory is the basis for geotechnical reliability analysis. In this chapter, the basics of the probability theory will be briefly introduced in the context of geotechnical engineering.

1.1 Set Theory 1.1.1 Elements of Set Theory The set theory provides the basis for formulation of probability problems. In probability theory, each possibility is considered as a sample point, the collection of all © Tongji University Press Co., Ltd. 2023 J. Zhang et al., Geotechnical Reliability Analysis, https://doi.org/10.1007/978-981-19-6254-7_1

1

2

1 Basics of Probability Theory

Fig. 1.1 Highways and slopes among cities

City 1

Slope 1

Slope 2 City 2

City 3 Slope 3

possibilities is called the sample space, and an event is a collection of some possibilities of the sample space. The sample space, events, and sample points can be modeled through the set theory, where a set is defined as a collection of elements without regard to their order. We often define sets by listing their contents within curly braces {}. For example, A = {1, 2, 3} is the set whose elements are the numbers 1, 2, and 3. If two sets have exactly the same elements, the sets are equal. For example, the set A = {1, 2, 3} are equal to the set B = {2, 1, 3}, and we write A = B. In probability theory, an event can be represented by a set, and the sample points in the event can be represented by the elements. The sample space can also be considered as a set, which is often denoted by S.

Example 1.1 As shown in Fig. 1.1, one can travel from City 1 to City 3 via City 2. City 1 and City 2 are connected through one highway, and City 2 and City 3 are connected through two highways. There is a slope along each of these three highways. Suppose if a slope fails, the adjacent highway will be blocked. Assume the accessibility of the highways is only affected by these three slopes. The possible status of the three slopes are as follows: SSS—the three slopes are all stable SSF—the first two slopes are stable but the third one is unstable SFS—the first and the third slopes are stable, but the second one is unstable SFF—the first slope is stable, but the second and the third slopes are unstable FSS—the first slope is unstable, but the rest slopes are stable FSF—the first and the third slopes are unstable, but the second slope is stable FFS—the first two slopes are unstable, and the third slope is stable FFF—all the three slopes are unstable. Each of the above possibility is a sample of the states (in failure or safety) of the three slopes. The sample space of the states of the three slopes has eight samples. Hence, the sample space in this example is S = {SSS, SSF, SFS, SFF, FSS, FSF, FFS, FFF}. Let E denote the event that at least one slope is unstable. By definition, the event of failure consists of seven samples, i.e., E = {SSF, SFS, SFF, FSS, FSF, FFS, FFF}. The sample space and events can be represented pictorially through Venn diagrams. In a Venn diagram, the sample space is often denoted by a two-dimensional region (usually a rectangle or a circle), and events are often denoted by closed regions

1.1 Set Theory

3

Fig. 1.2 Venn diagram for x∈A

within the sample space. Shading or highlighting is used in Venn diagrams to draw attention to special relationships or sets. Figure 1.2 shows a Venn diagram of the sample space S, a specific event A within S, and a sample point x ∈ A. As the sample point x is in the event A, we write x ∈ A, pronounced that x is an element of A. If x is not a sample point of A, we write x ∈ / A. The complement of the event A (with respect to the sample space S), denoted as A, is the collection of sample points in S but not in A. Figure 1.3 shows a Venn diagram illustrating the set A. The complement of the event A is pronounced that A complements or not A. For instance, if the sample space S denotes the slopes along the highway between City 1 and City 3 in Fig. 1.1, and if A comprises slopes along the highway between City 1 and City 2, A comprises slopes along the highways   between City 2 and City 3. The complement of a set is the original set: A = A. In Example 1.1, whether the highways are accessible are fully determined by the states of the slopes. Let S denote the sample space of all possible states of the slopes. If E denotes the event that the highway between City 1 and City 2 is accessible, i.e., E = {SSS, SSF, SFS, SFF}. The complement of E is then E = {FSS, FSF, FFS, FFF}. The event that has no elements is called the impossible event or empty event, which is denoted as ∅. The empty event is the complement of the sample space S, i.e., S = ∅ and ∅ = S. Suppose we have two events, A and B. If every sample point of A is also a sample point of B, we say that A is a subset of B and we write A ⊂ B. Figure 1.4 is a Venn diagram illustrating A ⊂ B. Fig. 1.3 Venn diagram for A and A

4

1 Basics of Probability Theory

Fig. 1.4 Venn diagram for A ⊂ B

The intersection of two or more events is the collection of sample points they all have in common. The intersection of the event A and the event B is sometimes written as A ∩ B or AB, and is pronounced “AB”, “A intersection B” or “A and B”. Figure 1.5 is a Venn diagram illustrating A ∩ B. In Example 1.1, suppose E 1 denotes the event that the highway between City 1 and City 2 is accessible, E 1 = {SSS, SSF, SFS, SFF}. Let E 2 denote the event that at least one of the highways between City 2 and City 3 is accessible, E 2 = {SSS, SSF, SFS, FSS, FSF, FFS}. The intersection of E 1 and E 2 , i.e., one can still access City 3 from City 1, is then {SSS, SSF, SFS}. The intersection of the impossible event and any other event is the impossible event, i.e., ∅ ∩ A = ∅. The intersection of the sample space S and any other event is that event, i.e., S ∩ A = A. Intersections are associative [A ∩ (B ∩ C) = (A ∩ B) ∩ C = A ∩ B ∩ C] and commutative [A ∩ B = B ∩ A]. If A ⊂ B, A ∩ B = A. Two events are disjoint or mutually exclusive if the two sets have no elements in common. In symbols, A and B are mutually exclusive if A ∩ B = ∅. Figure 1.6 is a Venn diagram illustrating that the sets A and B are mutually exclusive. For instance, suppose E 1 denote the event that the first slope is stable, i.e., E 1 = {SSS, SFS, SSF, SFF}. Suppose E 2 denote the event that the first slope is unstable, i.e., E 2 = {FSS, FFS, FSF, FFF}. As E 1 and E 2 have no sample points in common, they are mutually exclusive. The union of two or more events is the collection of samples that belong to at least one of the events. The union of the sets A and B is sometimes written as A ∪ B, A + B, or (A or B), and is pronounced “A union B” or “A or B”. Figure 1.7 is a Venn

Fig. 1.5 Venn diagram for A ∩ B

1.1 Set Theory

5

Fig. 1.6 Venn diagram for mutually exclusive events A ∩ B = ∅

diagram for A ∪ B. For instance, suppose E 1 denote the event that the first slope is stable, i.e., E 1 = {SSS, SFS, SSF, SFF}. Suppose E 2 denote the event that the first slope is unstable, i.e., E 2 = {FSS, FFS, FSF, FFF}. The union of E 1 and E 2 is {SSS, SFS, SSF, SFF, FSS, FFS, FSF, FFF}. The union of the empty event and any other event is the same event: ∅ ∪ A = A. The union of the sample space S and any other event is the sample space: S ∪ A = S. Unions are associative [A ∪ (B ∪ C) = (A ∪ B) ∪ C = A ∪ B ∪ C] and commutative [A ∪ B = B ∪ A]. If A ⊂ B, A ∪ B = B.

Fig. 1.7 Venn diagram for A ∪ B (dotted area)

6

1 Basics of Probability Theory

1.1.2 De Morgan’s Rule De Morgan’s rules are very helpful in untangling complicated relationships among sets. They characterize complements of unions and intersections: (A ∩ B) = A ∪ B

(1.1)

(A ∪ B) = A ∩ B

(1.2)

Unions and intersections are associative and commutative with themselves, but not with each other. For instance, A ∩ (B ∪ C) is not generally equal to (A ∩ B) ∪ C. However, there are rules for combining (distributing) unions and intersections: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(1.3)

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(1.4)

Example 1.2 A slope has two potential slip surfaces, as shown in Fig. 1.8. By assuming the landslide can only be triggered along either of these two slip surfaces, we can state that the slope failure contains two possible events: E 1 = landslides triggered by the instability of slip surface 1, and E 2 = landslide triggered by the instability of slip surface 2. Thus, for this slope system the failure means the occurrence of a landslide on either slip surface = E 1 ∪ E 2 . The safety of the slope system will be (E 1 ∪ E 2 ). Also, the safety of the slope system can only be ensured if no landslide takes place on both slip surfaces, i.e. E 1 ∩ E 2 . Thus, we can conclude that (E 1 ∪ E 2 ) = E 1 ∩ E 2 , which is a simple illustration of the De Morgan’s Rule for two events.

Example 1.3 In Example 1.1, let E 1 denote the event that the first slope is stable, i.e., E 1 = {SSS, SFS, SSF, SFF}. Let E 2 denote the event that the second slope is stable, i.e., E 2 = {SSS, SSF, FSS, FSF}. Prove the De Morgan’s rule.

1.2 Conditional Probability

7

Fig. 1.8 A slope comprising two potential slip surfaces

Solution The intersection of E 1 and E 2 is the first and the second slopes are both stable, i.e., E 1 ∩ E 2 = {SSS, SSF}, whose complement is (E 1 ∩ E 2 ) = {SFS, SFF, FSS, FSF, FFS, FFF}. The union of E 1 and E 2 is the first or the second slope is stable, i.e., E 1 ∪ E 2 = {SSS, SSF, FSS, FSF, SFS, SFF}, whose complement is (E 1 ∪ E 2 ) = {FFS, FFF}. The complement of E 1 is E 1 = {FSS, FFS, FSF, FFF}. The complement of E 2 is E 2 = {SFS, SFF, FFS, FFF}. The union of E 1 and E 2 is E 1 ∪ E 2 = {SFS, SFF, FSS, FSF, FFS, FFF}. The intersection of E 1 and E 2 is E 1 ∩ E 2 = {FFS, FFF}. As can be seen from the results, (E 1 ∩ E 2 ) = E 1 ∪ E 2 , and (E 1 ∪ E 2 ) = E 1 ∩ E 2 .

1.2 Conditional Probability 1.2.1 Axioms of Probability So far, the basic concepts of set theory have been discussed. The discussion will now proceed to the mathematics of probability, that is, using set theory to calculate the probability of failure or survival of an engineering system. Three basic assumptions or axioms for probability calculations are described next. Axiom 1.1 The probability of an event E, denoted hereafter as P(E), will always be nonnegative, that is, P(E) ≥ 0.

8

1 Basics of Probability Theory

Axiom 1.2 The probability of the sample space S is 1.0; that is, P(S) = 1.0. Axiom 1.3 For two mutually exclusive events E 1 and E 2 , the probability of their union is equal to the summation of their individual probability. Mathematically, this can be expressed as P(E 1 ∪ E 2 ) = P(E 1 ) + P(E 2 )

(1.5)

Based on these three axioms, several important observations can be made. Since an event E and its complement E are mutually exclusive, and E ∪ E = S, then using Axiom 1.2, P(S) = P(E ∪ E) = P(E) + P(E) = 1.0

(1.6)

P(E) = 1.0 − P(E)

(1.7)

or

According to Axiom 1.1, both P(E) and P(E) ≥ 0. Thus, P(E) ≤ 1.0. Axiom 1.3 can be generalized for n mutually exclusive events as P(E 1 ∪ E 2 ∪ . . . ∪ E n ) = P(E 1 ) + P(E 2 ) + . . . + P(E n )

(1.8)

Most events are not mutually exclusive; they have some common elements or overlapping, as shown in Fig. 1.9 for two events E 1 and E 2 . The third axiom can also be used in this case if all the elements can be broken into two mutually exclusive sets. In Fig. 1.9, although E 1 and E 2 are not mutually exclusive, E 1 and E 1 E 2 are mutually exclusive and their union contains the same elements as E 1 ∪ E 2 . Thus, P(E 1 ∪ E 2 ) = P(E 1 ∪ E 1 E 2 ) = P(E 1 ) + P(E 1 E 2 ) Fig. 1.9 Venn diagram for union of events

(1.9)

1.2 Conditional Probability

9

Using set theory and Fig. 1.9, we can subdivide E 2 into two mutually exclusive events as E2 = E1 E2 ∪ E 1 E2

(1.10)

Again, using the third axiom, P(E 2 ) = P(E 1 E 2 ) + P(E 1 E 2 )

(1.11)

If Eq. (1.11) is substituted into Eq. (1.9), a general expression for calculating the probability of the union of two events can be written as P(E 1 ∪ E 2 ) = P(E 1 ) + P(E 2 ) − P(E 1 E 2 )

(1.12)

Obviously, if the two events are mutually exclusive (the occurrence of one precludes the occurrence of the other), E 1 E 2 = ∅, or a null set, P(E 1 E 2 ) = 0, and Eq. (1.12) will reduce to Eq. (1.5).

Example 1.4 In Example 1.1, let E 2 and E 3 denote the failures of slope 2 and slope 3, respectively. E 2 E 3 will then denote the event that the two slopes fail simultaneously. Suppose the probability of failure of slope 2 is 0.01, and that the probability of failure of slope 3 is 0.02. The probability that the two slopes will fail simultaneously is 0.005. What is the probability that slope failure is observed along highways between City 2 and City 3? Solution Based on the above information, P(E 2 ) = 0.01, P(E 3 ) = 0.02, and P(E 2 E 3 ) = 0.005. Substituting these values into Eq. (1.12), the probability that slope failure is observed along highways between City 2 and City 3 can be calculated as follows: P(E 2 ∪ E 3 ) = P(E 2 ) + P(E 3 ) − P(E 2 E 3 ) = 0.01 + 0.02 − 0.005 = 0.025

1.2.2 Conditional Probability and Multiplication Rule In general, the probability of the intersection of events can be calculated using the multiplication rule. For two events, the multiplication rule is written as P(E 1 E 2 ) = P(E 1 |E 2 )P(E 2 ) = P(E 2 |E 1 )P(E 1 )

(1.13)

10

1 Basics of Probability Theory

where P(E 1 |E 2 ) and P(E 2 |E 1 ) are conditional probabilities, which define the probability of occurrence of one event given that the other has occurred. In a conditional probability calculation, the sample space is reduced by conditioning it with respect to the occurrence of one event. Sometimes, the occurrence of one event may depend on the occurrence of another event. If E 1 represents a rainstorm and E 2 represents the failure of a slope, then P(E 2 |E 1 ) means the probability of landslide given the occurrence of a rainstorm. However, in some other cases, the occurrence of one event may not depend on the occurrence of the other. For example, the occurrences of rainfall and earthquake at a site do not depend on each other, and they are called statistically independent events. If E 1 and E 2 are statistically independent events, then P(E 1 |E 2 ) = P(E 1 ). In this circumstance, Eq. (1.13) becomes P(E 1 E 2 ) = P(E 1 )P(E 2 )

(1.14)

The mathematics of probability for conditioned individual events must also be applicable to their complementary events, and their unions and intersections, as long as the condition is not changed. Thus, it can be easily shown that P(E 1 |E 2 ) = 1 − P(E 1 |E 2 )

(1.15)

P(E 1 ∪ E 2 |E 3 ) = P(E 1 |E 3 ) + P(E 2 |E 3 ) − P(E 1 E 2 |E 3 )

(1.16)

P(E 1 E 2 |E 3 ) = P(E 1 |E 2 |E 3 )P(E 2 |E 3 ) = P(E 1 |E 2 E 3 )P(E 2 |E 3 )

(1.17)

Generalizing the multiplication rule for n events, P(E 1 E 2 , . . . , E n ) = P(E 1 |E 2 , . . . , E n )P(E 2 |E 3 , . . . E n ) . . . P(E n−1 |E n )P(E n ) (1.18) The concept of multiplication rule is illustrated by the following example.

Example 1.5 Let E 1 , E 2 , and E 3 denote the failure of slope 1, slope 2, and slope 3 in Example 1.1, respectively. Suppose slope 1 is a rock slope while slope 2 and slope 3 are both soil slopes. The probability of failures of slope 1 is 0.01 and the failure of slope 1 is statistically independent from the failures of slope 2 and slope 3. The probability of failures of slope 2 is 0.01. The probability of failure of slope 3 given that slope 2 fails is 0.8. Calculate the probability of failure of all slopes.

