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ELSEVIER GEO-ENGINEERING BOOK SERIES VOLUME 1

Stability Analysis and Modelling of Underground Excavations in Fractured Rocks

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ELSEVIER GEO-ENGINEERING BOOK SERIES VOLUME 1

Stability Analysis and Modelling of Underground Excavations in Fractured Rocks Weishen Zhu Shandong University, Jinan, China

Jian Zhao Nanyang Technological University, Singapore

Geo-Engineering Book Series Editor

John A. Hudson Imperial College of Science, Technology and Medicine, University of London, UK

2004

Amsterdam – Boston – Heidelberg – London – New York – Oxford Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo

ELSEVIER Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB, UK ß 2004 Elsevier Ltd. All rights reserved. This work is protected under copyright by Elsevier, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-proﬁt educational classroom use. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (þ44) 1865 843830, fax: (þ44) 1865 853333; e-mail: [email protected] You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (þ1) (978) 7508400, fax: (þ1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (þ44) 207 631 5555; fax: (þ44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier’s Science & Technology Rights Department, at the phone, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made. First edition 2004 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for. British Library Cataloguing in Publication Data A catalogue record from the British Library has been applied for.

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Series Preface I am delighted to introduce the new Elsevier Geo-Engineering Book Series. Our objective is to publish high quality books on subjects within the broad Geo-Engineering subject area — e.g. on engineering geology, soil mechanics, rock mechanics, civil/mining/environmental/petroleum engineering, etc. The topics potentially include theory, ground characterization, modelling, laboratory testing, engineering design, construction and case studies. The principles of physics form a common basis for all the subjects, but the way in which this physics is manifested across the wide spectrum of geo-engineering applications provides a rich variety of potential book themes. Accordingly, we anticipate that an exciting series of books will be developed in the years ahead. We welcome proposals for new books. Please send these to me at the email address below. Professor John A Hudson FREng Series Editor [email protected]

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Preface It is well recognised that rock masses are very diﬀerent from common man-made engineering materials. The nature-created geological medium is a discontinuous, inhomogeneous, anisotropic and non-linearly elastic medium. The characteristics of rock masses vary with time, location and direction. Due to the nature of most of underground rock excavations, those excavated structures are usually of a large size and require long-term serviceability. Therefore, the way rock mass characteristics vary with time, location and direction will be reﬂected in the surrounding rock masses of the excavations. Such variation should, ideally, be captured in the modelling and analysis of rock masses. This book attempts to tackle the problems of modelling and analysis of excavations in rock masses, by taking into account discontinuity inﬂuence, time dependence behaviour and construction method. The content of this book is largely based on many years of research work by the senior author and his research group in China, supplemented by the contribution by the junior author and his research groups in Singapore and China. Much of the content of this book has been published in the form of technical papers, while some content is presented for the ﬁrst time. The physical and numerical modelling work described in this book is conducted with several research grants awarded to the senior author, including the China National Natural Science Foundation (NSF) program ‘‘Interaction between geomaterials and hydraulic structures’’ sub-project ‘‘Mechanical characteristics of jointed rock masses and construction mechanics of rock structures’’, other NFC projects (Numbers 59939190, 40272120 and 50229901), as well as the 7-5 National Programs on Science and Technology. Much of the work presented in this book results from the eﬀorts of the research groups headed by the authors. The accomplishment of this book would not have been possible without the great contributions by their colleagues and research students. The authors wish to thank Shiwei Bai, Kejun Wang, Rongming Pan, Xinping Li, Jingnan Xu, Guang Zhang, Ping Wang, Zuoyuan Liang, Bailin Wu, Suhua Li, Rui Ding, Haibin Xu, Haiying Bian, Jungang Cai, Shaogen Chen, Yuhui Zhao, Haibo Li, Yuyong Jiao, Xiaobao Zhao, Hongwei Song and Ashraf Hefny, for their contributions, comments and reviews. The authors also appreciate the encouragement and advice from Elsevier’s editorial team James Sullivan, Vicki Wetherell and Lorna Canderton. Weishen Zhu Jian Zhao

