*167*
*55*
*4MB*

*English*
*Pages 144
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*Year 2023*

Table of contents :

Foreword I

Foreword II

Preface

Contents

About the Authors

1 Introduction

1.1 Scope of the Book

1.2 Key Areas Covered

1.3 Book Outline

References

2 Terms and Notation

3 Performance-Based Seismic Design Framework of Slope

3.1 The PBSD Principle and Applications

3.1.1 Background

3.1.2 Evolution of PBSD for Slope Engineering

3.2 Global Framework of PBSD for Slope Engineering

3.2.1 Design Ground Motion of Slope Engineering Site

3.2.2 Seismic Performance Level and Criteria of Slope Engineering

References

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic Performance Design and Assessment

4.1 Determination of Seismic Ground Motion of Slope Site Based on Seismic Hazard Analysis

4.2 Determination of Seismic Excitation of Slope Site According to Historical Strong Earthquake Records

4.2.1 Correction of Historical Strong Seismic Record of Slope Site Based on IDA Theory

4.2.2 Modification of Seismic Strong Records Based on the Code’s Design Response Spectrum

4.3 Determination of Ground Motion of Slope Site According to the Strong Earthquake Database

4.3.1 International Specification for Ground Motion Determination

4.3.2 Selection of Strong Earthquake Ground Motion

4.4 Artificial Seismic Ground Motion Synthesis of Slope Site

4.4.1 Source-Based Ground Motion Model

4.4.2 Site-Based Ground Motion Model

Appendix

References

5 Deterministic Analysis Methods for Slope Seismic Dynamic Response

5.1 Seismic Response Analysis of Slope Based on the Quasi-static Method

5.2 Slope Dynamic Response Analysis Method Based on the Response Spectrum Method

5.3 Newmark Sliding Block Displacement Method for Seismic Dynamic Response Analysis of Slope

5.4 Slope Seismic Dynamic Response Analysis Based on Nonlinear Dynamic Time-History Analysis

5.5 Large Deformation Analysis Method of Slope

References

6 Probabilistic Performance-Based Seismic Design and Assessment for Slope Engineering

6.1 Source of Uncertainty and Its Description

6.1.1 Sources of Uncertainty in a Seismic Dynamic System of Slope

6.1.2 Establishment of a Random Field of Slope Geomaterials

6.2 Stochastic Seismic Dynamic Response Analysis Method of Slope

6.2.1 Pseudo-Excitation Method

6.2.2 Monte Carlo Stochastic Simulation

6.2.3 Stochastic Finite Element Method

6.2.4 Nonlinear Stochastic Seismic Dynamic Method

6.3 Seismic Dynamic Risk and Vulnerability Assessment of Slope

6.3.1 Seismic Dynamic Vulnerability Assessment of Slope

6.3.2 Seismic Dynamic Risk Assessment of Slope

6.4 Resilience-Based Seismic Performance Design

References

7 Case Study

7.1 Probabilistic Seismic Performance of Earth Dams

7.1.1 Seismic Performance Evaluation of Earth Dams

7.1.2 Seismic Probabilistic Risk Assessment of Earth Dams

7.2 Probabilistic Seismic Performance of Slope

7.2.1 Slope Probabilistic Seismic Performance Evaluation

7.2.2 Seismic Risk Assessment of Slope Retaining Systems

References

8 Conclusions and Prospects

8.1 Conclusions

8.2 Prospects

Foreword I

Foreword II

Preface

Contents

About the Authors

1 Introduction

1.1 Scope of the Book

1.2 Key Areas Covered

1.3 Book Outline

References

2 Terms and Notation

3 Performance-Based Seismic Design Framework of Slope

3.1 The PBSD Principle and Applications

3.1.1 Background

3.1.2 Evolution of PBSD for Slope Engineering

3.2 Global Framework of PBSD for Slope Engineering

3.2.1 Design Ground Motion of Slope Engineering Site

3.2.2 Seismic Performance Level and Criteria of Slope Engineering

References

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic Performance Design and Assessment

4.1 Determination of Seismic Ground Motion of Slope Site Based on Seismic Hazard Analysis

4.2 Determination of Seismic Excitation of Slope Site According to Historical Strong Earthquake Records

4.2.1 Correction of Historical Strong Seismic Record of Slope Site Based on IDA Theory

4.2.2 Modification of Seismic Strong Records Based on the Code’s Design Response Spectrum

4.3 Determination of Ground Motion of Slope Site According to the Strong Earthquake Database

4.3.1 International Specification for Ground Motion Determination

4.3.2 Selection of Strong Earthquake Ground Motion

4.4 Artificial Seismic Ground Motion Synthesis of Slope Site

4.4.1 Source-Based Ground Motion Model

4.4.2 Site-Based Ground Motion Model

Appendix

References

5 Deterministic Analysis Methods for Slope Seismic Dynamic Response

5.1 Seismic Response Analysis of Slope Based on the Quasi-static Method

5.2 Slope Dynamic Response Analysis Method Based on the Response Spectrum Method

5.3 Newmark Sliding Block Displacement Method for Seismic Dynamic Response Analysis of Slope

5.4 Slope Seismic Dynamic Response Analysis Based on Nonlinear Dynamic Time-History Analysis

5.5 Large Deformation Analysis Method of Slope

References

6 Probabilistic Performance-Based Seismic Design and Assessment for Slope Engineering

6.1 Source of Uncertainty and Its Description

6.1.1 Sources of Uncertainty in a Seismic Dynamic System of Slope

6.1.2 Establishment of a Random Field of Slope Geomaterials

6.2 Stochastic Seismic Dynamic Response Analysis Method of Slope

6.2.1 Pseudo-Excitation Method

6.2.2 Monte Carlo Stochastic Simulation

6.2.3 Stochastic Finite Element Method

6.2.4 Nonlinear Stochastic Seismic Dynamic Method

6.3 Seismic Dynamic Risk and Vulnerability Assessment of Slope

6.3.1 Seismic Dynamic Vulnerability Assessment of Slope

6.3.2 Seismic Dynamic Risk Assessment of Slope

6.4 Resilience-Based Seismic Performance Design

References

7 Case Study

7.1 Probabilistic Seismic Performance of Earth Dams

7.1.1 Seismic Performance Evaluation of Earth Dams

7.1.2 Seismic Probabilistic Risk Assessment of Earth Dams

7.2 Probabilistic Seismic Performance of Slope

7.2.1 Slope Probabilistic Seismic Performance Evaluation

7.2.2 Seismic Risk Assessment of Slope Retaining Systems

References

8 Conclusions and Prospects

8.1 Conclusions

8.2 Prospects

- Author / Uploaded
- Yu Huang
- Min Xiong
- Hongqiang Hu

- Similar Topics
- Technique

Yu Huang Min Xiong Hongqiang Hu

Guidelines for Probabilistic Performance-Based Seismic Design and Assessment of Slope Engineering

Guidelines for Probabilistic Performance-Based Seismic Design and Assessment of Slope Engineering

Yu Huang · Min Xiong · Hongqiang Hu

Guidelines for Probabilistic Performance-Based Seismic Design and Assessment of Slope Engineering

Yu Huang Department of Geotechnical Engineering, College of Civil Engineering Tongji University Shanghai, China

Min Xiong Department of Geotechnical Engineering, College of Civil Engineering Tongji University Shanghai, China

Hongqiang Hu Department of Geotechnical Engineering, College of Civil Engineering Tongji University Shanghai, China Center for Balance Architecture Zhejiang University Zhejiang, China

ISBN 978-981-19-9182-0 ISBN 978-981-19-9183-7 (eBook) https://doi.org/10.1007/978-981-19-9183-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte 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 translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, 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 publisher 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 publisher remains 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 I

I am very pleased to read the manuscript of this book. The Performance-Based Seismic Design (PBSD) theory was first applied to building structures, and then extended to civil engineering structures such as bridges. In recent years, more and more research results of PBSD on slope engineering come out. The performancebased seismic design and evaluation for slope engineering is developed as an important part of current seismic design of slope engineering. To date, remarkable research achievements have been made in the aspect of seismic performance design and evaluation of slope engineering in the world, but there is a lack of a guide specifically for PBSD and evaluation of slope engineering, which brings great difficulties to the front-line geotechnical engineers in the practice of PBSD of slope engineering. Professor Yu Huang, the leading author of this book, and his research team have worked hard in the field of seismic performance design of slope engineering for a decade. Professor Huang’s research results systematically reveal the stochastic propagation law of seismic dynamic excitation randomness and spatial variability of geotechnical parameters in slope nonlinear dynamic system. The first performancebased seismic design framework for slope engineering is developed in the world, which solves the long-standing problem of nonlinear seismic dynamic reliability calculation of slope, which has puzzled the international researchers for decades. These effective and exciting innovative concepts and methods are a major breakthrough in the field of reliability analysis and design of complex slope engineering systems. These researches represent the most important progress in the field of seismic performance design of slope engineering. I believe that the research achievements summarized in this book can form a milestone in the design and evaluation of seismic performance of slope engineering. With the support of the International Consortium on Geo-disaster Reduction (ICGdR), Professor Huang proposed and established the Committee on Seismic Performance-based Design for Resilient and Sustainable Slope Engineering (TC1) and has served as the founding chairman since 2018. This book is also the crystallization of the successful cooperation between Professor Huang’s research team and TC1 of ICGdR. I think this book can be used as an important reference standard

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Foreword I

for seismic design and assessment of slope engineering. It is believed that the publication of this book will fill the gap in the guide for PBSD of slopes, and promote the development of research on seismic performance design and evaluation of slope engineering.

Shanghai, China October 2022

Prof. Fawu Wang Foreign Fellow, The Engineering Academy of Japan President of International Consortium on Geo-disaster Reduction Chairholder of UNESCO Chair on Geo-environmental Disaster Reduction

Foreword II

The seismic design and evaluation of slope engineering has experienced the evolution of “three generations” to this day. The first generation of slope seismic design theory is based on the deterministic quasi-static method to determine the seismic stability limit state of slope engineering, that is, the safety factor method. On this basis, the second generation of slope engineering seismic design theory is a reliability method that partially considers the spatial variability of geotechnical parameters and the randomness of ground motions. In recent years, with the development of dynamic time-history analysis methods of slope seismic response considering the strong nonlinear mechanical behavior characteristics of rock and soil under seismic dynamic action, and the development of nonlinear stochastic dynamics theory, the seismic design method of slope engineering gradually transits to Performance-Based Seismic Design (PBSD) theory. For slope engineering, according to PBSD theory, the nonlinear dynamic time-history response, dynamic stability state, and permanent deformation of slope engineering under seismic dynamic action are evaluated. The real or practical seismic dynamic performance state of the slope is obtained, so that the geotechnical engineer can be liberated from the narrow design of determining the seismic dynamic safety factor of the slope engineering within the specified range. At the same time, no matter what kind of slope engineering, such as embankment, anchored slope or dam, and other generalized slope engineering, seismic dynamic safety design can be carried out according to the required performance objectives. In addition, if the geotechnical engineers who design or evaluate the seismic dynamic safety of slope engineering or slope retaining structure are different, the seismic dynamic safety design scheme of slope engineering will be different under different seismic excitation due to the intensity frequency non-stationary stochastic characteristics and the nonlinear dynamic deformation behavior of slope under seismic dynamic action. The previous anti-seismic design of slope engineering can only be characterized by the calculation of safety factor. In performance-based anti-seismic design of slope engineering, the performance goal has transcended the obstacle of simple mechanical calculation, and can also gradually develop from the anti-seismic performance state design to the most popular resilience-based anti-seismic design theory. In the aspect of seismic design of slope engineering, vii

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Foreword II

Professor Huang has taken the lead in establishing the disaster random response analysis method for large and complex slope projects in the world, which is the first framework for evaluating the seismic dynamic performance of slopes in the world. These fruitful and exciting innovative concepts and methods have solved the historical problems related to slope dynamic reliability and seismic performance design that have puzzled the international academic community for decades. By revealing the law of probability density evolution (propagation of randomness) of the nonlinear seismic dynamic response of the slope from the source, Professor Huang’s research systematically established the probability density function evolution analysis theory of the complex nonlinear dynamic system response of the slope, created the global seismic dynamic reliability design technology of the slope engineering, and formed a major breakthrough in the field of reliability analysis and design of complex slope engineering. This book is the first international monograph in the field of slope engineering that systematically introduces the performancebased seismic design and evaluation theory and specific framework process of slope engineering in detail. These studies represent the most significant advances in the seismic performance design of slopes. This work can be recommended as a reference manual for relevant engineering design. It is believed that the publication of this book will play a very positive role in improving the overall level of performance-based seismic design of slope engineering. At the same time, we hope that researchers, geotechnical engineers, and students can join in the research on performance-based seismic design of slope engineering, and make new contributions to further improve the seismic dynamic safety performance of slope engineering.

Aachen, Germany October 2022

Dr. Rafig Azzam Professor of RWTH Aachen University, Germany President of International Association for Engineering Geology and the Environment

Preface

Performance-based seismic design (PBSD) has been increasingly used in seismic dynamic design and superstructure assessment in recent years. However, the PBSD concept and method have not been widely applied in geotechnical problems, largely because a global framework and guideline for geotechnical engineering practice have not been fully established. Seismic design and the evaluation of slope engineering play a vital role in performance design. The geotechnical materials that constitute geotechnical mass (e.g., embankments, slopes, dams) are mainly natural products and exhibit notorious complexity. Compared with a superstructure, geotechnical engineering is generally a three-dimensional continuum with a wide spatial variability, and its stress–strain relationship has strong nonlinearity and dilatancy effects. Traditional design concepts and specifications do not prioritize performance as the design goal, which poses serious challenges to geotechnical engineers or relevant researchers for the design of geotechnical engineering projects such as slopes. More importantly, seismicity has significantly increased in recent years and the world may have entered an active seismic period, thus involving the frequent occurrence of strong earthquakes and increasingly high ground motion intensity. If the conventional limit design method is adopted, such large motion will often lead to intolerable results of slope engineering. It is therefore urgently needed to reconsider how to correctly design new buildings and new civil engineering structures, and how to transform existing structures from the perspective of their performance under increasing seismic loads. There is thus a critical need to develop and establish a practical and reliable PBSD framework and guide for slope engineering, which has become the focus of academic and industrial communities. The core difficulty of this problem is to establish the seismic performance fortification level of slope engineering, determine the seismic excitation and randomness of slope engineering sites under different performance levels, and evaluate the seismic dynamic stability of slope engineering according to seismic dynamic response analysis and different slope engineering performance criteria. Our joint research team has spent more than 10 years addressing the above-mentioned problems on the basis of free theory exploration combined with practical engineering experience. This work introduces a new theoretical tool of stochastic dynamics and aims to explore a unified technical route ix

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Preface

of earthquake dynamic performance design and assessment for generalized slope engineering. The research team has made a series of achievements including new ground motion modeling, slope seismic dynamic response analysis, a seismic performance index of slope engineering, a probabilistic performance evaluation method of slope engineering, preliminarily established a global framework of seismic probabilistic performance design and an assessment for generalized slope engineering based on the nonlinear stochastic dynamics approach, and carried out some research on practical applications. On the basis of summarizing the above research results, this book establishes a performance-based earthquake design and assessment guide for generalized slope from the perspective of nonlinear earthquake dynamic reliability. The global framework theory of this book is divided into several parts: the overall framework of PBSD and evaluation of slope engineering; the selection of strong earthquake records and synthetic ground motion models; time-domain dynamic nonlinear time-history analysis of slope earthquake dynamic response; and a probabilistic earthquake dynamic performance assessment of generalized slopes and specific engineering practice. This book includes eight chapters. Chapter 1 gives a general introduction to the performance-based earthquake design concept and procedure and the primary subjects of this book. Chapter 2 presents some of the terms and symbols used in this book. Chapter 3 outlines the basic framework analysis of PBSD in light of specific existing concerns, and introduces the basic performance design method including the main procedures and advantages of a global PBSD framework for the seismic dynamic performance assessment of slope engineering. Chapter 4 summarizes the methods that can be applied for random source input in the process of slope engineering probabilistic performance design. Chapters 5 and 6 present the numerical simulation framework of probabilistic PBSD in practical slope engineering. Chapter 7 highlights the application of probabilistic PBSD on dams and slopes based on the related design criteria mentioned above. The main conclusions and future development are discussed in Chap. 8. It is my hope that the global framework for probabilistic earthquake dynamic performance design and slope engineering assessment offers a useful reference for slope seismic assessment. This information can also benefit students, geotechnical engineers, and scientific researchers within fields such as earthquake engineering, geotechnical engineering, and structural dynamics. Several graduate students have conducted extensive work collating and editing manuscripts, to whom I express my sincere gratitude, especially Ph.D. student Cuizhu Zhao and masters students Wenwen Wang, Ying Luo, Mi Zhou, Zhiming Peng, and other group members who contributed to this comprehensive and innovative work. The research invested in the development of this book has received support from the Key Program of National Natural Science Foundation of China (Grant No. 41831291), the National Science Fund for Distinguished Young Scholars (Grant No. 41625011), the National Natural Science Foundation for Young Scientists of China (Grant No. 41902274), the Science and Technology Commission of Shanghai Municipality (Grant No. 2021DZ2207200), the National Key Research and Development Program of China (Grant No. 2017YFC1501304), the Sino-German

Preface

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Mobility Programme (NSFC/DFG) (Grant No. M-0129), and the fellowship of China Postdoctoral Science Foundation (Grant No. 2021M702791). In closing, I sincerely hope that the publication of this book will become a milestone for geotechnical engineers and researchers in the practical use of earthquake dynamic safety design and assessment of generalized slope engineering and the further development of the performance-based design method. The contents herein also provide a theoretical reference for the research and specific practice of scholars and geotechnical engineers in related fields such as earthquake dynamic safety assessment and generalized slope engineering design. The seismic performance design of slope is a large and developed subject, and it is also open and developing. I hope that readers can obtain not only knowledge regarding the seismic performance of slope engineering through this book, but also an essential understanding of the performance-based seismic design of slope, and then join the research efforts on the PBSD design and evaluation of generalized slope to collectively promote its development. This book was completed under the cooperation of the Committee on Seismic Performance-based Design for Resilient and Sustainable Slope Engineering (http:// www.icgdr.com/Home/Menu/303). I thank the Committee for its strong support and guidance.

Shanghai, China June 2022

Dr. Yu Huang

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Scope of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Key Areas Covered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Book Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 4 5 8

2 Terms and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

3 Performance-Based Seismic Design Framework of Slope . . . . . . . . . . . 3.1 The PBSD Principle and Applications . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Evolution of PBSD for Slope Engineering . . . . . . . . . . . . . . . 3.2 Global Framework of PBSD for Slope Engineering . . . . . . . . . . . . . . 3.2.1 Design Ground Motion of Slope Engineering Site . . . . . . . . 3.2.2 Seismic Performance Level and Criteria of Slope Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 13 16 19 25

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic Performance Design and Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Determination of Seismic Ground Motion of Slope Site Based on Seismic Hazard Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Determination of Seismic Excitation of Slope Site According to Historical Strong Earthquake Records . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Correction of Historical Strong Seismic Record of Slope Site Based on IDA Theory . . . . . . . . . . . . . . . . . . . . . 4.2.2 Modification of Seismic Strong Records Based on the Code’s Design Response Spectrum . . . . . . . . . . . . . . . 4.3 Determination of Ground Motion of Slope Site According to the Strong Earthquake Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 International Specification for Ground Motion Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 41 43 43 47 47 49 50 52

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Contents

4.3.2 Selection of Strong Earthquake Ground Motion . . . . . . . . . . 4.4 Artificial Seismic Ground Motion Synthesis of Slope Site . . . . . . . . 4.4.1 Source-Based Ground Motion Model . . . . . . . . . . . . . . . . . . . 4.4.2 Site-Based Ground Motion Model . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Deterministic Analysis Methods for Slope Seismic Dynamic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Seismic Response Analysis of Slope Based on the Quasi-static Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Slope Dynamic Response Analysis Method Based on the Response Spectrum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Newmark Sliding Block Displacement Method for Seismic Dynamic Response Analysis of Slope . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Slope Seismic Dynamic Response Analysis Based on Nonlinear Dynamic Time-History Analysis . . . . . . . . . . . . . . . . . . 5.5 Large Deformation Analysis Method of Slope . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54 59 60 62 64 67 69 69 73 75 76 78 81

6 Probabilistic Performance-Based Seismic Design and Assessment for Slope Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.1 Source of Uncertainty and Its Description . . . . . . . . . . . . . . . . . . . . . . 84 6.1.1 Sources of Uncertainty in a Seismic Dynamic System of Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.1.2 Establishment of a Random Field of Slope Geomaterials . . . 85 6.2 Stochastic Seismic Dynamic Response Analysis Method of Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2.1 Pseudo-Excitation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2.2 Monte Carlo Stochastic Simulation . . . . . . . . . . . . . . . . . . . . . 91 6.2.3 Stochastic Finite Element Method . . . . . . . . . . . . . . . . . . . . . . 92 6.2.4 Nonlinear Stochastic Seismic Dynamic Method . . . . . . . . . . 93 6.3 Seismic Dynamic Risk and Vulnerability Assessment of Slope . . . . 95 6.3.1 Seismic Dynamic Vulnerability Assessment of Slope . . . . . . 96 6.3.2 Seismic Dynamic Risk Assessment of Slope . . . . . . . . . . . . . 98 6.4 Resilience-Based Seismic Performance Design . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Probabilistic Seismic Performance of Earth Dams . . . . . . . . . . . . . . . 7.1.1 Seismic Performance Evaluation of Earth Dams . . . . . . . . . . 7.1.2 Seismic Probabilistic Risk Assessment of Earth Dams . . . . . 7.2 Probabilistic Seismic Performance of Slope . . . . . . . . . . . . . . . . . . . . 7.2.1 Slope Probabilistic Seismic Performance Evaluation . . . . . . 7.2.2 Seismic Risk Assessment of Slope Retaining Systems . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 108 117 121 121 125 128

Contents

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8 Conclusions and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.2 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

About the Authors

Prof. Yu Huang born in 1973, received his Ph.D. in geotechnical engineering from Tongji University, Shanghai, China. He is now a “Distinguished Professor of Changjiang Scholars of the Ministry of Education” in geological engineering in the College of Civil Engineering at Tongji University. Professor Huang’s primary research area includes earthquake geotechnical engineering, geologic disasters, computational geomechanics, foundation engineering, and environmental geology. He has authored more than 150 papers in international refereed journals. As first author, he has written four monographs entitled “Slope Stochastic Dynamics”, “Geo-disaster Modeling and Analysis: An SPH-based Approach”, “Hazard Analysis of Seismic Soil Liquefaction”, and “Social Infrastructure Maintenance Notebook” published by Springer. He now serves on the editorial board for Engineering Geology, Bulletin of Engineering Geology and the Environment, Geotechnical Research, and Geo-environmental Disasters. Professor Huang received the National Science Fund for Distinguished Young Scholars from the National Natural Science Foundation of China for his research on geological disasters triggered by earthquakes. Min Xiong born in 1986, received his Ph.D. from Tongji University under the supervision of Prof. Yu Huang. He received his bachelor’s and master’s degrees in civil engineering at Three Gorges University. He is currently working at Tongji University as a post-doctoral research fellow. Hongqiang Hu born in 1993, received his Ph.D. from Tongji University under the guidance of Prof. Yu Huang. He received his bachelor’s degree in geological engineering at Chang’an University. He is currently working at Zhejiang University and the Architectural Design & Research Institute of Zhejiang University Co., Ltd. as a post-doctoral research fellow.

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Chapter 1

Introduction

High-intensity earthquakes induce tremendous disasters, such as slope landslides, and cause huge economic losses and casualties. Landslides frequently interrupt the life lines of communication, which poses further difficulties for rescue work (Dunning et al. 2007; Keefer 2000; Sato et al. 2007; Yin et al. 2009). Such catastrophes have led to a reassessment of the current seismic design of practical slope engineering. Although the existing seismic dynamic design and evaluation of slope engineering have partially satisfied the purpose of maintaining the stability of slope subjected to seismic excitation in high-intensity seismic regions, it must be mentioned that the overall damage, instability, and failure of retaining structures and slopes have not been effectively and widely used or controlled. There are numerous situations in which slopes with aseismic design lose stability and slide under lower earthquake intensities than the seismic fortification intensity, which have accordingly caused high casualties and economic losses to countries worldwide. To minimize casualties and economic losses brought by slope disasters, it is indispensable to take reasonable and effective measures to reduce the deformation and damage of slope or structural engineering while preventing instability and failure, and to ensure that slope engineering can meet certain functional requirements and minimize damage under earthquake excitations. Performance-based seismic design offers an effective new concept and theoretical framework to ensure the seismic dynamic safety of slope engineering and resolve these problems, and has thus become a hot research topic in geotechnical seismic engineering as a relatively advanced seismic design method and important subject in slope engineering. Performance-based seismic design is based on the analysis of the seismic dynamic performance safety evaluation for geotechnical engineering such as slopes, embankments, and dams. For such geotechnical engineering, the seismic dynamic performance goals of generalized slope are completely different depending on the seismic fortification level. Geotechnical engineers select reasonable seismic performance targets in accordance with the economic and social development level for slope engineering design following the existing national and local safety standards and owner requirements. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Huang et al., Guidelines for Probabilistic Performance-Based Seismic Design and Assessment of Slope Engineering, https://doi.org/10.1007/978-981-19-9183-7_1

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The PBSD seismic theoretical framework for slope engineering should also mainly include the following three aspects: (1) Definition of the reference earthquake risks and corresponding seismic design levels of slope sites; (2) Definition of the seismic dynamic safety performance level of slope and its quantitative indicators, the seismic dynamic factor of safety or slope displacement, and the consequences of slope instability or failure/landslide owing to a certain earthquake excitation level; (3) The target performance of slope seismic design, which involves the level of seismic performance suitable for a certain seismic design standard or code. The establishment of the target seismic dynamic design performance must address various influencing factors, to include the construction site characteristics, slope function and importance, investment and benefits, and social and economic benefits. For example, the goal can be set to not be damaged or destabilized for slopes subjected to low-intensity earthquakes, to be reinforced under moderate earthquakes, and to not fail under high-intensity earthquakes according to the premise of meeting the above design standards. The target seismic dynamic performance of generalized slope engineering would then be to incur only minor damage or no damage after a small earthquake, and its normal in-service performance would remain unaffected. The slope engineering may be damaged or local instability may occur after a moderate earthquake, but its normal function would be unaffected after reinforcement. The slope engineering may destabilize after the dynamic impact of a high-intensity earthquake, but there would be no overall failure or landslide. Although slope engineering is taken as an example to introduce the aforementioned global framework and procedure of PBSD for seismic design and slope engineering evaluation, it is also applicable to other generalized slope works, such as dams and embankments. Performance-based seismic design is also applicable to supporting structures in generalized slope engineering, largely because the qualitative indicators are maneuverable and applicable for different slope works in geotechnical engineering under the global PBSD framework. It should also be emphasized here that seismic dynamic safety design and assessment require the formulation of different feasible fortification standards and the selection of performance criteria regarding the deterministic nonlinear seismic dynamic response based on the performance characteristics of different generalized slope projects. This is clearly the largest feature and advantage of the PBSD concept and method, which differs from traditional seismic design methods for slopes. It is important to note that although seismic dynamic safety design for slope engineering has achieved substantial development in a range of research fields and some specific engineering practice, it still lacks certain considerations in terms of uncertainty and randomness in the seismic design and assessment of slope engineering, which are inevitable when evaluating seismic reliability or performance. There are a host of uncertainty sources in the performance evaluation of slope stability assessment, such as the spatial variability of rock and soil parameters, and the randomness of seismic excitations (Hayashi and Ang 1992; Michael et al. 2016). This randomness and uncertainty seriously affect the accuracy of the seismic dynamic safety performance evaluation of slope engineering. In particular, it is impossible to accurately predict the possible future ground motions of a slope site. The nonlinear seismic

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dynamic time-history response of slope is therefore seriously affected by the coupling effects between the dynamic nonlinear characteristics of geomaterial (e.g., rock, soil) and randomness in seismic excitation. Certain challenges therefore remain owing to the aforementioned source of uncertainty in seismic dynamic performance design and risk assessment in slope engineering. This topic is also the research focus and characteristic innovation of this book.