1.3 Total Probability Theorem

11

Solution Based on Eq. (1.14), the probability of failure of all slopes can be written as: P(E 1 E 2 E 3 ) = P(E 1 )P(E 2 E 3 ) Then P(E 2 E 3 ) can be calculated as: P(E 2 E 3 ) = P(E 3 |E 2 )P(E 2 ) Finally, the probability of failure of all slopes can be calculated as: P(E 1 E 2 E 3 ) = P(E 1 )P(E 3 |E 2 )P(E 2 ) = 0.01 × 0.8 × 0.01 = 8 × 10−5

1.3 Total Probability Theorem As shown in Fig. 1.10, suppose the occurrence of an event A depends on the occurrence of other events E i , i = 1, 2, …, n. Suppose E i are mutually exclusive and collectively exhaustive. Let P(A| E i ) denote the conditional probability of occurrence of A given the occurrence of E i . Based the total probability theorem, the probability of the occurrence of A, i.e., P(A), can be calculated based on the conditional probability P(A| E i ) as well as the occurrence probability of P(E i ) using the following equation: P(A) =

n ∑ i=1

Fig. 1.10 Venn diagram for total probability theorem

P(A|E i )P(E i )

(1.19)

12

1 Basics of Probability Theory

Example 1.6 Suppose a passenger plans to travel from City 1 to City 3 in Example 1.1. If a slope failure occurs, the highway adjacent to the slope will be blocked. Let E 1 , E 2 , and E 3 denote the failure of slope 1, slope 2, and slope 3, respectively. Suppose the probabilities of failures of the three slopes are both 0.01 and the failures of the slopes are statistically independent. What is the probability that the passenger can travel from City 1 to City 3? Solution As mentioned previously, there are eight possible states of the three slopes. Let T denote the event that the passenger can reach City 3 from City 1. The conditional probabilities are as follows: P(T |SSS) = P(T |SSF) = P(T |SFS) = 1 P(T |SFF) = P(T |FSS) = P(T |FSF) = P(T |FFS) = P(T |FFF) = 0 The possibilities of the first three states are as follows:   P(SSS) = P E 1 E 2 E 3 = (1 − 0.01) × (1 − 0.01) × (1 − 0.01) = 0.970   P(SSF) = P E 1 E 2 E 3 = (1 − 0.01) × (1 − 0.01) × 0.01 = 0.010   P(SFS) = P E 1 E 2 E 3 = (1 − 0.01) × 0.01 × (1 − 0.01) = 0.010 Note the eight states are mutually exclusive and collectively exhaustive. As the conditional probabilities of the last five states are zero, their probabilities are not needed to calculate P(T ). Based on the total probability theorem, the chance that the passenger can travel from City 1 to City 3 can be written as follows: P(T ) = P(T |SSS)P(SSS) + P(T |SSF)P(SSF) + P(T |SFS)P(SFS) + P(T |SFF)P(SFF) + P(T |FSS)P(FSS) + P(T |FSF)P(FSF) + P(T |FFS)P(FFS) + P(T |FFF)P(FFF) = P(T |SSS)P(SSS) + P(T |SSF)P(SSF) + P(T |SFS)P(SFS) = 0.990

1.4 Discrete Random Variables A random variable is a numerical variable whose value cannot be predicted with certainty before an experiment [12]. The random variables can be used to represent uncertain phenomena quantitatively such that they can be further analyzed and

1.4 Discrete Random Variables

13

processed using probability theory. Discrete random variables are the random variables which take on only finite different values in a sample space [13]. For example, the number of rainstorms in a region that occurs in a year can be modeled as a discrete random variable. Let X denote a discrete random variable, and let x i (i = 1, 2, …, n) denote all possible values of X. For the discrete random variable X, its probability distribution can be described by its probability mass function (PMF), P(X = x i ) (i = 1, 2, …, n). The expected value of X, E(X), or the mean value of X, μX , can be computed using the following equation: μ X = E(X ) =

n ∑

xi P(X = xi )

(1.20)

i=1

The mean value reflects the central tendency of a discrete random variable. The variance is the expectation of the squared deviation of a discrete random variable from its mean, which measures the dispersion of the random variable relative to its mean. For a discrete random variable, its variance Var(X) can be computed using the following equation: Var(X ) =

n ∑

(xi − μ X )2 P(X = xi )

(1.21)

i=1

The standard deviation σ X is the square root of the variance. It has the same unit as the mean, and hence is a more convenient measure of dispersion of the random variable, which is defined as follows: √ σ X = Var(X ) (1.22) Sometimes it might be difficult to compare the degree of dispersion of different random variables based solely on the variance or the standard division. In such a case, the standard deviation may be normalized to the mean value, which results in the coefficient of variation (COV), another measure of the dispersion. Let CovX denote the COV of X. It can be calculated as follows: Cov X =

σX μX

(1.23)

Example 1.7 Figure 1.11 shows the PMF of the number of rainy days in a week during a rain season. The probabilities of 4 rainy days, 5 rainy days, and 6 rainy days in a week are 0.2, 0.6, and 0.2, respectively. Determine the mean, the variance, the standard deviation, and the COV of the number of rainy days in a week.

14

1 Basics of Probability Theory

Fig. 1.11 PMF for the number of rainy days in a week

0.6 PMF 0.2

0.2

4

5

6

Number of rainy days

Solution As can be seen from Fig. 1.11, the number of rainy days in a week can be modeled as a discrete random variable. Hence, the mean value can be computed based on its PMF using Eq. (1.20) as follows: μX =

3 ∑

xi P(X = xi ) = 0.2 × 4 + 0.6 × 5 + 0.2 × 6 = 5 days/week

i=1

The variance of the discrete random variable can also be obtained based on its mean value and PMF using Eq. (1.21): Var(X ) =

3 ∑

[X i − μ X ]2 P(X i ) = (4 − 5)2 × 0.2 + (5 − 5)2 × 0.6

i=1

+ (6 − 5)2 × 0.2 = 0.4 (days/week)2 The standard deviation of the number of rainy days in a week, i.e., can be calculated using Eq. (1.22): σX =

√

Var(X ) =

√ 0.4 = 0.63 days/week

The COV of the number of rainy days in a week can be calculated based on its mean value and standard deviation using Eq. (1.23): Cov X =

σ X 0.63 =0.126 = μX 5

X

1.4 Discrete Random Variables

15

1.4.1 Bernoulli Sequence and Binomial Distribution In probability theory, the Bernoulli process refers to a sequence of repeated trials based on the following assumptions [2]: (1) In each trial, there are only two possible outcomes; (2) The probability of occurrence of the event in each trial is constant; (3) The trials are statistically independent. Many phenomena can be modeled as the Bernoulli process. For example, during the service life of a building, if we consider each year as a trial, and define the two outcomes as the occurrence or non-occurrence of earthquakes in the year. If the occurrence probability of earthquakes in each year is constant and the occurrences of earthquakes are statistically independent, the above trials can be considered as a Bernoulli sequence. Let p denote the occurrence probability of an event in a Bernoulli sequence. The probability to observe k occurrences of the event out of n trials during the Bernoulli sequence is governed by the Binomial distribution with the following PMF P(X = k) = Cnk p k (1 − p)n−k (k = 0, 1..., n)

(1.24)

where Cnk is the binomial coefficient. The mean and the variance of a binomial random variable can be calculated as E(X ) = np

(1.25)

Var(X ) = np(1 − p)

(1.26)

Example 1.8 There are 100 slopes in a region. Suppose that failures of the slopes are statistically independent and that the failure probability of each slope is 0.001 per year. What is the probability to observe 3 slope failures in this region in one year? Solution The number of slope failures in one year follows the Binomial distribution with p = 0.001 and n = 100. The probability to observe three slope failures in one year in this region can be computed as follows: 3 P(n f ailur e = 3) = C100 0.0013 (1 − 0.001)100−3 = 0.000147

16

1 Basics of Probability Theory

1.4.2 The Poisson Process and Poisson Distribution In statistics, the Poisson process refers to a point process where an event can occur at random at any instant of time or any point in space based on the following assumptions [2]: (1) The occurrence of an event in a given time or space interval is statistically independent of that in any other nonoverlapping interval; (2) The probability of occurrence of an event in a small interval Δt is given by λΔt, where λ is the mean rate of occurrence of the event; (3) The probability of two or more occurrences of the event during Δt is negligible. Based on the above assumptions, it can be shown that the probability of k occurrences of the event during an interval of t is governed by the PMF of a Poisson distribution as follows P(X = k) =

(λt)k e−λt (k = 0, 1, ...) k!

(1.27)

where k! is the factorial of k. The expected value and the variance of X are as follows E(X ) = Var(X ) = λt

(1.28)

Example 1.9 Assume the occurrence of the typhoons in a city follows a Poisson distribution with 5 typhoons/year. Calculate (1) the probability of no typhoon in the next year; (2) the probability of 5 typhoons in the next year. Solution (1) In this example, λ = 5 typhoons per year. Based on Eq. (1.27), the probability of no typhoon in the next year can be computed as follows: P(X = 0) =

50 e−5 = 6.7 × 10−3 0!

(2) Similarly, the probability of 5 typhoons in the next year can be calculated as follows: P(X = 5) =

55 e−5 = 0.175 5!

The Poisson process is widely used to model the occurrence of earthquakes, based on which the performance-based framework for seismic design of buildings is built

1.5 Continuous Random Variables

17

[14]. In recent years, it has also been used for assessment of earthquake induced geohazards [15] and highway landslides [16].

1.5 Continuous Random Variables A random variable (X) is continuous if its possible values comprise either a single interval on the number line or a union of disjoint intervals [17]. For example, the undrained shear strength of the soil is a continuous random variable. The distribution of a continuous random variable can be represented by the probability density function (PDF). Let f (x) denote the PDF of a continuous random variable X. Its expectation or mean value (μX ) can be computed using an integration as follows: xU μ X = E(X ) =

x f (x)dx

(1.29)

xL

where f (x) is the PDF of X, and x L and x U are the lower and upper bounds of x in a sample space, respectively. The variance of a continuous random variable, denoted as Var(X), can be computed using an integration as follows: xU Var(X ) =

(x − μ X )2 f (x)dx

(1.30)

xL

For a continuous random variable (X), the probability of being smaller than x is represented by the cumulative distribution function (CDF), denoted as F(x), can be calculated from PDF as follows: x F(x) =

f (t)dt

(1.31)

xL

Example 1.10 Let f (cu ) denote the PDF of the undrained shear strength cu of a soil, which is uniformly distributed between 10 and 20 kPa as shown in Fig. 1.12. The PDF can be written as follows: f (cu ) = 0.1, 10 kPa ≤ cu ≤ 20 kPa

18

1 Basics of Probability Theory

PDF

Fig. 1.12 PDF for the undrained shear strength of a soil

0.2 0.1 0.0

10

20

cu (kPa)

Determine the mean, the variance, the standard deviation, and the CDF of the undrained shear strength. Solution The mean value of the undrained shear strength of a soil, denoted as μ, can be computed based on its PDF using Eq. (1.29) as follows: 20 μ=

20 f (cu )cu dcu =

10

0.1cu dcu = 15 kPa 10

The variance of the undrained shear strength can be computed based on its mean value and PDF using Eq. (1.30) as follows: 20

20 (cu − 15) f (cu )dcu =

Var(cu ) =

0.1(cu − 15)2 dcu = 8.33 kPa2

2

10

10

The standard deviation of the undrained shear strength denoted as σ, is equal to the square root of the variance, i.e., √ √ σ = Var(cu ) = 8.33 = 2.89 kPa Integrating the PDF of the undrained shear strength, its CDF, denoted as F(cu ), can be obtained as follows: cu f (cu )dcu =

F(cu ) = 10

⎧ ⎨

0 0.1(cu − 10) ⎩ 1.0

cu ≤ 10 kPa 10 kPa ≤ cu ≤ 20 kPa cu ≥ 20 kPa

1.5 Continuous Random Variables

19

1.5.1 Normal Distribution The normal distribution is also called the Gaussian distribution. The PDF and CDF for the normal distribution are as follows:

 1 (x − μ X )2 exp − (1.32) f (x) = √ 2σ X2 2π σ X x F(x) = −∞

 1 (t − μx )2 exp − dt √ 2σx2 2π σx

(1.33)

where μX and σ X are the mean and the standard deviation of the normal variable X, respectively. An example of a normal PDF is shown in Fig. 1.13. As can be seen from this figure, PDF is centered around the mean value, μX . The spread of the PDF is controlled by the standard deviation, σ X , i.e., the greater the value of σ X , the wider the spread. For the case of μX = 0 and σ X = 1, the normal distribution is often known as the standard normal distribution. Its PDF is shown as follows 2 u 1 (1.34) φ(u) = √ exp − 2 2π where φ() is the PDF of the standard normal distribution. In this book, the CDF of the standard normal distribution will be designated by ϕ(), which is given by: u ϕ(u) = −∞

Fig. 1.13 PDF for the normal distribution

2 t 1 dt √ exp − 2 2π

(1.35)

PDF

σX

μX

X

20

1 Basics of Probability Theory

A general normal random variable (X) with mean, μX , and standard deviation, σ X, can be related to a standard normal variable (U) through the following equation: X = μX + σX · U

(1.36)

It can be shown that the CDF of a general normal random variable X can be calculated through the CDF of a standard normal random variable as follows:

x − μX F(x) = ϕ σX

(1.37)

There are different ways for estimating the mean and the standard deviation of the normal distribution, such as the moment method, the percentile method, and the maximum likelihood method [17]. For example, when the moment method is used, we can use the sample mean and the sample standard deviation as the estimators of the mean and the standard deviation, which are shown below: ┌ | n n | 1 ∑  2 1 ∑ (i) √ X (i) − μ X μX ≈ X , σX ≈ n i=1 n − 1 i=1

(1.38)

where n denotes the number of samples and X (i) is the value of ith observed sample. In general, the mean and the standard deviation can be estimated quite accurately with data of a sample size between 30 and 100 [17].

Example 1.11 Suppose the undrained strength of soil at a site is normally distributed with a mean of 20 kPa and a standard deviation of 8 kPa. For a strip footing under undrained condition, its ultimate bearing capacity, denoted as qu , can be estimated using cu as follows [18]: qu = 5.14cu Assume the strip footing carries a uniform load, which is equal to 50 kPa. Determine the failure probability of the strip footing. Solution Based on the definition of normal distribution, (cu – 20)/8 follows the standard normal distribution. When the ultimate bearing capacity of the strip footing is smaller than the given load, the strip footing will fail, and the failure probability of the strip footing, denoted as P(qu < 50 kPa), can be calculated as follows:

1.5 Continuous Random Variables

21

P(qu < 50) = P(5.14cu < 50) = P

50/5.14−20 cu − 20 < 8 8

= ϕ(−1.284) = 0.10 Due to its simplicity, the normal distribution is widely used in geotechnical reliability analysis. Through techniques like the Johnson transformation, a non-normal random variable can be conveniently transformed to a normal random variable, based on which the joint distribution of multiple soil properties can be constructed [19]. As will be seen later in this book, in the first order reliability analysis, the reliability index can be calculated conveniently in the space of standard normal random variables.

1.5.2 Lognormal Distribution If lnX follows the normal distribution, X follows the lognormal distribution. By definition, X cannot be negative. In geotechnical engineering, there are many random variables which physically cannot take negative values. For example, the unit weight of the soil cannot be negative. The lognormal distribution has been widely used in geotechnical engineering to model non-negative random variables. The PDF of a lognormal random variable X can be written as follows: 

(ln x − λ X )2 f (x) = √ exp − 2ξ X2 2π ξ X x 1

(1.39)

where λX and ξ X are the mean and the standard deviation of ln X, respectively, which can be calculated based on the mean and the standard deviation of X using the following equations 1 λ X = ln(μ X ) − ξ X2 2 ┌  |

2  | σ X ξ X =√ln 1+ μX

(1.40)

(1.41)

where μX and σ X are the mean and standard deviation of X. The lognormal distribution has zero as its lower bound. The shifted lognormal distribution generalizes the lognormal distribution to account for nonzero lower bounds. If X is a shifted lognormal random variable with a lower bound value of bX , the relationship between X and the standard normal variable U is:

22

1 Basics of Probability Theory

λ + ξU = ln(X − b X )

(1.42)

where λ, ξ and bX are the parameters for the shifted lognormal distribution. The parameter bX is the lower bound of X and it is typically determined by physics. The shifted lognormal distribution was widely used in fields like transportation flow research [20].

Example 1.12 In Example 1.11, the undrained strength of soil follows the normal distribution. Suppose the undrained strength of soil in Example 1.11 follows the lognormal distribution. Determine the failure probability of the strip footing. Solution Based on the property of the lognormal distribution, ln(cu ) is normal with mean = λ and standard deviation = ξ, and these two parameters can be calculated as follows: ┌  | 2  | 8 1 ξ =√ln 1+ = 0.39, λ = ln(20) − × 0.392 = 2.92, 20 2 (ln(cu ) – 2.92)/0.39 follows the standard normal distribution. Then, the failure probability of the strip footing, denoted as P(qu < 50), can be calculated as follows: P(qu < 50) = P(5.14cu < 50)

ln(cu ) − 2.92 ln(50/5.14) − 2.92 =P < 0.39 0.39 = ϕ(−1.65) = 0.049 In geotechnical engineering, the lognormal distribution is widely used to model the uncertainty associated with bearing capacity prediction equation of deep foundations [21, 22], based on which reliability-based design of pile foundations can be performed conveniently. The shifted lognormal distribution was used in Najjar and Gilbert [23] to model the lower bound capacity of piles. Zhao and Ono [24] derived an efficient third moment reliability method for failure probability estimation based on the shifted lognormal distribution. In hydrologic analysis, the shifted lognormal distribution is frequently used in hydrologic analysis of extreme floods, seasonal flow volumes, duration curves for daily streamflow, and rainfall intensity duration [25].