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About the Authors Weishen Zhu is a professor of rock engineering and director of the Geotechnical and Structural Engineering Centre at Shandong University in Jinan, China. Dr Zhu graduated in construction engineering from the Beijing Institute of Mining and Technology in 1956 and obtained a PhD from the Institute of Mining and Metallurgy, Cracow, Poland in 1962. Between 1962 and 2001, Dr Zhu was a research scientist in the Chinese Academy of Science Wuhan Institute of Rock and Soil Mechanics, and he was also a director of the Institute. Dr Zhu is Editor-in-Chief of the Chinese Journal of Rock Mechanics and Engineering, President of the Chinese Society of Underground Engineering and Underground Space, and an editorial board member of the international journal Rock Mechanics and Rock Engineering. Dr Zhu has published 4 books and over 150 technical papers. Dr Zhu’s main research interests are mechanical properties of fractured rock masses, rock reinforcement by bolts, construction mechanics and stability analysis of large rock structures. Dr Zhu has more than 40 years experience in rock mechanics and engineering, and has been involved in numerous rock engineering and underground excavation projects in China. Jian Zhao chairs the underground technology and rock engineering program at Nanyang Technological University of Singapore, and is an adjunct CKSP professor of geotechnical engineering at China University of Mining and Technology. Dr Zhao graduated in civil engineering from the University of Leeds in 1983 and obtained a PhD from Imperial College, London University, in 1987. Dr Zhao is an Editor of the international journal of Tunnelling and Underground Space Technology, editorial board member of the International Journal of Rock Mechanics and Mining Sciences and of 5 other technical journals. Dr Zhao’s main research interests are rock joint properties, rock excavation and support, and rock structure stability under dynamic loads, and has published over 180 technical papers and 3 monographs. Dr Zhao also actively consults on rock engineering and tunnelling projects.

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Contents Series Preface Preface About the Authors

v vii ix

CHAPTER 11 INTRODUCTION

1.1. 1.2.

Rock Mechanics and Underground Excavation Stability Structure of the Book

1 3

CHAPTER 22 PHYSICAL MODELLING OF JOINTED ROCK MASS

2.1.

2.2.

2.3.

Modelling of Jointed Rock Masses Under Plane Stress State 2.1.1 Mechanical Properties of Equivalent Materials 2.1.2 Model Tests of Jointed Rock Mass 2.1.2.1. Effect of Joint Distribution Patterns 2.1.2.2. Strength of Typical Jointed Rock Masses 2.1.2.3. Anisotropy of Rock Strength 2.1.2.4. Relation Between Rock Mass Strength and Joint Persistence 2.1.3 Large Dimension Plane Strain Model Experiment 2.1.3.1. Material Properties and Experimental Methods 2.1.3.2. Testing Results Model Test of Rock Mass with Rock Bridges 2.2.1 Experiment Phase I 2.2.2 Experiment Phase II Model Tests on Stability of Surrounding Rock of Large-scale Cavity 2.3.1 Similarity Conditions of Modelling 2.3.2 Deformation of the Surrounding Rock Mass during Excavation 2.3.2.1. Deformation of the Surrounding Rock Mass during Excavation 2.3.2.2. Failure of the Surrounding Rock Mass during Excavation 2.3.2.3. Stability of the Surrounding Rock Mass

5 5 6 7 9 11 11 13 14 14 16 16 19 22 24 24 24 25 25

xii

Contents

CHAPTER 33 NUMERICAL MODELLING OF JOINTED ROCK MASS

3.1.

3.2.

3.3.

Equivalent Continuum Model for Jointed Rock Masses 3.1.1 Basic Principles 3.1.2 Deformation Equivalence 3.1.2.1. Deformation Equivalence with no Joint Dilation 3.1.2.2. Deformation Equivalence with Joint Dilation 3.1.3 Formula of Strength Equivalence 3.1.3.1. Strength Equivalence in the Case of a Single Joint 3.1.3.2. Strength Equivalence in the Case of Two Joint Planes 3.1.3.3. Analysis of Tensile Failure 3.1.4 Treatment of Elements with Non-persistent Joint 3.1.4.1. Variation of Joint Length 3.1.4.2. Variation of Joint Inclination 3.1.4.3. Scale Effect of Element 3.1.5 Verification of the Numerical Model by Physical Modelling 3.1.6 Examples of Engineering Applications 3.1.6.1. Prediction of Strength of Jointed Rock Mass 3.1.6.2. Stability Analysis of Jointed Rock Masses Equivalent Analysis for Rock Masses Containing Thick Joints 3.2.1 Equivalent Deformation Principle and Method 3.2.1.1. Equilibrium Condition 3.2.1.2. Displacement Compatibility Condition 3.2.1.3. Physical Equations 3.2.2 Basic Principle of Strength Equivalence 3.2.3 Engineering Applications Strength Characteristics of Fractured Rock Mass Under Compressive Shear Stress 3.3.1 Strength of Rock Mass Containing Collinear Cracks 3.3.1.1. Fracture Propagating at Tight Cracks 3.3.1.2. Determination of Shearing Strength of Crack Body 3.3.1.3. Verification Through Model Tests 3.3.2 Strength of Rock Mass Containing Multiple Cracks