1.1 Scope of the Book This book constructs a new design and evaluation framework based on slope stochastic dynamics theory for the probabilistic seismic performance of slope engineering. For the seismic dynamic stability and safety of slopes, it shifts from deterministic seismic dynamic analysis to quantitative analysis based on nonlinear stochastic dynamics, that is, from qualitative information to the description of stochasticity of earthquake excitation that meets the needs of related design specifications and establishes a performance standard. Stochastic dynamics theory is considered to be a new element to complete the performance design process. This process is presented in different models, such as the proposed ground motion power spectrum model and seismic excitation model. This book uses the design viewpoint based on the above random model as the basis for describing the seismic performance design process of slope engineering. This book also mainly focuses on the establishment of probabilistic performancebased seismic dynamic stability and safety evaluation procedures and design framework for slope engineering. This includes the application of a probabilistic PBSD framework in seismic dynamic safety design and the evaluation of generalized slope engineering such as slopes, embankments, and dams. Although the existing slope engineering performance design attempts to describe the key points involved in this book, a complete and uniform standard or framework system has not yet been formed. These existing schemes mainly focus on the evaluation criteria and risk assessment required for the deterministic slope engineering seismic design, without considering the randomness involved in the process in terms of the performance design scheme. In the nonlinear dynamic time-history analysis of slope subjected to seismic ground motion, the term “randomness” is used to express the uncertainty in the intensity and frequency of the earthquake excitation for slope engineering dynamic seismic performance. The spatial variability of geomaterial (e.g., rock, soil), which is mainly considered in the static stability analysis of slope, is also particularly significant for the safety evaluation of seismic dynamic stability of slope engineering. A widely accepted view is that the randomness of ground motion brings tremendous challenges for accurately predicting the possible destructive earthquake excitation of slope engineering in the future. For slope seismic dynamic stability performance assessment, earthquake ground motion is generally assumed to be a stochastic process. The rock and soil materials constituting the slope body are also highly nonlinear under the action of earthquake dynamics and highly sensitive to earthquake ground motion.

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This implies that the nonlinear seismic dynamic response of slope is highly sensitive to different ground motion conditions, which is referred to as chaos in the nonlinear field. These dynamic nonlinearities and dilatancy effects, spatial variability, and randomness are mutually coupled. Dynamic nonlinearity will therefore lead to randomness of the seismic time-history response of a slope, such as the displacement or safety factor, and randomness will inevitably aggravate nonlinearity. This book thus emphatically emphasizes the study of slope nonlinear dynamic stability performance based on complete probability information grounded in nonlinear stochastic dynamics theory. This mainly includes seismic design fortification standards, corresponding ground motion excitation, a performance index threshold, and slope deterministic nonlinear seismic dynamic response. Furthermore, seismic dynamic large deformation approaches address the entire process and comprehensive analysis for flow analysis after slope instability failure. The probabilistic seismic dynamic performance of slope engineering will eventually be characterized by nonlinear dynamic reliability.

1.2 Key Areas Covered This book provides a framework for slope engineering performance evaluation and analysis, and supports performance improvements in slope engineering seismic performance design. In particular, this book does the following: . Proposes fortification standards for slope seismic performance related to performance analysis. . Considers the randomness of seismic excitation in slope performance design to reasonably and effectively express the stochasticity of seismic ground motion and provide a category of external environmental excitation that meets the seismological and geological conditions of the slope site for slope probabilistic seismic performance analysis. . Provides a deterministic time-domain nonlinear dynamic analysis method for evaluating the performance of a nonlinear seismic dynamic system of slope engineering. . Uses stochastic performance modeling and nonlinear seismic dynamic reliability analysis requirements as an important means to improve slope performance evaluation methods.

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1.3 Book Outline Chapter 2 presents the terms and notation used in this book and after that, the remainder is mainly composed of the following three parts: Part 1: Theory Framework of Slope Probabilistic Design Performance (Chapters 3–6) . Chapter 3 outlines the basic framework analysis of slope seismic performance design under the existing specific concerns, and introduces the basic slope seismic dynamic performance design and assessment framework. The main differences between traditional deterministic seismic dynamic design and slope probabilistic performance design are analyzed and compared. The development of this theoretical framework provides valuable input for understanding and developing the proposed slope probabilistic performance design method. This approach mainly concerns the seismic design level and external earthquake excitation of the slope engineering site, deterministic nonlinear seismic dynamic response, dynamic reliability, performance criteria, and seismic dynamic vulnerability in the probabilistic seismic dynamic performance of generalized slope engineering. . Chapter 4 introduces how to choose earthquake ground motion input for slope engineering, including the seismic design standard according to the site category for slope engineering in different regions as a reference including China, Europe, the United States, Japan, and local seismic stations directly selected from the monitoring records. For situations without local earthquake records, sourcebased methods can be divided into three approaches: the deterministic method; stochastic method; and mixed method. . Chapter 5 proposes the seismic design methods in slope performance design, which include the quasi-static method, response spectrum method, time-domain dynamic nonlinear time-history analysis, and large deformation flow analysis method for slope instability failure. These deterministic response analysis methods are the core of seismic performance safety evaluation of slope engineering. This chapter therefore details the advantages and disadvantages of the above slope seismic response analysis methods. . Chapter 6 focuses on the main flow of nonlinear stochastic seismic dynamic response analysis in the entire generalized slope engineering process and varieties of uncertainty sources that affect the nonlinear seismic dynamic response of slope engineering. This includes the random field description and representative sample discretization of the spatial variability of geotechnical parameters. The probability density function of nonlinear seismic dynamic time-history response is achieved by slope stochastic dynamics. This is the core and basis of the probabilistic seismic dynamic performance for slope engineering based on seismic dynamic reliability criteria and time-domain nonlinear dynamic analysis.

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Part 2: Applications of Slope Probabilistic Design Performance (Chapter 7) . Chapter 7 highlights case studies of the performance design framework proposed above in slope engineering, combined with different performance design standards and a theoretical stochastic dynamics framework. Seismic dynamic risk assessments of slope and dam engineering are carried out as important references of a new and comprehensive perspective on seismic dynamic probabilistic performance design and the evaluation of generalized slope engineering. Part 3: Conclusions and Prospect of Slope Probabilistic Design Performance (Chapter 8) . Chapter 8 summarizes the important discoveries and achievements in this book and discusses potential development directions of the slope probabilistic performance evaluation framework in future engineering design and evaluation-probabilistic performance evaluation based on resilience and optimization design-and puts forward a future outlook. The logical relationship and procedure diagram among the main categories of this book are shown in Fig. 1.1.

1.3 Book Outline

Fig. 1.1 Structure of the book

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References Dunning SA, Mitchell WA, Rosser NJ et al (2007) The Hattian Bala rock avalanche and associated landslides triggered by the Kashmir Earthquake of 8 October 2005. Eng Geology 93(3–4):130– 144 Hayashi H, Ang AH-S (1992) Reliability of cut slopes under seismic loading. Soils Found 32(2):45– 56 Keefer DK (2000) Statistical analysis of an earthquake-induced landslide distribution-the 1989 Loma Prieta, California event. Eng Geol 58(3–4):231–249 Michael M, Al-Bittar T, Soubra A-H (2016) Effect of soil spatial variability on the dynamic behavior of a slope. In: Paper presented at the VII European congress on computational methods in applied sciences and engineering, Crete Island, Greece, 5–10 June 2016 Sato HP, Hasegawa H, Fujiwara S (2007) Interpretation of landslide distribution triggered by the 2005 Northern Pakistan earthquake using SPOT 5 imagery. Landslides 4(2):113–122 Yin Y, Wang F, Sun P (2009) Landslide hazards triggered by the 2008 Wenchuan earthquake, Sichuan, China. Landslides 6(2):139–152

Chapter 2

Terms and Notation

In various places in this guide, some more complicated words are used that may not be easy to understand for nonprofessionals. The terms in this book cover a wide range and are frequently applied. This chapter lists the definitions of the commonly used terms and explains the commonly used symbols in this guide to help nonprofessionals better understand the technical terms in other parts of this book or in the future, to facilitate possible interdisciplinary exchange in the future, and to realize the application potential of this book. However, it must be mentioned that some technical terms are presently limited to this book. The applicability of other books or texts remains to be studied. To help readers better understand and apply this seismic dynamic performance design and assessment guide for generalized slope engineering, and assist readers to quickly and more completely establish the relationship between this guide and various seismic design codes, this chapter introduces some essential terms that may be involved in the seismic dynamic safety assessment of slope engineering. Among them, there are some technical terms and special nomenclature in this book for the probabilistic seismic dynamic performance of slope engineering. . Performance-based seismic design (PBSD) The PBSD concept is the general name of a kind of method to realize the seismic dynamic safety assessment as much as possible on the basis of clarifying the performance goal. . Seismic performance fortification level The seismic performance fortification level refers to the seismic dynamic safety capacity requirements of the engineering structure determined by the owner and design engineer. . Performance index Seismic performance index refers to the physical quantity that describes the dynamic safety state of engineering structures subjected to earthquake excitation. For building © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Huang et al., Guidelines for Probabilistic Performance-Based Seismic Design and Assessment of Slope Engineering, https://doi.org/10.1007/978-981-19-9183-7_2

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structures, this can be inter-story drift, dynamic displacement, and acceleration. For slope engineering, this can be selected as, for example, the dynamic safety factor, or permanent horizontal and vertical deformation. . Peak ground acceleration (PGA) The PGA is the maximum absolute value of acceleration of ground motion during an earthquake. . Power spectrum density (PSD) The PSD is a method of probability statistics and also a measure of the mean square value of a stochastic process or random variable. It can generally be applied for stochastic dynamic analysis, and is defined as the “power” per unit frequency band (mean square value). . Seismic response spectrum The maximum seismic absolute response of a group of single-degree-of-freedom systems with the same damping and different natural vibration periods under the action of a certain earthquake ground motion time history. . Incremental dynamic analysis (IDA) A kind of nonlinear time-domain dynamic time-history analysis method of engineering structures subjected to seismic excitation by adjusting the peak ground acceleration. . Elastic response spectrum The relationship curve between the spectral acceleration and period. . Elastic demand spectrum The relationship curve between the spectral acceleration and spectral displacement. . Reliability The ability of slope engineering to complete the specified functions under the specified working conditions and within the specified service time. The reliability is essentially the probability measure of the dynamic stability of slope engineering under earthquake action during a service period. . Failure probability According to a certain recurrence period, the failure probability numerically corresponds to the reliability. Generally, it refers to the probability that the structure or component cannot complete the predetermined function. In this book, the failure probability refers to the probability of generalized instability and failure of slope engineering subjected to possible earthquake ground motion during the service life.

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. Resilience Resilience refers to the ability of the system (eg. slope, earth dam) to resist sudden external-environmental disturbances and quickly recover. In the field of disaster prevention and control, when a disaster occurs, the engineering structure can withstand the impact, respond, quickly recover, maintain the normal operation of its important functions, and better deal with future disaster risks through adaptation. In slope engineering, this means that when a disaster occurs, the slope engineering can withstand the impact, respond quickly, recover, maintain normal functioning, and better respond to future disaster risks through adaptation.

Chapter 3

Performance-Based Seismic Design Framework of Slope

The seismic performance design of slope engineering is the general design method aimed to achieve a given performance goal on the basis of clarifying the seismic performance goal. This may also be called performance-based, performance evaluation or performance design. It should be noted that the performance-based seismic dynamic safety of slope is only a goal, while the methodology for analysis and evaluation should be left to geotechnical engineers. In this way, a clear framework is required for the performance-based seismic design (PBSD) of slope engineering. This chapter mainly introduces the principles and process of slope seismic dynamic safety design based on the concept derived from structural seismic design. This principle instructs an effective methodology to deal with the challenge of ground motion amplification observations in a slope body, while the more direct purpose is to establish a global PBSD framework and guideline for slope engineering. This chapter therefore introduces the global PBSD framework of slope engineering, which mainly includes the following parts: (1) determination of the seismic excitation of a slope site according to different seismic performance fortification levels; (2) determination of the seismic dynamic response analysis method of slope; and (3) in combination with the randomness on the ground motion and variability of geotechnical materials, the seismic probabilistic performance of slope engineering is thus considered through stochastic dynamic reliability.

3.1 The PBSD Principle and Applications 3.1.1 Background An earthquake is a natural disaster triggered by the planet’s internal forces and stimulation of the lithosphere, and is manifested as the generation of sudden and stochastic excitations at the spot of the near-fault source. This excitation transmits its momentum and energy from depth toward the surface through rock mass using a © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Huang et al., Guidelines for Probabilistic Performance-Based Seismic Design and Assessment of Slope Engineering, https://doi.org/10.1007/978-981-19-9183-7_3

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form of acceleration-based or displacement-based waveform description. Although only a small number of earthquakes are perceived by people, destructive earthquakes have the potential to cause catastrophic damage to most engineering facilities and high casualties. Previous investigations on earthquake forecasting and the propagation of seismic waves in terms of the common precursors have not been particularly successful. The study on micro-seismic signals before and after strong earthquakes is an innovative and positive endeavor in the fields of seismology and earth science, but also brings challenges and calls for measurement developments because these signals are often slight or even invisible (i.e. lower than the instrument resolution). Although strong earthquakes are not common, their number is increasing and the development of measurement practice and theory has made long-term progress with the efforts of scientists and engineers. Earthquakes have stochastic spatiotemporal characteristics and can have tremendous intensities, causing high casualties of humans and social loss. With the purpose of improving the seismic design principles for professionals and popularization, this guideline provides a seismic design procedure that members involved in the process should develop, outline, and construct in agreement to mitigate these risks and reduce damage caused by catastrophic earthquakes. Traditional engineering philosophy considered the purpose of seismic design was to control the limit dynamic response, such as the peak acceleration and/or displacement of a construction, under a limited seismic load as empirically processed under the equivalent or quasi-static hypothesis. However, maximum quasi-static or periodic loading, simplified by regulations, is more than sufficient compared with the actual recordings, under which the overall structure remains operative under loads several times larger than the standard quasi-static value. Failure assessment in practical geotechnical slope engineering should also be restricted using parameters that are highly related with the experimental characteristics, such as the constitutive relations and physical model tests. Furthermore, none of the indicators should surpass their critical value of the corresponding failure state reported in the laboratory, and a factor greater than 1.0 should be applied to consider material inhomogeneity while using calibrations according to empirical explanations. Reinforced concrete materials are used to ensure the ductility of components, but the ductility extent of a structure is often difficult to measure under a long-term service life of the component, instead of considering the coupling and interaction analysis with other components. Under the circumstances of an insignificant conventional, quasi-static, and periodic load, the bearing capacity and deformation under different states should be calculated by considering the ductility as a coupling effect of each component because these loadings are deterministic. However, the ductility of components is generally considered and investigated independently in the stochastic seismic process, while the displacement of the important parts of a building and the ductility of components are discussed in terms of the performance indexes in different sets of systems. The PBSD principle has been put forward and adopted in building structures, including reinforced concrete frame structures, important railway bridges, super-high dams, and underground systems. In the late 1990s, some authoritative institutions

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reported conclusions on the performance-based design principles such as the Structural Engineers Association of California (SEAOC), the Applied Technology Council (ATC) and the Federal Emergency Management Agency (FEMA), describing PBSD as follows: . PBSD refers to a kind of deterministic design standard with an appropriate structural form, reasonable planning, and structural proportion. It considers the detailed structural design for structures and non-structures, and controls the quality during construction and over long-term maintenance operations so that any structural breakage can maintain the critical state (Vision 1995). . PBSD refers to structural design and a standard that is represented by a series of achievable structural performance objectives, and should be used in all kinds of structural designs (ATC 1996). . PBSD provides a new principle in which a series of performance purposes should be put forward that are subject to different use depending on hazard intensity on behalf of the joint benefit of the developers, owners, and engineers, and for the convenience of emergency planning for governments (FEMA 1996). As mentioned, strong earthquakes can sometimes cause high property losses, especially in population-saturated metropolises, even if most are not within the range of the high-risk area. The earliest objectives of seismic design and PBSD are used in high-rise buildings, followed by harbor facilities and geotechnical engineering projects such as slope engineering. The geotechnical materials, geometric topology, seismic geological conditions, and individual functional performance characteristics of each specific slope project account for the contradiction between the individuals’ privilege for life security and the inevitable loss owing to the impact of an earthquake or other force majeure. This is the same situation for governments and parties involved in the economy and physical environment when communal facilities suffer, such as slope engineering or harbor facilities. The prioritized people and their government account for the right to life and health for individuals, as well as the right to own and distribute property. Slopes are damaged in earthquakes, resulting in damage or loss of their “should have” performance. Although the current purpose of seismic fortifications is to ensure personnel safety and prevent structures from collapse under major earthquake disasters, slope instability failure has not been effectively controlled, resulting in huge social and economic losses. This shows that the seismic design of slope should not only prevent landslide disasters to achieve the goal of economic safety, but should also effectively ensure the seismic dynamic stability safety of slope engineering to meet the functional purposes and acceptable dynamic stability state of subsequent construction. Performance-based seismic design is the standardization of the empirical principle of multi-critical state seismic fortification from an engineering perspective. The core of its skeleton is that from the beginning of construction to the end of operation during its service life, there should be a hierarchy of performance critical states under seismic intensities to effectively minimize the total initial expenses and costs for future maintenance. New initiatives have been taken. Some follow practical routes to increase the specific technical details by adopting civil engineering over time. Most

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of the issues to be confronted in this process can be resolved by means of equipment progress made by professionals. Others are balanced with dialectical characteristics that must be achieved in slope engineering practice; a reflection of the prospect that governments and individuals take their social responsibility. . Multi-critical states with accordant rigorous fortification. Consideration of the hazard intensity variables and performance-based critical state hierarchy helps to ensure seismic dynamic stability safety even if it is in an inferior stage, while the seismic performance objectives in accordance with the performance design theorem include safety and direct instability failure losses of slope engineering. . Comprehensive quantification on economic and social benefits. Expectations matter. An adoption of the investment benefit principle should be made to determine the seismic fortification target in which every penny worth of quantifiable services and performance are achieved in slope engineering. The investment benefit principle means that in addition to technical factors, economic and social factors are also considered in the process. . Integration of standardization and personalization. In geological engineering, the seismic performance of slope engineering does not necessarily include every target specified in the regulations with the precondition that the area of the slope site is a hazardous spot with no expected property losses, but this should be determined on the basis of the demand and purpose of the owner. The procedures of design, construction, and measurement should be well-formulated by standardization for convenience and future checks for the owner if the necessity is evident, as they are custom-derived from the deterministic performance targets. . Technical efficiency and design flexibility. The geotechnical engineering seismic design codes subject to these performances still regulate the minimum allowable values for significant parameters such as nonlinear dynamic deformation and factor of safety. This gives the engineers greater flexibility and the initiative to nominate and explain the performance-based parameters and designed objectives that meet the seismic performance purposes required by the owner. The corresponding reinforcement measures for slope instability are conducive to the application of new materials and new structure-retaining forms.

3.1.2 Evolution of PBSD for Slope Engineering The principle of PBSD has been developing in structural design over the past two decades; however, few researchers have transferred the practical application of PBSD to slope engineering. This therefore requires the best efforts to introduce the framework from structural engineering toward practical use of slope engineering combined with the project investigation and expert experience. This process dates back that the earliest research based on performance design carried out by the Nordic Structure Regulations Committeein 1963, and initially formed a pyramid hierarchy of five critical states. In Europe, such design targeted at performance has become more restricted with the introduction of overseas regulations; however, it finally became a recognized

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but non-mandatory standard as reference for the over-rigidity reason. The earliest theorem on the formation of PBSD originated in the US and was first proposed by Bertero (1986) after the professionals summarized and reported statistics on the 1985 mega-earthquakes in Chile and Mexico. At almost the same time, the ATC issued the first probabilistic assessment guideline (ATC 1985) for seismic reinforcement expenses of various categories of structures, which filled the gap of international seismic design guide based on structural performance. The Ministry of Housing and Urban–Rural Development (MoHURD 1978) published the first design code in PRC based on lessons learned the devastating Tangshan mega-earthquake. MoHURD (1989) later put forward the three-critical states fortification goal of “no damage in small earthquakes, repairable in medium earthquakes, and no collapse in large earthquakes”. Combined with the two-stage design method in the US and Europe, the three-stage principle seems to show perfection on the description, but is actually short of the pertinence and enforceability of the three-critical state fortification standard. The Loma Prieta Earthquake (1989), Northridge Earthquake (1994), and Hyogo Earthquake (Hanshin 1995) exposed the imperfection of the traditional seismic design. Through research on the post-disaster vulnerability, the Architectural Research Institute of the Ministry of Construction of Japan (AIJ) concluded that in the Hyogo Earthquake, the dynamic safety of slope engineering and retaining structures satisfied the purposes of the first performance critical state and effectively reduced casualties. However, the damage to residences is hard to evaluate with uncertainties on the maintenance time and losses. The final statistics must give a total number of demolition expenses, showing that most residences have nearly zero resilience when strong earthquakes and their subsequent chain of hazards occur. The AIJ therefore carried out a project study of the “development of new structural system” in 1995–1997. The FEMA also issued a report (FEMA 1996). At the end of 2000s, the SEAOC constructed a fairly complete framework of PBSD to elaborate its global development trends by enriching the performance indexes on structural use corresponding to different performance critical states to balance the functional losses caused by force majeure (e.g., earthquakes, tsunamis, typhoons) under variable expenses (Vision 1995). Many governments have affirmed PBSD practice under the following joint efforts. For example, Australia issued the seismic code Australia 1996 on building fireproofing performance design in terms of PBSD, and New Zealand formed the first generation of New Zealand Building Code Handbook (King and Shelton 2004) after absorbing the unified architectural design principles. China carried forward the principle of three-critical state for seismic fortification and obtained inspiration from the two-stage PBSD approach, taking initiatives on the performance seismic design of individual structures on the basis of the quality and performance (MoHURD 1989). At the 12th World Conference on Seismic Engineering in 2000, research on PBSD application was discussed in plenty of articles, and issues of Earthquake Engineering and Structural Dynamics (EESD) were distributed. Researchers in seismic engineering and structural dynamic analysis circles have undertaken numerous discussion ideas on the principle of PBSD. This design concept of cognition and a series

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of problems to surround this concept have become some of the most central topics of seismic engineering and seismic design research. A portion of the contents of the performance-based seismic theorem became more abundant with practical applications according to the PBSD concept, which greatly contributed to the realization of PBSD concept of cognition for engineers. This roughly includes the following three stages: the conceptual design stage; the calculation design stage; and the performance evaluation stage. The PBSD application of slope engineering is carried out in sequence in accordance with these three stages. In the conceptual design stage, combined with the purpose of construction and specified use proposed by owners, geotechnical engineers should distinguish the seismic fortification targets in terms of seismological conditions, geomaterial properties, and multiple defense lines; and focus on the influence of reinforcement, preliminary seismic stability design, and determination on the retaining types. The seismic performance of slopes is therefore the result of multiple factors and the achievement on the seismic performance targets means controlling all factors in the meantime. When performing performance design, it is necessary to make a drawing outline and modify the design by checking until the predetermined fortification target is balanced. The above procedures make up the calculation design and performance evaluation stage. Upon encountering a large amount of practical use in the background construction boom, the beginning of twenty-first century has created a new environment for improving the PBSD theorem by engineering applications and transmission into other fields such as geotechnical infrastructure facilities design and slope engineering. On account of the contradiction between the seismic critical states in structural PBSD before the twentieth century and the performance concerned by owners and customers, a group of geotechnical engineers have urgently sought an approach to bridge the gap on the inequity of expectations and knowns to ensure they are incorporated with consistent information of the objectives and performance to ultimately be distinguished by both sides. These concepts are fortunate to also become success specifications after being proven necessary. The International Code Committee (ICC) issued the authority file Performance Code for Structures and Other Facilities that addresses the normal service performance, construction process, and long-term operation performance of the construction, which clearly stipulates significant regulations in accordance with the PBSD principle (Council 2003). The ATC signed a project entitled ATC 58 series projects in contract with FEMA to develop a new generation of PBSD guidelines for newly built and existing structures (ATC 2009). The purpose of the program was to evaluate the seismic performance of constructions and develop a PBSD guideline that quantifies uncertainty. Japan later formally incorporated the principle of seismic performance design into both non-structural and structural seismic reinforcement regulations. In 2003, the criteria subject to the critical state and performance was formulated into specification (Otani et al. 2003). Shortly thereafter, the Euro-concrete Association released a report on seismic design (Calvi et al. 2007). Japan set up the performance-based seismic design research board (PBSDRb). In agreement with the characteristics of multiple categories of popular domestic

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constructions, the committee presented the seismic design framework in accordance with these performances through a fairly effective statistical and empirical approach. For instance, to prove the rigor and reliability, a bridge case study illustrates in detail the relations between bridge seismic performance and the acceptable increased expenses and expected maintenance time. Detailed seismic performance objectives have been investigated for a sufficient number of random samples of residents and engineers from the devastating Hyogo Earthquake. In 2006, the PBSDRb issued the book PBSD: Current Situation and Topics (PBSDRb 2006) on the basis of current situations and determinable performance objectives of PBSD under evidence of various technical regulations. This book has made great contributions on the popularization of PBSD and interiorized by people. As reported by FEMA (2013) and ATC (2009), the average probability endangering personnel safety will then be used to express the performance goal to meet the higher purposes for seismic performance expression. For example, FEMA (2013) introduced the most crucial aspects as well as the methodology such as probability density description, with some performance indexes, uncertainty, categories of performance evaluation, and basic procedures of performance evaluation process concluded. Compared with the initial generation, the new-generation PBSD generally considered the seismic performance of different categories of single engineering structures from the view of probabilistic state and evolution of these performance indexes. It has therefore been widely investigated worldwide and prevailed to be considered one of the most useful guidelines in seismic design. Owing to the development of disaster prevention theory—especially the concept of resilient disaster prevention— the authors believe that the new generation of performance-based seismic dynamic safety evaluation and design of slope engineering should introduce the concept of disaster prevention based on resilience. The next generation of PBSD theory should therefore be established on the resilience-based framework. There is no doubt that resilience-based seismic dynamic safety performance evaluation should be developed for slope engineering. Chapter 6 (hyperlink to Chap. 6) of this book describes this point in detail.