1.6 Multivariate Distribution

23

1.6 Multivariate Distribution 1.6.1 Covariance and Correlation Coefficient In probability theory, covariance is a measure of the joint variability of two random variables [26]. Mathematically, the covariance between two random variables X 1 and X 2 , which is denoted as Cov(X 1 , X 2 ), is defined as follows:    Cov(X 1 , X 2 ) = E X 1 − μ X 1 X 2 − μ X 2 = E(X 1 X 2 ) − E(X 1 )E(X 2 )

(1.43)

When X 1 and X 2 are statistically independent, we will have E(X 1 X 2 ) = E(X 1 )E(X 2 )

(1.44)

Submitting Eq. (1.44) into Eq. (1.43) yields Cov(X 1 , X 2 ) = 0. In other words, if the two random variables are statistically independent, their covariance is zero. In general, the magnitude of the covariance is not easy to interpret, which depends on the magnitudes of the variables. To eliminate the scale effect, it is preferable to use the normalized covariance, which is often called the correlation coefficient. The correlation coefficient between X 1 and X 2 , denoted as ρ 12 , is defined as follows ρ12





X 2 − μX2 Cov(X 1 , X 2 ) X 1 − μX1 = =E σX1 σX2 σX1 σX2

(1.45)

The correlation coefficient is a dimensionless measure of linear dependence between two random variables. The value of ρ varies from −1 to 1. Figures 1.14a– 1.14f show the samples of two random variables as well as the correlation coefficient between the random variables. As can be seen, ρ = ±1 represents a perfect linear relationship (see Fig. 1.14a and Fig. 1.14b), ρ > 0 indicates a positive linear relationship (Fig. 1.14c), ρ < 0 means a negative linear relationship (Fig. 1.14d), and ρ = 0 implies that there is no linear relationship between X 1 and X 2 (Fig. 1.14e, f). A large absolute value of ρ indicates a strong linear relationship, and vice versa. Note ρ only measures the linear relationship between two random variables. As shown in Fig. 1.14f, although there is a deterministic non-linear relationship between the two random variables, the value of the correlation coefficient between the two random variables is still zero. Note various types of correlation coefficients have been developed in statistics. The correlation coefficient as described above is often called the Pearson productmoment correlation coefficient [27]. It can be estimated based on samples using the method of moments as follows:

24

1 Basics of Probability Theory

Fig. 1.14 Two random variables with different correlation coefficients

1.6 Multivariate Distribution ρ12 =

25

Cov(X 1 , X 2 ) σ1 · σ 2

∑ (k) (k) [1/(n − 1)] nk=1 (X 1 − m 1 ) · (X 2 − m 2 ) / ≈ / ∑n ∑ (k) (k) [1/(n − 1)] k=1 (X 1 − m 1 )2 × [1/(n − 1)] nk=1 (X 2 − m 2 )2

(1.46)

where n denotes the number of observed samples; the superscript (k) is the sample index; mi is the sample mean of X i . When there are multiple random variables, the covariance for these random variables can be written as a matrix, which is often called the covariance matrix. For a random vector of [X 1 X 2 … X M ], its covariance matrix can be written as follows: ⎤ Cov(X 1 , X 1 ) Cov(X 1 , X 2 ) · · · Cov(X 1 , X M ) ⎢ Cov(X 2 , X 2 ) · · · Cov(X 2 , X M ) ⎥ ⎥ ⎢ C=⎢ ⎥ .. .. ⎦ ⎣ . . symmetric Cov(X M , X M ) ⎡

(1.47)

where C is the covariance matrix. Likewise, the correlation coefficient for multiple random variables is often written in a matrix form, which is known as the correlation matrix. For a random vector [X 1 X 2 … X M ], its correlation matrix can be written as follows: ⎡

1

⎢ ⎢ R=⎢ ⎣

symmetric

ρ12 · · · 1 ··· .. .

⎤ ρ1M ρ2M ⎥ ⎥ .. ⎥ . ⎦

(1.48)

1

Based on Eq. (1.45), the covariance matrix in Eq. (1.47) can be written in terms of the correlation coefficient as follows: ⎡ ⎢ ⎢ C=⎢ ⎣

σ X2 1

symmetric

⎤ ρ12 σ X 1 σ X 2 · · · ρ1M σ X 1 σ X M σ X2 2 · · · ρ2M σ X 2 σ X M ⎥ ⎥ ⎥ .. .. ⎦ . .

(1.49)

σ X2 M

Table 1.1 The data of Example 1.13 Parameters

Measured values

cohesion c (kPa)

25.2 30.1 32.4 23.6 27.2 26.1 35.7 33.3 28.2 30.4 31.2 29.5 36.6 34.2 28.9

Friction angle ϕ (°) 20.3 24.1 17.1 23.6 20.5 23.4 18.1 19.9 23.4 16.3 19.9 20.1 15.1 17.8 19.9

26

1 Basics of Probability Theory

Example 1.13 There are 15 sets of experimental data of soil strength parameters, including the cohesion c and the friction angle ϕ as shown in Table 1.1. Esimate the correlation coefficient and covariance of the cohesion and the friction angle. Solution Based on the given information, we can estimate their covariance and correlation coefficient based on Eqs. (1.43) to (1.46). This can be done by using the two functions “cov” and “corr” in the MATLAB software. The MATLAB code is shown below: c = [25.2 30.1 32.4 23.6 27.2 26.1 35.7 33.3 28.2 30.4 31.2 29.5 36.6 34.2 28.9]; phi = [20.3 24.1 17.1 23.6 20.5 23.4 18.1 19.9 23.4 16.3 19.9 20.1 15.1 17.8 19.9]; cov_c_phi = cov(c,phi); cov_c_phi = cov_c_phi(1,2) rho = corr(c’, phi’,’type’, ‘Pearson’)

The output is cov_c_phi = −7.5717 rho = −0.7161.

Here, the covariance is −7.5717 (kPa·deg) and the correlation coefficient is −0.7161.

1.6.2 Multivariate Normal Distribution There might be multiple random variables involved in the same geotechnical problem. The joint distribution of multiple random variables can be modeled through multivariate distributions. The multivariate normal distribution is a generalization of the normal distribution in the case of multivariate random variables. Let X = [X 1 X 2 … X M ]T . If X follows the multivariate normal distribution, its joint PDF can be written as: 

1 (X − µ)T C−1 (X − µ) (1.50) exp − f (X) = √ M 2 (2π ) 2 · |C| where µ = [μ1 μ2 … μM ]T is the mean vector of X, C is a covariance matrix of X, and |·| indicates the determinant of a matrix. Consider a multivariate normal vector X = [X1 T X2 T ]T with X1 = [X 11 X 12 … X 1M1 ]T and X2 = [X 21 X 22 … X 2M2 ]T . Let µ = [µ1 T µ2 T ]T denote the mean of X, where µ1 and µ2 are the mean values of X1 and X2 , respectively. Suppose the covariance matrix of X can be written as

1.6 Multivariate Distribution

27



C11 C12 C= C21 C22

 (1.51)

where C11 and C22 are the covariance matrix of X1 and X2 , respectively; and C12 = C21 T are the matrix that contains the covariances between variables in X1 and X2 . Based on the property of the multivariate normal distribution, X1 or X2 also follows multivariate normal distribution [28]. For a random vector, the distribution of an element is called the marginal distribution. For a multivariate normal distribution, its marginal distributions are normal distributions [28]. Nevertheless, the reverse may not be true, i.e., even if each X i follows the normal distribution, the joint distribution of X may not be multivariate normal [28]. The covariance matrix of X specifies the linear dependence relationship between different random variables. When the values of some random variables are known, such information can be used to update the distributions of other random variables, which are often known as the conditional distribution. Given that the values of X1 are observed as X1 = a, the distribution of X2 conditional on X1 = a is also multivariate normal with the following mean vector and covariance matrix µ2|X1 =a = µ2 + C21 C−1 11 (a − µ1 )

(1.52)

C22|X1 =a = C22 − C21 C−1 11 C12

(1.53)

For the case where the dimension of the random variables is two, the multivariate normal distribution is reduced to the bivariate normal distribution. The above two equations can be further reduced to  σX2  x1 − μ X 1 σX1   = σ X2 2 1 − ρ 2

μ X 2 |X 1 =x1 = μ X 2 + ρ

(1.54)

σ X2 2 |X 1 =x1

(1.55)

where ρ is the correlation coefficient of X 1 and X 2 .

Example 1.14 Assume the cohesion, denoted as c, and the friction angle, denoted as ϕ, of a soil follow a bivariate normal distribution with a correlation coefficient, ρ = −0.5. The statistic of the random variables are as follows: μc = 8 kPa, σ c = 2 kPa, μϕ = 30°, σ ϕ = 3°. If the cohesion of the soil is measured to be 10 kPa, what is the chance that the friction angle is less than 30°?

28

1 Basics of Probability Theory

Solution Based on the property of the bivariate normal distribution, the mean of the friction angle as the cohesion is 10 kPa can be computed as follows: μϕ|c = 10 kPa = μϕ + ρσϕ

10 − μc σc

= 30 − 0.5 × 3 ×

10 − 8 2

= 28.5◦

The standard deviation of the friction angle as the cohesion is 10 kPa can be computed as follows: / √ σϕ|c = 10 kPa = σϕ 1 − ρ 2 = 3 × 1 − (−0.5)2 = 2.60◦ Based on the conditional mean and standard deviation of the friction angle, the chance that the friction angle is less than 30° is:

30 − 28.5 P(ϕ ≤ 30 |c = 10 kPa) = ϕ 2.60 ◦

= 0.718

In geotechnical engineering, the multivariate normal distribution has been widely used to develop the joint distribution of different soil properties [29]. When the original variables are not multivariate normal, transformation techniques like the Hermite Polynomials [30] and Johnson systems [19] can be used such that the transformed random variables can be modeled through the multivariate normal distribution. The multivariate normal distribution is also closely related to the Gaussian random fields and the kriging-based response surface model, which will be discussed in detail later in this book.

1.6.3 Multivariate Lognormal Distribution The multivariate lognormal distribution is a generalization of the lognormal distribution for the case of multiple random variables. Let X = [X 1 X 2 … X M ]T represents a vector containing a series of random variables. If their logarithm vector ln X = [ln X 1 ln X 2 … ln X M ]T follows the multivariate normal distribution, then X follows the multivariate lognormal distribution. Based on its definition, each element of X follows the lognormal distribution. Let λi and ξ i denote the mean and the standard deviation of ln X i . Let λ = [λ1 λ2 ... λ M ]T denote the mean vector of ln X = [ln X 1 ln X 2 … ln X M ]T . Let ρ ij denote the correlation coefficient between ln X i and ln X j . The covariance matrix of ln X can be written as follows

1.6 Multivariate Distribution

29

⎡

⎤ ρ12 ξ1 ξ2 · · · ρ1M ξ1 ξ M ⎢ ξ22 · · · ρ2M ξ2 ξ M ⎥ ⎢ ⎥ Cln = ⎢ ⎥ .. .. .. ⎣ ⎦ . . . 2 ξM ρ M1 ξ M ξ1 ρ M2 ξ M ξ2 · · · ξ12 ρ21 ξ2 ξ1 .. .

(1.56)

The joint PDF of X can be written as: f (x) =



1 M

x1 x2 ...x M (2π ) 2

(ln x − λ)T C−1 ln (ln x − λ) exp − √ 2 · |Cln |

 (1.57)

Let μ1 , μ2 , …, μM and σ 1 , σ 2 , …, σ M be the means and the standard deviations of X 1 , X 2 , …, X M , respectively. Then λ1 , λ2 , …, λM and ξ 1 , ξ 2 , …, ξ M can be calculated using Eqs. (1.40) and (1.41). Note that ρ ij in Eq. (1.56), which is the correlation coefficient between ln X i and ln X j , is generally not equal to the correlation coefficient between X i and X j , denoted as r ij here. The relationship between ρ ij and r ij can be expressed as the following equations [31]:  σσ ln 1 + ri j μii μjj ρi j = /

 2  σ ln 1+ μjj ln 1+ μσii



(1.58)

 exp ρi j ξi ξ j − 1 ri j = / $   2 #  exp ξi − 1 exp ξ 2j − 1

(1.59)

2

When the values of some of the elements of a lognormal random vector are observed, such information can also be used to update the distributions of other random variables. Such type of updating can be conveniently performed in the space of ln X, where the properties of the multivariate normal distribution can be used.

Example 1.15 Assume the cohesion, denoted as c, and the friction angle, denoted as ϕ, of a soil follow a bivariate lognormal distribution. The correlation coefficient between c and ϕ is r = -0.5. The statistics of the random variables are as follows: μc = 8 kPa, σ c = 2 kPa, μϕ = 30°, σ ϕ = 3°. If the cohesion of the soil is measured to be 10 kPa, what is the chance that the friction angle is less than 30°?

30

1 Basics of Probability Theory

Solution Based on the property of the lognormal distribution, the mean values and standard deviations of ln(c) and ln(ϕ) can be calculated using Eqs. (1.40) and (1.41) as follows: ┌  ┌  | 2  | 2  | | σ 2 c ξc =√ln 1+ = √ln 1+ = 0.2462 μc 8 1 1 λc = ln(μc ) − × ξc2 = ln(8) − × ξc2 = 2.0491 2 2 ┌  ┌  | 2  | 2  | | σ 3 ϕ √ √ ξϕ = ln 1+ = ln 1+ = 0.0998 μϕ 30   1 1 λϕ = ln μϕ − × ξϕ2 = ln(30) − × ξϕ2 = 3.3962 2 2 The correlation coefficient between ln(c) and ln(ϕ) can be calculated using Eq. (1.58) as follows:  σ σ ln 1 + r μcc μϕϕ ρ=/

  2 σ ln 1+ μσcc ln 1+ μϕϕ

2

= −0.5121

Based on the property of the bivirate normal distribution, the conditional distribution of the logarithm of the friction angle is normal. Its mean value given that the cohesion is 10 kPa, i.e., ln c = ln (10), can be computed as follows:

ln(10) − λc = 3.3436 λϕ|c = 10 kPa = λϕ + ρξϕ ξc The standard deviation of the logarithm of friction angle as the cohesion is 10 kPa can be computed as follows: √ ξϕ|c = 10 kPa = ξϕ 1 − ρ 2 = 0.0857

1.7 Summary and Further Readings

31

Based on the conditional mean and standard deviation of the friction angle, the chance that the friction angle is less than 30° is: P(ϕ ≤ 30◦ |c = 10kPa)



ln(30) − λϕ|c = 10 kPa = P(ln ϕ ≤ ln(30)|c = 30 kPa) = ϕ ξϕ|c = 10 kPa

= 0.7492 Due to its close relationship with the multivariate normal distribution, the multivariate lognormal distribution is also easy to apply and found many applications in geotechnical engineering. For example, it has been used to model the joint distribution of parameters of the soil–water characteristics curve [32–34] and load-settlement curves of piles [35–37].

1.7 Summary and Further Readings In this chapter, the basics of the probability theory have been introduced. The set theory provides a framework for the analysis of events and the relationships between events. It is based on logic alone. The probability theory provides a framework for analyzing the likelihoods of events and combinations of events. It is based on set theory and math. The random variables are tools to model and quantify uncertainties such that their effects can be assessed. In geotechnical engineering, the availability of data is one of the keys for calibrating the random variables. It has been argued that, in geotechnical engineering the amount of site-specific data is often quite limited [38–40]. Phoon [41] pointed out that a geotechnical generic database can be data-rich but not be directly and completely applicable while a fully applicable site-specific database is usually extremely datapoor. In such a case, the compilation of global or regional database can be very useful to complement the site-specific data [42, 43]. Recently, many efforts have been exerted to compile geotechnical database. Several techniques have also been developed on how to combine region-specific data with the site-specific data to develop site-specific distributions of the random variables [44–46]. The Bayesian method is the main tool for information combination, and it will be discussed in detail in Chap. 7 of this book. In modeling the distribution of multivariate variables, the correlation coefficient alone cannot capture the correlation structure between different random variables. In probability theory, the copula theory, which can model the correlation between two variables through a function, provides a more versatile tool to model the correlation structure among different random variables [47, 48]. The copula-based method provides a more flexible tool for modeling the multivariate distribution of geotechnical random variables [40, 49, 50].