28 28 29 30 33 33 33 35 36 36 37 37 38 39 39 39 42 45 45 45 46 46 50 52 54 55 55 56 59 59

CHAPTER 44 SENSITIVITY ANALYSIS OF ROCK MASS PARAMETERS

4.1.

Sensitivity Analysis of Commonly Used Parameters 4.1.1 Method of Sensitivity Analysis 4.1.2 Sensitivity Analysis of Stability of Underground Works

67 67 70

Contents

4.2.

4.3.

4.1.2.1. Computational Model 4.1.2.2. Analysing the Results 4.1.3 Application to Optimisation of Test Schemes Analysis of the Effect of Joint Parameters on Rock Mass Deformability 4.2.1 Application of Equivalent Model for Jointed Rock Mass 4.2.2 Basic Parameter for Sensitivity Analysis 4.2.3 Computational Results 4.2.3.1. Effects of Joint Elastic Moduli on Displacement 4.2.3.2. Effect of Joint Poisson’s Ratio on Rock Mass Deformation 4.2.3.3. Effect of Joint Dip Angle on Rock Mass Displacement 4.2.3.4. Effect of Joint Persistence on Rock Mass Deformations 4.2.3.5. Effect of Joint Aperture on Rock Mass Deformations 4.2.3.6. Comparison of Sensitivity of Different Parameters Sensitivity Analysis of Rock Mass Parameters on Damage Zones 4.3.1 Failure Criterion for the Equivalent Jointed Rock Mass 4.3.2 Sensitivity Analysis of an Underground Cavern Complex 4.3.3 Result and Analysis 4.3.3.1. Effect of Parameters on Damaged Zones 4.3.3.2. Comparison Between Sensitivities of Various Parameters 4.3.4 Summary

xiii 70 71 74 75 75 76 77 77 78 79 80 80 82 83 83 84 84 84 86 87

CHAPTER 55 STABILITY ANALYSIS OF RHEOLOGIC ROCK MASS

5.1. 5.2.

5.3.

5.4.

Rheological Mechanical Models for Rocks and Rock Masses Visco-elastic Surrounding Rock Mass and Supporting Problem 5.2.1 General Solution for Circular Visco-elastic Media 5.2.2 Interaction of Visco-elastic Surrounding Rock Mass and Elastic Lining 5.2.3 Interaction of Rock Mass and Lining of Different Visco-elastic Media 5.2.4 Two-dimensional Stress State in Surrounding Visco-elastic Rock Mass Interaction between the Visco-elastic–Plastic Surrounding Rock and Lining 5.3.1 Stress State in Plastic Zones of Rock Mass 5.3.2 Interaction between Surrounding Rock Mass and Lining Stress State in Visco-elastic–Visco-plastic Surrounding Rock Masses

89 91 92 95 100 105 111 111 113 116

xiv 5.5.

5.6.

5.7.

Contents Rheological Analysis with Dilation and Softening of the Rock Mass 5.5.1 Mechanical Model of Surrounding Rock Mass 5.5.2 Visco-plastic Model Considering Dilation and Softening 5.5.2.1. Physical Model 5.5.2.2. Geometric Equation 5.5.2.3. Equilibrium Equation 5.5.3 Stress Components in Each Zone 5.5.3.1. Visco-plastic Zone 5.5.3.2. Residual Strength Zone 5.5.4 Stress State without Lining 5.5.4.1. Visco-plastic Zone, R1 < r < R2 5.5.4.2. Residual Strength Zone, a < r < R1 5.5.4.3. Determination of Boundary R2 5.5.4.4. Determination of Boundary R1 5.5.5 Stress State with Lining Effect of Bolt Reinforcement in Visco-elastic Rock Mass 5.6.1 Stress State in Different Zones 5.6.2 Discussion and Application Rheological Damage Analysis of the Rock Mass Stability 5.7.1 Damage Evolution Equation 5.7.2 Viscoelastic–Viscoplastic-damage Constitutive Equation and FEM Method 5.7.2.1. Constitutive Equation 5.7.2.2. FEM Method 5.7.3 Application to Stability Analysis of an Underground Opening 5.7.3.1. Decomposition of the Rheological Deformation 5.7.3.2. Determination of Model’s Parameters 5.7.3.3. Comparison between Calculated and In Situ Measured Results