3.2 Global Framework of PBSD for Slope Engineering The PBSD uses the limit equilibrium method (LEM) for calculating bearing capacity and deformation, supplemented by second-order indicators such as ductility and dynamic stability, and global and local stability are characterized by the seismic dynamic safety factor. In practice, the minimum fortification target in the specifications is unable to reflect the degree of seismic dynamic stability of slope engineering to assess the economic losses. When an owner desires a higher seismic performance of a specific construction than the regulation-based design requirement, they should increase the initial quantitative expense to reduce the losses that would otherwise be caused by seismic influences. The owner may be aware of some conceptual indicators but it is impossible to know all. Construction corporations require only a

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tender offer for bidding on the expenses of materials, equipment, labor, and several sub-contract teams for sub-divisional work. It is generally not easy for geotechnical engineers to effectively help others clearly understand the seismic performance of slope engineering using descriptions such as strong nonlinear dynamic deformation and the global and local safety factors. If owners and geotechnical engineers fail to reach a negotiation on the expected and designed objectives in this process, it will eventually lead to potential issues that the owner may either have high expectations for seismic capacity of the slope engineering or spend a larger cost. The latter can be of little harm for the monomer slope, but the former may lose its function or cause casualties within the fortification. However, neither belong to mature behavior in modern society and ultimately do harm on the marketing ecology. The purpose of the negotiation process is to make both sides conceptualize how to deal with the investment-benefit relationship of seismic performance as quickly as possible. Among them, the technical problems will be simplified and the expression of seismic risk will adopt the deterministic method of seismic records or a method based on probability, which will be used to help the target decision-makers and asset managers, such as insurance companies and enterprises. Geotechnical engineers should make assessments on the slope seismic performance developed based on field investigations, which include site model experiments, numerical simulations and evaluation, and basic research on previous earthquaketriggered damage. These methods have been proven effective in geotechnical practice such as landslide disaster prevention and disaster reduction, but still face challenges to overcome. First of all, the results of seismic dynamic response numerical analysis of slope sometimes fail to predict the real nonlinear seismic dynamic of slope during a natural earthquake event below the seismic fortification intensity owing to the randomness of seismic excitation and spatial variability of geomaterials. Second, for a slope under given site conditions, there is a step increase in the cost caused by seismic engineering measures taken near the epicenter of a strong earthquake and large complex slopes, which will lead to a mismatch between the global stability safety, applicability, and repairability and the design cost. Technically, a means to reduce economic and social losses while ensuring personnel slope engineering safety (the lowest critical state) has become an urgent problem for designers. A way to accurately grasp the strong nonlinear seismic response of slope engineering under ground motion of different fortification levels is the core driving force of PBSD under the comprehensive premise of randomness of ground motion, three-dimensional spatial variability of geomaterial in slopes, the strong nonlinear stress–strain relationship, and shear dilatancy effects. According to the PBSD theorem, it is very significant to quantify randomness and uncertainties from seismic excitations under different fortification levels to the strong nonlinear seismic dynamic response of slope engineering based on the stochastic probabilistic levels. The PBSD theorem can be expressed probabilistically according to Eq. (3.1) (Günay and Mosalam 2013). . . . . E(DV ) =

p(DV |D M) p(D M|E D P) p(E D P|I M) p(I M)

(3.1)

3.2 Global Framework of PBSD for Slope Engineering

21

where the intensity measures (IM) represent the seismic intensity parameters, such as peak ground acceleration (PGA), corresponding to the site seismological conditions of slope engineering and the basis for decision-making of the fortification levels. The engineering demand parameter (EDP) represents the shape changes or damage to a group of components, such as detectable breaks on dams or slopes. In this guideline, the authors use a criterion of relevant classification by introducing different extreme state to quantify the observed variations for better demonstration, as applied in the form of fortification levels in many specifications but with more details. The demand management (DM) index describes the local seismic dynamic stability state of slope engineering (e.g., seismic dynamic factor of safety, horizontal or vertical seismic dynamic displacement, deformation) and assumes that performance-based indicators exist similar to these indexes in the aspects of each failure mode, and ultimately makes up the mentioned critical states or EDPs. The decision variable (DV) describes the reinforcement measure cost or the economic loss caused by instability failure of slope engineering caused by seismic dynamic excitation. In this assessment and design guideline, this step constitutes a linked ring in a chain for seismic dynamic stability performance evaluation and reinforcement design of slope engineering to which the governments or non-government institutions should pay attention. Instruction on statistical methodology is illustrated in the following chapters. This chapter generally introduces the framework and procedure of performancebased seismic dynamic safety evaluation and slope engineering design from the following aspects: (1) seismic fortification level; (2) ground motion excitation; (3) seismic dynamic response analysis; and (4) probabilistic performance evaluation of slope engineering. The outline of the PBSD theory and slope application is illustrated in Fig. 3.1, and each part is then illustrated in detail as follows. Part 1: Determination of seismic performance level of slope engineering In the process of slope seismic performance evaluation, the slope performance level can be determined by the following principles: (i) the seismic performance of the slope is determined by the owner and geotechnical engineer through consultation, and shall not be lower than the minimum requirements of the seismic design code of the site where the slope project is located; (ii) the seismic performance of the specific slope engineering should be determined according to the characteristics of each specific slope. The seismic performance of slope engineering is a measure to characterize its dynamic stability, reinforcement demand, and adaptability. It should not only meet the requirements of the owner, but also consider the social constraints, and should be determined by the owner and geotechnical engineer through consultation according to social and economic factors. The codes (MoHURD 2010, 2013) reflect the constraints of law and society through technical standards. The minimum standard of seismic performance of slope engineering shall be determined according to the needs, and the seismic performance level of each specific slope engineering shall not be lower than this standard. For the specific slope engineering, when determining its seismic performance level, we should consider the use, importance, service life, and change of load excitation with time, and comprehensively consider the technical and economic

22

3 Performance-Based Seismic Design Framework of Slope

Fig. 3.1 Principles and procedures of PBSD for slope engineering

3.2 Global Framework of PBSD for Slope Engineering

23

feasibility of improving the slope seismic performance level. It is also necessary to directly use the dynamic reliability and instability probabilistic description to evaluate the seismic dynamic stability of slope. The specific seismic performance level of slope is also introduced in this chapter. Part 2: Ground motion in slope seismic performance evaluation The ground motion in the seismic performance design and evaluation of slope engineering is described by the response spectrum or acceleration time history curve. The seismic response spectrum must be determined according to the seismic code of the site where the slope project is located. It should be defined on the bedrock and not affected by the topsoil. The amplification effect of surface overburden should be considered in the determination of the ground motion at the slope site. However, the ground motion acceleration of the site can be obtained by various means according to the above response spectrum. Factors such as seismicity, focal characteristics, and propagation path of a slope engineering site can also be considered. These factors can be considered through ground motion models based on the slope engineering site characteristics and focal mechanism. The specific selection and generation methods for seismic excitation of a slope engineering site are introduced in detail in the subsequent sections of this chapter and in Chap. 4 of this book. Part 3: Seismic dynamic response analysis approaches of slope The deterministic methods for slope seismic dynamic response analysis are divided into the quasi-static analysis method, Newmark sliding block displacement method, nonlinear dynamic time history analysis method, and large deformation analysis method according to the calculation efficiency, analysis accuracy, and applicable categories. These methods are proposed to predict the strong nonlinear dynamic response of slope engineering in strong earthquake scenarios. Nonlinear dynamic time history analysis is generally the analysis tool closest to the real results. The specific operation process and their characteristics are introduced in detail in Chap. 5. Part 4: Probabilistic seismic performance assessment of slope The seismic dynamic response of slope engineering clearly shows randomness and strong nonlinear dynamic coupling effects under random strong seismic dynamic excitation owing to the large spatial variability of slope geotechnical materials and the constitutive relationship. Therefore, to better grasp the nonlinear dynamic response of slope engineering under possible seismic dynamic excitation in the future, it is necessary to predict and analyze the nonlinear seismic dynamic response of slope engineering from the perspective of probability, especially the stochastic seismic dynamic response of slope engineering based on the theory of nonlinear stochastic dynamics. In this way, the performance-based seismic dynamic safety assessment and design of slope should be gradually transformed from the traditional deterministic level to the dynamic reliability level. Readers interested in nonlinear stochastic seismic dynamic response analysis of slope engineering are referred to another book (Huang and Xiong 2021) on slope stochastic dynamics for more detailed content.

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3 Performance-Based Seismic Design Framework of Slope

Previous regulations and reports based on historical slope engineering and some practical failures aimed to eliminate the explanation of many conceptual issues of great significance. From the published literature, there is a trend that PBSD should be applied to not only buildings and bridges but also to slope engineering, which must urgently catch up its progressive principle because the application of designing notion is oversimplified. Natural slopes are investigated as part of a geotechnical environment in terms of the design of soil embankments and earth dams, and are considered and reinforced in the ways the engineers commonly use them for ground treatment or liquification when it comes to slope engineering. Therefore, by virtue of the progress on the production and construction equipment and the elaboration and optimization of methodology upon which they are based, the new regulations are capable to be issued to acclimate to the reduction of initial expenses and performance convenience for constructors. More importantly, the project managers reorganize the construction procedures and make every step ahead of schedule for check and acceptance. This trend is irreversible with the high-efficiency management requirement, and it forces that the performance of slope reinforcement weigh heavier than the quantity of capital flow in this process, in particular within the stage of low-carbon and sustainable development and the “must-consider” extreme seismic loading conditions. Quasi-static simplification is commonly adopted for slope seismic and practical design in most countries, but the results do not fare very well and exhibit notable distinctions compared with a considerable number of slope engineering examples owing to the simplification and idealization of static models. The nonlinear dynamic time history analysis method was put forward to remedy the disadvantage of ignorance on the dynamical pulse effects on loadings, and has been verified to be a more reliable approach that takes the seismic excitation at a certain time as the dynamical input based on the historical strong earthquake observation records or artificial ground motion excitations to calculate the nonlinear dynamic response such as the factor of safety or deformation of the slope engineering. However, this approach depends heavily on the nonlinear stress–strain relationship of geotechnical materials and ground motion excitation, and its calculation amount is very high. Although it is presently considered to be a method with high prediction accuracy for the seismic dynamic response of slope, its strong nonlinear dynamic analysis results are too sensitive to ground motion excitation and the analysis efficiency is not particularly high. This method is thus not easily or quickly mastered by geotechnical engineers. Performance-based seismic design is more dependable for the principle that engineers must take measures to establish an optimized seismic performance design and assessment approach on safety, economy, and rationality for slope engineering (Cao 2015). This indicates that PBSD for slope engineering also covers the determination of the performance related to the research objectives, which includes the following five aspects: . The purpose of the determination of a framework that has similarity with PBSD on structural use but with different indicators on performance reflection. . The purpose of the definition of a seismic hazard assessment on the basis of regional risk mapping to distinguish the corresponding performance standard.

3.2 Global Framework of PBSD for Slope Engineering

25

. The purpose of certain performance levels and quantitative indicators for the target objectives to illustrate the loss degree of slope destruction and slope availability. . The purpose of some target performance of slope seismic design. . Undertake risk analysis and statistic instruction on direct or indirect losses caused by landslides. It should also be noted that the following seismic codes emphasize the seismic performance design and evaluation of the three-levels standard for engineering structures. For example, the specification (MoHURD 2013) regulates three-graded fortification levels to utilize the composite of the construction site and the function and significance of the targeting objectives, investment, profits, and socio-economic benefits. The calculation and checkouts on the basis of the bearing capacity and deformation of slope engineering are most commonly undertaken based on MoHURD (2010) and MoHURD (2013). When the resistance reaches the maximum bearing capacity of the materials, such as when the anchoring system is close to failure, the deformation unsuitable for continuous bearing or the slope instability meets the ultimate design expectation of the critical state of the bearing capacity. The slope engineering may reach an availability critical state when the deformation limit is reached, as specified by the normal function of slope engineering or the specified critical durability state.

3.2.1 Design Ground Motion of Slope Engineering Site The important premise of analyzing and obtaining the seismic dynamic safety performance of slope is to reasonably describe the seismic excitation that the slope engineering site may encounter in the future. Whether it is the essential randomness of ground motion excitations in the intensity spectrum or energy distribution or the lack of understanding of the essence of ground motion, earthquake motion with a deterministic physical mechanism is regarded as random. At present, the mechanism of earthquake occurrence and the corresponding ground motion excitation have random uncertainties. As mentioned, the nonlinear dynamic response of slope engineering under different ground motion excitations is an incomplete system owing to the spatial variability of geotechnical materials, sensitivity of slope to seismic dynamic excitation caused by the constitutive relationship, and dilatancy effects. Therefore, in this context, it is very necessary to consider the randomness of ground motion in the analysis and evaluation of seismic performance of slope engineering. In general, the randomness of seismic excitations is described using a probabilistic method for quantification, and the seismic dynamic reliability based on stochastic dynamics theory must also be introduced into the seismic dynamic safety performance evaluation of slope engineering. The amplification effect has been investigated according to seismic dynamic model experiments with deterministic inputs. Note that not only are the indicators

26 Table 3.1 Comparison of seismic fortification level of FEMA (1996), Vision (1995), and MoHURD (2010)

3 Performance-Based Seismic Design Framework of Slope Seismic fortification grade FEMA 273

VISION 2000

Average return period (years)

Level 1 (often occurs) 72 Level 2 (occasionally occurs)

225

Level 3 (seldom occurs)

474

Level 4 (rarely occurs)

2475

Frequent earthquakes

43

Few earthquakes

72

Rare earthquakes

475

Very rare earthquakes 970 Chinese codes

Small earthquakes

50

Moderate earthquakes 475 Great earthquakes

1641–2475

of an earthquake (e.g., seismicity, focal seismic characteristics, site-based propagation) considered when the engineers use intensity not magnitude, but there is also a focus on the personnel perception, construction losses, and casualties, in addition to future geo-disaster assessment. This probability is visualized by a zoning map in the design. The seismic fortification intensity of slope engineering is based on the basic seismic intensity of the region determined by this seismic zoning map according to MoHURD (2016). The earthquake used in this assessment is known as regional ground motion that statistically recurs over a regular period. Seismic fortification levels and exceeding probability based on FEMA (1996), Vision (1995), and MoHURD (2010) are generally subdivided by average return period or occurrence probability (Table 3.1). A strong correlation is assumed between seismic intensity and fortification levels. The seismic fortification grade of the slope is related to the seismic intensity and return period, in addition to the significance of the slope engineering. This includes the construction, maintenance, and repair cost of the slope, the severity of possible damage consequences, such as to threaten human life, economic losses, adverse social benefits, and slope geometric topology such as the category and height. The next procedure is to determine the seismic fortification criteria, which are significant in the PBSD framework. Engineers must be clear on the appearance and understanding of seismic hazard maps and the approach to generalize them using mathematical description matters. The importance of this step shows its legal insurance for engineers for protection from misconduct in slope engineering practices. The calculation of seismic excitation should be carried out in accordance with the current national standards. As reported in Table 3.2, in place with the regional seismic fortification level under intensity 6.25, the seismic excitation calculation may not be carried out for the supporting structures of slope engineering or taking a horizontal

3.2 Global Framework of PBSD for Slope Engineering

27

acceleration under 0.2 g, while the seismic structural measures shall be taken instead. In places with seismic fortification level over 6.25 intensity, the seismic condition should be resisted using a supportive structure for slope engineering. During seismic design, the combination of the seismic effect and load effect should be implemented based on the published seismic design regulations. The slope engineering in seismic areas shall consider the influence of earthquakes according to the following principles. An empirical method that engineers make the best use of ground motions and a numerical slope model to test the likelihood of earth dam failure over the return period, as shown in Fig. 3.2. Here, the probability of exceedance is used to describe the randomness seismic intensity with statistical rules as an instance. A more vivid diagram is illustrated, starting from the top of the restoring force characteristic envelope. The colored dots represent the applicability elastic state, applicability critical state, recoverable critical state, and safety critical state. Each state can be concentrated as a point with two layers of implications. On the one hand, the hazard levels of the seismic intensity based on government-issued intensity zonation documents and the probability of exceedance represent the tectonic settings that can lead to earthquakes, local environments, and population concentration. On the other, slope stability with seismic performance indicates the co-seismic capacity, and it depends on the scales of dams, available costs, and allowable risks and rebuilding expenses after the mainshock. The restoring force characteristic envelope decreases in the direction of the arrow.

3.2.2 Seismic Performance Level and Criteria of Slope Engineering (1) Seismic performance design and earth dam engineering assessment Earth dams are a kind of slope engineering in a broad sense. The dams’ function is to maintain local water balance by reservation and liberation in rainy or dry seasons, and of course, to reserve energy. The common categories of engineering dams can be generally divided as earth dams, rockfill dams, gravity dams, and arch dams, among which earth dams are the most commonly used type and their design theorem is the most simplified and developed. Earth dams can bear foundation shaking because the constructed materials are loose. However, water tends to seep into the dams through the clayey waterproof layer and reduce its stiffness and firmness. The prevailing number of earth dams is related to their high resilience to terrain and good adaptability to geological and complex climatic conditions. Earth dams are nearly the only choice for economic and applicability reasons under conditions such as extreme climate, complex engineering geological surroundings, and high-intensity earthquakes. Earth dams are the most abundant type worldwide, accounting for approximately 90% of the total number of dam constructions (USACE 2020). Moreover, earth dams have a long history and are associated with the largest number of accidents during operation. Since the 1970s, the FEMA has issued a series of national dam safety plans after

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3 Performance-Based Seismic Design Framework of Slope

Table 3.2 Seismic intensity scaling in Chinese specifications (code for Seismic Design of Buildings, GB 50,011–2010) For intensity > 5

Human perception

Construction damage

Damage on slope or ground

Horizontal acceleration factor

VI

Humans perceive shaking

Few are slightly Tiny cracks appear 0.45–0.89 damaged, and most in the river bank are intact and soft soil, and sand blasting and water gushing appear in the saturated sand layer

VII

Stand unsteadily in the beginning, but can walk outdoors

Few are seriously damaged, and the majority are moderately or slightly damaged

Some bank 0.90–1.77 collapse occurs, water spray and sand gushing are common in the saturated sand layer, and there are many ground cracks on the soft soil

VIII

Panic to run outdoors, and have difficulty standing without handrail

Many are seriously or moderately damaged, and the minority are slightly damaged

Cracks appear on the dry hard soil, and most of the saturated sand layers are sprayed with water that may cause liquefication

1.78–3.53

IX

Easily fall and have difficulty running outdoors

Vast majority are damaged (repairable) including some local landslide failure or damage of retaining structure (cannot repair)

Cut-through surface and bedrock cracks, dislocation, landslide, liquefication, and foundation collapse occur

3.54–7.07

X

Hard to escape Vast majority are outdoors since the destroyed doors cannot open including much global landslide or massive damage of retaining structure

Wide cut-through cracks occur and develop, and severe sand liquefication occurs

7.08–14.14

Note The quantifier definitions are listed here, but vary with the site differences. For reference, “few” refers to less than 10%, “minority” is 10%–45%, “many” is 40%–70%, “majority” is 60%–90%, and “vast majority” is more than 80%

3.2 Global Framework of PBSD for Slope Engineering

29

Fig. 3.2 Contingency diagram of the relationship between critical states and the seismic exceedance probability of dams 2010 (modified from MoHURD)

several terrible dam break accidents. During the 1950s to 1990s, the majority of dams in the US contributed up to 50,000 dams in use for agricultural irrigation, transportation, and energy conservation (USACE 2020). The design life of these dams is typically over 70 years or even longer, but few of them can reach this standard without specialized maintenance. Earth dam safety has thus became such a vital matter of national strategy that the FEMA and US Army Corps of Engineers annually published a white paper report based on an investigation of the current conditions, hazards, and risks since 1996 (FEMA 1996) to reduce the benefits of individuals’ properties. In response to the recent Oroville dam damage (USACE 2017) and Edenville dam break (USACE 2020), the US government manifested an efficient emergency management plan for dams, but some notable issues that are not only concerned with techniques and engineering have also been revealed. The International Commission On Large Dams (ICOLD 1989) therefore recommended in early specification Bulletin 72 that the seismic design of dams should be based on a hierarchy of critical states of fortification to different categories of constructions, including structural or non-structural ones and under different levels of earthquakes, which are divided into operational basic earthquakes and the safety assessment earthquakes. The Institute of Water Resources and Hydropower (IWHR 2000) issued the Specifications for Seismic Design of Hydraulic Structures, explicitly stipulating the seismic fortification for large earth dam projects with two fortification levels. However, researchers reckon that a detailed plan on the expansion of a fortification division should be an urgent requirement because, as far as they are concerned, the current specifications never paid sufficient attention to the societal economic benefits and scarcely considered the sustainability of other facilities, which may cause inevitable damage to human safety and potentially ruin reservoir

30

3 Performance-Based Seismic Design Framework of Slope

constructions under emergency conditions because of the compromised expected performance. Seismic hazard analysis and assessment based on dam performance is therefore necessary to strengthen the applicability and avoid dam breaks. The seismic safety estimation of a single earth dam construction generally includes three aspects: seismic input; seismic response; and its resistance. By undertaking a review of numerous dam break cases worldwide, it is concluded that overflow, also called reservoir water overtopping, contributes the most. Following previous investigations on these failures, the International Dam Association made a statement that the reasons directly related to the explanation of reservoir overtopping are as follows: . The main reason is due to the settlement and permanent deformation of an earth dam caused by earthquake excitation. The settlement and permanent deformation of the dam crest should not exceed the height difference formed by the pool level in the normal state. Otherwise, the reservoir dam should be considered in the overtopping limit state. The overestimation of the foundation is half of the explanation regarding the overall settlement, which should not have occurred. . Some special attention should be paid to the reduction of shear strength of earth dam material owing to excess pore water pressure or liquefaction. Even though overtopping will not happen within the service of an account with good foundation stiffness, the landslide of dams is probable under conditions of long-term seepage with entrainment and water erosion. . The final influencing factor that may be caused by the slippage of nearby faults and torrential rainfall is the sudden rise of water level, which may also lead to a rapid rock slide. Only upon completion of hazard history statistics and emergency plans can the damage can be avoided within an expected range. Misestimations of the accumulative deformation and extreme weather may cause earth dams to break in service. Emergency management departments therefore require sharp indicators that reflect and forecast dam breaks with reliability. Earth dams can withstand the same reservoir water pressures as during dam construction under static conditions, but the dam crest is vulnerable to erosion owing to waterflow under dynamical excitation. Special geotechnical materials are required to eliminate the entrainment of dam crest material in the rapid flow slide. However, from the perspective design planning, the larger factor of safety should be used to improve safety reserves for earth dams, particularly those over 100 m and controllable in lifeline projects. The ICOLD summarized that of the direct breakage of earth dams, 39% were related to overtopping and floods, 18% were caused by seepage failure of the dam body, and 12% were caused by seepage in the foundation and dam body owing to liquefication. Over 80% of these were dams of over 15 m in elevation. This implies that conventional regulations are insufficient to provide detailed and substantial design references. Different seismic intensities can also cause direct changes on the degree of abrupt overflow risks. Once the reservoir water overflows the top of an earth dam or the foundation settles too low for the top below the reservoir surface, a flood can scour and destroy the crest beginning with a crack, followed by entrainment of its height because of the flow-made rapid mass loss and reduced height lead

3.2 Global Framework of PBSD for Slope Engineering

31

more reservoir water to flow through the dam crest. This malignant cycle makes the crest subject to further erosion and damage. Overflow precautions should therefore address all of the relevant details. The first procedure is to make site-specific seismic intensity zonation. The purpose of seismic zoning is to formulate different seismic fortification standards in different areas by dividing the territory into several areas with different seismic risk degrees expressed through mapping. The technical method of seismic zoning research is commonly probability-based statistics of seismic risk analysis. The zoning conception was first proposed by Professor Cornell in combination with seismic tectonics and seismicity on the basis of some hypotheses to achieve the description of all these possible causes: . With reports of surrounding faults and seismicity, a division is made of the potential source areas that can be line sources or other irregular source formations, and the possibility is specified that an internal earthquake is the same. . The average occurrence rate of earthquakes in the potential focal area is a spatial constant. . Earthquake occurrence conforms to random independent Poisson distributions. . The number of earthquakes in the potential focal area decreases exponentially with increasing magnitude, which can be expressed by the magnitude frequency relationship of Gutenberg Richter. . Site ground motion parameters follow a function of epicenter distance (or source) and magnitude. The general conceptions of seismic zoning map in the US and China are almost same. The US government divides the territory into different subregions using the boundaries inferred from the attenuation characteristics of ground motion. Some sets of seismic hazard analysis models are then established according to different seismological parameters of the relationships in the Next Generation Attenuation program. The 2002 US zoning map gives the PGA isolines with a 50-year exceedance probability of 10% and 2% (interval = 0.02 g), and an acceleration response spectrum isoline of 0.2 s and 1.0 s (interval = 0.02 g) with a damping ratio of 0.5%. Unified hazard spectra and hazard curves are also provided for periods of 0.1, 0.3, 0.5, 1.0, and 2.0 s acceleration spectra. It is worth noting that the zoning map is based on specific hard rock sites, so that the average equivalent shear wave velocity within 30 m is beyond 760 m/s, which is equivalent to the boundary between grade B and C sites specified in National Earthquake Hazards Reduction Program (NEHRP). By means of multi-disciplinary comprehensive research and long-run preparation, the seismic ground motion zoning map in China fully considers the regional distinctions of its surroundings, tectonics, and focal seismicity. The seismic prediction results of different time scales consider the uncertainty of each procedure to reflect the influence of the site conditions. MoHURD (2016) summarizes the zonation with two maps and one diagram. The two maps are drawn on the basis of different indicators: one zoning map uses the peak acceleration with a 50-year exceedance probability of 10% for class II sites and divided into seven grades of less than 0.05 g, 0.10 g, 0.15 g, 0.20 g, 0.30 g, and over 0.40 g; and the other map uses a damping

32

3 Performance-Based Seismic Design Framework of Slope

ratio of 0.05 for the class II site characteristic period zoning map of the acceleration response spectrum, divided into three grades of 0.30, 0.35, and 0.40 s. The two maps emphasize the control of the seismic environment to shape the response spectra. One diagram is a characteristic period adjustment of the ground motion response spectra in different sites. With ongoing seismic research and the accumulation of relevant data and economic improvements, the investigation on seismic zoning should be updated and developed to reflect the conception of people-oriented engineering to reduce casualties as the most important consideration, and to make anti-collapse measures the primary consideration aim of zoning. In the 1980s, the China Seismological Bureau reviewed and approved the earthquake intensity scaling table derived from 1883 Mercalli intensity scaling, which also gives consideration to human perception and building damage (Table 3.2). The empirical relationship between magnitude (M) and intensity (I 0 ) is M = 0.58I0 +1.5 in the epicenter region. The European Earthquake Commission made an additional classification on the basis of structural vulnerability and earthquake damage, which is divided into six grades from grades A to F. The Japan Meteorological Agency divides the territory into intensity 0 to 7, with weighing categories of buildings including wooden houses, reinforced concrete houses, lifeline constructions, and ground and slope constructions. In terms of the ground and slopes, the document says that “cracks appear in some soft soil, and some mountain slopes are damaged and rockfall” under intensity 5 and that “the ground is distorted by a large number of cracks, hillsides are damaged and landslides occur, and the overall landform may also change” under intensity 7. For slope seismic fortification levels, the performance levels should at least include the allowable or ultimate damage of the slopes under the specifically designed earthquakes that may be encountered. Countries worldwide agree to make divisions on the critical states on the basis of their regional seismic zonation. The result of the division may be distinct but the principle refers to the Mercalli intensity scale mapping. The slope critical states should therefore be specified according to these factors, e.g. structure categories, slope components, significance, economic indicators, and population factor. The performance hierarchies should meet the following conditions. . Integrity. This includes the minimum number requirement of the performance level and understanding of the stages a building that may suffer. The principle of life cycle and index quantification method should run through this. The number of the levels should be more than the number for fortifications, and each performance level should include quantitative indicators at the stages from establishment to collapse. . Applicability. The performance indexes should match the fortification requirement so that engineers and owners can make choices for specific deals in the range of given levels. These performance levels should be differentiated on a gradient, that is neither too far nor too close, to conveniently distinguish their explanation. . Pertinence. The categories fairly differ even for geotechnical slopes. The stability and performance of the slopes are subject to the site conditions, thus some specific

3.2 Global Framework of PBSD for Slope Engineering

33

sites should be regarded as special targeted projects. Some engineers consider the site condition as an indicator of the hazard assessment for regional earthquakes and “intensity-based” fortification, but ignore the importance of earth dam foundation and amplification influence that contributes to seismic excitation. Site conditions are extremely significant and contain certain boundaries. After the Hyogo Earthquake (1995), multi-dimensional nonlinear dynamic nonlinear analysis was used to reevaluate the seismic performance of Japanese dams according to the seismic degree analysis method in the guideline. Water storage is the basic function of dams, thus the functionality related to waterproofing is of significance. The critical states of seismic performance of a dam at a certain site under stochastic seismic excitation are divided into the following grades, considering their function and following the analogy of the division of structural critical states. . Seismic performance I. The earth dam construction is completely free of damage and its function does not need to be reevaluated. . Seismic performance II. The earth dam is slightly damaged, cracks are generated on the body but do not penetrate, its function is basically slightly reduced, and it should be reassessed. . Seismic performance III. The body of the earth dam is seriously damaged and has through cracks. Considering the continuous impact of water flow at the leakage point on the dam, it can maintain the water storage function, but this function will decline over a long time. . Performance critical state. The water storage function of the earth dam is instantaneously lost to a great extent, but there is only a small probability of secondary disasters. The reservoir operation must be stopped immediately and the loss assessment must be carried out. Some performance-based objectives and descriptions are demonstrated in Table 3.3. Several indicators that reflect the performance when the critical states are unyielding are discussed here. The settlement and bearing capacity are two parameters nominated in priority for slopes and subgrades. As mentioned, the purpose of PBSD for slope engineering includes both the availability for slope engineering and personnel safety. The current guidelines and reports aim to add additional critical states through empirical interpolation, which is quite effective for options when engineers face a large performance distinction with the owners’ requirements. The settlement and bearing capacity are good quantitative indicators for the foundation and subgrades, which may not be that precise for building structures or bridges. In general, the design of the foundation is based on the elastic foundation beam hypothesis, but it may be hard to accurately consider the load bearing characteristics and nonlinearity of the ground, especially under dynamic and seismic circumstances. Performance-based seismic design for slope engineering should mirror the material constitutive models based on elastic–plastic theory and introduce nonlinearity and ductility that are adopted in structural PBSD, not only for mechanical components but also for slope engineering.

34

3 Performance-Based Seismic Design Framework of Slope

Table 3.3 Performance-based critical states for dam objectives with different categories (2006) (Modified from PBSDC-JAEE) Performance-based critical state

Performance objectives and descriptions

Seismic performance I

The earthquake will not cause damage to the earth dam Check item of earth dam: No sliding failure No residual deformation

Seismic performance II

The structural damage caused by the earthquake can be repaired and the functions of water storage and drainage can be maintained Check items of earth dam: No sliding failure No seepage failure No overflow of water storage No break Can be maintained discharge function

Seismic performance III

The dam body is damaged and the water storage function is slightly affected

Performance critical state

The dam body is seriously damaged and the function is affected, it is able to prevent secondary disaster and not collapse

Some detailed parameters such as cracks are listed in Table 3.4. The skeleton curve supported by Table 3.4 is shown in Fig. 3.2. To ensure the structural factor of safety considering the initial imperfection of materials and randomness for dynamic loads, the load is multiplied by an amplification coefficient and the material resistance is multiplied by a reduction factor. For loads in multiple forms with variable probabilities, different partial factors are required to reflect the amplification effect of these loads based normally on engineering experience. Table 3.4 Critical states, damage state, and repair goals on moderate earth dams

Critical state

Damage state

Description

Available

I

Unseen cracks No need to ( 1, the acceleration is increasing. • The monotonically varying ground motion intensity magnitude (IM) level can be the PGA, peak ground velocity (PGV), or EPA. Here, the IM is a function of the scale factor η, IM = f (a(t), η). When the acceleration a(t) is constant, the IM changes monotonically with η, and there are many variables that describe IM, which can be characterized by the EPA, PGA, acceleration spectrum corresponding to PGA, PGV, or the standardized parameter λ = η/ηyield , where ηyield represents the minimum proportional strength when a given strong earthquake record and slope begin to appear in the critical instability state, and the scale is numerically equal to the strength reduction factor (η). These variables are proportional to the scale factor, i.e. IM = η · f (a(t), η).