32

1 Basics of Probability Theory

Exercises Exercise 1.1 The annual failure probability of a slope subjected to rainfall is 0.0002. Suppose the failure events of the slope during different years are statistically independent. What is the probability that the slope may fail during a service life of 50 years? Exercise 1.2 Suppose the occurrence of earthquakes in a region can be modeled through a Poisson process with a mean rate of v = 0.05. When an earthquake occurs, the failure probability of a tunnel is 0.01. What is the failure probability of this tunnel during a service life of 50 years? Exercise 1.3 Suppose the vertical bearing capacity of two adjacent piles follow a bivariate normal distribution. The mean and the standard deviation of the bearing capacity of the first pile are 100 kN and 20 kN, respectively. The mean and the standard deviation of the bearing capacity of the second pile are 130 kN and 30 kN, respectively. The correlation coefficient between the two bearing capacities is 0.6. (1) What is the probability that the bearing capacity of the second pile is less than 90 kN? (2) A load test is conducted on the first pile and it is found its capacity is 110 kN. What is the probability that the bearing capacity of the second pile is less than 90 kN? Exercise 1.4 What are the answers to the previous problem if the bearing capacity of the two piles follow the bivariate lognormal distribution?

References 1. Ang AHS, Tang WH (2007) Probability concepts in engineering planning and design: emphasis on application to civil and environmental engineering. John Wiley and Sons, New York 2. Ang AHS, Tang WH (1984) Probability concepts in engineering planning and design, Vol. II: decision, risk, and reliability. John Wiley and Sons, New York 3. Vanmarcke E (1983) Random fields: analysis and synthesis. MIT Press, Cambridge, MA, USA 4. Fenton GA, Griffiths DV (2008) Risk assessment in geotechnical engineering. John Wiley and Sons, New York 5. Baecher GB, Christian JT (2005) Reliability and statistics in geotechnical engineering. John Wiley and Sons, New York 6. Phoon KK, Ching J (2015) Risk and reliability in geotechnical engineering. CRC Press, Boca Raton, FL, USA 7. Juang CH, Lee DH, Sheu C (1992) Mapping slope failure potential using fuzzy sets. J Geotech Eng 118(3):475–494

References

33

8. Tang WH (1993) Recent developments in geotechnical reliability. In: KS Li and SCR Lo (Eds.) Probabilistic methods in geotechnical engineering (pp 3–27). CRC Press, Rotterdam 9. Whitman RV (2000) Organizing and evaluating uncertainty in geotechnical engineering. J Geotech Eng 126(7):583–593 10. Christian JT (2004) Geotechnical engineering reliability: how well do we know what we are doing? J Geotech Geoenviron Eng 130(10):985–1003 11. Phoon KK (2020) The story of statistics in geotechnical engineering. Georisk: Assess Manage Risk Eng Syst Geohazards 14(1):3–25 12. Benjamin JR, Cornell CA (1970) Solutions manual to accompany probability, statistics, and decision for civil engineers. McGraw-Hill, New York 13. Stewart WJ (2009) Probability, Markov chains, queues, and simulation: the mathematical basis of performance modeling. Princeton University Press, Princeton 14. Baker JW, Cornell CA (2006) Correlation of response spectral values for multicomponent ground motions. Bull Seismol Soc Am 96(1):215–227 15. Kramer SL, Mayfield RT (2007) Return period of soil liquefaction. J Geotech Geoenviron Eng 133(7):802–813 16. Lu M, Zhang J, Zhang L, Zhang L (2020) Assessing the annual risk of vehicles being hit by a rainfall-induced landslide: a case study on Kennedy Road in Wan Chai, Hong Kong. Nat Hazard 20(6):1833–1846 17. Devore JL (1995) Probability and statistics for engineering and the sciences. Duxbury Press, Belmont 18. Craig RF (2004) Craig’s soil mechanics. CRC Press, Boca Raton, Florida 19. Ching J, Phoon KK (2014) Correlations among some clay parameters—the multivariate distribution. Can Geotech J 51(6):686–704 20. Mei M, Bullen AGR (1993) Lognormal distribution for high traffic flows. Transp Res Rec 1398:125–128 21. Tang C, Phoon KK, Chen YJ (2019) Statistical analyses of model factors in reliabilitybased limit-state design of drilled shafts under axial loading. J Geotech Geoenviron Eng 145(9):04019042 22. Zhang J, Hu J, Li X, Li J (2020) Bayesian network based machine learning for design of pile foundations. Autom Constr 118:103295 23. Najjar SS, Gilbert RB (2009) Importance of lower-bound capacities in the design of deep foundations. J Geotech Geoenviron Eng 135(7):890–900 24. Zhao YG, Ono T (2001) Moment methods for structural reliability. Struct Saf 23(1):47–75 25. Singh VP (1998) Entropy-based parameter estimation in hydrology, vol 30. Springer Science & Business Media, New York 26. Rice JA (2006) Mathematical statistics and data analysis. Cengage Learning, Boston 27. Galton F (1886) Regression towards mediocrity in hereditary stature. J Anthropol Inst G B Irel 15:246–263 28. Gut A (2013) Probability: a graduate course. Springer Science & Business Media, New York 29. Ching J, Phoon KK (2015) Constructing multivariate distributions for soil parameters. In: KK Phoon and J Ching (Eds.) Risk and reliability in geotechnical engineering (pp 3–76). CRC Press, Boca Raton, Florida 30. Wang DY (2002) Development of a method for model calibration with non-normal data. PhD Thesis. The University of Texas, Austin 31. Rowcroft JE (1975) Distributions in statistics: continuous multivariate distributions. J Roy Stat Soc: Ser D (The Statistician) 24:146–147 32. Phoon KK, Santoso A, Quek ST (2010) Probabilistic analysis of soil-water characteristic curves. J Geotech Geoenviron Eng 136(3):445–455 33. Santoso AM, Phoon KK, Quek ST (2011) Probability models for SWCC and hydraulic conductivity. In Proc., 14th Asian Regional Conf. on Soil Mechanics and Geotechnical Engineering. Hong Kong Polytechnic University, Hong Kong SAR, China 34. Tan X, Wang X, Khoshnevisan S, Hou X, Zha F (2017) Seepage analysis of earth dams considering spatial variability of hydraulic parameters. Eng Geol 228:260–269

34

1 Basics of Probability Theory

35. Li DQ, Tang XS, Zhou CB, Phoon KK (2015) Characterization of uncertainty in probabilistic model using bootstrap method and its application to reliability of piles. Appl Math Model 39(17):5310–5326 36. Zheng JJ, Liu Y, Pan YT, Hu J (2018) Statistical evaluation of the load-settlement response of a multicolumn composite foundation. Int J Geomech 18(4):04018015 37. Wu XZ, Xin JX (2019) Probabilistic analysis of site-specific load-displacement behaviour of cement-fly ash-gravel piles. Soils Found 59(5):1613–1630 38. Duncan JM (2000) Factors of safety and reliability in geotechnical engineering. J Geotech Eng 126(4):307–316 39. Uzielli M (2008) Statistical analysis of geotechnical data. In: AB Hang and PW Mayne (Eds.) Geotechnical and geophysical site characterization (pp 173–194). CRC Press, London 40. Zhang J, Huang HW, Juang CH, Su WW (2014) Geotechnical reliability analysis with limited data: consideration of model selection uncertainty. Eng Geol 181:27–37 41. Phoon KK (2020) The Goldilocks dilemma—too little or too much data? Geostrata Mag 24:14– 16. 42. Phoon KK, Kulhawy FH (1999) Characterization of geotechnical variability. Can Geotech J 36(4):612–624 43. Phoon KK, Kulhawy FH (1999) Evaluation of geotechnical property variability. Can Geotech J 36(4):625–639 44. Zhang J, Juang CH, Martin JR, Huang HW (2016) Inter-region variability of Robertson and Wride method for liquefaction hazard analysis. Eng Geol 203:191–203 45. Ching J, Phoon KK (2019) Constructing site-specific multivariate probability distribution model using Bayesian machine learning. J Eng Mech 145(1):04018126 46. Bozorgzadeh N, Bathurst RJ (2022) Hierarchical Bayesian approaches to statistical modelling of geotechnical data. Georisk 16(3):452–469 47. Clemen RT, Reilly T (1999) Correlations and copulas for decision and risk analysis. Manage Sci 45(2):208–224 48. Nelsen RB (2006) An introduction to copulas. Springer Science & Business Media, New York 49. Li DQ, Tang XS, Phoon KK, Chen YF, Zhou CB (2013) Bivariate simulation using copula and its application to probabilistic pile settlement analysis. Int J Numer Anal Meth Geomech 37(6):597–617 50. Prakash A, Hazra B, Sekharan S (2020) Probabilistic analysis of soil-water characteristic curve of bentonite: multivariate copula approach. Int J Geomech 20(2):04019150

Chapter 2

First Order Reliability Methods

Due to the existence of uncertainties in geotechnical engineering, the performance of a geotechnical system can hardly be predicted deterministically. In such a case, the geo-system performance can be measured by the probability of failure, i.e., the probability that the designed performance cannot be achieved. The objective of geotechnical reliability analysis is to quantify the uncertain geo-system performance subject to failure risk based on reliability theory. Of particular concern is how likely the geotechnical system may fail in terms of failure probability. There are two common types of probabilistic methods for geotechnical reliability analysis: the first type is known as Monte Carlo simulations, as will be introduced in the next chapter; and the second type is known as numerical approximation method for reliability analysis. Depending on the complexity of geo-system performance and the prediction accuracy requirement, there are several numerical approximation methods being developed for geotechnical reliability analysis over the last decades. Among them, the first order reliability methods which are based on first order approximation of a geotechnical system are widely used, probably because these methods are easy to follow and in many cases are reasonably accurate. The focus of this chapter is on the first order reliability methods.

2.1 Concept of Geotechnical Reliability In geotechnical engineering, the uncertainty associated with geotechnical parameters is one of the typical sources of geo-system performance uncertainty. In addition, environmental factors such as precipitation and earthquakes are also highly uncertain. Due to the complex behavior of soils, any geotechnical model may only be an abstraction of the real world, and model uncertainty always exists. For the above reasons, exact prediction of a geo-system performance is not realistic, and probabilistic methods can be used to assess the uncertainty associated with the geotechnical performance.

© Tongji University Press Co., Ltd. 2023 J. Zhang et al., Geotechnical Reliability Analysis, https://doi.org/10.1007/978-981-19-6254-7_2

35

36

2 First Order Reliability Methods

Fig. 2.1 Concept of LSF between the safe and unsafe design regions

For the sake of simple illustration, let the relevant uncertain variables related to a geosystem be denoted by a vector of M random variables x = [x1 x2 . . . x M ]T . Suppose that the functional relationships corresponding to the system performance can be described by z = g(x) with g(x) < 0 denoting failure and g(x) > 0 denoting safety. In reliability analysis, z = g(x) is often called the performance function, and g(x) = 0 which divides the design parameter space into safe and unsafe regions, is called the limit state function (LSF). As an example, Fig. 2.1 shows a general performance function in the space with two random variables x 1 and x 2 denoting resistance and load, respectively. The line separating the safe domain from the failure domain, is the LSF, i.e., g(x 1 , x 2 ) = 0. Considering the uncertainty of design parameters (random variables), the probability of failure, Pf , can be calculated by the integral { Pf =

{ ...

f (x1 , x2 , ..., x M ) dx1 dx2 . . . dx M

(2.1)

g(x) 0 is a penalty parameter, e.g., c = 100 suffices for most engineering problems. The Armijo rule as a backtracking line search technique can be used to determine an optimal (maximum) step length, as follows: { | } λk = max b j |m(xk + b j dk ) − m(xk ) ≤ −ab j j

(2.51)

where a, b ∈ (0, 1) are prescribed parameters, j is an integer for optimal solution. The Armijo rule is equivalent to applying a shrinkage factor b on the step length until the merit function m(xk + b j dk ) is sufficiently reduced. Based on numerical tests, the Armijo rule would find the optimal solution with j up to 5 when a and b are set to be 0.5.

Example 2.9 The Clarence Cannon Dam completed in 1983 is located in the Salt River in northeastern Missouri and forms Mark Twain Lake [10]. The dam is part of a multi-purpose project which provides flood control, recreation, water supply, fish and wildlife conservation, and hydropower. The embankment section analyzed in this example is located at survey station 12 + 75 and is representative of the dam near its maximum height. The embankment geometry at this location is shown in Fig. 2.11. The section includes a compacted clay foundation cutoff trench through an abandoned glacial channel. The cutoff trench and embankment base are constructed of Phase I fill materials and the remainder of the embankment is constructed of Phase II fill materials. The stability of Cannon Dam has been previously studied by Hassan and Wolff [10] and many others. Ji and Liao [35] presented a probabilistic study of the Cannon Dam by using the HLRF-x algorithm for reliability analysis in combination with strengthreduction finite element code Plaxis for the embankment stability evaluation. The detailed soil properties are listed in Table 2.6 and Table 2.7. The results of the probabilistic strength-reduction finite element stability analysis of Cannon Dam are compared with others using limit equilibrium methods, as shown in Table 2.8. Hassan and Wolff [10] and Bhattacharya et al. [36] used MVFOSM in combination with limit equilibrium methods of slices. Thus the reliability indexes β reported by the two studies are in closer agreement. In contrast, a relatively smaller value of β was found in Ji and Liao [35].

2.3 Advanced First Order Reliability Method (AFORM) Function HLRF_x(VarSeq, PerFunc, V_CheckingPoint, V_EqvMean, V_EqvSD, V_PerFuncGradient, M_Corr) '%%%%% Notations: 'VarSeq = Sequence of random variables, e.g. 1, 2, 3, etc; 'PerFunc = Performance function; 'V_CheckingPoint = Vector of the checking point xk; 'V_EqvMean = Vector of the equivalent mean values; 'V_EqvSD = Vector of the equivalent standard deviation values; 'V_PerFuncGradient = Vector of the performance function gradient values; 'M_Corr = Correlation matrix; Dim xv, mv, sv, P, Pv, R xv = V_CheckingPoint: mv = V_EqvMean: sv = V_EqvSD: P = PerFunc: Pv = V_PerFuncGradient: R = M_Corr: n = VarSeq Dim i, j, m As Integer m = UBound(xv, 1) 'to compute the denominator Denominator = 0 For j = 1 To m Sum = 0 For i = 1 To m 'note: T(i, j)= sv(i, 1) * R(i, j) * sv(j, 1) Sum = Sum + Pv(i, 1) * sv(i, 1) * R(i, j) * sv(j, 1) Next i Denominator = Denominator + Sum * Pv(j, 1) Next j 'to compute the Numerator Sum = 0 For i = 1 To m Sum = Sum + Pv(i, 1) * (xv(i, 1) - mv(i, 1)) Next i Numerator1 = Sum - P Sum = 0 For j = 1 To m Sum = Sum + sv(n, 1) * R(n, j) * sv(j, 1) * Pv(j, 1) Next j Numerator2 = Sum HLRF_x = mv(n, 1) + Numerator1 * Numerator2 / Denominator End Function

Fig. 2.10 VBA code of user-defined function for HLRF-x algorithm

65

66

2 First Order Reliability Methods 0

20

40 m

Scale 221.92 199.34 3 1

Upstream 201.92

1

181.92 1

161.92

3.5

Phase ΙΙ Clay Fill

3

Downstream 186.23 176.17 169.16

Sand Filter

156.97 Phase Ι Clay Fill Foundation Sand

Fill

Foundation Sand

141.92 Limestone Foundation 121.92

Firm Base

Fig. 2.11 A section of the Cannon Dam with results from strength-reduction finite element stability analysis (adapted from Ji and Liao [35]) Table 2.6 Soil properties of Cannon Dam used for strength-reduction finite element stability analysis (adapted from Hassan and Wolff [10]) Poisson ratio ν Cohesion c (kPa)

Friction angle ϕ Unit weight γ (°) (kN/m3 )

Phase II clay 200 fill

0.3

143.64

15

22

Phase I clay fill

200

0.3

117.79

8.5

22

Sand filter

200

0.3

0

35

22

Foundation sand

200

0.3

5

18

20

Fill

200

0.3

5

35

25

Limestone

200

0.3

100

35

20

Layer description

Young’s modulus E (MPa)

Table 2.7 Statistical information of strength parameters of the Cannon Dam (adapted from Hassan and Wolff [10]) Layer description

Strength parameter

Mean value

Standard deviation

Coefficient of variation

Correlation coefficient

Phase II clay fill

c2 (kPa)

143.64

79

0.55

−0.55

ϕ 2 (°)

15

9

0.6

Phase I clay fill

c1 (kPa)

117.79

58.89

0.5

ϕ 1 (°)

8.5

8.5

1.0

0.1

2.4 System Reliability Analysis

67

Table 2.8 Comparison between reliability analyses of the Cannon Dam Method of analyses

Probabilistic critical failure mechanism

Reliability index β

MVFOSM and simplified Bishop method (Hassan and Wolff [10])

Non-circular surface

2.664

MVFOSM and Spencer method of slices (Bhattacharya et al. [36])

Non-circular surface

2.674

Probabilistic strength-reduction finite element analysis (HLRF-x)

Non-circular band

2.491

2.4 System Reliability Analysis Many engineering systems are composed of multiple subsystems. The failure probability of a system is related not only to the failure probability of a single subsystem, but also to the relationship between these subsystems. If the system fails only after all subsystems have failed, the system is called a parallel system. For instance, the multi-strands wire rope in Fig. 2.12a can resist tension until all single strands have failed. In contrast, if the failure of a single subsystem causes the failure of the whole system, it is called a series system. For example, for a single-strand wire rope in Fig. 2.12b, failure of any single strand will cause the failure of the whole system. Many geotechnical reliability problems are of the second type, i.e. the series system. For instance, for a gravity retaining wall, the failure of the foundation can be caused by insufficient bearing capacity, insufficient sliding resistance, and overturning. Each failure mode could result in the failure of the retaining wall; hence it is a series system problem. In this textbook, the focus is on series systems.