122 122 124 124 124 125 125 125 127 127 127 128 129 129 130 132 132 140 144 145 146 146 148 150 150 152 153

CHAPTER 66 BACK ANALYSIS AND OBSERVATIONAL METHODS

6.1.

Elastic Back Analysis and Stress Distribution Analysis 6.1.1 Elastic Back Analysis 6.1.1.1. Basic Formulation 6.1.1.2. Deformation Monitoring and Back Analysis 6.1.2 Back Analysis of In Situ Stress Distribution

159 159 159 160 163

Contents

6.2.

6.3.

6.4.

6.5.

6.6.

6.7.

6.1.2.1. Computational Zone and Monitored In Situ Stress 6.1.2.2. Determination of Stress Function Visco-elastic Back Analysis and Its Engineering Applications 6.2.1 Method of Site Deformation Monitoring and Its Application Results 6.2.2 Visco-elastic Back Analysis 6.2.2.1. Computation Method 6.2.2.2. Visco-elastic Analysis Results Back Analysis and Optimised Methods in Transverse Isotropic Rock 6.3.1 Basic Formulae of Transverse Isotropic Mechanics 6.3.2 Optimisation Analysis Method 6.3.3 Examples of Engineering Applications 6.3.4 Discussions Back Analysis of Jointed Rock Mass and Stability Prediction 6.4.1 Description of the Project and Monitoring 6.4.1.1. Description of the Project 6.4.1.2. Data Processing and Modification 6.4.2 Back Analysis Using Pure Shape Acceleration Method 6.4.2.1. Computational Procedure 6.4.2.2. Computational Results 6.4.3 Stability Prediction of Powerhouse and Transformer Chamber Three-dimensional Back Analysis of Anisotropic Rock 6.5.1 Displacement Monitoring in Trial Tunnel and Results 6.5.1.1. Set-up of Displacement Monitoring 6.5.1.2. Monitoring Results 6.5.2 Back Analysis Three-dimensional Back Analysis of Jointed Rock Mass and Stability Analysis 6.6.1 Mechanic Model 6.6.2 Summary of Site Monitoring Data 6.6.3 Finite Element Back Analysis of Underground Powerhouse Complex 6.6.4 Stability of Powerhouse and Transformer Chamber Applications of Statistics Model in Deformation Prediction 6.7.1 Non-linear Regression Model 6.7.2 Grey System Theory Model 6.7.3 Engineering Application 6.7.4 Discussion

xv 163 163 166 166 170 170 171 172 172 176 177 179 181 181 181 181 182 183 183 185 190 190 190 191 191 194 194 194 197 200 201 202 203 207 207

xvi

Contents

CHAPTER 77 CONSTRUCTION MECHANICS AND OPTIMISATION OF EXCAVATION SCHEMES

7.1.

7.2.

7.3.

7.4.

Basic Principles of Interactive Construction Mechanics 7.1.1 Basic Principles 7.1.2 Engineering Applications 7.1.2.1. Description of the Project 7.1.2.2. Computational Implementation and Results for Different Excavation Sequences 7.1.2.3. Discussions Applications of Interactive Programming in Optimisation of Cavern Construction 7.2.1 Principles of Interactive Programming 7.2.2 Applications to the Optimisation of Cavern Construction 7.2.2.1. Discussions Artificial Intelligence Techniques in Construction Optimisation 7.3.1 Artificial Intelligence Language Prolog 7.3.2 Problem Solving Algorithm in Cavern Construction Optimisation 7.3.2.1. Automatic Determination of Cavern Excavation Sequences 7.3.2.2. Automatic Generation of Data Files for Finite Element Computation 7.3.2.3. Implementation of Excavation Scheme Optimisation 7.3.3 Discussions Engineering Applications of Artificial Intelligence Optimisation Methods 7.4.1 Description of the Project 7.4.2 Layout of Cavern Group and Arrangement of Step Excavation 7.4.3 Optimisation of Excavation Sequence 7.4.3.1. Rock Mass Assumed as Isotropic Medium 7.4.3.2. Rock Mass Assumed as Layered Isotropic Medium 7.4.3.3. Summaries

212 212 214 215 216 221 221 222 224 227 228 229 229 229 234 237 238 238 238 239 239 240 243 244

CHAPTER 88 REINFORCEMENT MECHANISM OF ROCK BOLTS

8.1.