4.2 Determination of Seismic Excitation of Slope Site According …

49

4.2.2 Modification of Seismic Strong Records Based on the Code’s Design Response Spectrum For slope engineering sites with measured historical strong earthquake records, its response spectrum may not be in good agreement with the site response spectrum commonly recommended in the seismic code. There is therefore a need to modify the site’s measured strong earthquake records to minimize the fitting error between its response spectrum and that recommended by the standard. This modification process mainly involves fitting the code response spectrum with that of the measured strong ground motion records. The results (i.e., a modified ground motion time history) retains the non-stationary characteristics and duration of the original time history to a certain extent. There are generally at least four kinds of methods to estimate slope site seismic ground motion by modifying the site-measured strong earthquake records and comparing their response spectrum with the code-recommended response spectrum. Two common adjustment practices are introduced here. (1) Proportional adjustment method Significant differences between the response spectrum of existing seismic records and standard response spectrum, when present, can be greatly reduced by an overall scaling of the entire acceleration time history. During the specific operation, the sum of squares of the error between the response spectrum and standard response spectrum is minimized by linearly scaling the seismic acceleration record. Let Sa (ωi )(i = 1, 2, · · · , M) be the acceleration response spectrum value of a(t), and ωi is the natural vibration frequency of the single degree of freedom vibrator. If η is the scaling coefficient, the response spectrum value of ηa(t) is ηSa (ωi ). If the target acceleration response spectrum value is SaT (ωi ), the sum of squares of relative errors (SSRE) is: | | ΣM | ηSa (ωi ) − S T (ωi ) |2 a | | SS R E = | i=1 | S T (ω ) a

(4.9)

i

Deriving from Eq. (4.9) to obtain the extreme value, the value of η that minimizes SSRE is given as: ΣM

Sa (ωi )/SaT (ωi ) η = Σ i=1 2 T M 2 i=1 Sa (ωi ) /S a (ωi )

(4.10)

(2) Adjusted Fourier amplitude spectrum method The Fourier amplitude spectrum of actual seismic records is iteratively adjusted to fit the standard response spectrum curve, during which the phase spectrum of seismic records remains unchanged. The artificial acceleration time-history records based on the actual earthquake records can be obtained through several iterations, which can

50

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic …

then meet the requirements with a certain degree of accuracy. This method completely retains the phase characteristics of actual ground motion and is an ideal adjustment method for actual ground motion. The synthesis process is as follows: (i)

Fourier transform is performed on the selected ground motion record a(t), and the Fourier amplitude spectrum A(ωi ) and phase spectrum .(ωi ) of discrete points are obtained. (ii) The site code response spectrum SaT (ωi ) is determined according to the slope seismic geological conditions. (iii) The response spectrum Sa (ωi ) is calculated from the time history a(t) of the measured strong earthquake records on the slope site. (iv) The response spectrum control point is selected and the allowable error is specified: ε=

M Σ Sa (ωi ) − SaT (ωi ) ≤ εr SaT (ωi ) i=1

(4.11)

in which εr is the relative error. (v) The Fourier amplitude spectrum A(ωi ) is adjusted by numerical iteration. Ak+1 (ωi ) =

SaT (ωi ) k A (ωi ) Sa (ωi )

(4.12)

Here, Ak (ωi ) and Ak+1 (ωi ) are the results of iterations k and k +1, respectively. (vi) According to the modified Fourier spectrum A(ωi ) and the original phase spec˜ can be obtained by the inverse trum .(ωi ), a new acceleration time history a(t) Fourier transform. Check whether the requirements are met. If not, return to step (ii) until the allowable error is met (ε ≤ εr ). This method category seems to be generally simple and easy to operate, but it can hardly be satisfied without a certain number of historical measured strong earthquake records of the slope site. Unfortunately, owing to the long earthquake return period and limited number strong earthquake stations, it is difficult to obtain the actual strong earthquake records at a given slope site. Even if they are available, the strong earthquake records of each time tend to substantially differ, thus this method is difficult to realize.

4.3 Determination of Ground Motion of Slope Site According to the Strong Earthquake Database When there are no available earthquake data for the analyzed slope site, the selection of seismic excitations for input can be referred to other nearby sites or a site with similar categories with strong earthquake ground records. For a slope engineering site

4.3 Determination of Ground Motion of Slope Site According to the Strong …

51

without historical measured strong earthquake records, the site ground motion can be determined from the well-known global strong earthquake record database (e.g., United States Geological Survey, USGS; Pacific Earthquake Engineering Research, PEER) according to the site response spectrum recommended in the seismic design code. The ground motion and geotechnical parameter design values, which must be carefully considered in the design of slope random seismic performance, are summarized here to evaluate the site categories. Some slope factors in the ground motion design for slope engineering should be considered, starting with the topographic amplification effect. Slope bodies gradually magnify the ground motion along the elevation, and both horizontal or vertical earthquakes can occur. Topographic site effects must also be considered, in addition to the progressive failure effect. The reflection effect of seismic waves leads to the tensile and fracture failure of rock mass. The main factors affecting the response spectrum of ground motion are soil medium heterogeneity, slope layer structure, unconsolidated slope layer thickness, local geology, and geomorphology. The seismic performance evaluation of a slope and selection of ground motion in the design process correspond to different analysis methods of the dynamic response (Chaps. 5 and 6). Seismic vulnerability analysis is widely used in modern seismic engineering, and there are two main types: (1) the global analysis method and (2) the narrow domain analysis method. The difference between the two methods is only the size of the considered failure domain. The global analysis method pays more attention to the failure domain of the structure, such as multi-stripe analysis and the IDA method, while the narrow domain analysis method pays more attention to the failure domain of the structure, such as cloud map analysis (CMA), single stripe analysis (SSA), and double stripe analysis (DSA). The SSA within the narrow domain analysis method actually belongs to the category of scaling methods, and involves the scaling of selected seismic acceleration time-history samples set to the same PGA for analysis and obtains multiple engineering demand parameters (EDPs) in the same ground motion set. These EDPs form a fringe that is used to fit the mean and variance of the probability distribution, as shown in Fig. 4.3a. The SSA does not provide the statistical law between the EDPs and ground motion parameters, thus the DSA was developed. This method actually requires more stripe data than the SSA, as shown in Fig. 4.3b. If it is assumed that the relationship between the EDP and ground motion parameters meets the linear law (a logarithmic relationship is also acceptable), the data with the linear relationship can be obtained through the DSA. But, unfortunately, the DSA cannot calculate the variance of the corresponding data, and the calculated variance is not accurate. The CMA was developed to scatter the ground motion parameters within a considered range, and then analyze and obtain the corresponding cloud map data, as shown in Fig. 4.3c. The CMA overcomes the shortcomings of the above two methods, can be used to calculate the statistical law between the EDP and ground motion parameters and the corresponding data variance, and is a widely used theoretical method for vulnerability analysis. For the global analysis method, the multi-fringe method generates multi-fringe data under different ground motion parameters, as shown in

52

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic …

Fig. 4.3d. Because the IDA method must also obtain multi-fringe data when obtaining the IDA curve, the main difference between the two is that the seismic wave used in the IDA method is fixed during scaling, whereas the multi-fringe method can adopt different seismic wave sets under different ground motion parameters. Many scholars have developed methods to reduce the large calculation amount of the two methods. One is the maximum likelihood estimation analysis (MLEA), as shown in Fig. 4.3e, and the other is the application of the soft computing method. We introduce here the developing endurance time analysis (ETA) method, which was actually developed from the IDA method but greatly reduces the corresponding amount of calculation. In general, this approach integrates the considered seismic wave set into three waves for calculation. From the theoretical level, while reducing the amount of calculation, the uncertainty of ground motion in the whole seismic wave set is also reduced, thus it still has a great impact on the vulnerability curve.

4.3.1 International Specification for Ground Motion Determination For the selection of seismic excitation for slope engineering, the most important thing in the actual implementation of the above selection principles and methods of seismic excitation is to consider the seismic geological conditions and slope site types of the site where the slope is located. The principles for determining the site category are briefly introduced here according to the relevant provisions in the Chinese national seismic design codes. The seismic design codes of buildings in China, America, and Europe are briefly introduced from the aspects of site category, seismic response spectrum, and design calculation principles, with the aim to provide references for the design of related projects. Important ground motion parameters in the model can be selected by referring to the site type. The site effect should therefore be considered when calculating the seismic effect. (1) Chinese seismic code The “Code for Seismic Design of Buildings” (GB 50011-2010 2016) divides the construction site into five site categories: I0 , I1 , II, III, and IV, referring to the equivalent shear wave velocity of the soil layer and thickness of the site cover layer, of which category I sites include two subcategories. The “Code for Seismic Design of Railway Engineering” (GB50111-2006 2009) stipulates that the site category is divided into four categories according to the equivalent shear wave velocity Vse value of the soil layer within the calculated site depth. The “Code for Seismic Resistance of Highway Engineering” (JTGB02-2013 2013) stipulates that the site category is divided into four categories according to the average shear wave velocity and site cover thickness. The “Code for Seismic Design of

4.3 Determination of Ground Motion of Slope Site According to the Strong …

IM

53

IM

EDP

EDP (b)

(a) IM

IM

EDP

EDP

(c)

(d)

IM

EDP

EDP (e)

Time(IM) (f)

Fig. 4.3 Selection principle and method for ground motion of a slope site based on historical strong earthquake records: a SSA, b DSA, c CMA, d IDA, e MLEA, f ETA

Hydraulic Structures for Hydropower Engineering” (NB35047-2015 2015) stipulates that the site category is based on the site soil type and site cover thickness, and is divided into five categories. (2) European seismic code The “Design of Structures for Earthquake Resistance—Part 1: General Rules, Seismic Sections and Rules for Buildings” (EN1998-1 2004) is the statutory standard for engineering seismic design in EU countries. According to the different nature of rock and soil, the site is divided into seven site categories, including five basic categories, A–E. Compared with the two special categories, S1 (highly plastic soil)

54

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic …

and S2 (liquefiable soil, sensitive clay), the A–E site categories in the European code are similar to the Chinese code categories of I0 –IV. Special studies on seismic action are required for sites with S1 or S2 site conditions to establish the relationship among the (i) response spectra, thickness of the soft clay layer, and silt shear velocity (Vs) value; (ii) response spectra and the layer and the lower stratum; and (iii) stiffness differences. The possibility of soil damage under earthquake action should be considered for S2 sites. For S1 type sites, it should be noted that such soils usually have very low shear wave velocity VS values, low internal damping, and an abnormally expanded linear performance range, which may cause abnormal seismic site amplification and soil structure interaction effects. (3) United States Seismic Code The United States Seismic Code divides the venue categories into five categories: A, B, C, D, and E. The soil is initially assessed as hard rock, rock, soft rock, hard soil, and soft soil based on the shear wave velocity 30 m below the surface as an index. The code then considers the standard penetration hammer number and undrained shear strength to determine the venue category (ASCE/SEI 7-16 2017). Through comparative analysis, it can be found that: • The site classification methods of European and American standards are basically the same. Among them, the limit values of the number of hammer strokes in the standard penetration test are the same, whereas those of the undrained shear strength slightly differ. • There are great differences in the equivalent shear wave velocity limit values between Chinese, European, and American standards. The United States has a relatively large range of limits, and the classification of hard and weak sites is relatively safe. • Weak site classification: European and American regulations have specific site classification methods for soft soils. The European regulations are more detailed than the American regulations. For the hard site divisions, the American regulations are more detailed than the Chinese and European regulations.

4.3.2 Selection of Strong Earthquake Ground Motion At present, the latest version of the ASCE/SEI 7-16 (2017) specification clearly stipulates that no less than 11 ground motions should be selected for each target response spectrum. Prior to ASCE/SEI 7-16 (2017), the specification required the use of at least three ground motions for nonlinear response time-history analysis. If three ground motions are used, the maximum results obtained from all ground motions must be used to evaluate whether the structure is qualified. If seven or more ground motions are used, the average result can be used for evaluation. However, neither three nor seven ground motions are sufficient to accurately describe the average response or record the recorded response changes. In ASCE/SEI 7-16 (2017), the minimum

4.3 Determination of Ground Motion of Slope Site According to the Strong …

55

number of ground motions has been increased to 11. This number of ground motions is not based on detailed statistical analysis, but is given to balance the competitive goal of a more reliable average structural response evaluation (by using more ground motions) and the amount of calculation (reduced by using less ground motions) to determine the choice. One advantage of using more ground motions is that if more than one of the 11 ground motions produce an unacceptable response, it means that a certain risk level structure will very likely fail to meet the target collapse seismic dynamic performance. For the actual number of ground motions in the actual response analysis process, there is currently no uniform standard across China. A reasonable value can be selected when the regulation requirements are met. This guide does not give a specific criterion for the selection for actual ground motions, but only proposes a selection criterion for use. This case provides a reference in the practical application of subsequent chapters. According to current seismic design codes (Code for Seismic Design of Buildings, GB50011-2010; IBC, 2018; EN-Eurocode 8, 2005) (Fig. 4.4), the design response spectrum of the corresponding site can be determined by the following formula (Table 4.1). In the expression of the recommended response spectrum in the current seismic design codes of the countries mentioned above, the response spectrum is generally divided into 3–4 sections (mostly four): a rising section; a stable section; an attenuation section; and a long-period section. In Eq. (4.13), S DS is the short-period design acceleration response spectrum value, S D1 is the design acceleration value when the period T is equal to 1 s, and TL > 6s is the long period. In Eq. (4.14), αmax is the seismic influence coefficient, η1 and γ are the shape coefficients, η2 is the damping attenuation coefficient, and Tg is the characteristic period of the site corresponding to site categories. In Eq. (4.15), G s (T ) is the site magnification factor for different site types. In Eq. (4.16), TB and TC are the range limits of the periodic constant of the acceleration spectrum. These response spectra recommended by the seismic design codes are the basis and reference for selecting ground motion excitation for nonlinear dynamic time-history analysis of engineering structures such as slopes. Of course, the selection of ground motion excitation is also related to the seismic geological conditions and near and far earthquakes of the site where the slope project is located, but the response spectra (Fig. 4.5) recommended in the seismic design code partially consider these factors. The following example illustrates how to select the ground motion excitation that meets the seismic and geological conditions of the slope site based on the above ground motion selection principles and methods. An example of the modified ground motion model based on response spectrum is introduced. Here, it is assumed that the ground motion of the site is group 2 and the site is category 2. After the design response spectrum of a specific site is determined, the design response spectrum can be uploaded to the PEER seismic database to select a suitable site-measured acceleration time history. Table 4.4 lists some of the basic properties of 50 ground motions selected from the PEER seismic database. Figure 4.6 compares the selected measured response spectra

56

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic …

α

Fig. 4.4 Response spectrum curves recommended in various seismic design codes

4.3 Determination of Ground Motion of Slope Site According to the Strong …

57

Table 4.1 Response spectra in different seismic codes (code for seismic design of buildings, GB50011-2010; IBC, 2018; EN-Eurocode 8, 2005)

USA (a)

( ⎧ S DS 0.4 + ⎪ ⎪ ⎪ ⎪ ⎨ S DS Sa1 (T ) = S D1 ⎪ ⎪ ⎪ ⎪ ⎩ STD1 TL T2

China (b)

Japan (c)

European (d)

Sa2

0.6T T0

)

0 ≤ T < T0 T0 ≤ T ≤ TS TS < T ≤ TL TL < T

⎧ η −0.45 2 ∗ T ∗ αmax + 0.45αmax ⎪ ⎪ 0.1 ⎪ ⎪ ⎨ η2 αmax = ( Tg )γ ⎪ ⎪ T η2 αmax ⎪ ⎪ ( )] ⎩[ η2 ∗ 0.2γ − η1 T − 5Tg αmax

0 < T ≤ 0.1 0.1 < T ≤ Tg Tg < T ≤ 5Tg

Sa4

g

[ ·S· 1+

T TB

(4.14)

T > 5Tg

Sa3 (T ) = Z G s (T )S0 (T ), ⎧ ⎪ T ≤ 0.16 ⎪ ⎨ 3.2 + 30T S0 (T ) = 8.0 0.16 < T ≤ 0.64 ⎪ ⎪ ⎩ 5.12/T 0.64 < T ⎧ ⎪ a ⎪ ⎪ g ⎪ ⎪ ⎨a g = ⎪ a g ⎪ ⎪ ⎪ ⎪ ⎩a

(4.13)

(4.15)

] (2.5η − 1) 0 ≤ T ≤ TB

· S · η · 2.5 [ ] · S · η · 2.5 · TTC [ ] · S · η · 2.5 · TCTT2 D

TB < T ≤ TC TC < T ≤ TD

(4.16)

TD < T

and the target response spectrum. The dark blue upper line represents the mean value plus sigma and the light blue lower line represents the mean value minus sigma. These two lines give a reasonable scope of the selected ground motion spectrum acceleration. The mean value of the selected ground motion spectrum acceleration basically coincides with the designed response spectrum, indicating that the selection of ground motion samples is reasonable. In this book, the PGA is selected as the measurement index (IM) of ground motion intensity. The selection of the observed historical strong ground motion is an effective method to determine the seismic excitation. However, the nonlinear stochastic seismic dynamic response analysis of slope is highly difficult, especially the probabilistic performance-based seismic dynamic stability performance evaluation and analysis of slope presented in this book. Lacking the suitable ground motion samples that meet the condition of the same set system makes it difficult to carry out strict statistical analysis of the nonlinear stochastic seismic dynamic response of a slope based on nonlinear stochastic dynamics. It should be noted that the condition of the same set of seismic ground motions here refers to the ground motion excitation that meets the seismic and geological conditions of the slope site. This difficulty

58

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic … 0.7 USA Seismic Code China Seismic Code Japan Seismic Code European seismic Code

0.6

Sa(g)

0.5

0.4

0.3

0.2

0.1

0

0

1

2

3

4

5

6

7

8

9

10

Period(sec) Fig. 4.5 Illustration of the target response spectrum

exists in the nonlinear seismic dynamic response analysis of slope, as well as in other fields of seismic processing. This is because earthquakes are rare events, which are not controlled or produced by humans. As another example, when studying the stochastic seismic dynamic response of engineering structures, if only the linearity or weak nonlinearity of engineering structures are considered, it is possible to obtain the theoretical solution of the stochastic seismic response. However, for geotechnical materials in slopes that exhibit strong nonlinear mechanical behavior under seismic dynamic action, it is necessary to consider their nonlinear dynamic behavior under seismic action, thus it is difficult to obtain a better theoretical solution. At this time, it is necessary to determine the nonlinear stochastic seismic dynamic response of slope based on the method of nonlinear stochastic dynamics (see Chaps. 3 and 6). For these methods, a large number of earthquake ground motion excitation samples (e.g., acceleration, velocity, displacement time histories) of the same set system meeting the given statistical characteristics (seismic and geological conditions of the same site) are required. When there are not many measured strong earthquake records in the same set that meet the requirements, there is only artificial ground motion. There is also no alternative approach because most of the existing ground motion records do not meet the requirements of a single set system. In the nonlinear seismic dynamic stability analysis of important or special, super-large slope engineering, or in seismic response shaking table (centrifuge shaking table) tests of slope engineering, it is also necessary to select the ground motion input suitable for the seismic and geological conditions of the specific site, and the existing ground motion records sometimes do

4.4 Artificial Seismic Ground Motion Synthesis of Slope Site

59

0

10

-1

Spectral Acceleration(g)

10

-2

10

-3

10

-4

10

Mean Mean+Sigma Mean-Sigma Target

-5

10

-2

10

-1

0

10

10

1

10

Period(sec) Fig. 4.6 Comparison of the acceleration spectra from the selected measured ground motion record and the target response spectrum

not meet the requirements. It is therefore also necessary to use the synthetic ground motion time random process. There are many ways to make ground motion. When testing on site, subtle continuous multiple blasting, industrial blasting, or nuclear blasting can be used. Numerical or simulated ground motion excitation is produced in slope shaking table tests and nonlinear dynamic time-history analysis. This book only discusses the latter. There are two main approaches to this method, namely, the ground motion model based on the source and ground motion model based on the site seismic code.

4.4 Artificial Seismic Ground Motion Synthesis of Slope Site There are two common methods of ground motion simulation: one is the seismological method based on the seismic source; the other is the synthetic seismic waves based on the site.

60

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic …

4.4.1 Source-Based Ground Motion Model According to the different methods of source model building and propagation path characterization, the seismological method includes three categories: the deterministic method; seismic dynamic performance method; and mixed method. Based on the elastic dislocation theory, the displacement of a point in the site caused by fault rupture and slip is expressed as the convolution of the source time function and the space–time convolution of Green’s function, which characterizes the propagation path effect of the seismic wave. The deterministic method is based on the physical mechanism of the seismic source. The simulation of ground motion can be mainly divided into two parts: the acquisition of the source time function and solution of the Green function. A comparison of q is shown in Table 4.2. The stochastic method is a semi-empirical and semi-theoretical method, which simplifies the effects of the source rupture, path propagation, and site effect on ground motion into different function forms, thus the simulation accuracy of ground motion is mainly controlled by the source, path, and site parameters. On the basis of establishing the finite fault model, it is necessary to calculate the ground motion generated by each sub-source at the site. The Fourier spectrum generated by each point source FA (M0 , f, R) can be expressed as: FA (M0 , f, R) = S(M0 , f ) · P(R, f ) · G( f ) · I ( f )

(4.17)

where M0 is the seismic moment, f is the frequency, R is the distance between the sub-source and the site, S(M0 , f ) is the source spectral function, P(R, f ) is the distance decay function, which can be represented as P(R, f ) = Z (R) · D(R, L), Z(R) is the geometric attenuation associated with R, D(R, L) is anelastic attenuation related to R and f , and G( f ) is the site effect function, which can be represented as G( f ) = A( f ) · K ( f ), where A( f ) is a near surface amplitude magnification factor and K ( f ) is the high-frequency cutoff filter. I ( f ) is the ground motion type factor, Table 4.2 Common deterministic methods Methods

Advantages

Limits

Spectral element method

Using 3D crustal velocity structure Simulation effect is better at low frequency

Mesh size should not be too small Long calculation time

Frequency-wavenumber method

The calculation of high-frequency ground motion is efficient No need to divide grid

Difficult to calculate high-frequency ground motion Long calculation time Adopts the horizontal stratification velocity model (different from the actual situation) Excludes the high-frequency random part

4.4 Artificial Seismic Ground Motion Synthesis of Slope Site

61

Table 4.3 Model simulation parameters Earthquake Source parameters magnitude Fault Fault Fault burial length width depth

Fault strike

Fault dip angle

Sliding angle

Sliding Sub-fault value size

Stress drop Path parameters Path quality factor Q

Crustal Medium Radiation Window Geometric shear density intensity function attenuation wave factor velocity

which can determine the data type of synthetic ground motion, mainly including acceleration, velocity, and displacement. The model simulation parameters are listed in Table 4.3. (i) Among these parameters, the main control parameters are introduced below in detail. Fracture velocity and shear wave velocity. For the ratio of rupture velocity to shear wave velocity, it is generally set as 0.8 in practical application. The shear wave velocity of a medium is usually obtained by the borehole method and surface wave method. The shear wave velocity has an effect on both the source term and propagation path. Larger shear wave velocities are associated with smaller amplitude spectra and reduced ground motion intensity. (ii) Stress drop. The stress drop controls the magnitude of dislocation after rupture, and also the intensity of radiation stress waves in a dynamic rupture. For the point source model, the stress drop refers to the difference value between the final stress after the rupture and the initial stress before the rupture, while the initial stress in the crustal medium and the stress after the rupture cannot be determined. For the fault as a whole, the stress drop is the average stress difference on the fault before and after the earthquake. (iii) In the stochastic method, the stress drop is not determined by calculation, but by the inversion of seismic records or experience. There is no fixed method for the stress drop value in the current literature, and is very random in an earthquake. Because the relationship between the stress drop and magnitude (seismic moment) is unclear, the stress drop is generally assumed to be constant for large earthquakes. (iv) High-frequency attenuation parameter: Kappa factor. The Kappa factor is mainly related to the site condition and fault spacing. Softer sites are associated with larger Kappa values, which reflects that smaller peak accelerations are associated with more seismic energy concentrated in the low-frequency part. Smaller fault spacing is associated with a more significant effect of the Kappa value on the peak of ground motion. The Kappa value also affects the response spectrum, and the effect is particularly notable in the high-frequency part. Larger Kappa values are associated with peak values in the high-frequency part of the response spectrum. More attention is paid to high-frequency seismic waves in engineering, and the seismic observation results show that the source radiation and seismic wave propagation effect tend to be random in the high-frequency range, thus the random synthesis

62

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic …

method is often used. However, to acquire broad-band ground motion, the various ground motion generation methods can be applied for low- and high-frequency ground motion. The deterministic method is used in the low-frequency part, while the stochastic is used in the high-frequency part to simulate the ground motion considering the combination of seismic wave propagation and scattering effect. In the low-frequency part, the empirical green function is adopted to characterized the seismic propagation path, while in the high-frequency part, the path transfer function is the product of the geometric diffusion function and anelastic attenuation function. The ground motion generated by the two parts is ultimately matched with a cross frequency and added up in the time domain to acquire the wide-band ground motion. The main task of this method is to reproduce the overall characteristics of the observed motion over a wide frequency range. The common ground motion parameters are horizontal peak ground acceleration (PGA), peak ground velocity (PGV), and response spectrum acceleration (PSA).

4.4.2 Site-Based Ground Motion Model Because earthquake records have the characteristics of non-stationarity in both the time and frequency domain, the seismic design of engineering often pays attention to the frequency spectrum characteristics of the earthquake. With the accumulation of strong earthquake data, random vibration theory has become a means of analyzing the ground motion. The engineering method of stochastic seismic ground motion mainly goes through three stages: stationary model, intensity non-stationary model and intensity-frequency non-stationary model. Here, in this book, for nonlinear stochastic seismic dynamic response analysis of slope only the intensity-frequency fully non-stationary seismic model is introduced. People have long realized that earthquake intensity is non-stationary, and recently, the frequency domain non-stationary has been gradually recognized and concerned. The physical factors causing the non-stationarity in the frequency domain of ground motion are relatively complex, such as the dispersion effect of seismic wave propagation and the response of soil layer at the site, which will all lead to the intensity variation of each frequency component of ground motion. The intensity envelope of each frequency component of the ground motion is different, which means the superior frequency of the ground motion differs in different stages, and the time at which each frequency component reaches the maximum value also differs. This phenomenon is called frequency non-stationarity. To establish the stochastic seismic ground motion model for a slope engineering site, the first thing is to select the classic ground motion power spectrum density model in Eq. (4.18). The approximate conversion relationship between the response spectrum and power spectrum can also be used to convert the target response spectrum into a power spectrum. This must also be determined following the seismic geological conditions and corresponding parameters of the slope engineering site according to

4.4 Artificial Seismic Ground Motion Synthesis of Slope Site

63

the site categories mentioned in this book. 1 + 4ξg2 ωω2 g ) ·( S X¨ (ω) = ( 2 ω2 1 − ω2 + 4ξg2 ωω2 1− g g 2

ω4 ω14 ) 2 2

ω ω12

+ 4ξ12 ωω2 2

· S0 .

(4.18)

1

Equation (4.18) is the classical Clough-Penzien power spectral density function (Clough and Penzien 1975), in which ωg and ξg are the excellent circular frequency and damping ratio of site soil respectively, S0 is the white noise spectrum intensity of bedrock, and the frequency parameter ω1 and damping parameter ξ1 are selected to give the required filtering characteristics. Sa (ω) =

ξ 1 ]. S X¨ (ω) [ −π πω ln ln(1 − r )

(4.19)

ωT0

In Eq. (4.19), Sa (ω) is the target spectrum simulated by the acceleration response spectrum, S X¨ (ω) is the power spectrum density, T0 is the duration of the stochastic process, and ξ is the damping ratio. For characterizing the intensity-frequency nonstationary characteristics of ground motion, the time–frequency modulation function A(t, ω) needs to be introduced: ⎧

[ ( ) ( )] A(t, ω) = I0 exp −b1 ωωta ta − exp −b2 ωωta ta · f (t) S X¨ (t, ω) = |A(t, ω)|2 · S X¨ (ω)

(4.20)

where I0 is the intensity factor, and b1 and b2 are shape parameters of the frequency modulation function. The parameters ωa and ta can be taken according to the ground motion site seismo-geological conditions, and f (t) is an intensity envelope function independent of the frequency ω. Furthermore, the acceleration time-history samples ai (t) of seismic ground motion with rich assigned probability characteristics of the slope engineering site can be obtained by the stochastic process spectrum representation method or orthogonal expansion method (Fig. 4.7). ⎧ ΣN ⎪ ⎨ ai (t) = k=1 At,k [cos(ω1 k t)X k + sin(ωk t)Yk ] [ ] At,k =√2S X¨ (tk , ω).ω 2 , ωk = √ k.ω(k = 1,2, · · · , N ) ⎪ ⎩ X k = 2cos(n. + ϕ), Yk = 2sin(n. + ϕ)(n = 1, 2, · · · , N )

(4.21)

64

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic …

2 0 -2

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15 Time(sec)

20

25

30

Acceleration(g)

2 0 -2 2 0 -2 1 0 -1

Fig. 4.7 Typical representative seismic ground motion acceleration time-history samples

Appendix Refer Table 4.4.