2.4.1 Ditlevsen’s Bounds for System Reliability Analyses In geotechnical engineering, many systems have multiple subsystems or multiple failure modes. These subsystems/failure modes may be mutually correlated. In such a case, it is often difficult to accurately calculate the failure probability of a system. In

Fig. 2.12 Concept of parallel and series systems

68

2 First Order Reliability Methods

practice, computing the possible range of system failure probability can still provide very useful information for decision making. In this section, Ditlevsen’s bounds [37] are introduced to provide an estimate of series system failure probability considering correlation coefficients between failure modes. Consider a series system consists of M subsystems and use E and E i to represent the system failure event and the failure of ith subsystem, respectively. The system failure event can be decomposed as follows: E=

M | |

ei

(2.52)

i=1

where e1 = E 1 e2 = E 2 ∩ E 1 e3 = E 3 ∩ E 2 ∩ E 1 .. .( ) i−1 ∩ Ej ei = E i ∩

(2.53)

j=1

where e1 denotes failure event of the first subsystem, ei denotes failure event of the ith subsystem provided that the other i − 1 subsystems are safe. Figure 2.13 shows this concept with i equal to 4. The series system failure probability can be rewritten as: P(E) =

M ∑ i=1

Fig. 2.13 Contribution of ith subsystem to the system failure event

P(ei )

(2.54)

2.4 System Reliability Analysis

69

According to the definition of ei , P(ei ) denotes the extra contribution of the event, E i , to the system failure probability. Since the direct computation of P(ei ) is difficult, a rational bounds may be used as a compromise. The inequality for the bound solution is given as follows [15]: ⎤ [ ] i−1 ∑ ) ) ( ( ⎣ P(E i ) − P E i E j ⎦ ≤ P(ei ) ≤ P(E i ) − max P E i E j ⎡

j 0.01), the computed Ditlevsen’s bounds might be wide; (2) all the failure modes (or subsystems) must be ranked in increasing reliability indexes to produce reasonably narrow failure bounds.

2.4.2 Linearization Approach Besides the aforementioned Ditlevsen’s bounds, Hohenbichler and Rackwitz [47] developed an approximate approach to solve series reliability problems in a simple

2.4 System Reliability Analysis

73

and efficient manner, in which a certain value of system failure probability can be obtained. Zeng and Jimenez [46] introduced this approach to solve geotechnical system reliability problems, and named it the linearization approach. As shown in Fig. 2.17, LSFs of subsystems can often be linearly approximated in the vicinity of the design points using AFORM. Then, a two-dimensional u-space with two linear approximations can be divided into four subdomains (i.e., A, B, C and D). In which, the union of subdomains C and D denotes the failure domain of g1 (i.e., g1 < 0); the union of subdomains A and C denotes the failure domain of g2 (i.e., g2 < 0); the subdomain B denotes the safe domain (i.e., g1 > 0 and g2 > 0). Thus, the system failure event is: P(E) = P(g1 ≤ 0 ∪ g2 ≤ 0) = P({C ∪ D} ∪ {A ∪ C}) = P(A ∪ C ∪ D) (2.60) which can be expressed simply as: P(E) = P(A ∪ C ∪ D) = 1 − P(B)

Y

Allowable extreme nodes intervals Representative slip surface 1 Representative slip surface 2 Representative slip surface 3 Representative slip surface 4

15 3

γ1 = γ2 = γ3 = 19.5 kN/m

10

Elevation (m)

(2.61)

Layer 1

5 Layer 2

0 Layer 3

-5 -40

-35

-30

-25

-20 -15 -10 Horizontal distance (m)

-5

0

5

10

X

Fig. 2.15 Geometry of a three-layer soil slope and the identified representative slip surfaces with genetic algorithm (adapted from Zeng et al. [41])

Table 2.11 Reliability indices and correlation matrix of Example 2.11 (adapted from Zeng et al. [41]) β

R

1

2

3

4

2.1696

1

1.0000

0.8327

0.8457

0.4073

2.5546

2

0.8327

1.0000

0.6057

0.8447

2.9174

3

0.8457

0.6057

1.0000

0.1813

3.0773

4

0.4073

0.8447

0.1813

1.0000

74

2 First Order Reliability Methods

beta=[2.1696 2.5546 2.9174 3.0773]; co_matrix=[1 0.8327 0.8457 0.4073; 0.8327 1 0.6057 0.8447; 0.8457 0.6057 1 0.1813; 0.4073 0.8447 0.1813 1;]; N=length(beta); %number of subsystems %%-----------lower bound -------------Pf=zeros(1, N); for i=1: N Pf(i)=normcdf(-beta(i)); end sum=0; for i=2: N sum1=0; for j=1:(i-1) sum1=sum1+probFiFj(beta(i),beta(j),co_matrix(j,i),1); end sum=sum+max(Pf(i)-sum1,0); end Pf_lower=Pf(1)+sum %%-----------upper bound -------------sum=0; sum1=0; Pf=zeros(1, N); for i=1: N Pf(i)=normcdf(-beta(i)); end for i=1: N sum=sum+Pf(i); end for i=2: N for j=1:(i-1) tem=probFiFj(beta(i),beta(j),co_matrix(j,i),0); if j==1 maxP=tem; end if tem>maxP maxP=tem; end end sum1=sum1+maxP; end Pf_upper=sum-sum1

Fig. 2.16 MATLAB script for Example 2.11

%%-----------probFiFj function-------------function pij=probFiFj(beta_i,beta_j,rho,index) if rho==1 rho=0.99999999; end if rho== 1 rho=-0.99999999; end para_a=normcdf( -beta_i)*normcdf( -(beta_j rho*beta_i)/sqrt(1 -rho^2)); para_b=normcdf( -beta_j)*normcdf( -(beta_irho*beta_j)/sqrt(1 -rho^2)); if rho>=0 && index==0 pij=max(para_a,para_b); end if rho=0 && index==1 pij=para_a+para_b; end if rho −βi }

i=1

{−α i u < βi } = 1 − ϕ M (β, R)

(2.62)

i=1

where gi (x) is the performance function in the x-space, gi (u) is the performance function in the (transformed) u-space and, g i (u) is the first order approximation (linearization) of gi (u) at the design point. ϕ M (β, R) is the CDF of the M-dimensional standard normal distribution evaluated for the vector of reliability indices β = [β1 β2 · · · β M ], with correlation matrix R, given by Rij = α Ti α j , and α i is the unit direction vector at the design point of the i th LSF. The probability of event-intersection given by ϕ M can be computed by the MATLAB function mvncdf .

76

2 First Order Reliability Methods

Example 2.12 The retaining wall system in Example 2.10 is reviewed again with the linearization approach. Solution According to previous results, the reliability indexes of the two failure modes are 2.491 and 3.102, respectively; the correlation coefficient between the two failure modes is 0.533. Thus, [

1 0.533 R= 0.533 1

]

Then, Eq. (2.62) can be directly used to compute the series system failure probability: ( [ P(E) = 1 − ϕ2 [2.491, 3.102],

1 0.533 0.533 1

]) = 0.713%

The above failure probability is within the Ditlevsen’s bounds obtained previously.

Example 2.13 The linearization approach is used to estimate the system failure probability of the three-layer soil slope with four representative slip surfaces given in Example 2.11. Solution The reliability indexes and correlation matrix of the four representative slip surfaces are given in Table 2.2. MATLAB script is created for solving this example problem, which is given in Fig. 2.18. The computed system failure probability is 1.76%, which is within the Ditlevsen’s bounds obtained in Example 2.11.

clear;clc %% input parameter beta=[2.1696 2.5546 2.9174 3.0773]; %reliability indices of RSSs co_matrix=[1 0.8327 0.8457 0.4073; 0.8327 1 0.6057 0.8447; 0.8457 0.6057 1 0.1813; 0.4073 0.8447 0.1813 1;]; %correlation matrix N=length(beta); %number of subsystems %% compute system failure probability Pf=1-mvncdf(beta,zeros(1,N),co_matrix)

Fig. 2.18 MATLAB code for Example 2.13

2.4 System Reliability Analysis

77 o θ

R

x

Soil properties: cu

= cu

W

T

Fig 2.19 An undrained slope with a cylindrical slip surface

Table 2.12 Distribution and statistical parameter of intensity variables Variable

Distribution type Mean

Standard deviation

Internal friction angle ϕ

Normal

35°

3.5°

Interface friction angle δ between the wall and Normal the soil

20°

2°

Base cohesion ca

100 kPa 15 kPa

Table 2.13 Parameters of each soil layer

Normal

Soil layer

Thickness (m)

Shaft resistance (kPa)

End bearing capacity (kPa)

Silt clay 1

17

20

–

Silt clay 2

7

26

–

Fine sand

24

60

5000

Table 2.14 Statistical characteristics of random variables Design parameters

Mean value

COV

Distribution

Ultimate shaft resistance qs1 (kPa)

20

0.12

Normal

Ultimate shaft resistance qs2 (kPa)

26

0.14

Normal

Ultimate shaft resistance qs3 (kPa)

60

0.10

Normal

Ultimate end bearing capacity qp (kPa)

5000

0.2

Lognormal

Uncertainty factor of the calculated model η

1.1

0.15

Normal

Load effect of pile top Q (kN)

4000

0.07

Normal

The linearization approach is also widely used in geotechnical system failure probability estimation [see e.g., 41, 45, 48–52]. However, when a large number of subsystems (or failure modes) exist, the mvncdf function in MATLAB may fail to provide accurate result. Because the correlation matrix, R, might not be a positive definite matrix due to some strongly correlated subsystems (i.e., some correlation

78 Table 2.15 Correlation matrix for failure modes

2 First Order Reliability Methods R

1

2

3

1

1.0000

0.0746

0.5902

2

0.0746

1.0000

0.8490

3

0.5902

0.8490

1.0000

coefficients are close to 1). In such a circumstance, the conditional probability-based method could be employed [53].

2.5 Summary and Further Readings This chapter presents simple yet useful first order reliability methods for geotechnical reliability analysis. The first order reliability methods and their numerical implementations are substantially demonstrated through worked examples. The AFORM can provide fast solutions to problems with performance functions of low nonlinearity. If the performance function is highly nonlinear and/or if there are multiple local minimums in terms of reliability index, special care must be exercised in ensuring the robustness of the AFORM [54–58]. In such cases, a second order reliability method or importance sampling method can be employed to improve the probability of failure [59–64]. Also, the reliability index and design point values for random variables contain important information resulted from the reliability analysis, and they constitute a formula for further conducting reliability-based design optimization [65–69]. In some recent geotechnical applications, reliability analysis has to deal with high dimension random variables, e.g., for the modeling of random fields (see Chap. 5). In such cases, one may consider to reduce the dimension of the random variables in order to use the AFORM [70–72]. On the other hand, if a geotechnical system by itself consists of multiple failure modes, a system reliability analysis might be required. In this regard, the Ditlevsen’s bounds and linearization approach introduced in this chapter are both simple and efficient. AFORM can provide inputs for these methods for system reliability analysis. The accuracy of these methods, however, will also depend on AFORM. For geotechnical problems with highly nonlinear LSFs, the second order reliability method could be used to refine the reliability index estimation (see e.g., [38, 73]). Meanwhile, it should be noted that, prior to applying the Ditlevsen’s bounds and the linearization approach to system failure probability estimation, one must identify all the possible failure modes (and their reliability indices and correlation matrix) for a given geotechnical system. Sometimes, this might be a difficult task. Then, response surface methods (see e.g., [74–76]) and simulation methods (see e.g., [77, 78–80]) could be used to obtain a better estimate of the system failure probability but often with more computational cost.

2.5 Summary and Further Readings

79

Exercises Exercise 2.1 Consider the following performance function: g(x) = x1 x2 − 1140 where x 1 and x 2 are lognormal variables, and the statistics parameters are x 1 ~ LN (38.0, 3.8), x 2 ~ LN (54.0, 2.7), respectively. Please compute the reliability index and design point values for random variables x 1 and x 2 . Exercise 2.2 Examine the effect of the statistical distributions parameters of random variables on the results of reliability analysis, where the performance function with four random variables is as follows: g(x) = x12 − 4x2 − 2x3 x4 (a) x 1 ~ x 4 are normal variables, the statistics parameters are x 1 ~ N (4.104, 0.41), x 2 ~ N (2.162, 0.43), x 3 ~ N (1.783, 0.18), x 4 ~ N (1.0, 0.1), respectively. (b) x 1 is normal variable while the others are lognormal variables, the statistics parameters are x 1 ~ N (4.104, 0.41), x 2 ~ LN (2.162, 0.43), x 3 ~ LN (1.783, 0.18), x 4 ~ LN (1.0, 0.1), respectively. Please compute the reliability index and design point values for random variables x 1 and x 2 . Exercise 2.3 An undrained slope with a potential cylindrical slip surface is shown in Fig. 2.19, the FOS of homogeneous soil slope is given below: Fs =

τLR Wx

where the length of arc L = Rθ, τ is average shear stress along the slip surface, the weight of soil block W = γ V, γ is the soil unit weight, and the undrained soil shear strength cohesion is cu . Suppose the volume of the soil block V is 175 m3 , the radius of the sliding surface R is 18 m and the rounded corner θ is 120°, respectively. The horizontal distance x between the circle center of the sliding surface and the mass center of the soil block is 5 m. Compute the reliability index and design point values for random variables cu and γ . Suppose cu and ϕ to be random variables: cu obeys the lognormal distribution, and γ obeys the normal distribution, the mean value

80

2 First Order Reliability Methods

and standard deviation of cu are 35 kPa and 8.75 kPa, respectively, the mean value and standard deviation of γ are 18.5 kPa and 2.775 kPa, respectively. Exercise 2.4 By infinite slope stability model, the FOS of homogeneous soil slope is given below: Fs =

c tan ϕ + γ H sin ψ cos ψ tan ψ

where the depth of sliding H is 5.0 m, the slope inclination ψ is 30°, the soil unit weight, γ is 17 kN/m3 , the effective soil shear strength cohesion is c and friction angle is ϕ. Suppose c and tan ϕ to be random variables. The mean value and standard deviation of c are 10 kPa and 3 kPa, respectively. The mean value and standard deviation of tan ϕ are 0.5774 and 0.1154, respectively. For the following three scenarios, compute the first order reliability index and design point values for random variables c and tan ϕ (a) The parameters c and tan ϕ all obey normal distribution. (b) The effective cohesion c obeys lognormal distribution, and the tanϕ obeys normal distribution. (c) The parameter distribution information is the same as the scenario (b), but further consider a negative correlation between c and tan φ, ρ = −0.3. Exercise 2.5 The retaining wall as shown in Fig. 2.14 is revisited. For the sliding failure mode with performance function: gsliding = bca − Pa cos δ where Pa = 21 K a γs H 2 . The Coulomb active coefficient K a can be calculated as: ⎡

⎤2

sin(α − ϕ)/ sin α ⎦ / Ka = ⎣ √ / sin(α + δ) + sin(ϕ + δ) sin(ϕ − λ) sin(α − λ)

where α is the inclination of wall back and α = 90° in this exercise. The geometry parameters are H = 6 m, a = 0.4 m, b = 1.8 m and λ = 10°. The unit weights of the retaining wall γ w and the retained soil γ s are 24 kN/m3 and 18 kN/m3 , respectively. The statistical parameters of the soil internal friction angle ϕ, the interface friction angle δ between the wall and the soil, and the base cohesion ca are shown in Table 2.12. In addition, ϕ and δ are

2.5 Summary and Further Readings

81

positively correlated with ρ = 0.8. Use AFORM to compute the reliability index and the design point. Exercise 2.6 In this earth retaining system example, the back of the retaining wall is straight, the base and fill surfaces are horizontal, and the FOS values against sliding F S1 and overturning F S2 are calculated as: (w1 + w2 ) Pa (w1 · Ar m 1 + w2 · Ar m 2 ) = Pa

FS1 = FS2

where w1 and w2 are the self-weights of the left and right parts of the retaining wall, respectively; Pa is the main dynamic earth pressure; Arm1 , Arm2 and H 0 are the distances from the line of action to the toe of the wall for w1 , w2 and Pa respectively; and μ is the coefficient of friction at the base of the retaining wall. In addition, w1 = γwall bH/2, w2 = γwall a H , Ar m 1 = 2b/3, Ar m 2 = b+a , a and b are the geometrical dimensions of 2 the wall base, H is the height and γ wall the weight of the retaining wall. The active earth pressure Pa is calculated based on Rankine theory: Pa =

√ 1 2c2 γsoil H 2 K a − 2cH K a + 2 γsoil

where γ soil is the weight of the fill soil, c and ϕ are the cohesion and internal friction angle, respectively, K a is the coefficient of active earth pressure being calculated by K a = tan2 and H0 =

(π 4

−

ϕ) 2

) ( 2c 1 H− √ 3 γsoil K a

Note that c and ϕ will be taken as lognormal random variables. The mean values and COVs of c and ϕ are respectively 14 kPa and 0.4, and 25° and 0.2, respectively. The remaining parameters H, γ wall , γ soil , μ, a and b are taken as constants to be 6.5 m, 24 kN/m3 , 18 kN/m3 , 0.50 m and 1.45 m, respectively. (a) Calculate the reliability index against sliding based on F s1 ; (b) Calculate the reliability index against overturning based on F s2 ; (c) Calculate the system failure probability of the retaining system.

82

2 First Order Reliability Methods

Exercise 2.7 Consider a cast-in-place pile foundation problem. The pile diameter is 1.0 m, and the depth of pile is 24.0 m. The pile resistance parameters in each soil layer are listed in Table 2.13, and the random variables are listed in Table 2.14. The corresponding performance function for pile bearing capacity is: z = η(u

n ∑

qsi li + q p A p ) − Q

i=1

where u, Ap are the perimeter of the cross-section of the pile and the area of the pile bottom (m, m2 ), respectively, l i is the thickness of each soil layer on the side of the pile, qsi is the unit ultimate shaft resistance of the ith layer of soil on the side of the pile (kPa), qp is the unit ultimate end bearing capacity, η is a random variable describing the uncertainty of the calculated model, Q is the load effect on the top of the pile (kN). Calculate the reliability index and designed point for the random variables with AFORM. Exercise 2.8 Three representative slip surfaces are identified for an embankment with two layers adapted from Zeng et al. [41]. Their reliability indices are found to be β = [0.7127, 0.8064, 1.6414]T . The correlation matrix for the three failure modes is given in Table 2.15. Assess the system failure probability using Ditlevsen’s bounds and linearization approach, respectively.

References 1. Freudenthal AM (1956) Safety and the probability of structural failure. Trans Am Soc Civ Eng 121(1):1337–1375 2. Cornell CA (1967) Bounds on reliability of structural systems. J Struct Div ASCE 93(ST1):171– 200 3. Ditlevsen O (1973) Structural reliability and the invariance problem. Solid Mechanics Division, University of Waterloo, Waterloo, Canada 4. Dolinski K (1983) First-order second-moment approximation in reliability of structural systems—Critical review and alternative approach. Struct Saf 1(3):211–231 5. Bjerager P (1990) On computation methods for structural reliability analysis. Struct Saf 9(2):79–96 6. Rackwitz R (2001) Reliability analysis: A review and some perspectives. Struct Saf 23(4):365– 395 7. Shinozuka M (1983) Basic analysis of structural safety. J Struct Eng ASCE 109(3):721–740 8. Christian JT, Ladd CC, Baecher GB (1994) Reliability applied to slope stability analysis. J Geotech Geoenviron Eng 120(12):2180–2206 9. Malkawi AIH, Hassan WF, Abdulla FA (2000) Uncertainty and reliability analysis applied to slope stability. Struct Saf 22(2):161–187

References

83

10. Hassan AM, Wolff TF (1999) Search algorithm for minimum reliability index of earth slopes. J Geotech Geoenviron Eng 125(4):301–308 11. Wong FS (1985) Slope reliability and response surface method. J Geotech Eng ASCE 111(1):32–53 12. Hasofer AM, Lind NC (1974) Exact and invariant second moment code format. J Eng Mech ASCE 100(EM1):111–121 13. Rackwitz R (1976) Practical probabilistic approach to design. Comite European du Beton, France, Paris 14. Haldar A, Mahadevan S (2000) Probability, reliability and statistical methods in engineering design. Wiley, New York 15. Ang AH-S, Tang WH (1984) Probability concepts in engineering design, vol 2: decision, risk and reliability. Wiley, New York 16. Der Kiureghian A, Liu PL (1986) Structural reliability under incomplete probability information. J Eng Mech ASCE 112(1):85–104 17. Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23(3):470–472 18. Hohenbichler M, Rackwitz R (1981) Non-normal dependent vectors in structural safety. J Eng Mech Div 107(6):1227–1238 19. Nataf A (1962) Determination des distributions dont les marges sont donnees. Comptes Rendus de l’ Academie des Sciences 225:42–43 20. Rackwitz R, Fiessler B (1978) Structural reliability under combined random load sequences. Comput Struct 9:484–494 21. Low BK, Tang WH (2004). Reliability analysis using object-oriented constrained optimization. Struct Saf 26:69–89 22. Chen X, Lind NC (1983) Fast probability integration by three-parameter normal tail approximation. Struct Saf 1(4):269–276 23. Liu P-L, Der Kiureghian A (1986) Multivariate distribution models with prescribed marginals and covariances. Probab Eng Mech 1(2):105–112 24. Low BK, Tang WH (2007) Efficient spreadsheet algorithm for first-order reliability method[J]. J Eng Mech 133(12):1378–1387 25. Li D-Q Zheng D, Cao Z-J, Tang X-S, Phoon K-K (2016). Response surface methods for slope reliability analysis: review and comparison. Eng Geol 203:3–14 26. Ji J, Low BK (2012) Stratified response surfaces for system probabilistic evaluation of slopes. J Geotech Geoenviron Eng ASCE 138(11):1398–1406 27. Zhang J, Huang H, Phoon K (2013) Application of the Kriging-based response surface method to the system reliability of soil slopes. J Geotech Geoenviron Eng 139(4):651–655 28. Zhang W, Goh AT (2012) Reliability assessment on ultimate and serviceability limit states and determination of critical factor of safety for underground rock caverns. Tunnel Undergr Space Technol 32:221–230 29. Liu P-L, Der Kiureghian A (1991) Optimization algorithms for structural reliability. Struct Saf 9(3):161–177 30. Ji J, Kodikara JK (2015) Efficient reliability method for implicit limit state surface with correlated non-Gaussian variables. Int J Numer Anal Meth Geomech 39(17):1898–1911 31. Ramesh RB, Mirza O, Kang W-H (2017) HLRF-BFGS-based algorithm for inverse reliability analysis. Math Probl Eng 2017:1–15 32. Wang Z, Broccardo M, Der Kiureghian A (2016) An algorithm for finding a sequence of design points in reliability analysis. Struct Saf 58:52–59 33. Ji J, Zhang C, Gao Y, Kodikara J (2018) Effect of 2D spatial variability on slope reliability: a simplified FORM analysis. Geosci Front 9:1631–1638 34. Ji J, Zhang C, Gao Y, Kodikara J (2019) Reliability-based design for geotechnical engineering: an inverse FORM approach for practice. Comput Geotech 111:22–29 35. Ji J, Liao HJ (2014) Sensitivity-based reliability analysis of earth slopes using finite element method. Geomech Eng 6(6):545–560 36. Bhattacharya G, Jana D, Ojha S, Chakraborty S (2003) Direct search for minimum reliability index of earth slopes. Comput Geotech 30(6):455–462

84

2 First Order Reliability Methods

37. Ditlevsen O (1979) Narrow reliability bounds for structural systems. J Struct Mech 7(4):453– 472 38. Low BK, Zhang J, Tang WH (2011) Efficient system reliability analysis illustrated for a retaining wall and a soil slope. Comput Geotech 38(2):196–204 39. Low BK (2005) Reliability-based design applied to retaining walls. Géotechnique 55:63–75 40. Zhang J, Huang HW, Juang CH, Li DQ (2013) Extension of Hassan and Wolff method for system reliability analysis of soil slopes. Eng Geol 160:81–88 41. Zeng P, Jimenez R, Jurado-Piña R (2015) System reliability analysis of layered soil slopes using fully specified slip surfaces and genetic algorithms. Eng Geol 193(2):106–117 42. Oka Y, Wu T (1990) System reliability of slope stability. J Geotech Eng 116(8):1185–1189 43. Chowdhury RN, Xu DW (1995) Geotechnical system reliability of slopes. Reliab Eng Syst Saf 47(3):141–151 44. Lü Q, Chan C, Low B (2013) System reliability assessment for a rock tunnel with multiple failure modes. Rock Mech Rock Eng 46(4):821–833 45. Liu H, Low BK (2017) System reliability analysis of tunnels reinforced by rockbolts. Tunnel Undergr Space Technol 65:155–166 46. Zeng P, Jimenez R (2014) An approximation to the reliability of series geotechnical systems using a linearization approach. Comput Geotech 62:304–309 47. Hohenbichler M, Rackwitz R (1982) First-order concepts in system reliability. Struct Saf 1(3):177–188 48. Ma JZ Zhang J, Huang HW, Zhang LL, Huang JS (2017) Identification of representative slip surfaces for reliability analysis of soil slopes based on shear strength reduction. Comput Geotech 85:199–206 49. Xiao T, Li D-Q, Cao Z-J, Tang X-S (2017) Full probabilistic design of slopes in spatially variable soils using simplified reliability analysis method. Georisk: Assess Manage Risk Eng Syst Geohazards 11(1):146–159 50. Huang HW, Wen SC, Zhang J, Chen FY, Martin JR, Wang H (2018) Reliability analysis of slope stability under seismic condition during a given exposure time. Landslides 15(11):2303–2313 51. Juang CH Zhang J, Shen M, Hu J (2018) Probabilistic methods for unified treatment of geotechnical and geological uncertainties in a geotechnical analysis. Eng Geol 248:149–161 52. Duan X, Zhang J, Huang H, Zeng P, Zhang L (2020) System reliability analysis of soil slopes through constrained optimization. Landslides 18:655–666 53. Xiao Z-P, Lü Q, Zheng J, Liu J, Ji J (2020) Conditional probability-based system reliability analysis for geotechnical problems. Comput Geotech 126:103751 54. Periçaro GA, Santos SR, Ribeiro AA, Matioli LC (2015) HLRF–BFGS optimization algorithm for structural reliability. Appl Math Model 39(7):2025–2035 55. Ghohani Arab H, Rashki M, Rostamian M, Ghavidel A, Shahraki H, Keshtegar B (2019) Refined first-order reliability method using cross-entropy optimization method. Eng Comput 35(4):1507–1519 56. Lee I, Choi KK, Gorsich D (2010) Sensitivity analyses of FORM-based and DRM-based performance measure approach (PMA) for reliability-based design optimization (RBDO). Int J Numer Meth Eng 82(1):26–46 57. Zhu S-P, Keshtegar B, Chakraborty S, Trung N-T (2020) Novel probabilistic model for searching most probable point in structural reliability analysis. Comput Methods Appl Mech Eng 366:113027 58. Keshtegar B, Chakraborty S (2018) A hybrid self-adaptive conjugate first order reliability method for robust structural reliability analysis. Appl Math Model 53:319–332 59. Ching J, Phoon KK, Hu YG (2009) Efficient evaluation of reliability for slopes with circular slip surfaces using importance sampling. J Geotech Geoenviron Eng ASCE 135(6):768–777 60. Huang J, Griffiths DV (2011) Observations on FORM in a simple geomechanics example. Struct Saf 33(1):115–119 61. Zeng P, Jimenez R, Li T (2016) An efficient quasi-Newton approximation-based SORM to estimate the reliability of geotechnical problems. Comput Geotech 76:33–42

References

85

62. Lü Q, Low BK (2011) Probabilistic analysis of underground rock excavations using response surface method and SORM. Comput Geotech 38(8):1008–1021 63. Zhao Y-G, Ono T (1999) A general procedure for first/second-order reliability method (FORM/SORM). Struct Saf 21(2):95–112 64. Koyluoglu HU, Nielsen SRK (1994) New approximations for SORM integrals. Struct Saf 13(4):235–246 65. Phoon K-K, Kulhawy Fred H, Grigoriu Mircea D (2003) Development of a reliability-based design framework for transmission line structure foundations. J Geotech Geoenviron Eng 129(9):798–806 66. Low BK, Phoon K-K (2015) Reliability-based design and its complementary role to Eurocode 7 design approach. Comput Geotech 65:30–44 67. Lü Q, Xiao ZP Ji J, Zheng J (2017) Reliability based design optimization for a rock tunnel support system with multiple failure modes using response surface method. Tunnel Undergr Space Technol 70:1–10 68. Mahmood Z (2020) Reliability-based optimization of geotechnical design using a constrained optimization technique. SN Appl Sci 2(2):1–11 69. Javankhoshdel S, Bathurst RJ (2014) Simplified probabilistic slope stability design charts for cohesive and cohesive-frictional (c-φ) soils. Can Geotech J 51(9):1033–1045 70. Lacasse S, Nadim F (2007) Probabilistic geotechnical analyses for offshore facilities. Georisk: Assess Manage Risk Eng Syst Geohazards 1(1):21–42 71. Low BK (2014) FORM, SORM, and spatial modeling in geotechnical engineering. Struct Saf 49:56–64 72. Luo Z, Atamturktur S, Cai YQ, Juang CH (2012) Simplified approach for reliability-based design against basal-heave failure in braced excavations considering spatial effect. J Geotech Geoenvironm Eng 138(4):441–450 73. Zeng P, Li T, Jimenez R, Feng X, Chen Y (2018) Extension of quasi-Newton approximationbased SORM for series system reliability analysis of geotechnical problems. Eng Comput 34(2):215–224 74. Jiang S-H, Li D-Q, Zhang L-M, Zhou C-B (2014) Slope reliability analysis considering spatially variable shear strength parameters using a non-intrusive stochastic finite element method. Eng Geol 168:120–128 75. Kang F, Han S, Salgado R, Li J (2015) System probabilistic stability analysis of soil slopes using Gaussian process regression with Latin hypercube sampling. Comput Geotech 63:13–25 76. Metya S, Mukhopadhyay T, Adhikari S, Bhattacharya G (2017) System reliability analysis of soil slopes with general slip surfaces using multivariate adaptive regression splines. Comput Geotech 87:212–228 77. Ching J, Phoon K, Hu Y (2009) Efficient evaluation of reliability for slopes with circular slip surfaces using importance sampling. J Geotech Geoenviron Eng 135(6):768–777 78. Wang Y, Cao Z, Au S-K-K (2011) Practical reliability analysis of slope stability by advanced Monte Carlo simulations in a spreadsheet. Can Geotech J 48(1):162–172 79. Jiang S-H, Li D-Q, Cao Z-J, Zhou C-B, Phoon K-K (2015) Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo simulation. J Geotech Geoenviron Eng 141(2):04014096 80. Li D-Q, Xiao T, Cao Z-J, Zhou C-B, Zhang L-M (2016) Enhancement of random finite element method in reliability analysis and risk assessment of soil slopes using subset simulation. Landslides 13(2):293–303

Chapter 3

Simulation-Based Methods

Although first-order reliability methods introduced in the previous chapter are efficient and easy to apply, they may not work well for problems with high dimensionality, nonlinear performance functions, and multiple failure modes (e.g., Schuëller et al. [1]; Song et al. [2]), which are not uncommon in geotechnical engineering particularly when the spatial variability of soil properties is considered (e.g., Xiao [3]), which will be introduced in Chap. 5. For such problems, simulation-based (or sampling-based) reliability analysis methods would be useful, sometimes even be the only feasible methods. These methods aim to repeatedly simulate possible realizations (i.e., random samples) of system configurations, aka random sampling, and estimate the failure probability by calculating the ratio of failure samples to total samples. This chapter will first introduce widely-used random sampling techniques for univariate and multivariate variables. Then, several simulation-based methods for failure probability estimation are provided, starting from Monte Carlo simulation, followed by three common variance reduction techniques, including Latin hypercube sampling, importance sampling, and subset simulation.