Appendix

65

Table 4.4 Basic properties of 50 ground motions selected from PEER seismic database (new NGA-West 2 ground motion database and new NGA-East ground motion database) Seismic events

Year

Station

Magnitude

Seismic mechanism

Rrup (km)

Humboldt Bay

1937

Ferndale City Hall

5.8

Strike slip

71.57

Imperial Valley-01

1938

El Centro Array #9

5

Strike slip

34.98

Northwest Calif-01

1938

Ferndale City Hall

5.5

Strike slip

53.58

Imperial Valley-02

1940

El Centro Array #9

6.95

Strike slip

6.09

Northwest Calif-02

1941

Ferndale City Hall

6.6

Strike slip

91.22

Northern Calif-01

1941

Ferndale City Hall

6.4

Strike slip

44.68

Borrego

1942

El Centro Array #9

6.5

Strike slip

56.88

Imperial Valley-03

1951

El Centro Array #9

5.6

Strike slip

25.24

Northwest Calif-03

1951

Ferndale City Hall

5.8

Strike slip

53.77

Kern County

1952

LA-Hollywood Stor FF

7.36

Reverse

117.75

Kern County

1952

Pasadena-CIT

7.36

Reverse

125.59

Kern County

1952

Santa Barbara Courthouse

7.36

Reverse

82.19

Kern County

1952

Taft Lincoln School

7.36

Reverse

38.89

Northern Calif-02

1952

Ferndale City Hall

5.2

Strike slip

43.28

Southern Calif

1952

San Luis Obispo

6

Strike slip

73.41

Imperial Valley-04

1953

El Centro Array #9

5.5

Strike slip

15.64

Central Calif-01

1954

Hollister City Hall

5.3

Strike slip

25.81

Northern Calif-03

1954

Ferndale City Hall

6.5

Strike slip

27.02

Imperial Valley-05

1955

El Centro Array #9

5.4

Strike slip

14.88

ElAlamo

1956

El Centro Array #9

6.8

Strike slip

Central Calif-02

1960

Hollister City Hall

5

Strike slip

121.7 9.02 (continued)

66

4 Seismic Ground Motion Excitations for Slope Seismic Dynamic …

Table 4.4 (continued) Seismic events

Year

Station

Magnitude

Seismic mechanism

Rrup (km)

Northern Calif-04

1960

Ferndale City Hall

5.7

Strike slip

57.21

Hollister-01

1961

Hollister City Hall

5.6

Strike slip

19.56

Hollister-02

1961

Hollister City Hall

5.5

Strike slip

18.08

Parkfield

1966

Cholame—Shandon Array

6.19

Strike slip

17.64

Parkfield

1966

Cholame—Shandon Array

6.19

Strike slip

9.58

Parkfield

1966

Cholame—Shandon Array

6.19

Strike slip

12.9

Parkfield

1966

San Luis Obispo

6.19

Strike slip

63.34

Parkfield

1966

Temblor pre-1969

6.19

Strike slip

15.96

Northern Calif-05

1967

Ferndale City Hall

5.6

Strike slip

28.73

Northern Calif-06

1967

Hollister City Hall

5.2

Strike slip

37.69

Borrego Mtn 1968

El Centro Array #9

6.63

Strike slip

45.66

Borrego Mtn 1968

LA-Hollywood Stor FF

6.63

Strike slip

222.42

Borrego Mtn 1968

LB—Terminal Island 6.63

Strike slip

199.84

Borrego Mtn 1968

Pasadena-CIT

6.63

Strike slip

207.14

Borrego Mtn 1968

San Onofre—So Cal Edison

6.63

Strike slip

129.11

Lytle Creek

1970

Castaic—Old Ridge Route

5.33

Reverse

103.39

Lytle Creek

1970

Cedar Springs Pumphouse

5.33

Reverse

22.94

Lytle Creek

1970

Cedar Springs_Allen 5.33 Ranch

Reverse

19.35

Lytle Creek

1970

Colton—So Cal Edison

Reverse

30.11

5.33

Lytle Creek

1970

Devil’s Canyon

5.33

Reverse

20.24

Lytle Creek

1970

LA—Hollywood Stor FF

5.33

Reverse

73.67

Lytle Creek

1970

Lake Hughes #1

5.33

Reverse

90.42

Lytle Creek

1970

Puddingstone Dam

5.33

Reverse

30.02

Lytle Creek

1970

Santa Anita Dam

5.33

Reverse

42.52 (continued)

References

67

Table 4.4 (continued) Seismic events

Year

Station

Magnitude

Seismic mechanism

Rrup (km)

Lytle Creek

1970

Wrightwood—6074 Park Dr

5.33

Reverse

12.14

San Fernando

1971

2516 Via Tejon PV

6.61

Reverse

55.2

Helena_ Montana-01

1935

Carroll College 6

6

Strike slip

2.86

Helena_ Montana-02

1935

Elena Fed Bldg

6

Strike slip

2.92

References ASCE/SEI 7-16 (2017) Minimum design loads and associated criteria for buildings and other structures. C.F.R. Clough RW, Penzien J (1975) Dynamics of structures. McGraw-Hill, USA EN1998-1 (2004) Design of structures for earthquake resistance—part 1: general rules, seismic actions and rules for buildings. C.F.R. GB 50111-2006 (2009) Code for seismic design of railway engineering. C.F.R. GB 50011-2010 (2016) Code for seismic design of buildings. C.F.R. JTGB02-2013 (2013) Specification of seismic design for highway engineering. C.F.R. NB35047 (2015) Code for seismic design of hydraulic structures for hydropower engineering. C.F.R. Wang GY, Cheng GD, Shao ZM, Chen HQ (1999) Optimal fortification intensity and reliability of aseismic structures. Science Press, Beijing

Chapter 5

Deterministic Analysis Methods for Slope Seismic Dynamic Response

5.1 Seismic Response Analysis of Slope Based on the Quasi-static Method The pseudo-static method was the first approach used to analyze slope dynamic stability. Owing to its simplicity, practicality, and rich engineering experience, it remains the most commonly used method to evaluate the seismic stability of slope. Terzaghi (1950) first applied the quasi-static method to seismic slope stability analysis. The quasi-static method simplifies seismic action to a horizontal or vertical constant acceleration that generates inertia forces acting on an unstable mass center. According to limit equilibrium theory, all of the forces acting on the potential sliding body are decomposed along the sliding surface, and the safety factor along the sliding surface is obtained. The factor of safety is related to the slope material characteristic value, the shape and location of the failure surface, and the magnitude of the seismic force. In quasi-static analysis, the slope material characteristics can be measured by field or laboratory tests. The shape and location of the failure surface are usually determined by experience or engineering analogy according to the geological conditions of the slope, and can be simplified as straight, circular, or non-circular. To calculate the magnitude of the seismic force, the effect of the seismic dynamic force on the structure is considered to be equivalent to the inertia force directly and statically loaded on the structure. Earthquakes produce vibrational forces, which are usually characterized by oscillations and transient behavior. However, regardless of the complex reaction, the quasi-static method applies the static force instead of the dynamic force to analyze earthquake-induced slope damage. Acceleration can produce inertial forces, which in turn cause ground motion effects. These forces act on the center of mass of each strip in the form of horizontal and vertical forces, and the pseudo-static force F is the seismic coefficient k times the gravity of the sliding body G. In the analysis process, the seismic equivalent static force is usually decomposed into horizontal and vertical directions. Existing studies and seismic codes of slopes © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Huang et al., Guidelines for Probabilistic Performance-Based Seismic Design and Assessment of Slope Engineering, https://doi.org/10.1007/978-981-19-9183-7_5

69

70

5 Deterministic Analysis Methods for Slope Seismic Dynamic Response

mainly consider the horizontal seismic inertia force in the slope seismic response analysis. The formula for calculating the horizontal seismic inertia force is: Fh =

ah G = kh G. g

(5.1)

The formula for calculating the seismic inertia force in the vertical direction is: Fv =

av G = kv G. g

(5.2)

In Eqs. (5.1) and (5.2), ah and av are the pseudo-static seismic accelerations in the horizontal and vertical directions, respectively, G is the gravity of the investigated strip, g is the acceleration of gravity, and kh and kv are seismic coefficients in the horizontal and vertical directions, respectively. When the quasi-static method is used to analyze slope stability, the soil strips are divided into vertical strips to allow gravity to act on the sliding surface along the vertical direction. This process is usually combined with the limit equilibrium theory for slope stability analysis to obtain the safety factor (Fig. 5.1). The formula to calculate the safety factor (Fs ) using the Swedish article points method is as follows: ) ] ∑ ∑ [( 1 i cl i i G i ± Fv,i cosα − ubi cosα − Fh,i sinα tanϕ + ] ) Fs = . (5.3) ∑ [( i G i ± Fv,i sinα + Mc,i /R The formula to calculate Fs using the Bishop method is as follows:

Fig. 5.1 Calculation diagram of the Swedish article points method

5.1 Seismic Response Analysis of Slope Based on the Quasi-static Method

∑ Fs =

1 ma

) ] ∑ G i ± Fv,i cosα − ubi tanϕ + cbi ) ] . ∑[( G i ± Fv,i sinα + Mc,i /R

71

[(

(5.4)

In Eqs. (5.3) and (5.4), for the ith strip, G i is the weight of soil strip, bi is the width of the soil strip, α is the included angle between the gravity direction of the strip and the radius of the midpoint passing through the bottom of the strip, ϕ is the internal friction angle of soil, c is the cohesive force of the soil, u is the pore pressure acting on the bottom of soil strip, Fh,i is the representative value of the horizontal seismic inertial force acting at the center of gravity on the corresponding ith strip, Fv,i is the representative value of the vertical seismic inertia force acting on the center of gravity of the corresponding ith strip, li is the length of the slope, generally taken as a unit length for analysis, and Mc,i is the sliding moment caused by the horizontal seismic inertial force Fh,i . The Swedish article points method is established in the slope sliding body vertical section, and its physical significance is very clear. However, it is unreasonable to calculate the anti-sliding moment caused by the horizontal seismic force and inertia force when using a vertical bar. The Bishop method does not consider the antisliding moment generated by the horizontal seismic force, which may be to avoid the unreasonable calculation of the anti-sliding moment in the Swedish article points method. In addition to the limit equilibrium method for quasi-static analysis, the finite element method can also be adopted for numerical simulation and the strength reduction method for slope stability calculation. The stress and strain relation of the soil is not considered when dividing the slope, and the displacement and stress field of the slope under unstable conditions cannot be reflected. The characteristic of numerical simulation analysis method is to consider the relationship between soil stress and strain, which overcomes the shortcoming of the limit equilibrium method that does not consider the relationship between soil stress and strain. The finite element method not only simulates the overall slope failure as does the limit equilibrium method, but also simulates the local slope failure and integrates the overall and local failure into a unified system. The strength reduction method reduces the shear strength indexes c and ϕ of the soil using a reduction factor Fs , and then replaces the original shear strength indexes c and ϕ with the reduced shear strength indexes c F and ϕ F , and continuously decreases until the slope surface is damaged and the safety factor Fs is obtained. The safety factor method is the most direct quantitative calculation method to express slope stability, and its expression is as follows: cF =

tanϕ , Fs

tanϕ F =

tanϕ . Fs

(5.5) (5.6)

72

5 Deterministic Analysis Methods for Slope Seismic Dynamic Response

The Mohr–Coulomb yield criterion is adopted in the traditional limit equilibrium method of slope stability. The safety factor is defined as the ratio of the shear strength of the sliding surface to the actual shear force of the sliding surface, which can be expressed as: ∫l Fs =

0 (c

+ σ tanϕ)dl . ∫l 0 τ dl

(5.7)

When the left side of Eq. (5.6) is equal to 1, this indicates that the slope has reached the ultimate equilibrium state after the shear strength parameter is reduced by F s . The idea of the finite element method is essentially the same as that of traditional methods, but the advantages of the finite element strength reduction method are shown in the following aspects: (i) The nonlinear elastoplastic constitutive relation of soil is considered. (ii) It can dynamically simulate the instability process of slope and shape of the slip surface. (iii) When solving the safety factor, it is not necessary to assume the shape of the slip surface or the interaction force between soil strips. Because the earthquake force in the equivalent pseudo-static method is considered to be a static effect, the conventional static stability analysis method can be used for analysis. This greatly simplifies the sliding body during the seismic dynamic failure process and the calculation and theory are relatively simple. Using the pseudo-static method to analyze the slope seismic response analysis also has the advantage of fast speed and high efficiency. However, the quasi-static method itself has defects, mainly including the following aspects: (i)

The quasi-static method assumes that slopes are absolutely rigid bodies with the same acceleration at any point, whereas in practice most slopes are deformable rather than rigid bodies. (ii) The quasi-static method assumes that the quasi-static force does not change. However, the quasi-static force is not constant in size, nor unidirectional, but rapidly fluctuates in magnitude and direction, and is characterized by alternation and pulsation. The pseudo-static method cannot reflect the intensityspectrum characteristics of ground motion. (iii) The pseudo-static method assumes that slope failure is the only failure mode and only occurs when the safety factor is less than 1. However, overall slope instability may not occur even if the safety factor is less than 1, but may only lead to a certain permanent deformation of the slope. (iv) The seismic dynamic coefficient in the pseudo-static method is generally selected according to experience, which is arbitrary and lacks theoretical basis. This is not consistent with the actual situation to use a specific seismic dynamic coefficient to calculate and analyze the whole slope in the horizontal or vertical direction. The seismic dynamic coefficient may be a function of lithology, time,

5.2 Slope Dynamic Response Analysis Method Based on the Response …

73

distance from the epicenter, or something else, which must be determined by relevant research. In general, the quasi-static method cannot reflect the strong nonlinear dynamic behavior of soil–rock mixtures in slope under seismic dynamic action.

5.2 Slope Dynamic Response Analysis Method Based on the Response Spectrum Method Compared with the quasi-static method, the dynamic characteristics of the structure and ground during an earthquake can be considered in the response spectrum. This indicates that the slope seismic analysis method has, to some degree, formally entered the stage of dynamic research. The main idea of the response spectrum method is that a multiple-degree-of-freedom structure can be simplified into a combination of multiple single-degree-of-freedom structures. Using the single-degree-of-freedom structures allows the earthquake response to be more reasonably calculated. Biot (1941) and Housner (1941) proposed the concept of the response spectrum to calculate the seismic response in an elastic state. Housner (1941) calculated the first batch of response spectrum curves by selecting a large number of representative strong earthquake acceleration records and processing the accelerations. The response spectrum theory was adopted in the 1956 California Anti-seismic Design Code. The emergence of this theory has played an important role in promoting the development of structural seismic design. The seismic response spectrum refers to the relationship curve between the maximum seismic response under a single-degree-of-freedom system, including the acceleration, velocity, displacement, and the natural vibration period of the structure. Among them, the acceleration response spectrum is the most commonly applied. The response spectrum theory considers the dynamic relationship between the structural dynamic characteristics and ground motion characteristics and maintains the original static theory form. The maximum seismic base shear force on the structure is written as: V0 = kβ(T )W,

(5.8)

where β(T ) is the ratio of acceleration response spectrum Sa (T ) to the maximum acceleration of ground motion a, which represents the amplification of the acceleration of the structure. This can be written as: β(T ) =

Sa (T ) . a

(5.9)

The design response spectrum is plotted based on the relationship between the seismic influence coefficient and the natural vibration period T of the structure, and the horizontal inertial force of the structure is calculated.

74

5 Deterministic Analysis Methods for Slope Seismic Dynamic Response

The response spectrum theory can be described as follows. The structure can be simplified as a multi-degree-of-freedom system, and the seismic response of the multi-degree-of-freedom system can be decomposed into a combination of multiple single-degree-of-freedom system responses according to the vibration mode. The maximum response of each single-degree-of-freedom system can be obtained from the response spectrum. Because the maximum value of each mode reaction of the same reaction quantity does not occur at the same time and may be positive or negative, it is necessary to study how to determine the system reaction after the mode superposition from the maximum reaction of each mode reaction. With the continuous development of response spectrum theory in the 1960s and 1970s, Newmark (1979) proposed methods for estimating the linear response spectrum and nonlinear response spectrum of ground motion and obtained the tripartite spectrum. Fajfar and Fischinger (1990) later directly obtained the elastic–plastic response spectrum through elastic response spectrum reduction. Chopra and Goel (1999) reduced the elastic response spectrum to an elastoplastic response spectrum using the improved capacity spectrum method, and transformed it into a demand spectrum. The modal combination method commonly used in engineering field is the square root of the sum of squares (SRSS), which assumes that each mode response of the structure does not affect each other. When the structure has a similar natural frequency, the maximum response value calculated by this method may be too large or too small. According to the random vibration theory of structures, the complete quadratic grouping method was then presented based on the SRSS method, namely the conventional modal decomposition reaction spectrum method. After extensive study, there are now a variety of mode-combination methods of the reaction spectrum. The most commonly used methods are the SRSS method and the complete quadratic combination (CQC) method. The equation calculated by the SRSS method is: F=

/ Σn i=1

Si 2 ,

(5.10)

where F is the seismic effect of the structure, Si is the seismic effect of the structural mode i, and n is the mode order. The SRSS method has high accuracy in the seismic calculation of structures with large period differences of the principal modes, and is suitable for single-direction seismic effect problems. The equation calculated by the CQC method is: ρT = F=

Ti 0.1 , ≥ Tj 0.1 + ξ

/Σ Σ n n i=1

j=1

Si ri j S j ,

(5.11)

(5.12)

5.3 Newmark Sliding Block Displacement Method for Seismic Dynamic …

ri j =

8ξ 2 (1 + ρT )ρT 2/3 2

((1 + ρT )2 ) + 4ξ 2 ρT (1 + ρT )2

.

75

(5.13)

Equation (5.11) is the conditional equation applicable to the CQC method, ρT is the cycle ratio, T j and Ti (T j ≥ Ti ) are the periods of two adjacent modes, ξ is the damping ratio, Si and S j are the seismic effects of order i and j, respectively, and ri j is the correlation coefficient of the mode combination. The seismic influence coefficient of a slope is analyzed by the response spectrum method, and the distribution characteristics of the different amplification coefficients of the above parameters are studied by taking the dynamic peak acceleration under seismic action obtained by the seismic safety evaluation as the basic parameter to obtain the slope vibration mode. On the basis of the calculated earthquake influence coefficient in different positions of the slope and the seismic inertia force, the seismic inertia force is superposed onto each cell in the body of the slope stability analysis. The slope under seismic dynamic modal and stress is then calculated according to the strength subtraction and the slope stability coefficient can be obtained, which is a more advanced method of quasi-dynamic analysis. The response spectrum analysis method is based on the response of an elastic system with a single mass point in the actual earthquake process to analyze the structural response. According to this theory, the response of a slope can be calculated according to the actual ground motion using the seismic spectrum curves. The response spectrum method partly reflects the spectral characteristics of ground motion and can reflect the weakly linear dynamic mechanical behavior. However, the nature of the intension-spectrum characteristics of ground motion is still not addressed, and can thus not reflect strong nonlinear mechanical behavior.

5.3 Newmark Sliding Block Displacement Method for Seismic Dynamic Response Analysis of Slope In the Rankine lecture in 1965, Newmark proposed a seismic permanent displacement analysis method (Newmark 1965). The method of using permanent displacement to evaluate the seismic performance of a slope is called the Newmark slider displacement method, also known as the rigid slider method. The basic idea is that when the ¨ of the sliding body exceeds the yield acceleration a y , permanent acceleration δ(t) deformation occurs in the ith period according to: ¨ δi =

] [ ¨ − a y dtdt. δ(t)

(5.14)

The seismic acceleration is reciprocating and the total permanent deformation is the accumulation of deformation in each period:

76

5 Deterministic Analysis Methods for Slope Seismic Dynamic Response

δ=

Σn i=1

δi .

(5.15)

More seismic permanent displacement methods have been proposed, such as the decoupled seismic permanent displacement method (Makdisi and Seed 1978) and the coupled seismic permanent displacement method (Bray and Travasarou 2007). The most primitive Newmark slider method assumes that the sliding body is a rigid body, and that the sliding body itself will not deform under earthquake action. The decoupling method separates dynamic analysis from permanent displacement calculation and can be more complex and accurate. The decoupling method is combined with the finite element method for dynamic analysis, and the permanent displacement is then calculated by the stress field obtained from the slope dynamic analysis. The influence of the seismic response on the permanent displacement and the plastic deformation on seismic dynamic response can both be considered. The Newmark method believes that even if the instantaneous value of the slope safety factor is less than 1, it will not necessarily be unstable, but will only produce a certain permanent displacement. This method partly considers the time-domain characteristics of ground motions and the nonlinear behavior of soil. However, this method assumes that the slope soil is a rigid plastic body during an earthquake and that the strength of the soil body will not significantly decrease during an earthquake, which is quite different from the actual situation. It is also difficult to reflect the true dynamic behavior of the slope under earthquake action.

5.4 Slope Seismic Dynamic Response Analysis Based on Nonlinear Dynamic Time-History Analysis Around the 1960s, with the continuous advancement of science and technology, great computational improvements, and rapid development of the level of experimental technology, the collective understanding of the laws of earthquakes has continued to deepen. Structural nonlinear dynamic time-history analysis technology has also achieved rapid development and is a direct dynamic method. Because the actual seismic records are directly adopted to analyze the seismic response of the structure, the three elements that characterize the ground motion (amplitude, frequency spectrum, and duration of the seismic design) are fully considered. The elastoplastic properties of structures or components also use a more reasonable full-process restoring force curve model. The calculation results thus provide a detailed and specific full process of the elastoplastic seismic response of the structure, which has very important scientific significance. Nonlinear dynamic time-history analysis is also a very accurate method for slope seismic systems composed of soil and a retaining structure. In the whole dynamic history, the slope seismic system will produce cumulative damage, which changes its properties and soil structure. This results in the residual deformation and strength loss of soil and stiffness changes may lead to slope resonance, thus resulting in damage.

5.4 Slope Seismic Dynamic Response Analysis Based on Nonlinear …

77

Slope aseismic systems are a nonlinear structure system that includes material nonlinearity, geometric nonlinearity, and state nonlinearity. Various analysis methods of a nonlinear system can be summarized by establishing the balance equation of internal work and external force, which is the nonlinear equation of node displacement, and solving the balance equation. For a nonlinear dynamic system of slope subjected to seismic ground motion, the equilibrium equation must first be established based on the Hamiltonian principle, as follows: ( ) ¨ + C X˙ + f X, X˙ = −M J U¨ g (t), MX

(5.16)

where M and C are the total mass matrix and total damping matrix, respectively, ¨ X, ˙ and X are the node acceleration, velocity and displacement arrays, respecX, ( ) tively, f X, X˙ is the nonlinear restoring force model, U¨ g (t) is the input seismic acceleration array, and J is the indicator matrix of each ground motion component. The rock and soil parameters for a large complex slope are highly uncertain owing to the joint action of complex conditions, such as the geological environment of the slope engineering site, slope geometry, and surrounding underground water level. The rock and soil materials of the slope body are generally highly nonlinear materials and have different deformation characteristics under different stress levels. The slope engineering structure seismic design and performance evaluation require an accurate forecast of the ground motion excitation and a reasonable slope material constitutive description. The seismic dynamic response analysis method has high precision and can accurately represent the slope under the action of an earthquake dynamic response, as well as the dynamic stability of the key factors, including the selection of a suitable dynamic constitutive model. Many kinds of soil constitutive relations have been developed in slope analysis. The dynamic constitutive model of soil includes the dynamic viscoelastic plastic model, equivalent linear dynamic viscoelastic model, and dynamic plastic model of soil. The nonlinear dynamic constitutive models, including the bilinear model (Granger and Andersen 1979), Ramberg–Osgood model (Ramberg and Osgood 1943), and Davidenkov model (Martin and Boltonseed 1983), which describe the stress–strain relationship of a slope body, are widely adopted in the dynamic analysis of soil mass, and can accurately determine the shear stress and shear strain response of soil mass under earthquake action. However, they cannot calculate permanent deformation nor consider the influence of the stress path and soil anisotropy, and also have large errors under high deformation. On the basis of the nonlinear dynamic constitutive model, the elastic–plastic dynamic constitutive model (e.g., multi-yield surface model) and internal time model have been proposed, which can consider the plastic cumulative deformation of soil. In the solution process, the equilibrium equation of the system should be continuously modified according to the nonlinear characteristics of the system to obtain the incremental solution of the equation. The increment is calculated by the direct integral method, central difference method, Wilson-θ, and Newmark method. The

78

5 Deterministic Analysis Methods for Slope Seismic Dynamic Response

stiffness matrix K of the system should be constantly modified and iterated at every calculation moment according to the nonlinearity of the system. At present, the common balanced iterative methods include the modified Newton iterative method and the Broyden–Fletcher–Goldfarb–Shanno method (Head and Zerner 1985). After each iteration, the result should be checked and assessed according to the selected convergence criterion. If the convergence condition is satisfied, the iteration on this time step ends. Otherwise, the iteration continues. The dynamic time-history analysis method starts from the selected input seismic wave (seismic wave acceleration time history), then uses the multi-node and multidegree-of-freedom slope finite element dynamic calculation model to establish the dynamic equation, followed by the step integral method to solve the equation and obtains the displacement, then finally to calculate the velocity and acceleration response of each instantaneous slope in the earthquake process. The change of the internal force and the entire process of slope instability under earthquake action are thus analyzed. This situation reflects the direction and characteristics of ground motion and the influence of continuous action on the final result. This represents the spectral characteristics of ground motion and can reflect the strong nonlinear mechanical behavior of rock and soil under seismic dynamic action. Although this method is more accurate than the response spectrum method, it can better represent the actual situation of the structure, has higher calculation accuracy, and can easily calculate the response of the slope in the nonlinear stage. But this method requires a lot of work, the calculation time of the model is long, and the post-processing work is complicated. At the same time, the sensitivity of the ground motion in this method is very high, and the selection of the seismic wave has a clear influence on the change of the final result.

5.5 Large Deformation Analysis Method of Slope Soil is a special accumulation of granular material. In the process of soil instability, granular material moves rapidly, its state changes from elastic to plastic, and the stress–strain relationship no longer conforms to the assumption of small deformation, but enters the stage of large deformation. Disasters caused by the large deformation of slope soil have very severe consequences, such as long sliding distance, large disaster intensity, and very serious losses. An effective way to reduce the risk and loss is to determine the possible disaster range and disaster intensity by analyzing the large deformation of slope (Fig. 5.2). Many scholars have systematically studied the large deformation analysis methods of soil flow hazards and compared these analysis methods, which are divided into three categories: empirical statistical model; simplified analytical model; and numerical analysis model. The experienced statistical model uses field investigations and the remote sensing image analysis method for the large deformation of soil flow disaster geomorphologic

5.5 Large Deformation Analysis Method of Slope

79

Soil and ash layer North

Old landslide deposit

South Lake

Initial failure of lacustrine clay

Major motion

Large massive clods

200m Fig. 5.2 Entire slope evolution process from plastic deformation to crack emergence and development, and then instability to large deformation flow (Modified from Hu 1988)

geometric features (e.g., slope height, volume) and statistical analysis, and geometric features and flow slide landforms parameters (e.g., sliding distance, flow depth) for the regression analysis to obtain a series of empirical statistical models. The empirical statistical model is relatively simple and easy to use, but is strongly dependent on statistical data. Geological conditions and geomorphic environments substantially differ from region to region, thus leading to a strong specificity and high limitations. The physical mechanism of landslides is also not considered in the model building process, which lowers the model accuracy. The simplified analytical model can reasonably simplify the movement process of landslides and establish a physical model based on the particle dynamics principle to solve the flow slide parameters. The simplified analytical model can be divided into two categories: a particle motion model and a continuous or discontinuous analysis model. The particle motion model of a landslide is simplified as a particle. Along the curved path of movement, the friction angle of the sliding surface remains unchanged throughout the entire movement process, and movement in the process of friction power loss of potential energy is used to overcome all slide surfaces based on the assumption of the final sliding distance and velocity solutions. The continuous or

80

5 Deterministic Analysis Methods for Slope Seismic Dynamic Response

discontinuous analysis model considers the deformation of rock and soil mass to a certain extent, such as the equivalent fluid model and particle flow model. Compared with the empirical statistical model, the simplified analytical model can simplify the landslide movement process to a certain extent and accurately predict the flow slip parameters of soil with large deformation. However, the existing simplified analytical models lack the comprehensive analysis of the large deformation flow process of soil. Numerical methods can simultaneously consider the geometrical information of complex slopes, such as nonlinear soil, elastic–plastic soil, and pore water. Numerical simulations have the advantages of low cost and good repeatability, and have therefore become an important means to study the flow hazards of soil with large deformation. There are two kinds of numerical analysis methods in common use at present: numerical methods based on a discontinuous medium; and numerical methods based on a continuous medium. The numerical methods based on discontinuous media mainly include the discrete element method (DEM), discontinuous deformation analysis (DDA), and numerical manifold method. Numerical models based on a continuum mainly include the finite element method (FEM), smooth particle hydrodynamics (SPH), and finite difference method (FDM), among others. The FEM and FDM have high computational efficiency, but face problems in the analysis of large deformation, such as mesh winding and distortion, which reduce the computational accuracy and even errors. Smooth particle hydrodynamics is derived from computational fluid dynamics (CFD). In the whole computational process, the motion information of each particle is tracked at all times to analyze the motion and deformation process of the whole problem domain, which can avoid the problems of insufficient accuracy caused by mesh entanglement and distortion during large deformation. It is of great significance to design slope seismic analysis accurately and reasonably, because the slope instability caused by an earthquake often causes huge casualties and property losses. Slope seismic analysis has been treated with the quasi-static method, response spectrum method, and dynamic time-history analysis method, and is gradually advancing to the dynamic analysis method based on probability. The slope large deformation analysis method has also gradually improved. In the slope response analysis, the method should be chosen according to the actual situation to meet the needs of the practical engineering. For the large deformation analysis of seismic dynamic response of slope based on SPH, the dynamic differential equation must be rewritten as follows: ( αβ ) αβ ΣN σj σi dviα ∂ Wi j αβ = mj + 2 + δ αβ Πi j + f n Ri j + Fi . (5.17) β j=1 dt ρi2 ρj ∂ xi The numerical manifold method (NMM) is also a new numerical method created by Shi (1991) using finite coverage technology. It has two sets of independent grids, namely, a mathematical grid reflecting the numerical accuracy and a physical grid representing the geometric characteristics. The NMM can solve continuous and discontinuous deformation problems and can be understood as a generalized

References

81

numerical method including finite element and discontinuous deformation analysis methods. The NMM has been successfully applied to elastoplastic analysis, anchorage support, and seismic dynamic response analysis of slope engineering. For the seismic dynamic stability performance evaluation of slope by NMM, the dynamic differential equation can still be written in the form of Eq. (5.14). Using NMM to solve Eq. (5.17) requires the introduction of damping to dissipate the kinetic energy generated by the system.