3.1 Random Sampling for Univariate Variable The first task in simulation-based methods is to generate random samples from given probability distributions, i.e., PMF for a discrete random variable and PDF for a continuous random variable. Because generating samples for discrete variables is simple, this book will focus on the generation of samples for continuous variables, including the univariate case (i.e., a single variable) in this section and the multivariate case (i.e., multiple variables or a random vector) in the next section. In general, the random samples of target PDFs will be produced from the samples of two standard PDFs, namely, standard uniform distribution bounded between 0 and 1, and standard normal distribution with zero mean and unit variance.

© Tongji University Press Co., Ltd. 2023 J. Zhang et al., Geotechnical Reliability Analysis, https://doi.org/10.1007/978-981-19-6254-7_3

87

88

3 Simulation-Based Methods

3.1.1 Inverse Transformation Method Let F(x) and F −1 (x) denote the CDF and the corresponding inverse CDF of a random variable x, respectively. Given a random sample υ* from a standard uniform distribution between 0 and 1, it can be used to generate a random sample x* that follows F(x) using the inverse transformation method as (e.g., Robert and Casella [4]):   x ∗ = F −1 υ ∗

(3.1)

The CDF of x* is exactly equal to F(·):         P x ∗ ≤ x = P F −1 υ ∗ ≤ x = P υ ∗ ≤ F(x) = F(x) Considering the relation ϕ(u) = υ for the standard normal distribution, Eq. (3.1) can be revised to link x* and u* from a standard normal distribution as:    x ∗ = F −1 ϕ u ∗ Samples of the standard uniform distribution and standard normal distribution can be generated in MATLAB using the built-in functions “rand” and “randn”, respectively. Uniform samples can be generated in MS EXCEL using the built-in function “rand()”. The inverse transformation method is easy-to-use when a closed-form F −1 (x) exists. For illustration, the transformation expressions for seven commonly-used PDFs, including normal, lognormal, uniform, exponential, triangular, Gumbel, and Weibull distributions, are provided in Table 3.1.

Example 3.1 Suppose the width (x) of cracks in soils follows the exponential distribution with the following CDF:  F(x) =

1 − exp(−λx), x ≥ 0 0, x < 0

where λ = 0.2 mm−1 . Generate samples of x using the inverse transformation method. Solution According to Eq. (3.1), we can establish the transformation relation between a standard uniform random variable υ and x as: 1 x = F −1 (υ) = − ln(1 − υ) λ

3.1 Random Sampling for Univariate Variable

89

Table 3.1 Transformation from standard uniform or normal variable to others (adapted from Au and Wang [5]; Li et al. [6]) Distribution

Transformation υ → x or u → x

Parameter

PDF, f (x)

Normal

μ = mean σ = standard deviation σ >0

f (x) = √ 1 exp − 21 2π σ

Lognormal

λ = mean of ln(x) ζ = standard deviation of ln(x) ζ >0

f (x) = √ 1

x = exp(λ + ζ u)

Uniform

a = lower limit b = upper limit a0

  f (x) = a1 exp − ax , x ≥ 0

x = −a ln(1 − υ)

⎧ 2(x − a) ⎪ ⎪ ⎨ (b − a)(c − a) , a ≤ x ≤ c f (x) = ⎪ 2(b − x) ⎪ ⎩ ,c0 Weibull a = location (Extreme value parameter b = scale Type III) parameter c = shape parameter b, c > 0

% Example_3_1 nsamples = 1e4; lamda = 0.2;





x−μ 2 σ



  2

exp − 21 ln x−λ ζ 2π ζ x

f (x) = 



1 x−a x−a b exp − b − exp − b

f (x) =  c−1

c b

x−a b

 , x ≥a

x = μ + σu

x = a − b ln(− ln υ)

1

x = a + b[− ln(1 − υ)] c   c  exp − x−a , x ≥a b

U = rand(nsamples, 1); X = -log(1-U)/lamda; hist(X, 1:2:50);

Fig. 3.1 MATLAB code for the inverse transformation method

90

3 Simulation-Based Methods

Fig. 3.2 Histogram of the crack width with samples drawn using the inverse transformation method

Figure 3.1 shows the MATLAB code to generate 10,000 samples of x using the inverse transformation method. The histogram of crack width is shown in Fig. 3.2.

3.1.2 Acceptance-Rejection Method When the inverse CDF does not have a closed form, the inverse transformation method could be computationally intensive. In such a case, the acceptance-rejection method can be applied to generate samples according to the PDF of a random variable. Let f (x) and s(x) be the target PDF and a user-defined auxiliary sampling PDF from which the samples x* can be generated conveniently, respectively. The acceptancerejection method can be implemented as follows (e.g., Robert and Casella [4]): (1) Generate x* from s(x) and υ* from a standard uniform distribution; (2) Accept x* as a sample of f (x) if υ* ≤ f (x*)/[K·s(x*)]; (3) Repeat these two steps until sufficient samples are obtained. In the abovementioned procedure, K is a constant to ensure that f (x) ≤ K·s(x) always holds. Note that the probability of x* being accepted is P(acceptance|x*) = f (x*)/[K·s(x*)]. According to the total probability theorem, the unconditional acceptance rate, P(acceptance), can be estimated as:  P(acceptance) =

   P acceptance|x s x ∗ dx ∗ = 

∗



1 f (x ∗ )  ∗  ∗ s x dx = K · s(x ∗ ) K

This means a smaller constant K will contribute to a higher acceptance rate and hence higher efficiency. Improving similarity between the sampling PDF s(x) and the target PDF f (x) helps to reduce the constant and improve the efficiency of the acceptance-rejection method. Finally, the PDF of the accepted samples conditioned

3.1 Random Sampling for Univariate Variable % Example_3_2 nsamples = 3e4; a = 10; b = 20; q = 6; r = 2;

91 X = rand(nsamples, 1)*(b-a)+a; U = rand(nsamples, 1); fx = (X-a).^(q-1).*(b-X).^(r-1)/(b-a)^(q+r1)/beta(q, r); sx = 1/(b-a); ind = U0); nn = length(ni); if nn==0 X(i) = U(i); else X(i) = nb_w{nn}*X(ni)+nb_s(nn)*U(i); end end plot(z, X);

Fig. 5.25 MATLAB code for the sequential Gaussian simulation

Fig. 5.26 Random field realizations using the sequential Gaussian simulation

202

5 Spatial Variability of Soils

Example 5.8 Generate one realization of the random field described in Example 5.5 using the sequential Gaussian simulation method. Solution The neighborhood with a size of 1 m contains five data points. For simplicity, a forward-moving path from z = 0 m to z = 9.8 m is applied. The process of a sequential Gaussian simulation is shown in Table 5.4, in which x i , i = 1, 2, 3, …, is the ith sample along the path. Starting with i = 1 at the location of z = 0 m, the first sample (x 1 ) is drawn from a standard normal distribution. Then, it is treated as a known data to generate a sample (i.e., x 2 ) at the location of z = 0.2 m using Eq. (5.24). The sample (i.e., x 3 ) at the location of z = 0.4 m is then generated using Eq. (5.24) conditioned on both x 1 and x 2 . As more and more known data are involved, the conditional standard deviation σ p|k continuously decreases. Starting from x 6 , five previous samples are used to predict the current sample, and the conditional standard deviation σ p|k decreases to a constant of 0.145. Figure 5.25 shows the MATLAB code for implementing the sequential Gaussian simulation, in which the conditional standard deviation values from one neighbor to five neighbors are determined at the beginning to avoid repeated calculation. Figure 5.26 demonstrates two realizations of the random field using the code. In addition to the abovementioned four random field simulation methods, many other random field simulation methods have been developed in literature, such as local average subdivision (e.g., Fenton and Vanmarcke [47]), spectral representation (e.g., Shinozuka and Deodatis [48]), turning bands method (e.g., Fenton [49]), fast Fourier transform (e.g., Fenton [49]), circulant embedding (e.g., Dietrich and Newsam [50]), Fourier series method (e.g., Jha and Ching [51]), and modified linear estimation method (e.g., Liu et al. [52]). Their pros and cons have been summarized and compared by Fenton [49], Sudret and Der Kiureghian [41], and Xiao [8]. Table 5.4 Process of the sequential Gaussian simulation z (m)

xp

xk in neighborhood

μp|k

σ p|k

Sample

0

x1

−

0

1

−0.649

0.2

x2

x1

−0.572

0.471

−0.016

0.4

x3

x1 , x2

0.480

0.296

0.256

0.6

x4

x1 , x2 , x3

0.121

0.216

−0.118

0.8

x5

x1 , x2 , x3 , x4

−0.696

0.172

−0.841

1.0

x6

x1 , x2 , x3 , x4 , x5

−1.323

0.145

−1.406

1.2

x7

x2 , x3 , x4 , x5 , x6

−1.536

0.145

−1.617

1.4

x8

x3 , x4 , x5 , x6 , x7

−1.557

0.145

−1.531

…

…

…

…

…

…

5.4 Multidimensional and Multivariate Random Field

203

5.4 Multidimensional and Multivariate Random Field The preceding section focuses on the simulation of a 1-D univariate random field. In reality, the spatial variability of soils is more often to be multidimensional in space and multivariate regarding various soil properties. All the described methods can be extended to simulate multidimensional and multivariate random fields. For brevity, this section only describes how this can be achieved using the covariance matrix decomposition method. Again, the soil property x in the following sections is modeled by a stationary standard normal random field with zero mean and unit variance.

5.4.1 Spatial Correlation Modeling with Separable Correlation Functions Similar to the 1-D univariate random field, the correlation in the multidimensional and multivariate random field is also characterized by correlation function. The correlation function in higher dimensions can be extended from the 1-D correlation function listed in Table 5.1, in which the separable correlation function is the simplest way to construct the multidimensional correlation function. Consider a 3-D random field varying in the x-, y- and z-directions. A separable correlation function can be expressed as the product of multiple 1-D correlation functions in different directions. It has two forms, i.e., fully separable form [Eq. (5.25)] or partially separable form [Eq. (5.26)]: ( ) ( ) ρ τx , τ y , τz = ρ(τx )ρ τ y ρ(τz )

(5.25)

) ( ) ( ρ τx , τ y , τz = ρ τx , τ y ρ(τz )

(5.26)

where τ x , τ y , and τ z are separation distances between two locations in the x-, y-, and z-directions, respectively. Taking the single exponential correlation function as an example, the corresponding fully separable, partially separable, and inseparable forms can be, respectively, written as: ) ( ) ( ) ( ) ( 2τ y 2τz 2τx exp − exp − ρ τx , τ y , τz = exp − δx δy δz ⎛ / ⎞ ( )2 ( )2 ) ( ( ) τy ⎠ τx 2τz ⎝ ρ τx , τ y , τz = exp −2 + exp − δx δy δz ⎛ / ⎞ ( )2 ( )2 ( )2 ) ( τy τx τz ⎠ ⎝ ρ τx , τ y , τz = exp −2 + + δx δy δz

204

5 Spatial Variability of Soils

Fig. 5.27 Example of fully separable correlation function (adapted from Li et al. [53]): a 3-D lattice; b 1-D correlation matrix; c 3-D correlation matrix

where δ x , δ y , and δ z are scales of fluctuation in the x-, y-, and z-directions, respectively. Figure 5.27 presents an example of a fully separable correlation function. The 3-D lattice consisting of eight points, i.e., P1, P2, …, P8, is organized in a particular sequence, namely along the x-, y-, and z-directions successively, as shown in Fig. 5.27a. Assume correlation coefficients between P1 and P2, P1 and P3, and P1 and P5 are ρ x , ρ y , and ρ z , respectively. According to Eq. (5.25), the correlation coefficient between P1 and P8 equals ρ x ρ y ρ z . Similarly, the 3-D correlation matrix R can be eventually formed, as shown in Fig. 5.27c, which is a multilevel block matrix. Mathematically, the Kronecker product ⊗ is defined as follows: if A = [a]m×n is an m × n matrix and B is a p × q matrix, the Kronecker product A ⊗ B is a pm × qn block matrix as: ⎤ a11 B · · · a1n B ⎥ ⎢ A ⊗ B = ⎣ ... . . . ... ⎦ am1 B · · · amn B ⎡

With the help of the Kronecker product, the 3-D correlation matrix R can be written as the product of three 1-D correlation matrices as: R = Rz ⊗ R y ⊗ R x

(5.27)

where Rx , Ry , and Rz = 1-D correlation matrices in the x-, y-, and z-directions, respectively, as shown in Fig. 5.27b. It should be highlighted that Eq. (5.27) holds only when the fully separable correlation function [i.e., Eq. (5.25)] is adopted. For multivariate random fields, such as the cohesion (c) and friction angle (ϕ) of the same soil, two different types of correlation exist, as shown in Fig. 5.28a. The first type is auto-correlation, which represents the spatial correlation among the values of a given soil property at different locations, such as ci and cj at locations i and j, and it is characterized by an auto-correlation function ρ A (·) and scales of fluctuation, as discussed in previous sections. Subject to the same natural process in history, correlation functions and scales of fluctuation of different properties of the same soil

5.4 Multidimensional and Multivariate Random Field

205

Fig. 5.28 Multivariate random field (adapted from Li et al. [53]): a auto-correlation and crosscorrelation; b global correlation matrix

are often assumed to be the same for simplicity. The second type is cross-correlation, denoting the correlation between different soil properties at the same location, such as ci and ϕ i . It is usually described by cross-correlation coefficients ρ C , as discussed in Sect. 3.2. Regarding a stationary random field, the cross-correlation coefficients can be assumed constant at different locations. For simplicity, the correlation among multiple random fields at different locations is also assumed separable and conventionally decoupled as the product of autocorrelation and cross-correlation, i.e., ρ = ρ A ρ C (e.g., Ji et al. [54]; Voˇrechovský [55]; Xiao et al. [15]; Zhu et al. [34]), as shown in Fig. 5.28a. That means ρ(ci , ϕ j ) = ρ(ci , cj )·ρ(cj , ϕ j ) = ρ(ϕ i , ϕ j )·ρ(ci , ϕ i ). Considering all spatial locations for all soil properties, the global correlation matrix RG that characterizes both auto- and cross-correlation can be formed, as shown in Fig. 5.28b. Again, with the aid of the Kronecker product, it can be rewritten as: RG = RC ⊗ R A

(5.28)

where RA and RC are auto- and cross-correlation matrices, respectively. If a separable 3-D auto-correlation function is considered, RA in Eq. (5.28) can be substituted with R in Eq. (5.27), forming a four-dimensional upgrade of Eq. (5.27), where the type of soil property is the fourth dimension in addition to the three space dimensions. By this means, the global correlation matrix for a multidimensional and multivariate random field can be constructed in a unified form of the Kronecker product with separable correlation functions.

5.4.2 Simulation of Multidimensional Random Field Covariance matrix decomposition can be easily extended to simulate 3-D random fields. The procedure is generally the same with that of the 1-D random field as described in Sect. 5.3.1, in which the key issue is to construct and decompose the 3-D correlation matrix.

206

5 Spatial Variability of Soils % Example_5_9 x = 0:0.5:10; nx = length(x); y = 0:0.5:10; ny = length(y); z = 0:0.5:10; nz = length(z); deltax = 10; deltay = 10; deltaz = 2; corrFun1 = @(d, delta) exp(-2*d/delta); %%%%%%%%%%%%%%%%%%%% % Fill in 3-D correlation matrix R in Fig. 5.30 %%%%%%%%%%%%%%%%%%%%

L = chol(R, 'lower'); U = randn(nx, ny, nz); X = L*U(:); X = reshape(X, [nx, ny, nz]); [xg, yg, zg] = meshgrid(x, y, z); Xg = permute(X, [2, 1, 3]); slice(xg, yg, zg, Xg, x([1, nx]), y([1, ny]), z([1, nz]));

Fig. 5.29 MATLAB code for 3-D random field simulation using the covariance matrix decomposition % Method 1: traditional way to form R [xg, yg, zg] = ndgrid(x, y, z); Dx = abs(bsxfun(@minus, xg(:), xg(:)')); Dy = abs(bsxfun(@minus, yg(:), yg(:)')); Dz = abs(bsxfun(@minus, zg(:), zg(:)')); R = corrFun1(Dx, deltax).*corrFun1(Dy, deltay).*corrFun1(Dz, deltaz);

% Method 2: with the help of Kronecker product Dx = abs(bsxfun(@minus, x', x)); Dy = abs(bsxfun(@minus, y', y)); Dz = abs(bsxfun(@minus, z', z)); Rx = corrFun1(Dx, deltax); Ry = corrFun1(Dy, deltay); Rz = corrFun1(Dz, deltaz); R = kron(Rz, kron(Ry, Rx));

Fig. 5.30 MATLAB code for constructing a 3-D correlation matrix R

Fig. 5.31 Examples of 3-D random fields: a 21 × 21 × 21 in size; b 201 × 201 × 201 in size

Example 5.9 Suppose a 3-D standard normal random field distributes on a 3-D lattice in space with x = y = z = [0, 0.5, 1, …, 10] m. Its spatial correlation is modeled by a 3-D fully separable single exponential correlation function

5.4 Multidimensional and Multivariate Random Field

207

as given by Eq. (5.25) with scales of fluctuation [δ x , δ y , δ z ] = [10, 10, 2] m. Generate one realization of the 3-D random field using the covariance matrix decomposition method. Solution The number of locations in each direction is 21, leading to a total of 21 × 21 × 21 = 9261 locations. Therefore, the correlation matrix among all locations is 9261 × 9261 in size. We will generate a 9261 × 1 vector first using Eq. (5.16) and then reshape it as a 21 × 21 × 21 array of the 3-D random field according to spatial coordinates. Figure 5.29 shows the MATLAB code for 3-D random field simulation using the covariance matrix decomposition. Two equivalent ways can be used to construct the 3-D correlation matrix R, as provided in Fig. 5.30. The first one is a traditional way and can be applied to both separable and inseparable correlation functions, while the second one follows Eq. (5.27) and it is applicable only for separable correlation functions with the aid of the Kronecker product. Figure 5.31a demonstrates one realization of the random field using the code. As illustrated in Example 5.9, a small-scale 3-D random field (i.e., only 21 points in each direction) has already corresponded to a large 3-D correlation matrix (i.e., 9261 × 9261 in size) that approaches the upper limit of the covariance matrix decomposition method. This is far from being desired in practice. To address this issue, a stepwise covariance matrix decomposition method (Li et al. [53]) can be used for simulating a large-scale 3-D random field. Let N x , N y , and N z be the numbers of locations in the x-, y-, and z-directions, respectively, of a 3-D random field, and N = N x N y N z is the total number of locations. Particularly for separable correlation functions, the decomposition of the 3-D correlation matrix, i.e., Eq. (5.27), can be realized, according to the mixed-product property of the Kronecker product (e.g., Horn and Johnson [56]), as: L = Lz ⊗ L y ⊗ L x

(5.29)

where Lx , Ly , and Lz = decomposition matrices satisfying Lx Lx T = Rx , Ly Ly T = Ry , and Lz Lz T = Rz , respectively. The direct decomposition of a large 3-D correlation matrix is replaced by the decomposition of three 1-D correlation matrices, the latter of which is a much easier task. Substituting Eq. (5.29) into Eq. (5.16) gives: ) ( x = Lz ⊗ L y ⊗ L x u

(5.30)

where u and x are N x N y N z × 1 vectors and L = Lx ⊗ Ly ⊗ Lz is an N x N y N z × N x N y N z matrix. Although the decomposition of a large matrix is bypassed, the multiplication of two large matrices is still needed in Eq. (5.30). To further overcome this difficulty, a matrix-array multiplication operation “ ×i ” is developed, which performs piecewise matrix multiplication over the ith dimension

208

5 Spatial Variability of Soils

of an array. Consider that C = A ×i B for example, in which A = [a] is a square matrix, and B = [b] and C = [c] are arrays with the same array size and the length of the ith dimension of B equals the length of the second dimension of A. In the matrixarray multiplication operation, each entry of C is given by multiplying all entries along the second dimension of A by the entries along the ith dimension of B and summing up the results. Intuitively, it appears that the first dimension of the matrix takes the place of the ith dimension of the array. ∑ Particularly with respect ∑ to a 3-D is calculated as c j pq = k a jk bkpq if i = 1, c pjq = k a jk b pkq array B, C = A ×i B ∑ if i = 2, and c pq j = k a jk b pqk if i = 3; with respect to a 2-D array B (or a matrix B), A ×1 B = AB and A ×2 B = BAT . With the aid of the matrix-array multiplication operation, a stepwise strategy can be applied to rewrite Eq. (5.30), referred to as stepwise covariance matrix decomposition method (Li et al. [53]), as: [ ( )] x = L z ×3 L y ×2 L x ×1 u

(5.31)

where u and x = N x × N y × N z arrays reshaped from N x N y N z × 1 vectors u and x, respectively. Concretely, u = [uijk ], u = [up ], and uijk = up for p = i + (j − 1)N x + (k − 1)N x N y (i = 1, 2, …, N x ; j = 1, 2, …, N y ; k = 1, 2, …, N z ). Figure 5.32 illustrates diagrammatically the matrix-array multiplication for Eq. (5.31). Firstly, Lx is multiplied over the first dimension of u to produce the spatial correlation in the xdirection. Similarly, Ly and Lz are multiplied over the second and third dimensions of u, successively and respectively, to produce the spatial correlation in y- and zdirections, respectively. During the whole process, the maximum size of involved arrays and matrices is N x × N y × N z . Li et al. [53] has proved that Eqs. (5.30) and (5.31) are mathematically equivalent. This guarantees the identical 3-D random fields generated by the general and stepwise covariance matrix decomposition methods given separable correlation functions. Fig. 5.32 3-D random field simulation using the stepwise covariance matrix decomposition (adapted from Li et al. [53])

5.4 Multidimensional and Multivariate Random Field % Example_5_10 x = 0:0.5:10; nx = length(x); y = 0:0.5:10; ny = length(y); z = 0:0.5:10; nz = length(z); deltax = 10; deltay = 10; deltaz = 2; corrFun1 = @(d, delta) exp(-2*d/delta); Dx = abs(bsxfun(@minus, x', x)); Dy = abs(bsxfun(@minus, y', y)); Dz = abs(bsxfun(@minus, z', z)); Rx = corrFun1(Dx, deltax); Ry = corrFun1(Dy, deltay); Rz = corrFun1(Dz, deltaz); Lx = chol(Rx, 'lower'); Ly = chol(Ry, 'lower'); Lz = chol(Rz, 'lower');

209

U = randn(nx, ny, nz); X = Lx*reshape(U, nx, ny*nz); X = reshape(X, nx, ny, nz); X = permute(X, [2, 3, 1]); X = Ly*reshape(X, ny, nz*nx); X = reshape(X, ny, nz, nx); X = permute(X, [2, 3, 1]); X = Lz*reshape(X, nz, nx*ny); X = reshape(X, nz, nx, ny); X = permute(X, [2, 3, 1]); [xg, yg, zg] = meshgrid(x, y, z); Xg = permute(X, [2, 1, 3]); slice(xg, yg, zg, Xg, x([1, nx]), y([1, ny]), z([1, nz]));

Fig. 5.33 MATLAB code for 3-D random field simulation using the stepwise covariance matrix decomposition

Example 5.10 Generate one realization of the 3-D random field described in Example 5.9 using the stepwise covariance matrix decomposition method. Solution Recall the second method to construct the 3-D correlation matrix R in Fig. 5.30. Three 1-D correlation matrices, i.e., Rx , Ry , and Rz , are obtained with sizes of N x × N x , N y × N y , and N z × N z , respectively. They are directly decomposed and multiplied by u using Eq. (5.31), avoiding matrices in size of N x N y N z × N x N y N z during the whole process, compared with the general covariance matrix decomposition method. Figure 5.33 shows the MATLAB code for 3-D random field simulation using the stepwise covariance matrix decomposition. This code generates exactly the same random field as that from the code shown in Fig. 5.29, but requires less computational memory with higher efficiency. For easy implementation, the matrix-array multiplication operation is transformed into multiple matrix multiplication operations. Table 5.5 compares the performance of general and stepwise covariance matrix decomposition methods in 3-D random field simulation. The general method requires O[(N x N y N z )3 ] and O[(N x N y N z )2 ] floating point operations for matrix decomposition and random field realization, respectively, while the stepwise method spends O(N x 3 + N y 3 + N z 3 ) and O[N x N y N z (N x + N y + N z )] floating point operations on the two steps, respectively. For a 100 × 100 × 100 random field, computational efforts of the stepwise method are dramatically decreased by 12 and 4 orders of magnitude. The stepwise covariance matrix decomposition method significantly reduces the computational efforts in terms of both time and memory space and, hence, is very suitable for large-scale 3-D random field simulation. Figure 5.31b demonstrates a

210

5 Spatial Variability of Soils

Table 5.5 Comparison of covariance matrix decomposition methods (adapted from Li et al. [53]) Method

Computational complexity Matrix decomposition

General covariance matrix decomposition

Stepwise covariance matrix decomposition

=R

Random field realization x = Lu

Formula

LLT

FLOPsa

O[(N x N y N z )3 ] (≈ 1018 )c

O[(N x N y N z )2 ] (≈ 1012 )c

Memory spaceb

O[(N x N y N z )2 ] (≈ 7.3 TB)c

O(N x N y N z ) (≈ 7.6 MB)c

Formula

Lx Lx T = R x , Ly Ly T = R y , Lz Lz T = R z

x =Lz ×3 [Ly ×2 (Lx ×1 u)]

FLOPsa

O(N x 3 + N y 3 + N z 3 ) (≈ 3 × 106 )c

O[N x N y N z (N x + N y + N z )] (≈ 3 × 108 )c

Memory spaceb

O(N x 2 + N y 2 + N z 2 ) (≈ 0.2 MB)c

O(N x N y N z ) (≈ 7.6 MB)c

Note a FLOPs = floating point operations; b required memory space for R and u; c evaluated for N x = N y = N z = 100

201 × 201 × 201 random field using the code shown in Fig. 5.33 by changing the upper bounds of x, y, and z to 100, which is generated within one second on a personal desktop.

5.4.3 Simulation of Multivariate Random Field As mentioned in Sect. 5.4.1, the type of soil properties can be considered as an additional dimension of the random field. For example, a multivariate 2-D random field is mathematically like a univariate 3-D random field. Therefore, the multivariate random field can also be simulated using the stepwise covariance matrix decomposition with relative ease. Again, with the aid of the Kronecker product, the decomposition of global correlation matrix RG , i.e., Eq. (5.28), can be realized as: LG = LC ⊗ L A

(5.32)

where LC and LA are decomposition matrices satisfying LC LC T = RC and LA LA T = RA , respectively. Then, the multivariate random field can be generated through the stepwise covariance matrix decomposition as (Li et al. [53]): x = L A uLCT

(5.33)

where u and x are N × N p matrices; N is the total number of locations (e.g., N = N x N y N z for 3-D random field); and N p is the number of soil properties. Regarding a

5.4 Multidimensional and Multivariate Random Field % Example_5_11 x = 0:0.5:100; nx = length(x); y = 0:0.5:50; ny = length(y); rhoc = -0.5; nc = 2; deltax = 10; deltay = 2; corrFun1 = @(d, delta) exp(-2*d/delta); Dx = abs(bsxfun(@minus, x', x)); Dy = abs(bsxfun(@minus, y', y)); Rx = corrFun1(Dx, deltax); Ry = corrFun1(Dy, deltay); Rc = [1, rhoc; rhoc, 1]; Lx = chol(Rx, 'lower'); Ly = chol(Ry, 'lower'); Lc = chol(Rc, 'lower');

211

U = randn(nx, ny, nc); X = Lx*reshape(U, nx, ny*nc); X = reshape(X, nx, ny, nc); X = permute(X, [2, 3, 1]); X = Ly*reshape(X, ny, nc*nx); X = reshape(X, ny, nc, nx); X = permute(X, [2, 3, 1]); X = Lc*reshape(X, nc, nx*ny); X = reshape(X, nc, nx, ny); X = permute(X, [2, 3, 1]); [xg, yg] = meshgrid(x, y); subplot(2, 1, 1); pcolor(xg, yg, X(:, :, 1)'); subplot(2, 1, 2); pcolor(xg, yg, X(:, :, 2)');

Fig. 5.34 MATLAB code for multivariate random field simulation using the stepwise covariance decomposition method

Fig. 5.35 Example of negatively correlated random fields: a random field I; b random field II

3-D multivariate random field with a fully separable 3-D correlation function, Eqs. (5.31) and (5.33) can be combined as: { [ ( )]} x = LC ×4 Lz ×3 L y ×2 Lx ×1 u

(5.34)

where u and x are N x × N y × N z × N p arrays. Equation (5.34) is the upgrade of Eq. (5.31) in four-dimensional space. Therefore, simulating a multivariate random field is generally the same as simulating a multidimensional random field using the stepwise covariance matrix decomposition.

Example 5.11 Suppose two 2-D standard normal random fields distribute on a 2-D grid in space with x = [0, 0.5, 1, …, 100] m and y = [0, 0.5, 1, …, 50] m. Their spatial correlations are modeled by the same 2-D fully separable single exponential correlation function with scales of fluctuation [δ x , δ y ] = [10, 2] m. The two random fields are negatively correlated with ρ C = −0.5. Generate a realization of two 2-D random fields using the stepwise covariance matrix decomposition method.

212

5 Spatial Variability of Soils

Solution Figure 5.34 shows the MATLAB code for simulating multivariate random field using stepwise covariance matrix decomposition, modified from the code shown in Fig. 5.33. Figure 5.35 gives an example generated by the code. Graphically, the two random fields share the same spatial correlation structure and have an apparent negative cross-correlation.

5.5 Effects of Spatial Variability on Geotechnical Reliability The spatial variability of geotechnical materials has significant influences on geotechnical reliability, including both failure probability and failure mechanism. In the early stage, simplified reliability methods are often used to evaluate the reliability of geotechnical systems, in which the spatial variability is modeled approximately by spatial averaging technique (e.g., Christian et al. [12]; El-Ramly et al. [13]; Li and Lumb [57]; Li et al. [58]; Vanmarcke [11]). However, different from a uniform geological unit, there might be several weak paths at the same time in the spatially varying geo-materials. What is worse is that the irregular paths may have conflicts with the failure mode assumptions adopted in traditional geotechnical calculations so that they cannot be directly used in reliability analysis. For example, many slope stability analysis methods based on circular slip surfaces may be invalid owing to irregular slip surfaces; and traditional bearing capacity analysis of footings may not work due to non-uniform settlement. To overcome these difficulties caused by spatial variability, analysis methods without the need of assuming the failure mechanism in advance (e.g., finite element method) are preferable. In this aspect, stochastic finite element method (e.g., Beacher and Ingra [59]; Ghanem and Spanos [35]; Ishii and Suzuki [60]; Vanmarcke et al. [61]) and random finite element method (e.g., Griffiths and Fenton [62]; Fenton and Griffiths [7]) are two typical methods to incorporate the spatial variability rigorously into finite element analysis. Comparison between the two methods can be referred to Griffiths and Fenton [63]. These methods have also been extended with many other numerical methods, such as the finite difference method (e.g., Srivastava et al. [64]) and the material point method (e.g., Wang et al. [65]). Thanks to the advance of computational power in recent years, their applications increasingly boost. Specifically, the random finite element method that adopts simulation-based method, particularly the MCS, is widely applied to various geotechnical problems owing to its simplicity, such as slope stability (e.g., Griffiths and Fenton [62]; Hicks and Spencer [66]; Huang et al. [67]; Li et al. [32]; Xiao et al. [42]), bearing capacity or settlement of foundation and retaining wall (e.g., Al-Bittar and Soubra [44]; Cho and Park [45]; Fenton and Griffiths [68]; Gholampour and Johari [69]; Papaioannou and Straub [33]), infiltration and seepage (e.g., Santoso et al. [70]; Srivastava et al. [64]; Tan et al. [71]), and tunneling (e.g., Gong et al. [31]; Huang et al. [72]; Song et al. [73]).

5.5 Effects of Spatial Variability on Geotechnical Reliability

213

Performing reliability analyses considering the spatial variability is a challenging task, at least due to three aspects (Xiao [8]): (1) high uncertainty dimension, (2) low failure probability, and (3) multiple failure modes. Advanced simulation-based reliability analysis methods, such as subset simulation, are desired for geotechnical reliability analysis considering the spatial variability of geo-materials. To demonstrate the effects of spatial variability on geotechnical reliability, three reliability problems of slopes in spatially varying soils are investigated. For simplicity, the limit equilibrium method with a predefined slip surface, instead of the finite element method, will be used to perform the deterministic analysis of slope stability in the following examples. Fig. 5.36 Example of an infinite slope in spatially varying soils

% Example_5_12 Cm = 25; Cs = 5; corrFun = @(dz, delta) exp(-2*dz/delta); H = (0.01:0.01:2)'; n = length(H); dH = abs(bsxfun(@minus, H, H')); delta = [0.1, 1, 10, 100, 1000]; nd = length(delta); Pfsys = zeros(1, nd); [Pf, Nf] = deal(zeros(n, nd)); nSim = 1e4; for i = 1:nd R = corrFun(dH, delta(i)); L = chol(R, 'lower'); U = randn(n, nSim); C = Cm+Cs*L*U; C(C