References Biot MA (1941) A mechanical analyzer for the prediction of earthquake stresses. Bull Seismol Soc Am 31(2):151–171 Bray JD, Travasarou T (2007) Simplified procedure for estimating earthquake-induced deviatoric slope displacements. J Geotech Geoenviron Eng 133(4):381–392 Chopra AK, Goel RK (1999) Capacity-demand-diagram methods for estimating seismic deformation of inelastic structures: SDF systems. Report No. PEER1999/02. Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA Fajfar P, Fischinger M (1990) Earthquake design spectra considering duration of ground motion. Paper presented at the proceedings 2nd US national conference on earthquake engineering Granger CW, Andersen AP (1979) An introduction to bilinear time series models. J Am Stat Assoc 74(368):927 Head JD, Zerner MC (1985) A Broyden–Fletcher–Goldfarb–Shanno optimization procedure for molecular geometries. Chem Phys Lett 122(3):264–270 Housner GW (1941) Calculating the response of an oscillator to arbitrary ground motion. Bull Seismol Soc Am 31(2):143–149 Hu YX (1988) Earthquake engineering. Seismological Press, Beijing Makdisi FI, Seed HB (1978) Simplified procedure for estimating dam and embankment earthquakeinduced deformations. J Geotech Eng Div-ASCE 104(7):849–867 Martin PP, Boltonseed H (1983) One-dimensional dynamic ground response analyses. Int J Rock Mech 20(1):A9–A9 Newmark NM (1965) Effects of earthquakes on dams and embankments. Géotechnique 15(2):139– 160 Newmark N (1979) Earthquake resistant design and ATC provisions. Paper presented at the proceedings, 3rd WCEE Ramberg W, Osgood WR (1943) Description of stress-strain curves by three parameters. National Advisory Committee for Aeronautics, Washington, D.C. Shi G (1991) Manifold method of material analysis[C]//Transactions of the 9th army conference on applied mathematics and computing. Minneapolis, USA Terzaghi (1950) Mechanisms of landslide. Eng Geol

Chapter 6

Probabilistic Performance-Based Seismic Design and Assessment for Slope Engineering

There is a considerable amount of randomness and uncertainties in the seismic systems of slope engineering, including the spatial variability of geotechnical physical properties and the random non-stationary properties of ground motion frequency and intensity. Under the influence of these uncertain factors, the performance status of slope is also uncertain. This makes it difficult to accurately predict the slope dynamic response. These uncertainties and randomness must therefore be considered in the design of the slope seismic performance to quantitatively characterize these uncertain effects. The biggest advantage of the probabilistic analysis method is that it cannot only quantitatively characterizes the uncertainty of the impact on the system, and provides more objective evaluation indicators, but can also combine almost all existing stability analysis methods of slope engineering (e.g., pseudo-static method, limit analysis method, Newmark slider displacement method). The probabilistic analysis method is an improvement of the traditional deterministic analysis method, rather than a change or rejection. There are two key points in the performance-based seismic engineering method: (1) the selection of suitable structural seismic performance evaluation indicators; and (2) quantification of the uncertainty and its influence (Cornell et al. 2002). After quantitatively characterizing the uncertainty of the slope system, this chapter introduces the stochastic dynamic analysis method and accordingly establishes a risk assessment framework for the slope seismic performance. Human society also has increasingly high requirements for slope engineering performance alongside the continuous development of technology and economy. This chapter therefore also briefly introduces the design of the seismic performance based on the concept of resilience.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Huang et al., Guidelines for Probabilistic Performance-Based Seismic Design and Assessment of Slope Engineering, https://doi.org/10.1007/978-981-19-9183-7_6

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6.1 Source of Uncertainty and Its Description Accurate ground motion predictions and reasonable slope material constitutive description are the basic factors to accurately predict the dynamic response for slope seismic design and safety performance evaluation. However, owing to the randomness of ground motion and inability to fully obtain the spatial distribution of site geotechnical parameters due to engineering constraints, a stochastic quantitative characterization based on statistics can establish more realistic ground motion and random field models. This section explains the effects of rock and soil parameter variability and the nonstationary properties of earthquake excitation intensities and frequencies on slope stability from the perspective of stochastic dynamic equations. The basic methods of random field establishment and stochastic ground motion generation are presented.

6.1.1 Sources of Uncertainty in a Seismic Dynamic System of Slope The influence of the uncertainty factors of the slope dynamic system itself is first considered based on the dynamic equation. Here, based on D Alembert’s principle (D’Alembert 1743), the dynamic equation of slope engineering under the action of ground motion x¨ g (t) can be rewritten as: ¨ + C X˙ + f ( X, ˙ X) = −M I x¨ g (t) MX

(6.1)

where M and C represent the mass and damping, respectively, which are expressed ˙ X) represents the nonlinear as matrixes for a multi-degree-of-freedom system, f ( X, dynamic behavior of slope rock and soil under seismic excitation, I is the unit vector ¨ X, ˙ and X are the acceleration, velocity, and displacement, respectively, matrix, X, and x¨ g (t) is the ground motion acceleration. (1) Spatial variability of geotechnical parameters There are three main sources of uncertainty in rock and soil parameters. (a) Rock and soil are natural products and their physical properties exhibit spatiotemporal variability owing to weathering, denudation, transport, deposition, and different loading histories. (b) The size of a slope body is generally considerably larger than that of the superstructure. (c) The rock and soil materials of a slope have strong nonlinearity and are highly sensitive to ground motion. This implies that slopes have very different dynamic responses under stochastic ground motion. (2) Randomness of ground motion Although most earthquakes occur between the earth’s plates, it is impossible to accurately determine the location, time, and magnitude of future earthquakes. The

6.1 Source of Uncertainty and Its Description

85

three elements of ground motion are also highly random, as summarized here. (a) The intensity of ground motion is usually expressed by the peak acceleration (PGA). The peak value of ground motion is generally very high but the damage is not serious, thus the effective peak acceleration (EPA) and other parameters appear instead. (b) The spectral characteristics of ground motions (ω) are highly random, as is the (c) earthquake duration (T ). Using a series of stochastic vectors .i to express the uncertainty of the abovementioned geophysical properties and the randomness of ground motions, the general dynamic differential equation of the slope system can be expressed as: ) ( ˙ X = −MI x¨ g (.2 , t) ¨ + C(.1 )X ˙ + f .1 , X, M(.1 )X

(6.2)

where .i (i = 1, 2) is a stochastic vector that represents the uncertainty of the rock and soil parameters, the randomness of ground motion parameters, and other stochastic factors in the slope dynamic system. The rock and soil materials constituting a slope have strong nonlinear behavior under ground motion action. This results in rock–soil materials having a high sensitivity to ground motions, in which the same slope may have very different seismic responses under different earthquakes. This highlights the importance of choosing a nonlinear constitutive relation to study the dynamic response behavior of slope. The randomness and uncertainty of ground motion can also not be ignored in the analysis of slope seismic stability. It is highly essential to study the dynamic response behavior and seismic reliability evaluation of slopes from the perspective of stochastic dynamics.

6.1.2 Establishment of a Random Field of Slope Geomaterials The method for generating stochastic ground motion is introduced in Chap. 5 and is not repeated here, but rather a basic introduction to the random field establishment method is provided, which is used to characterize the variability of geotechnical parameters. The soil parameters of a slope often have spatial variation characteristics owing to the complexity of the local geological evolution processes. This demonstrates that soil parameter values change with spatial position and have a certain correlation, in which two points have increasingly similar soil properties with decreasing distance to one another. The random field theory proposed by Vanmarcke (1977) describes the uncertainty of soil parameters. The establishment of a random field must first consider the spatial correlation between multiple variables to generate random variables, discretize them over the entire study area, and then assign them to the corresponding units. Discrete methods include the point discrete method, average discrete method, and series

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expansion method (Sudret and Der Kiureghian 2000). This book recommends using the Karhunen-Loève (K-L) series expansion method because it has the advantages of small information loss and fast efficiency (Ghiocel and Ghanem 2002). The basic points of the establishment of related non-normal random fields are given below. (1) The random field is(characterized by the mean μ(x), variance σ (x), and autocor') relation function ρ x, x . Among them, the autocorrelation function describes the spatial correlation of soil parameters. In the discrete simulation of a random field of slope engineering, the theoretical autocorrelation function is generally used, and the exponential type and Gaussian type are commonly used. The expression using a simple exponential function as the autocorrelation function is: ⎧ ( || ' || ) |x−x | ⎪ ⎪ exp − l , (a) ) ⎨ ( ' ( || ' || || ' || ) (6.3) ρ x, x = |x−x | |y−y | ⎪ ⎪ ⎩ exp − l − l , (b) x y '

In Eq. (6.3), (a) represents the one-dimensional case where x and x are the coordinates of any two points, and l is the relative distance of the ( ' random ') field; and (b) represents a two-dimensional system where (x, y) and x , y are coordinates, and l x and l y are the relative distances in the horizontal and vertical directions, respectively. The random field parameters are not completely independent in space, the correlation decreases with increasing distance, thus the autocorrelation function is bounded. The eigenvalue λi and eigenfunction ϕi (x) can be solved by: .

) ( ') ( ' ρ x, x ϕi x d.x ' = λi ϕi (x)

(6.4)

.

(2) Consider the correlation between parameters. When considering the uncertainty of multiple soil parameters, the cross-correlation between parameters should be considered. For example, there is often a negative correlation between the cohesion c and friction angle ϕ (Cho 2010; Tang et al. 2013). In this process, the independent normal random field must be converted into a correlated nonnormal random field. We take a two-dimensional normal random field as an example: H k (x, θ ) = μk (x) +

n Σ

. σk (x) λi ϕi (x)ξk,i (θ )k = c, ϕ

(6.5)

i=1

where ξk,i (θ ) is an independent standard normal stochastic variable. The first n items are usually intercepted and H k (x, θ ) is used to approximate H k (x, θ ) in practical applications under the premise of ensuring the calculation accuracy. .

6.2 Stochastic Seismic Dynamic Response Analysis Method of Slope

87

A consideration of correlation can first use the Cholesky decomposition to convert ξk,i (θ ) into the relevant standard normal random variable χk,i (θ ) (Jiang et al. 2015; Vorechovsky 2008). Through the equal probability transformation in probability theory (e.g., Nataf transformation), the relevant normal random field is then converted into the relevant non-normal random field: ]} { [ C CNN H k (x, θ ) = G i−1 φ H k (x, θ ) k = c, ϕ (6.6) .

.

C

.

where H k (x, θ ) represents the correlated normal random field replaced by χk,i (θ ), G i−1 is the inverse function of the marginal cumulative distribution C

.

function of H k (x, θ ), and .(·) is the cumulative distribution function of the standard normal variable. Assuming that c and ϕ obey a lognormal distribution as an example, the display expression of the above formula is: ( C H k (x, θ )

.

= exp μ ln k(x) +

n Σ

.

)

σ ln k(x) λi ϕi (x)χ k, i (θ ) k = c, ϕ

i=1

(6.7) where μlnk (x) and σlnk (x) are the mean and standard deviation of χk,i (θ ), respectively. Several two-dimensional slope random fields can be established based on the above method, as shown in Fig. 6.1.

6.2 Stochastic Seismic Dynamic Response Analysis Method of Slope The main contents of performance-based stochastic dynamic analysis of slopes include: stochastic dynamic response analysis; seismic vulnerability analysis; risk assessment; and performance optimization design. The main methods of slope dynamic analysis are given in Chap. 5. This section introduces the stochastic dynamic analysis method, focusing at the uncertainty problem in slope systems.

6.2.1 Pseudo-Excitation Method Chinese scholar Jiahao Lin proposed the pseudo-excitation method (PEM) (Lin et al. 2001). This method makes the power spectrum calculation of the structural dynamic response more convenient, efficient, and accurate. This section briefly introduces the application principle of the PEM for slope systems.

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6 Probabilistic Performance-Based Seismic Design and Assessment …

Cohesion(kPa)

10

5

0 0

(a)

Friction angle(o)

10

5

0

(b)

Fig. 6.1 a Random field of cohesion; b Random field of internal friction angle

(1) Stationary stochastic response of the PEM Under single point stationary random excitation X(t) with the self-power spectral density function S X X (ω), the self-power spectral density function SY Y (ω) of the random dynamic response Y (t) of the linear stationary random system can be expressed as: SY Y (ω) = |H |2 S X X (ω)

(6.8)

6.2 Stochastic Seismic Dynamic Response Analysis Method of Slope

89

where H expresses the frequency response function. If the unit harmonic excitation ei ωt is used to replace the original excitation X(t), i ωt it is easy to obtain the simple √ harmonic dynamic response as Y (t) = H e . If this is multiplied by a constant S X X , then a pseudo-excitation x(t) ˜ can be constructed and its expression is: x(t) ˜ =

.

Sx x eiωt

(6.9)

. Its corresponding response is y˜ (t) = (S x x )H eiωt , which is multiplied by the same constant compared with the simple harmonic dynamic response. The following relationship can therefore be obtained through analysis and calculation: y˜ ∗ y˜ = | y˜ |2 = |H |2 Sx x = S yy x˜ ∗ y˜ = y˜ ∗ x˜ =

. .

Sx x e−iωt ·

.

Sx x H e−i ωt ·

Sx x H ei ωt = Sx x H = Sx y

.

Sx x ei ωt = H ∗ Sx x = S yx

(6.10) (6.11) (6.12)

In Eqs. (6.10)–(6.12), Sx y and S yx are cross-power spectral density functions. The displacement U, internal force F, stress σ, strain ε, and other random seismic virtual responses of a slope system can therefore be obtained using Eq. (6.9), and their self-power spectral density can be directly obtained: | |2 | |2 | ∼|2 |∼|2 | | | | | | | | SUU = |U˜ | , S F F = | F˜ | , Sσ σ = |σ | , Sεε = | ε |

(6.13)

Or its cross-power spectral density: ∼ SU F = U˜ ∗ ε, S yU = y˜ ∗ U˜

(6.14)

(2) Non-stationary stochastic response of the PEM In the dynamic response analysis of the stationary random system, the pseudoexcitation in Eq. (6.9) can be directly brought into Eq. (6.2) to obtain the slope displacement, velocity, acceleration response, and its self-power spectral density. The action process of non-stationary ground motion can generally be written as: X (t) = g(t)x(t)

(6.15)

where x(t) is a stationary stochastic process with a mean of zero, and g(t) is a modulation function. For slope engineering, the initial condition is static, and the linear dynamic response y(t) under the influence of excitation X (t) can be written in the following

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6 Probabilistic Performance-Based Seismic Design and Assessment …

form according to the Duhamel integral: .t y(t) =

h(t − τ ) f (τ )dτ

(6.16)

0

Under a zero-mean non-stationary random ground motion excitation, the autocorrelation function of the seismic dynamic response y(t) can be expressed as: R yy (t1 , t2 ) = E[y(t1 )y(t2 )] t1 t2

= ∫ ∫ h(t1 − τ1 )h(t2 − τ2 )E[ f (τ1 ) f (τ2 )]dτ1 dτ2 0 0 t1 t2

= ∫ ∫ h(t1 − τ1 )h(t2 − τ2 )g(τ1 )g(τ2 )E[x(τ1 )x(τ2 )]dτ1 dτ2

(6.17)

0 0

Here, E[x(τ1 )x(τ2 )] is equal to Rx z (τ), and τ = τ1 − τ2 . According to the WinnerKhintchine theorem: .+∞ E[x(τ1 )x(τ2 )] = Rx x (τ ) = Sx x (ω)exp{iω(τ2 − τ1 )}dω

(6.18)

−∞

Substituting Eq. (6.18) into Eq. (6.17) yields: .+∞ Rx x (t1 , t2 ) = Sx x (ω)I ∗ (ω, t1 )I (ω, t2 )dω

(6.19)

−∞

In Eq. (6.19): .t I (ω, t) =

h(t − τ )g(τ )exp{iωτ }dτ

(6.20)

0

In addition, t1 = t2 = t, and Eq. (6.19) is the variance of y(t): σ y2 (t)

.+∞ = Rvy (t, t) = Sx x (ω)I ∗ (ω, t)I (ω, t)dω

(6.21)

−∞

The integrand in Eq. (6.21) is the self-power spectral density function of the response function:

6.2 Stochastic Seismic Dynamic Response Analysis Method of Slope

S yy (ω, t) = Sx x (ω)I ∗ (ω, t)I (ω, t)

91

(6.22)

It can be seen from Eq. (6.20) that I (ω, t) is the dynamic response of the elastic initial static system at time t of the deterministic excitation g(t)exp{iωτ }. If constructing the incentive: f˜(t) =

.

Sx x (ω)g(t) exp{iωt}

(6.23)

√ The response at time t under the above excitation is therefore y˜ (ω, t) = Sx x (ω)I (ω, t), which is: y˜ ∗ (ω, t) y˜ (ω, t) = Sx x (ω)I ∗ (ω, t)I (ω, t)

(6.24)

Contrasting Eqs. (6.22) and (6.24) shows that: S yy (ω, t) = y˜ ∗ (ω, t) y˜ (ω, t) = | y˜ (ω, t)|2

(6.25)

In summary, the analysis of non-stationary stochastic response (mainly referring to power spectrum here) can be transformed into a dynamic time√history response calculation under the action of deterministic external excitations Sx x (ω)g(t)exp{iωt}.

6.2.2 Monte Carlo Stochastic Simulation Monte Carlo simulations (MCS) are also known as a random sampling or statistical testing method. This is a technique for solving replication problems using random numbers and probabilities, and is generally accepted to be the only accurate method of generalizability (Schuëller 1997, 2006) and is therefore widely used in reliability analysis. Using MCS for slope reliability analysis can comprehensively consider the randomness of ground motion and the spatial uncertainty of geophysical properties. It can input the cohesion, friction angle, and yield strength of the geotechnical materials, among others, as random variables into the system. Second, several deterministic ground motions can be imported for batch dynamic time history calculations. The slope seismic response probability is then obtained according to the MCS analysis, so that the system reliability can be calculated (Hayashi and Tang 1994). The use of MCS to calculate system reliability or failure probability also refers to the concept of frequency. Under N times of a deterministic response calculation, the case where the function Z is less than zero is M times (For example, the safety factor is used as the judgment index of system failure, and if it is less than 1, the system fails), then according to the theorem of large numbers, the failure frequency can be approximately estimated by:

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6 Probabilistic Performance-Based Seismic Design and Assessment … .

P f = P(Z < 0) =

M N

(6.26)

.

where P f represents the estimated value of the failure probability P f . Through N calculations, the probability density function of the system dynamic response physical quantity can be obtained by further fitting. The computational efficiency can be improved by random sampling methods, including importance sampling, Latin hypercube sampling, and directional sampling. Choosing an appropriate sampling method can greatly improve the MCS calculation speed. As a stratified sampling technique, Latin hypercube sampling has a good sampling effect and high efficiency, which leads to its wide application in slope reliability analysis (Li et al. 2013). Based on this method, the probability distribution function of the random variables of a slope system are divided into N uniform intervals of equal probability. A point is chosen in each interval to form a random variable sample containing N with a certain probability (Olsson and Sandberg 2002). Owing to its high calculation accuracy, MCS have become a common method for reliability analysis. The method can also consider the nonlinearity and discreteness of the limit state, which provides conditions for its generality. However, this method requires extensive computation, especially for the problem of an implicit function solution, which makes it difficult to be successfully applied.

6.2.3 Stochastic Finite Element Method The stochastic finite element method (SFEM) was developed on the basis of MCS. It combines the stochastic analysis theory with a finite element, uses a finite element program to repeatedly calculate the samples of random variables, and ultimately quantifies the results. As a numerical analysis method, the SFEM mainly focuses on the establishment and solution of governing equations. Because it involves the representation and processing methods of random factors, this approach includes the SFEM based on random variables and random field discretization. The former treats random parameters as random variables, and the latter treats random parameters as random processes. This section mainly briefly introduces these two points. (1) Stochastic finite element method based on random variables The SFEM based on random variables mainly includes the Taylor expansion stochastic finite element method (TSFEM), perturbed stochastic finite element method (PSFEM), and Neumann expansion stochastic finite element method (NSFEM) (Benaroya and Rehak 1988). Among them, the PSFEM is mainly used in the field of stochastic dynamic analysis of slopes (Gao et al. 2009). The PSFEM assumes that underlying random variables produce tiny perturbations at the mean point. Using a Taylor series to represent random variables as a

6.2 Stochastic Seismic Dynamic Response Analysis Method of Slope

93

deterministic part and perturbation-induced part, the finite element governing equation (Eq. 6.27) is transformed into a set of linear recursive equations. The statistical properties of responses such as displacement and stress are thus obtained. The governing equation of the finite element static analysis expressed in matrix form is: KU = F

(6.27)

where U is the displacement, F is the equivalent nodal load array, and K is the overall stiffness. Suppose αi is the tiny perturbation of random variable X i at mean point X i , in which αi = X i − X i , thus: K ≈ K0 +

n Σ i=1

αi

n ∂2 K ∂K 1 Σ + αi α j ∂αi 2 i, j=1 ∂αi ∂α j

(6.28)

From this, the displacement response and its statistical characteristics can be obtained. (2) Stochastic finite element method based on a random field In the first section of this chapter, the establishment process of the slope random field was briefly introduced, through which it can be known that the main problem of the SFEM based on the random field is the dispersion of the random field. The current methods of random field discretization are listed in Table 6.1. There is also a local averaging method. In the derivation of the element stiffness matrix, the weighted integral method uses the weighted integral at the element Gaussian point to characterize the random field. The local average method can be regarded as a special case of the weighted integration method, in which the weight coefficients are all the same. Therefore, this section does not specifically give the discrete form of the local average method. The random field representation based on the K-L series expansion method is also given in Eq. (6.5). At present, in the stochastic finite element analysis of slopes, the non-invasive method relies on existing finite element numerical simulation software for batch stochastic analysis, and is widely used because of its relatively simple operation. The finite element numerical simulation software that supports batch operations for stochastic analysis currently includes Abaqus, FLAC3D, and Geo-Studio.

6.2.4 Nonlinear Stochastic Seismic Dynamic Method Slope rock and soil materials exhibit a strong, nonlinear dynamic behavior and are sensitive to seismic dynamics. The nonlinear dynamic response of slope earthquakes therefore requires extensive research attention.

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Table 6.1 Random field discrete methods Methods

Discrete form

Remark

References

Center point method

X u = X u ci u ∈ Di

The random field element Di is characterized by a random variable u c at the element centroid

Der Kiureghian and Ke (1988)

Local average X T (t) = method

Interpolation method

Orthogonal expansion method

Xu =

q Σ

1 T

t+T ∫ /2 t−T /2

X (t)dt

Ni (u)X (u i )

i=1

⎧ ⎪ ⎪ Hk (x, θ ) = μk (x) + .H k (x, θ ) ⎪ ⎪ n √ Σ ⎨ .H k (x, θ ) = σk (x) λi ϕi (x)ξk,i (θ ) i=1 ⎪ ) ( ') ⎪ ∫ ( ' ⎪ ⎪ ⎩ ρ x, x ϕi x d.x ' = λi ϕi (x) .

[ ] t − T2 , t + T2 : discrete element; T : local average element length

Vanmarcke et al. (1986)

Ni (u): Zhou et al. Interpolation shape (2005) function; X (u i ): the value of the field at node i

μk (x) and σk (x): mean and standard deviation of a standard normal random variable; ξk,i (θ ): independent standard normal random variable; λi and ϕi (x): eigenvalue and eigenfunction of the autocorrelation ) ( ' function ρ x, x

Spanos and Ghanem (1989)

In the previous introduction, various methods were shown to still have certain limitations. Although the PEM better considers the spectral characteristics of ground motions, it is more aimed at linear or equivalent nonlinear slope systems. For nonlinear slope stochastic dynamic systems, it is difficult or even impossible to obtain the probability density function (PDF) of the non-Gaussian stochastic seismic dynamic response using the PEM, thus making it hard to obtain an accurate seismic stochastic response of the slope (Chen et al. 2011). While the Monte Carlo method is widely used in slope nonlinear stochastic dynamic system (Fattahi et al. 2018; Huang and Xiong 2017; Pang et al. 2021), the seismic reliability analysis of slopes requires thousands or even hundreds of thousands of deterministic dynamic calculations to make the calculation results converge.

6.3 Seismic Dynamic Risk and Vulnerability Assessment of Slope

95

When a nonlinear system is subjected to stationary or non-stationary stochastic excitation, the Fokker–Planck equation (FPK) is often used to analyze the response results. The probability density evolution method (PDEM) developed from the FPK equation serves as the theoretical basis, which facilitates the analysis of the dynamic response of the slope (Rosenbluth et al. 1957). The FPK description of the slope stochastic dynamic system is: ] [ [ ] n n n Σ Σ Σ ∂ Ai ( y, t) pY ∂ 2 pY σi j ( y, t) ∂ pY ( y, t) =− + ∂t ∂t ∂ yi ∂ y j i=1 i=1 j=1

(6.29)

where ( pY ( y, t)) is the PDF, which can also be understood as the transition PDF pY y, t | y0 , t0 of the slope dynamic system. Clearly, it is difficult to solve the FPK of an n-dimensional multiple-degree-of-freedom stochastic dynamic system of slope. The PDEM is based on the principle of conservation of probability. This implies that the probability of a conservative stochastic system is conserved during evolution. The method can comprehensively consider randomness and uncertainty of initial conditions, system parameters, and external excitations. It can be seen from the generalized probability density evolution equation (GPDEE): ) ( m ) ∂ pY. y, θq , t ∂ pY. (y, θ, t) Σ ( + y j θq , t =0 ∂t ∂yj j=1

(6.30)

In Eq. (6.30), the probability space formed by the discrete random vector . is .. , and θ q (q = 1, 2, . . . , n sel ) is a series of sample points. Compared with FPK, it can be seen that the dimension of the probability density evolution equation is unrelated to the system’s degree of freedom, but is only related to the dimension of the concerned physical quantity. The PDEM method overcomes the defect that FPK cannot solve. The PDEM can also be combined with the stochastic finite element method, which can improve the calculation speed of the stochastic dynamic response of the seismic slope. The main steps include: determining the basic random variables of the slope system and their statistical characteristics; probabilistic space partitioning of elementary random variables; the establishment and dispersion of random fields; stochastic ground motion sample generation; nonlinear dynamic time history watch computation and extraction; GPDEE solution and analysis.

6.3 Seismic Dynamic Risk and Vulnerability Assessment of Slope On the basis of the above, this chapter establishes an evaluation method of the slope seismic vulnerability performance based on the probabilistic method, considering the

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randomness of ground motion and uncertainty of geotechnical physical properties, and considers the seismic risk and possible economic loss, and establishes a risk assessment methodological framework.

6.3.1 Seismic Dynamic Vulnerability Assessment of Slope Vulnerability refers to the possibility that a system reaches a certain damage level under different disaster intensities. Correspondingly, the seismic vulnerability refers to the probability of a slope reaching different failure states under the action of different earthquake intensities. The vulnerability can provide the conditional probability of the system exceeding different failure states under the action of different intensity earthquakes. This is a very effective way to evaluate the seismic performance of structures under earthquake action, and it is also the most important part of a performance-based seismic probability risk assessment. This approach has been widely used in earthquake disaster prediction, earthquake risk assessment, earthquake restoration plan optimization, and decision-making. Relatively speaking, seismic vulnerability analysis can provide more systematic, comprehensive, and richer probability analysis results than traditional reliability analysis. This is because the results of vulnerability analysis include multiple curves, while the results of reliability analysis have only a single curve (Schultz et al. 2010; Wu 2015). The seismic performance of structures under different ground motion intensities can be obtained through seismic vulnerability analysis to establish the relationship between the ground motion intensity and structure that exceeds different performance levels or failure states. The methods of seismic vulnerability analysis are mainly divided into the following categories: Probabilistic statistical methods based on post-earthquake survey data or experimental data; vulnerability analysis based on empirical models; vulnerability analysis based on expert systems or expert judgment; analytical vulnerability analysis methods; and mixed methods. Among them, the analytical vulnerability analysis methods are widely used. This is suitable for seismic vulnerability analysis of different site conditions and structural forms because it can combine different numerical analysis methods and the analyst can control the initial conditions and target states of the analysis object (Baker 2015). At present, there are two analytical vulnerability analysis strategies in seismic engineering. The first assumes that the seismic dynamic response and vulnerability curve of the analyzed object are lognormally distributed. The vulnerability curve is then obtained through parameter estimation methods, such as regression analysis or maximum likelihood estimation. The vulnerability defined using this method is: ] [ ln a − λ F =. ζ

(6.31)

6.3 Seismic Dynamic Risk and Vulnerability Assessment of Slope

97

where a is the disaster intensity (e.g., PGA, EPA), . stands for lognormal distribution, and λ and ζ are the mean and standard deviation of the ground motion parameter lna required to achieve a certain degree of damage. The other strategy involves the use of nonlinear stochastic dynamics to directly obtain the dynamic response failure probability under different ground motion intensities. The seismic vulnerability curve obtained based on the lognormal distribution has the defect that it cannot reflect the true structure of the data and the validity is therefore questionable. In the design of slope seismic performance, it is necessary to reasonably consider uncertain factors such as geotechnical parameters and external loads that affect the seismic performance of slopes and supporting structures (Argyroudis et al. 2013). It is therefore generally necessary to adopt probability analysis methods or reliability analysis methods to quantitatively characterize the influence of these uncertain factors and perform vulnerability analysis. The vulnerability analysis method based on reliability analysis is: β=

μZ σZ

F =1−β

(6.32) (6.33)

where μ Z and σ Z are the mean and standard deviation of the functional function Z , respectively, and β represents the reliability index. The functional function of slope engineering can be built in light of the solution method of the slope safety factor, and MCS can also be used to solve the probability of occurrence under different performance levels to solve the reliability index. The safety factor is generally selected as the slope performance index to assess slope stability under earthquake action, which is controversial. The dynamic response behavior is controlled by the slope deformation (Seed 1973). Liu et al. also considered it unreasonable to use the minimum safety factor in the earthquake duration to evaluate the slope stability (Liu et al. 2003). This guide therefore suggests that permanent displacement can also be used as the performance index of slope resistance. The specific performance level division is given in Chap. 3. Reliability-based analysis methods can quantitatively characterize the uncertainty of geotechnical parameters, ground motion, and their influence on the vulnerability of the slope system, as well as the failure probability of the system under different ground motion intensities to directly construct the seismic vulnerability analysis curve. The fragility under different performance index requirements can be obtained and calculated according to the requirements. This method is very suitable for seismic slope performance analysis and design under random dynamic response. Slope seismic vulnerability analysis is similar to dynamic reliability analysis, but more design ground motions and performance levels need to be calculated. Generally speaking, six degrees (0.05 g), seven degrees (0.1 g), eight degrees (0.2 g), and nine degrees (0.4 g), or even more batches of seismic activity can be calculated separately (Kramer and Paulsen 2004). For stochastic analysis, each ground motion peak corresponds to a batch of calculation samples. Samples with different ground motion

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peaks can be processed by simple amplitude modulation or regeneration. In addition, vulnerability analysis also needs to comprehensively consider the engineering site conditions and seismic fortification level of the slope project, and reasonably determine the different performance levels (Kramer and Paulsen 2004; Tsompanakis et al. 2010).

6.3.2 Seismic Dynamic Risk Assessment of Slope Seismic probabilistic risk assessment (SPRA) provides reasonable and effective guidelines for earthquake risk management, risk decision-making, disaster mitigation measures, and engineering restoration from the perspective of a probability assessment of the possible damage and loss of important structures and infrastructure under earthquake disasters. For example, for important structures such as reservoirs and dams, most of the existing water conservancy dams were built in the last century, and most of the design methods are outdated or cannot meet the requirements of existing codes. A reasonable seismic probability risk assessment of dams can provide effective reinforcement and repair strategies. This section takes the seismic probability risk assessment as the goal, and constructs a performance-based probabilistic risk assessment framework for slope earthquakes. The seismic probability risk assessment is divided into three parts: site seismic risk analysis; seismic vulnerability analysis; and seismic loss analysis (Ichii 2002). The site seismic risk analysis is given in Chap. 4, and the slope vulnerability analysis method is introduced in the previous section. This section presents a performance-based framework for seismic risk assessment and design of slopes. For an important structure, its seismic probability risk considers the ground motion intensity that the project may suffer and the response probability. This can be expressed in the following formula (Ellingwood 2001; Yegian et al. 1991): R=H·F·L

(6.34)

where R is the earthquake risk, H stands for site seismic risk, which refers to the probability of ground motion of different intensities occurring in the engineering site within a certain period of time, F is the seismic vulnerability, which indicates the conditional probability of the system exceeding different damage states under the action of earthquakes with different intensities, and L is the earthquake damage. The first two terms of H · F in Eq. (6.21) can be expressed as the following relation: P(Ls) = P(Ls | A = a)P(A = a)

(6.35)

where Ls represents the seismic performance level or the damage level of the structure, A is a measurement index of the ground motion intensity, P(Ls | A = a) is the seismic vulnerability, which represents the conditional probability under the

6.4 Resilience-Based Seismic Performance Design

99

action of ground motion intensity a, P(A = a) is field ground motion intensity for probability a, P(Ls) is the total probability of a system that is more than a certain level of performance, which can be achieved by the field seismic vulnerability curve of the system and is regarded as the earthquake risk probability (Zentner et al. 2017). Seismic vulnerability P(Ls | A = a) is the core part of earthquake probability risk assessment systems. In the previous chapter, we proposed the seismic vulnerability analysis method, which can be coupled into the framework of seismic probabilistic risk assessment to establish the seismic probabilistic risk assessment method of a slope system. Combined with the nonlinear stochastic finite element analysis method, Fig. 6.2 shows a flow chart of the performance-based framework for SPRA of slopes.

6.4 Resilience-Based Seismic Performance Design Traditional concepts and methods of earthquake disaster assessment and prevention, led by the performance-based seismic design method, are guided by a single goal of ensuring no loss of life. However, these methods face great difficulties in dealing with the challenges posed by the uncertainty and complexity inherent in earthquakes. Furthermore, with advancing society and technology, in terms of preventing and responding to disasters, mankind increasingly expects that the material and social systems on which people depend for survival will not collapse even under extreme disaster conditions and without relying on external rescue can still quickly recover from disasters, thereby ensuring the safety of life and property and the normal operation of social functions to the greatest extent. The concept of “resilience” originated from the concept of “ecological resilience” proposed by Canadian ecologist Holling (1973). The 2005 World Conference on Disaster Reduction confirmed the importance of resilience in disaster prevention and mitigation. Resilience refers to a system’s ability to resist disasters, reduce disaster losses, and reasonably allocate resources to quickly recover from disasters. This requires that the engineering system has strong robustness and rapid recovery (Burton and Deierlein 2016). The rapidly developing theory of resilience provides brand new research ideas and technical approaches for the realization of the goal of disaster prevention and mitigation projects. As a new method of disaster prevention and mitigation, it puts forward new requirements for engineering construction and also provides a new way to achieve sustainable development of disaster prevention and mitigation (Xiong and Huang 2019). The resilience-based approach to disaster prevention and mitigation is multiobjective. It considers that the safety and long-term resistance of engineering are equally important, and requires that accidents can be avoided as much as possible in the face of unpredictable risks. Although the resilience-based disaster prevention concept has been widely used in the seismic design of many above-ground engineering structures (Chang and Shinozuka 2004; D’Lima and Medda 2015; Kim et al. 2017; Rose 2007; Zhang et al. 2017), the quantification of its resilience to disasters still faces tremendous challenges, especially in the field of slope design.

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Fig. 6.2 Performance-based framework for the SPRA of slopes

In slope seismic engineering systems, resilience means the resistance, cushioning, and recovery of the slope engineering system when affected by uncertain disasters (for example, earthquakes). This means that the slope system is required to be able to resist and disperse risks as much as possible, delay the damage time as much as possible when damaged, and restore the slope support function as quickly as possible in the later period, as shown in Fig. 6.3. The resilience-based slope disaster prevention and mitigation design is guided by the multi-objective function of the

6.4 Resilience-Based Seismic Performance Design

101

slope system, and also puts forward higher requirements for social development. Whether it is slope engineering anti-seismic technology or community response and rescue, further improvements are needed. From the above basic conceptual introduction, as shown in the Fig. 6.4, engineering based on resilience design has the highest risk aversion ability. Based on the vulnerability assessment, the introduction of resilient disaster prevention design can greatly reduce the slope seismic risk, which represents the highest level of safety. Based on the requirements for engineering seismic resistance, Bruneau et al. (2003) introduced four characteristics of resilience as a measure of the resilience of

Fig. 6.3 Schematic diagram of seismic resilience of slope system

Fig. 6.4 Importance of “resilience” (Modified from Qiu 2018)

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6 Probabilistic Performance-Based Seismic Design and Assessment …

the infrastructure in the assessment framework for quantitatively assessing the earthquake resilience. In slope seismic engineering, the four characteristics of toughness can be specifically expressed as follows: Robustness: When faced with external disturbances (e.g., earthquakes), the slope engineering itself has a certain strength to maintain stability. Redundancy: Considering the safety and reliability of a slope engineering system, the structural design refers to the repeated configuration of the supporting structure of key parts. When a disaster occurs, the structure is damaged and invalid, and the redundant configuration can be used as a backup to bear the disaster in time. In the design of seismic slope engineering, redundancy is mainly included in the analysis of regional landslide hazards. By evaluating the possible future disasters of slope and such factors as its scope, scale, strength, and economic losses, reasonable slope reinforcement and disaster prevention planning and design can be carried out to reduce the possibility of danger. Relatively speaking, higher strengths of slope reinforcement are associated with more disaster prevention resources, and higher performance levels adopted in disaster prevention planning, higher system redundancy, and stronger seismic toughness of the slope engineering. Intelligence: When disasters such as earthquakes occur, the slope system can identify risks through monitoring and quickly make intelligent decisions. This characteristic requirement of resilience covers a number of key technical and scientific issues in slope seismic engineering. This includes various links such as slope monitoring data generation, data conversion and calculation, storage and transmission, and intelligent early warning: (i)

In the data generation stage, it is necessary to have advanced and complete monitoring equipment to monitor the entire development process of the slope project, and to achieve remote monitoring and real-time data update, thereby reducing labor. This is the main direction of the current development of geotechnical engineering monitoring equipment. (ii) In the data conversion and calculation stage, the system must be able to determine the occurrence of disasters in time. This requires considerable experience in earthquake landslide disasters to guide the engineering design of earthquake slopes and the prevention and control of landslide disasters. At present, artificial intelligence-based slope seismic engineering design research is becoming increasingly extensive. As shown in Fig. 6.5, through machine learning of a large amount of earthquake landslide data, the establishment of an early warning model for the long-term performance of slope seismic resistance can better realize a timely disaster warning. (iii) In the data storage and transmission stage, as well as the realization of the final intelligent warning, it is necessary to rely on a comprehensive risk management data platform. To build an ideal data cloud platform, it is necessary to integrate an advanced machine learning platform, a data storage and computing platform, and a client that can realize real-time data update and disaster warning, as shown in Fig. 6.6. By building a comprehensive risk prevention and control data platform, users can obtain information such as slope project monitoring

6.4 Resilience-Based Seismic Performance Design

103

Historical data

Related data

Geotechnical parameters,

Slope supporting structure

slope stress history,

and its materials,

performance status, etc.

monitoring data, etc.

Real-time data update

Forecast visualization

CS Displacement and stress data prediction

Machine term platform TensorFlow, Keras, PyTorch, Pandas, etc.

Predictive model RNN, CNN, ANN, LSTM, etc. Instability warning

Fig. 6.5 Slope early warning model based on machine learning

data, a performance status evaluation, and a timely disaster warning at any time and any place, to more quickly and easily achieve seismic risk management and control of slope projects. (iv) Rapidity: When a disaster occurs, it can play a role in time to realize the stability of the slope system and control the loss. At the same time, it has the ability to quickly recover to avoid future danger from happening again. Based on the above introduction, it can be determined that slope projects must not only be able to withstand external interference, but also be able to cope with its own interference and not cause large-scale damage over time (Das et al. 2018). It is also best to return to a stable state in the shortest possible time. It is possible to carry out regional geological disaster assessment and post-disaster reconstruction based on resilience assessment to mitigate the economic cost while ensuring human lives and property losses remain low.

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Fig. 6.6 Comprehensive safety risk management and control data platform for slope engineering

References Argyroudis S, Kaynia AM et al (2013) Development of fragility functions for geotechnical constructions: Application to cantilever retaining walls. Soil Dyn Earthq Eng 50:106–116 Baker JW (2015) Efficient analytical fragility function fitting Using dynamic structural analysis. Earthq Spectra 31(1):579–599 Benaroya H, Rehak M (1988) Finite element methods in probabilistic structural analysis: a selective review. Appl Mech Rev 41(5):201–213 Bruneau M, Chang SE, Eguchi RT (2003) A framework to quantitatively assess and enhance the seismic resilience of communities. Earthq Spectra 19(4):733–752 Burton HV, Deierlein G (2016) Framework for incorporating probabilistic building performance in the assessment of community seismic resilience. J Struct Eng 142(8) Chang SE, Shinozuka M (2004) Measuring improvements in the disaster resilience of communities. Earthq Spectra 20(3):739–755 Chen J, Peng Y et al (2011) A note on the pseudo-excitation method. Chin J Comp Mech 28(2):163– 167 Cho SE (2010) Probabilistic assessment of slope stability that considers the spatial variability of soil properties. J Geotech Geoenviron Eng 136(7):975–984 Cornell CA, Jalayer F (2002) Probabilistic basis for 2000 SAC federal emergency management agency steel moment frame guidelines. J Struct Eng-ASCE 128(4):526–533 D’Alembert JL (1743) Traité de dynamique: David l’aîné Das JT, Puppala AJ et al (2018) Sustainability and resilience analyses in slope stabilisation. Proc Inst Civ Eng-Eng Sustain 171(1):25–36

References

105

Der Kiureghian A, Ke J-B (1988) The stochastic finite element method in structural reliability. Probab Eng Mech 3(2):83–91 D’Lima M, Medda F (2015) A new measure of resilience: an application to the London Underground. Transp Res Part a Policy Pract 81:35–46 Ellingwood BR (2001) Earthquake risk assessment of building structures. Rel Eng Sys Safe 74(3):251–262 Fattahi H, Babanouri N et al (2018) A Monte Carlo simulation technique for assessment of earthquake-induced displacement of slopes. J Min Environ 9(4):959–966 Gao R-X, Gong W-H et al (2009) Analysis of stability and reliability of bedding rock slope by stochastic finite element method. Rock Soil Mech 30(4):1165–1169 Ghiocel DM, Ghanem RG (2002) Stochastic finite-element analysis of seismic soil–structure interaction. J Eng Mech 128(1):66–77 Hayashi H, Tang WH (1994) Probabilistic evaluation on progressive failure in cut slopes. Struct Safe 14(1–2):31–46 Holling CS (1973) Resilience and stability of ecological systems 4(1):1–23 Huang Y, Xiong M (2017) Dynamic reliability analysis of slopes based on the probability density evolution method. Soil Dyn Earthq Eng 94:1–6 Ichii K (2002) A seismic risk assessment procedure for gravity type quay walls. Struct Eng Earthq Eng 19 Jiang S, Li D et al (2015) Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo simulation. J Geotech Geoenviron Eng 141(2):04014096 Kim B, Shin SC et al (2017) A resilience loss assessment framework for evaluating flood-control dam safety upgrades. Nat Hazards 86(2):805–819 Kramer S, Paulsen SB (2004) Seismic performance evaluation of reinforced slopes. Geosynth Int 11(6):429–438 Li D, Jiang S et al (2013) Reliability analysis of slopes considering spatial variability of soil parameters using non-intrusive stochastic finite element method. Chin J Geotech Eng 35(8):1413– 1422 Lin J, Zhao Y et al (2001) Accurate and highly efficient algorithms for structural stationary/nonstationary random responses. Comp Methods Appl Mech Eng 191(1–2):103–111 Liu H, Fei K et al (2003) Time history analysis method of slope seismic stability. Rock Soil Mech 24(4):553–556 Olsson AMJ, Sandberg GE (2002) Latin hypercube sampling for stochastic finite element analysis 128(1):121–125 Pang R, Xu B et al (2021) Seismic time-history response and system reliability analysis of slopes considering uncertainty of multi-parameters and earthquake excitations. Comp Geotech 136 Qiu BX (2018) Methods and principles of designing resilient city based on complex adaptive system theory. Urban Dev Stud 25(10):1–3 Rose A (2007) Economic resilience to natural and man-made disasters: multidisciplinary origins and contextual dimensions. Enviro Mehine 7(4):383–398 Rosenbluth MN, MacDonald WM et al (1957) Fokker-Planck equation for an inverse-square force. Phys Rev 107(1):1 Schuëller GI (1997) A state-of-the-art report on computational stochastic mechanics. Probab Eng Mech 12(4):197–321 Schuëller GI (2006) Developments in stochastic structural mechanics. Arch Appl Mech 75(10):755– 773 Schultz MT, Gouldby B et al (2010) Beyond the factor of safety: developing fragility curves to characterize system reliability Seed HB (1973) Stability of earth and rockfill dams during earthquakes. Wiley Sons, Incorporated Spanos PD, Ghanem R (1989) Stochastic finite element expansion for random media. J Eng Mech 115(5):1035–1053 Sudret B, Der Kiureghian A (2000) Stochastic finite element methods and reliability: a state-ofthe-art report: Dept Civil Environ Eng, University of California ….

106

6 Probabilistic Performance-Based Seismic Design and Assessment …

Tang X, Li D et al (2013) Impact of copula selection on geotechnical reliability under incomplete probability information. Comp Geotec 49:264–278 Tsompanakis Y, Lagaros ND et al (2010) Probabilistic seismic slope stability assessment of geostructures. Struct Infrastr Eng 6(1–2):179–191 Vanmarcke EH (1977) Probabilistic modeling of soil profiles. J Geotech Eng Div 103(11):1227– 1246 Vanmarcke E, Shinozuka M et al (1986) Random fields and stochastic finite elements. Struct Safe 3(3–4):143–166 Vorechovsky M (2008) Simulation of simply cross correlated random fields by series expansion methods. Struct Safe 30(4):337–363 Wu X (2015) Development of fragility functions for slope instability analysis. Landslides 12(1):165– 175 Xiong M, Huang Y (2019) Novel perspective of seismic performance-based evaluation and design for resilient and sustainable slope engineering. Eng Geol 262 Yegian MK, Marciano EA et al (1991) Seismic risk analysis for earth dams. J Geotech Eng-ASCE 117(1):18–34 Zentner I, Guendel M et al (2017) Fragility analysis methods: review of existing approaches and application. Nucl Eng Des 323:245–258 Zhang W, Wang N et al (2017) Resilience-based post-disaster recovery strategies for road-bridge networks. Struct Infrastruct Eng 13(11):1404–1413 Zhou X, Zhou X et al (2005) Application study of spatial interpolation method in geological random field. Rock Soil Mech 26(2):221–224

Chapter 7

Case Study

7.1 Probabilistic Seismic Performance of Earth Dams Earth dams belong to generalized slope engineering and are useful for water storage and flood mitigation. Earth dams are the largest type of constructed dams worldwide and account for more than 90% of the total number of dams. The construction of earthrock dams also has a long history with the highest number of accidents. So-called dam safety therefore mainly refers to the safety of earth dams. A study of global dam breaks shows that earth dam flooding leads to the most accidents and is the main cause of earth dam failure. The seismic performance of earth dams must be analyzed to better understand their behavior under earthquake action. Furthermore, most old dams were designed using outdated seismic analysis and design methods, which are unlikely to meet the current seismic design criteria (Hariri-Ardebili and Saouma 2016). Seismic dynamic evaluation of existing dams therefore also requires reliable tools to ensure effective reinforcement and reconstruction strategies (Peng et al. 2018). Performance-based seismic design (PBSD) evaluation is a recently developed method that involves the performance of slope engineering under seismic loading. As an inherent feature, the random analysis of earth dams largely differs from what is considered deterministic. The seismic response of earth dams should be analyzed from a random perspective. Most previous studies were also limited to the vulnerability modeling of dams, and do not combine the vulnerability model with seismic hazard and seismic loss analysis within the seismic probabilistic risk assessment (SPRA) framework of the entire dam. Seismic performance including seismic risk analysis of earth dams considering the PBSD criterion and nonlinear dynamic time-history method mentioned above are evaluated and taken as an example according to the design guideline introduced above.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Huang et al., Guidelines for Probabilistic Performance-Based Seismic Design and Assessment of Slope Engineering, https://doi.org/10.1007/978-981-19-9183-7_7

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7.1.1 Seismic Performance Evaluation of Earth Dams (1) Performance-based seismic design of earth dams We select a single earth dam as an engineering case. The dam body is 44-m high with a homogeneous soil composition. Table 7.1 lists information regarding the model layers and thickness, which lie in a weak soil area with high seismic intensity. The layering of the dam foundation and embankment is complicated, and the soil is in a state of consolidation or secondary consolidation. Serious damage can be caused by strong earthquake activity to the dam. The performance-based seismic evaluation is very important to the safety of the dam and people living in the surrounding areas. According to the probabilistic seismic performance design framework of slope engineering mentioned in Chap. 3, the overall evaluation process of earth dam can be summarized in the following points: a. Seismic fortification level and performance indicator of an earth dam The first step is to determine the seismic performance level and quantify the performance targets under different seismic levels. Guidelines entitled “Selecting seismic parameters for large dams” (ICOLD 1989) can be used as a reference with two levels of earth dam fortification: operating basis earthquakes (OBE) and safety evaluation earthquakes (SEE). The hydraulic seismic design code also requires two-level seismic fortification for large-scale dam projects in China (IWHR 2018). Two levels of fortification for earth dams are considered in this chapter in combination with the ICOLD guidelines and Chinese guideline “Standard for seismic design of hydraulic structures” (GB51247-2018). (i) OBE The performance goal of a dam under OBE action is no damage, or only minor damage that does not affect the basic water retention and storage function of the dam. According to the dam anti-seismic code and related research, this study sets the vertical displacement to less than 0.10 m under OBE conditions as the Table 7.1 Main soil mechanics parameters of earth dams

Number of layers

1

2

3

Unit weight γ (kN/m3 )

18

18

20

Effective cohesion c (kPa)

5

0

5

Effective 34 friction angle ϕ (°)

34

36

Poisson’s ratio υ

0.4

0.4

Compacted fill Hydraulic fill Alluvial layer

0.4

7.1 Probabilistic Seismic Performance of Earth Dams

109

basic performance criterion (Shao et al. 2011). The dam is assumed to lie in an area with a seismic intensity of VII and corresponding peak ground acceleration (PGA) of 0.1 g. (ii) SEE A SEE is used for upper limit seismic analysis and is a higher dam safety index with a higher seismic intensity than an OBE (Huang and Xiong 2017; ICOLD 1989). The dam is assumed to have no structural damage or uncontrolled overflow, such that the ultimate load state of the earth dam never occurs. Here, the performance index is set as the peak value of permanent settlement is less than 0.30 m. The PGA of the SEE for the secondary fortification standard is assumed as 0.2 g. b. Intensity-frequency, non-stationary seismic excitations The key issue for the second step is to determine the ground motion input. However, the considered site lacks sufficient seismic records, and most are small earthquakes and remote site seismic records. For the earth dam engineering site studied in this chapter, there is an insufficient amount of strong earthquake records, which brings great difficulties to the stochastic seismic response and anti-vibration reliability assessment of the earth dam. Therefore, in this chapter, prior to the stochastic seismic response and reliability assessment of earth dam, it is necessary to establish a suitable ground motion model to generate acceleration samples with rich probability characteristics and to better facilitate stochastic seismic response research of earth dams based on the nonlinear dynamic time-history method. The existing intensity-frequency, non-stationary evolution power spectrum model is taken as an example and mainly includes the following two methods (Douglas and Edwards 2016): • Multiply the envelope function by the stationary power spectrum model, and the envelope function reflects the evolution of intensity and frequency; • Use the envelope function to consider the non-stationary characteristics of the seismic intensity and the frequency variety versus time in the power spectrum model parameters. Using the modified second spectrum representation method and idea of random function (Liu et al. 2016), seismic acceleration time-history samples are generated according to the seismic geological conditions, as shown in the Fig. 7.1. c. Finite element model setting of an earth dam Geo-studio software was used for the model setting prior to obtaining the dynamic time-history results of the dam. A finite element mesh model with a height of 44 m is shown in Fig. 7.2. Equal displacement boundaries are used at both sides of the model.

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Fig. 7.1 Typical stochastic seismic acceleration time-history sample El 349

350

El 338

Elevation - m

340

El 333

330 320 310 300 290

0

50

100

150

200

250

300

350

Distance - m

Fig. 7.2 Model of earth dam section

(2) Nonlinear seismic dynamic response of earth dam The seismic dynamic response (vertical displacement) of the model for a typical seismic sample is obtained based on the finite element dynamic time-history method, as shown in Figs. 7.3 and 7.4. We designed the relevant ground motions following to the OBE and SEE standards according to the content related to the dam performance design, and obtained the stochastic seismic response results based on the SPRA framework in Chap. 6. This provides a reliable case reference for relevant engineers to carry out the stochastic dynamic stability design of dam engineering. a. Nonlinear dynamic response of earth dam under OBE Under the action of an OBE load, combined with the performance design framework of Chap. 3, the PDEM equation is solved and the overall displacement response is obtained. The mean and standard deviation of the results are depicted in Fig. 7.5.

7.1 Probabilistic Seismic Performance of Earth Dams

111

0.003

Displacement(m)

0.002

0.001

0

-0.001

-0.002

-0.003

0

5

10

15

Time(s)

Fig. 7.3 Time-history curve of the vertical displacement of an earth dam under OBE action 0.2

Acceleration(g)

0.1

0

-0.1

-0.2

0

5

10

15

Time(s)

Fig. 7.4 Acceleration time-history curve of an earth dam under OBE action

Figure 7.5 shows the gradual change of the mean permanent seismic subsidence value of the dam under OBE action. The standard deviation initially increases with time and then decreases, but the overall trend is increasing. This shows that the variability of dam crest subsidence gradually increases because of the nonlinear behavior of the soil. The attenuation of the standard deviation of the vertical subsidence occurs

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Fig. 7.5 Second-order statistical information of the permanent seismic subsidence of a dam crest under OBE action

because the permanent crest subsidence gradually decreases owing to the attenuation of the ground motion. Figure 7.6 shows the surface of the probability density function (PDF) of the permanent seismic subsidence of the dam crest under OBE action. Figure 7.7 describes the change of the PDF curve of the permanent seismic subsidence of the dam body over time. The evolution surface of the PDF appears as a rolling peak, which indicates that the amount of dam crest subsidence fluctuates and evolves over time. This demonstrates that the permanent seismic subsidence of the dam crest has great variability with time. The probabilistic flow process is also complicated. Under normal circumstances, this is non-stationary flow with a large number of vortices. Figure 7.8 is a typical probability density curve of the permanent seismic subsidence at the top of the dam, which also confirms this point. Here, the PDF has a large variation between 4 and 8 s, which shows that the evolution of the instantaneous PDF changes with time is very complex, which affects the dynamic reliability of the earth dam engineering structure. b. Nonlinear dynamic response of an earth dam under SEE action Similar to the random seismic response analysis of an earth dam under OBE action, the deterministic dynamic time-history response of an earth dam under SEE action is regarded as a generalized velocity and then substituted into the probability density evolution equation. Solving this equation allows abundant probability density information to be obtained regarding the dynamic behavior of the earth dam under SEE loads. The evolution process of the PDF of dam crest seismic subsidence under SEE action shows that the fluctuation characteristics of the probability density evolution surface (Fig. 7.9) are more severe than those under OBE action. This is because with

7.1 Probabilistic Seismic Performance of Earth Dams

113

Fig. 7.6 Probability density function (PDF) evolution surface of the permanent seismic subsidence of a dam crest under OBE action

Fig. 7.7 Typical equal probability density curve of dam crest subsidence under OBE action

increasing earthquake peak acceleration, an earth dam will more strongly enter the nonlinear state under earthquake action. This also indicates the necessity to grasp the seismic dynamic response analysis of an earth dam from the perspective of stochastic dynamics.

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Fig. 7.8 Typical probability density curve of dam crest subsidence under OBE action

Fig. 7.9 Probability density function of dam crest seismic subsidence under SEE action

(3) Seismic performance evaluation of earth dams The safety and performance evaluation of a dam is carried out following the above two-level seismic fortification level and evaluation method. The performance level is divided into four categories according to the reliability results, as listed in Table 7.2.

7.1 Probabilistic Seismic Performance of Earth Dams Table 7.2 Performance level classification according to reliability

115

Reliability (R)

Performance level

0.7 ≤ R

Safe

0.4 ≤ R < 0.7

Comparatively safe

0.1 ≤ R < 0.4

Comparatively dangerous

R < 0.1

Dangerous

a. Stochastic dynamic reliability of an earth dam under OBE action The performance of a dam under OBE action is evaluated corresponding to the fortification criterion of this level based on the calculation results of the dynamic finite element analysis in the previous section. By introducing the basic theory of PDEM and equivalent extreme value events, the PDF and cumulative distribution function (CDF) of the distribution of the extreme value of the dam crest seismic subsidence can be obtained (Fig. 7.10 and Fig. 7.11). The seismic subsidence threshold of the dam crest yields Settlementthreshold = 0.1 m, Reliability = 1, and the corresponding seismic safety level is safe. b. Stochastic dynamic reliability of an earth dam under SEE action To evaluate the state of a dam under SEE loading, reliability analysis and its safety level are evaluated under SEE action. As above, the PDF and CDF of the extreme dam crest subsidence value can be obtained (Figs. 7.12 and 7.13). It can be seen that the seismic subsidence threshold of the dam crest is Settlementthreshold = 0.3 m, Reliability = 1.0, and the corresponding seismic safety level is safe. In summary, the seismic performance and safety assessment of the dam are all regarded as safe under OBE and SEE action. Fig. 7.10 PDF of an extreme value event of dam crest subsidence under OBE action

116 Fig. 7.11 CDF of an extreme value event of dam crest subsidence under OBE action

Fig. 7.12 PDF of an extreme value event of dam crest subsidence under SEE action

Fig. 7.13 CDF of an extreme value event of dam crest subsidence under SEE action

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7.1.2 Seismic Probabilistic Risk Assessment of Earth Dams In addition to performance evaluation, risk evaluation is also an important engineering issue. After a reasonable selection of the indicators for performance evaluation, the results can also be used to carry out related risk assessment research. In this section, the above dam is selected to illustrate the application process of probabilistic performance assessment results in risk assessment for reference according to the SPRA framework presented in Chap. 6. Similarly, a dam model for probabilistic risk analysis according to the process in Sect. 7.1.1 is chosen here. The same earth dam dynamic model and the same parameter information and nonlinear constitutive model in GEO-SLOPE software were used for stochastic dynamic analysis. The stochastic dynamic method, as a PDEM example that can better consider the uncertainty and non-stationarity of seismic activity, is introduced for the dam performance evaluation and stochastic ground motions are also artificially generated following the process described in Sect. 7.1.1 (Huang and Xiong 2017; Liu et al. 2016). (1) Seismic fragility assessment of earth dams The relative dam crest settlement (ratio of the crest settlement to dam height) can be also used obtained using different safety assessments and grading standards (Darbre 2004; Swaisgood 2003), similar to the seismic performance assessment standard of earth dams used here. The critical damage states and damage criteria were selected for the example: Level 1: 0.01% and Level 2: 0.1%. The PDF and CDF of the maximum vertical dam settlement under different peak ground motions (0.1 g, 0.2 g, 0.4 g, 0.8 g) are shown in Figs. 7.14 and 7.15. The peak value of the PDF function decreases with increasing PGA and the range increases, indicating an overall change of the dam settlement. The CDF also shifts over this time, the maximum displacement value becomes smaller when reaching 1, and the overall dynamic reliability decreases with increasing seismic intensity. Under the same framework, dynamic reliability information of the dam under the action of ground motions of 0.1 g, 0.2 g, 0.4 g, 0.8 g is obtained in sequence. The overall reliability information of the dam is obtained using three different levels of performance design standards. The seismic fragile analysis here reflects the probability of the state of the system to exceed a certain limit value. The performance level evaluation and fragile results of the dam under different earthquake levels can be obtained in the stochastic dynamic reliability analysis. The seismic vulnerability curve of the dam is shown in Fig. 7.16 in combination with the above analysis results using a designed limit state. The results show that the dynamic reliability of the dam changes with ground motion amplitude, indicating a synchronous change that reflects the influence of the seismic excitation uncertainty on the overall dam stability.

118 Fig. 7.14 PDF of the maximum vertical displacement of the dam crest under different seismic intensities

Fig. 7.15 CDF of the maximum vertical displacement of the dam crest under different seismic intensities

Fig. 7.16 Earthquake vulnerability curve of an earth dam

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(2) Seismic probabilistic risk assessment of earth dams In probabilistic seismic hazard analysis, the PGA is selected as the intensity value. The probability distribution of the earthquake PGA can be assumed to be a type II extreme value distribution (Ellingwood 2001). It can be considered that (Shen et al. 2008): / FA (a) = exp (−(a ag )−k )

(7.1)

where a represents the PGA at a certain point, ag represents the probability of PGA exceeding 63.2% within a period of time, and K is the shape parameter. The values of ag and K can be determined from the seismic risk analysis data of the determined location. In existing studies, ag and K have been assumed to be 1.14 m/s2 and 3.0834, respectively, under typical site conditions (Shen et al. 2008). Figure 7.17 shows the seismic hazard curve for a 100-year period. The seismic vulnerability results of each performance level in the PDEM are combined with the seismic risk probability analysis results to obtain the seismic risk probability of the earth dam shown in Fig. 7.18. Within a 100-year cycle, the three performance levels of earth dams have different seismic risks under the influence of different intensities. The damage probability is the highest under the seismic conditions of 0.4 g intensity and performance Level 1. Earthquake loss analysis includes the direct economic loss, indirect economic loss, and non-economic loss. It can also be regarded as an indirect economic loss caused by the interruption of power generation and upstream water supply utilities. Disaster relief costs are another form of indirect economic loss. Non-economic losses include casualties and psychological and spiritual effects caused by earthquakes. Fig. 7.17 Seismic hazard curve of an earth dam

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Fig. 7.18 Seismic risk probability of an earth dam with different performance criteria

(a)

(b)

Comprehensive earthquake loss analysis should include all these parts, but this can be very complicated. Thus, only direct losses from the destruction and its auxiliary facilities are considered in this study. The construction costs of a hydraulic dam are very high. The cost and workload of repair and reinforcement are also very high, even if only a small part of the dam is destroyed after an earthquake. The direct economic loss caused by the destruction of the dam and its auxiliary facilities can be calculated as (Shen et al. 2008): L(Ls) = VrLs

(7.2)

where L(Ls) is the direct economic loss when the damage state is Ls, V is the total value of the dam, and rLs is the loss rate of the dam body when the failure state Ls

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occurs, and is represented as the ratio of the repair cost and the total value of the dam. In this study, we assume that the total value of V of the dam is 500 million USD and rLs1 is set to 0.1 (Wang et al. 2012). Considering that the probability of dam damage is higher in Level 1, we only consider the direct economic loss of the dam in this level. When the earthquake PGA is 0.4 g, the seismic loss can reach 50 million USD. The seismic risk probability P(Ls) in this level is 0.01218, thus the whole risk loss can be 609,000 USD. The earthquake risk probability and risk loss obtained in SPRA can provide basic analysis and decision-making basis for risk management. This includes dam reconstruction, risk mitigation strategies, anti-seismic optimization of existing old dams that do not meet the current seismic requirements, and the determination of earthquake risk reduction measures.

7.2 Probabilistic Seismic Performance of Slope Retaining structures are useful and effective means for reinforcement and prevention of slope engineering. When a strong earthquake occurs, high damage like traffic system paralysis, can seriously delay the rescue work after the disaster. Deterministic theory is widely used for anti-seismic performance design of retaining structures. However, the running state of slopes will be affected by some uncertain factors during its service life, its performance and influence of these uncertainties on the seismic performance of slope can therefore not be considered using the conventional deterministic analysis method. Thus, the probabilistic performance-based seismic design is introduced and applied to slope engineering.

7.2.1 Slope Probabilistic Seismic Performance Evaluation (1) Slope seismic dynamic reliability based on horizontal displacement A slope model supported by an anchor-pile slab wall is taken as an example, and the probabilistic evaluation method based on nonlinear stochastic dynamics proposed above is applied to the evaluation of slope seismic dynamic reliability considering the randomness of ground motion. Figure 7.19 is a slope model supported by a pile slab wall with two rows of anchor cables. The pile wall is 12-m high with 3-m embedded. The pile body is regarded as a beam element. The upper anchor cable is approximately 8.4 m and has a dip angle of 17.3°; and the lower anchor cable is approximately 7.3 m with an angle of 15.8°. The anchorage section is simulated as a beam element and can be subjected to tension and compression. The free section is simulated by a rod element and can only be strained. The soil material is assumed to be uniformly distributed and the main physical and mechanical parameters include

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bulk density (γ) = 18 kN/m3 , cohesion = 3 kPa, internal friction angle (ϕ) = 34°, and Poisson’s ratio (μ) = 0.4. The Mohr–Coulomb constitutive model is used and, as mentioned, a nonlinear stochastic dynamic method such as PDEM is used here for the transformation of the slope stochastic seismic dynamic response into nonlinear time-history results, and also for solving tools of the generalized probability density evolution equation. Considering the randomness of seismic excitations, a spectral representation of the random function method is used to generate 144 seismic acceleration samples in Fig. 7.20. A series of random earthquakes were input into the model. Deterministic nonlinear dynamic time-history analysis was used to obtain the dynamic response of the slope using Geo-Studio software. Considering that the horizontal displacement value is a common and reasonable seismic performance evaluation index of slope, the seismic displacement time histories of 144 pile tops were derived and solved by introducing the results into the generalized probability density evolution equation (GPDEE). The stochastic seismic Upper anchor Lower anchor Free por tion of anchor Sheet pie wall

Fig. 7.19 Model of the slope anchor-pile wall 0.1

Fig. 7.20 Time history of an earthquake sample Acceleration(g)

0.05

0

-0.05

-0.1

0

5

10 Time(s)

15

20

7.2 Probabilistic Seismic Performance of Slope

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Fig. 7.21 Mean and standard deviation of the horizontal displacement of the pile tip

dynamic response and reliability of the dynamic system can be obtained. Figure 7.21 shows the mean value and second-order statistics such as the standard deviation of the horizontal displacement of the pile tip. The displacement of the pile tip has great variability under the action of random earthquakes. The standard deviation increases gradually and then decreases with increasing time. The potential reason is the permanent displacement of the pile tip gradually increases when the earthquake intensity increases, but decreases with the attenuation of the earthquake, thus increasing the variability of the displacement. Figure 7.22 shows the PDF curves of the horizontal displacement of the pile tip at three typical times. The displacement of the PDF is found to change over time. Figure 7.23 shows the PDF surface of displacement of the anchor-pile top at 5.2–5.8 s to characterize the change process of the PDF displacement with time. It can be seen that the displacement of the pile top is mostly in the range of ±0.03 m owing to the small ground motion intensity. The PDF surface looks like an undulating mountain with a peak, and the peak represents the point at which the probability is greatest. This surface shows the evolution process of the PDF with time, and also reflects the transformation and evolution process of randomness in the slope stochastic dynamic system. The displacement time history of a series of models is viewed as a random process and input into the GPDEE for the solution, but the extreme value of displacement is not a random process. According to the research of Li et al. (2007), a virtual stochastic process is introduced based on the equivalent extreme value event principle, and then brought into the GPDEE to obtain the stochastic response results of the extreme values and reliability of the system. Figure 7.24 is the CDF curve of the extreme displacement of the anchor-pile tip.

124

Fig. 7.22 PDFs of the horizontal displacement of the pile top at different times

Fig. 7.23 PDF surface of horizontal displacement of pile tip at 4–6 s Fig. 7.24 CDF of extreme value event of the pile tip

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If a small damage level of 0.02 m is regarded as the evaluation index, then the seismic reliability of this model under random ground motions with PGA at 0.1 g is 0.9935, which averages the failure probability to 0.0065. If the other standard performance requirements of evaluation indexes are used, the corresponding seismic reliability of the model can also be obtained. According to the study of the relationship between failure probability and performance, it can be seen that the structure was safe and reliable under the action of random earthquakes.

7.2.2 Seismic Risk Assessment of Slope Retaining Systems Uncertain factors such as ground motion will affect the dynamic stability and performance of slope retaining structures, while the probability analysis method can be used to quantitatively represent the influence of these uncertain factors, which provide more practical, more reasonable, and more credible seismic evaluation results. (1) Slope seismic fragility analysis Fragile analysis is a kind of probabilistic analysis methods that refers to the possibility of the system reaching different damage degrees under different disaster intensities. Correspondingly, the seismic fragile nature of retaining structures refers to the probability of the slope retaining structure system reaching different failure states suffering seismic excitations with different amplitudes, which can provide the conditional probability of the system exceeding different failure states. It is an effective method to evaluate the seismic performance of structures under earthquake action, and also the most important part of the probabilistic seismic risk analysis methods. In this section, the seismic vulnerability analysis method of the supporting structure based on PDEM is used in the seismic vulnerability analysis of slope. There are many studies on the fragile analysis of slope related disasters (Akbas et al. 2009; Fotopoulou and Pitilakis 2017; Uzielli et al. 2008). Included among the vulnerability analysis methods based on a specific distribution hypothesis are incremental dynamic analysis (IDA), strip methods, and cloud map methods. Among them, the IDA has been widely applied in seismic vulnerability assessment of large structures such as buildings, bridges, and dams (Baker 2015; Günay and Mosalam 2013; Vamvatsikos and Cornell 2004) by modulating the measured ground motion amplitude. The nonlinear stochastic dynamic theory proposed in Chap. 3 can quantitatively characterize the uncertainties from different sources and their influences on the performance level of slope to obtain the seismic dynamic reliability of the slope system. Seismic analytical fragile analysis of slope can be carried out based on this framework, and the conditional probability of slope system exceeding different failure states under different ground motion intensities can be calculated using the main idea of the IDA method to construct the seismic fragile analysis curve.

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It is pointed out in the foregoing section that the displacement of slope with supporting structures is useful as the evaluation index of seismic performance of slope, and they can also be applied to the evaluation of the seismic vulnerability of slope. To better reflect the performance level of the retaining structure under different earthquake intensities, the horizontal displacement can be used to characterize its seismic performance level under an earthquake. According to the assessment criteria of displacement of a retaining structure, if the small damage level of 0.02 m is regarded as the evaluation index, then the seismic reliability of this model under random ground motions with PGA of 0.1 g is 0.9988. The corresponding seismic reliability of the model can also be obtained when using standard performance requirements of evaluation indexes. The sum of reliability and failure probability is 1. Therefore, failure probability under different ground motion intensities and different failure states can be obtained through the nonlinear stochastic dynamic method, which means that the seismic vulnerability of a supporting structure considering random factors can be obtained. For example, the amplitude of the 144 random ground motion generated above can be adjusted to 0.2 g, 0.3 g, and 0.4 g with an interval of 0.1 s according to the IDA method. The CDF curves of the extreme model displacement under random ground motion loads of different intensities can be obtained through the same process mentioned above, as shown in Fig. 7.25. The CDF curve moves to the right with increasing ground motion intensity, which means that with the increase of seismic amplitude, the seismic reliability decreases and the seismic performance decreases. Taking the evaluation criteria of the three displacement criteria (0.02, 0.03, and 0.04 m) as an example, the seismic reliability of the model in different performance states can be obtained under the action of different ground motion intensities according to Fig. 7.25 and the three performance state evaluation criteria. The failure probability of the model under different conditions can be obtained by subtracting the reliability value from 1. The seismic vulnerability curve of the retaining wall can accordingly be obtained, as shown in Fig. 7.26. The horizontal coordinate is the ground motion peak, whereas the vertical axis is the exceeding probability of the model exceeding different performance states. (2) Seismic risk assessment of slope engineering structures In the previous chapter, based on the PDEM method, we obtained the seismic vulnerability analysis curve of a dam under the influence of composite random factors (Fig. 7.26). The seismic risk probability can be obtained when combining the site seismic hazard analysis in Sect. 7.1.2, and Fig. 7.27 shows the seismic risk probabilities of the three performance levels of the supporting structure under the earthquake with different intensities. The risk probability curves of different performance levels vary in shape, and the peaks corresponding to seismic intensity points are also different. It can be seen from the figure that the 0.2 g ground motion causes the model to reach the performance level of 0.01 m with the highest risk probability.

7.2 Probabilistic Seismic Performance of Slope Fig. 7.25 CDF of settlement for the different PGAs

Fig. 7.26 Seismic fragility curves of slope

Fig. 7.27 Seismic risk probabilities of the three performance levels of the supporting structure under the earthquake with different intensities

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There is little research on the direct economic loss of the damage of the supporting structure caused by earthquakes. In this section, the direct economic loss of a supporting structure is calculated by referring to the calculation method of direct loss in dam engineering. In dam engineering, the costs of dam construction, repair, and fixation are very high. Even if the damage caused by an earthquake is small, the cost of repair is large. According to relevant domestic studies (Shen et al. 2008), the direct economic loss caused by the dam failure under the influence of earthquake is calculated by Eq. 7.2. Referring to the definition of direct economic loss in the dam, we use the same definition to obtain the seismic loss of slope with a supporting structure. Considering that the probability of the dam damage is bigger in the performance level of 0.02 m, we only consider the direct economic loss of the dam in this level. Suppose the overall cost of the retaining structure is ~0.5 million USD and rLs1 is 0.1. When PGA of the earthquake is 0.2 g, seismic loss in this level can reach 50,000 USD. Seismic risk probability P(Ls) in this level is 0.1021, and the whole risk loss can then be 5,105 USD. Although this section only presents the vulnerability and risk analysis results of a simple supporting structure model, this framework is also applicable to other geological engineering and geotechnical engineering structures. We can also determine the seismic probabilistic risk result of the slope under the stochastic coupling effect of ground motion and rock and soil parameters with spatial variability. The basic process is nearly the same. Readers can refer to the above introduction and our other book, Slope Stochastic Dynamic (Huang et al. 2022).

References Akbas SO, Blah˚ut J, Sterlacchini S (2009) Critical assessment of existing physical vulnerability estimation approaches for debris flows. In: Malet JP, Remaitre A, Bogaard T (eds) Proceedings of landslide processes: from geomorphologic mapping to dynamic modelling. Strasburg, France, vol 67. Strasbourg, pp 229–233 Baker JW (2015) Efficient analytical fragility function fitting using dynamic structural analysis. Earthq Spectra 31(1):579–599 Darbre GR (2004) Swiss guidelines for the earthquake safety of dams. Paper presented at the 13th world conference on earthquake engineering, Vancouver Douglas J, Edwards B (2016) Recent and future developments in earthquake ground motion estimation. Earth-Sci Rev 160:203–219 Ellingwood BR (2001) Earthquake risk assessment of building structures. Reliab Eng Syst Safe 74(3):251–262 Fotopoulou SD, Pitilakis KD (2017) Probabilistic assessment of the vulnerability of reinforced concrete buildings subjected to earthquake induced landslides. Bull Earthq Eng 15(12):5191– 5215 Guidelines for seismic parameters of dams. Bulletin 72 C.F.R. (1989) Günay MS, Mosalam KM (2013) PEER performance-based earthquake engineering methodology, revisited. J Earthq Eng 17(6):829–858 Hariri-Ardebili MA, Saouma VE (2016) Probabilistic seismic demand model and optimal intensity measure for concrete dams. Struct Safe 59:67–85

References

129

Huang Y, Xiong M, Zhao L (2022) Theoretical framework of slope stochastic dynamics. In: Huang Y, Xiong M, and Zhao L (eds), Slope stochastic dynamics. Springer, Singapore, pp 27–52 Huang Y, Xiong M (2017) Probability density evolution method for seismic liquefaction performance analysis of earth dam. Earthq Struct Dyn 46(6):925–943 Li J, Chen J, Fan W (2007) The equivalent extreme-value event and evaluation of the structural system reliability. Struct Safe 29(2):112–131 Liu Z, Liu W, Peng Y (2016) Random function based spectral representation of stationary and non-stationary stochastic processes. Probabilistic Eng Mech 45:115–126 Peng J, Tong X, Wang S, et al. (2018) Three-dimensional geological structures and sliding factors and modes of loess landslides. Environ Earth Sci 77(19) Shao L, Chi S, Li H et al (2011) Preliminary studies of ultimate aseismic capacity of high core rockfill dam. Rock Soil Mech 32(12):3826–3882 Shen H, Jin F, Zhang C (2008) Performance-based aseismic risk analysis model of concrete gravity dams. Rock Soil Mech 29(12):3323–3328 Standard for seismic design of hydraulic structures, GB 51247-2018 C.F.R. (2018) Swaisgood J (2003) Embankment dam deformations caused by earthquakes. Paper presented at the Pacific conference on earthquake engineering Uzielli M, Nadim F, Lacasse S et al (2008) A conceptual framework for quantitative estimation of physical vulnerability to landslides. Eng Geol 102(3–4):251–256 Vamvatsikos D, Cornell CA (2004) Applied incremental dynamic analysis. Earthq Spectra 20(2):523–553 Wang D, Liu H, Yu T (2012) Seismic risk analysis of earth-rock dam based on deformation. Rock Soil Mech 33(5):1479–1484

Chapter 8

Conclusions and Prospects

8.1 Conclusions This book mainly focuses on the key subjects of the global framework of earthquake dynamic safety performance within generalized slope engineering. This starts from the quantitative characterization of randomness and spatial variability on the earthquake excitation and geomaterial properties, in combination with the time domain nonlinear dynamic analysis and stochastic dynamics, performance-based earthquake engineering, and dynamic reliability, to propose a comprehensive evaluation framework for earthquake dynamic performance of generalized slope engineering based on nonlinear stochastic dynamics. The specific framework content is as follows: (1) A summary of the existing development status of slope engineering seismic performance design is proposed by quantitatively characterizing the randomness and uncertainty and their influence on the stochastic dynamic analysis methods for slope probabilistic earthquake dynamic performance according to the time domain nonlinear dynamic analysis. This includes a stochastic earthquake excitation model to quantitatively characterize the randomness of the earthquake excitation on the slope site, and to synthesize a series of acceleration time-history samples that meet the engineering site conditions. The spatial uncertainty of geomaterials is then described using the random field model. Nonlinear stochastic dynamic analysis is introduced, and the generalized evolution equation of slope engineering seismic performance indicators is established, which quantitatively characterizes the influence of variability on the seismic dynamic safety of slope engineering. (2) According to the PBSD concept, appropriate evaluation indicators for the seismic performance of slope engineering are selected, combined with the proposed probabilistic dynamic analysis framework, respectively, the effects of randomness of earthquake excitation or spatial variability of geomaterial on the earthquake dynamic performance of slope engineering are studied. The impact reveals the probability evolution and migration of uncertainty in the stochastic © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Huang et al., Guidelines for Probabilistic Performance-Based Seismic Design and Assessment of Slope Engineering, https://doi.org/10.1007/978-981-19-9183-7_8

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dynamic system of slope engineering from multiple levels such as first-order statistics (mean or standard deviation), the probability density function (PDF), and the seismic dynamic reliability of generalized slope. Performance indicators are also a stochastic process under the influence of random factors, and the nonlinear stochastic dynamic response results such as nonlinear safety of factor and displacement time histories have strong variability. (3) In combination with the above reliability evaluation indicators, slope engineering seismic vulnerability analysis and hazard analysis research were also carried out. The advantage of the physical simulation-based vulnerability analysis method that is not based on the assumption of a specific probability distribution is presented, and the proposed method based on the PDF is proposed. The nonlinear stochastic vibration analysis method is applied to the seismic dynamic vulnerability assessment of slope engineering. It studies and quantitatively characterizes the impact of uncertain factors on the seismic dynamic vulnerability and risk analysis of generalized slope. Under the conditional probability of exceeding different seismic performance levels, the slope engineering seismic vulnerability analysis curve is obtained through the probability distribution assumption, and the method is coupled to the seismic probability risk assessment framework, combining the site seismic risk assessment and seismic loss analysis to obtain the seismic risk probability and seismic loss results of the slope support system.

8.2 Prospects Great research improvements have been made in the earthquake dynamic safety evaluation of generalized slope in recent years. As for how to develop and perfect the global procedure framework for the earthquake performance assessment of generalized slope engineering with new disaster prevention concepts (e.g., resilience) must still be further promoted and studied. Here, this book develops a probability-based method for the seismic performance of slope engineering, which can be used as a basis for evaluating the reliability of slope engineering under earthquake action, and is perfected and improved on this basis. The above research results are expected to provide a scientific and reasonable theoretical basis and an effective analytical tool for the earthquake dynamic risk assessment and emergency management of geological disasters such as landslides. Probability-based analysis and design methods can optimize the seismic performance of slope engineering for ground motion with a complete set of probabilities, making the designed and optimized structure more robust to any future ground motions to achieve the goal of resilience and disaster prevention and build resilient urban and rural areas. This book introduces the concept of resilience to explore this possible development direction.

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(1) Resilience-based slope seismic design The concepts of resilience design and disaster prevention are based on the probabilistic analysis method and developed after the geological disaster prevention concept based on the factor of safety, reliability, and performance. Here, we build on the existing research foundation of this book. A brief description of the resiliencebased seismic performance design method of slope engineering is given. The probabilistic earthquake dynamic performance assessment procedure proposed in this guideline is considered to be coupled to the framework of resilience design and disaster prevention, to achieve the goal of earthquake disaster prevention resilience of generalized slope, and to build resilient cities and rural areas. Design and evaluation of toughness-based slope seismic performance should also consider the various uncertainties and randomness faced by the system during the entire disaster resistance process, including rock and soil properties, ground motion strength, supporting structure, among others. The randomness of these physical properties during the restoration process is very important to the functional status of the entire slope engineering system. These should be presented in the resilience recovery model through random variables and stochastic processes. In general, seismic design based on resilience considers many aspects. In addition to basic slope stability, structural design, reliability and vulnerability analysis, it also covers remote automatic detection systems for slopes and artificial intelligence. As an emerging PBSD slope seismic design method, the concept of resilience should be introduced. The disaster prevention design based on the concept of resilience not only improves the seismic robustness of generalized slope engineering, but also reduces the time and economic cost of post-disaster recovery or reconstruction. (2) Optimization seismic design of slope engineering In performance-based generalized slope engineering, with the development of nonlinear seismic stochastic dynamic analysis and earthquake dynamic response control theory, the former two are mutually integrated and promote each other. Earthquake dynamic response control technology and measures have been widely applied and developed in generalized slope engineering. Previous studies pointed out the challenge of precisely grasping the earthquake dynamic performance of generalized slope according to the traditional deterministic vibration control theory owing to the uncertainty of external loads and system physical and mechanical parameters. The control of the earthquake dynamic response and dynamic optimization of the seismic performance of slope requires reasonable quantitative characterization. The randomness in the system realizes the optimal control of the system from the probabilistic level, and proposes the optimal control strategy. In geotechnical engineering and geological engineering research, the randomness of geotechnical parameters is greater than that of concrete materials and steel, and the geotechnical structure is also more strongly affected. Therefore, it is momentous to optimize the design and control of geotechnical engineering structures in the sense of

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probability description level. Based on the probabilistic earthquake dynamic performance of generalized slope obtained in this book, the optimal design of the supporting structure at the probabilistic level can be realized from the following three perspectives: ensemble average, probability density function, and reliability. The pile-anchor supporting structure includes different anti-slide pile stiffnesses, anchor cable and lock head arrangements, pile body and anchor cable layout positions, anchoring angles, and anchoring depths, which are, for the most part, the primary tasks for earthquake dynamic performance of anchor cable anti-slide piles. The impact of adding expanded polystyrene material behind the pile and adding shock-absorbing and energy-consuming devices at the pile-anchor joints will also have an impact. Therefore, in the design of specific parameters and layout schemes, the stochastic optimization design method can be used in nonlinear earthquake dynamic optimization for the seismic performance of the pile-anchor supporting system from the probabilistic level.