Effect of Bolts on Supporting the Rock Mass 8.1.1 Effects of Rock Bolts 8.1.1.1. Reinforcement 8.1.1.2. Post Effect of Pre-stressing 8.1.1.3. Prompt Prevention

247 247 247 248 248

Contents

8.2.

8.3.

8.4.

8.1.1.4. Good Match to Deformation 8.1.1.5. Flexibility in Construction 8.1.2 Reinforcement Mechanism of Rock Bolts Physical Modelling of Rock Bolts 8.2.1 Similarity of Model Materials 8.2.2 Comparison of Different Bolting Methods 8.2.3 Analysis of Test Results Numerical Modelling of Bolt 8.3.1 Basic Parameters of the Numerical Model 8.3.2 Models with Far Field Stresses s1 ¼ s2 ¼ 20 MPa 8.3.3 Models with Far Field Stresses s1 ¼ 2s2 ¼ 20 MPa Scaled Engineering Model Test

xvii 248 248 248 250 250 251 255 255 255 256 258 258

REFERENCES

263

SUBJECT INDEX

287

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Chapter 1

Introduction This book attempts to provide techniques for solving the problems in modelling and analysis of excavations in fractured rock masses, by taking into account the discontinuity inﬂuence, time-dependent behaviour and construction method dependent phenomenon. This chapter provides an overview on the rock mechanics issues related to underground excavation and stability.

1.1.

ROCK MECHANICS AND UNDERGROUND EXCAVATION STABILITY

Rock structures are generally excavated within comparatively competent rock masses. However, rock masses usually consist of various discontinuity features, such as faults, joints and fractures. They signiﬁcantly inﬂuence the stability of rock masses surrounding an excavation opening. The deformation and stability of the surrounding rock masses is controlled by the mechanics of the rock masses subjected to the change of conditions, which is a function of in situ condition before the excavation and disturbance due to the construction activities. The instability of underground works is mainly caused by the redistribution of stresses in the surrounding rock masses due to the excavation activity, due to excessive stress or excessive deformation. The process and result involves the interaction among the rock masses, the boundary conditions, and the engineering activity [1–3]. In order to understand the behaviour of the rock masses surrounding the excavations, it is necessary to understand the basic behaviour of the fractured rock masses subjected to loading and unloading conditions [4–7]. Common methods to study such behaviour are through theoretical analysis when the problems are simple, and through physical and numerical modelling when analytical solutions are not readily available [8–16]. Physical modelling is one of the basic tools to understand the behaviour and mechanism of jointed rock masses subjected to various boundary conditions, primarily loading conditions [17–19]. Whenever possible, physical modelling and testing should be conducted to provide the direct observation and basic understanding of engineering behaviour. It is diﬃcult, often impossible, to quantitatively predict the mechanical properties of jointed rock masses by physical modelling due to their complexity and large scale. Numerical modelling oﬀers wide applications to simulate jointed rock mass

1

2

Chapter 1

behaviours including the eﬀects of loading and time, and the behaviour of rock material, rock joints and rock masses [20–26]. The rationality and reliability of the results from numerical methods depend, to a great extent, upon the appropriate selection of computational model and mechanical and mathematical parameters [27–30]. Once the computational model is determined, the key to success hinges on the rational selection of the computing parameters. There are many factors and parameters that aﬀect the rock mass behaviour and stability. One has to identify the order of importance of all the parameters [31,32]. In terms of computation, the limited resource may be the primary restriction. One of the common methods is the sensitivity analysis of various parameters within a system [33,34]. Rocks and rock masses often exhibit time-dependent behaviour, especially weak and soft rocks or highly fractured rock masses [35–45]. In underground excavation, the time-dependent phenomenon can be found that the loading on support elements gradually increases, leading to the ﬁnal failure of the excavated structure [46–48]. Study of underground excavation in rock masses usually involves various modelling techniques and analysis approaches. Those methods can be physical tests, numerical modelling, observation and back analysis. The ultimate goal is to optimise excavation and support [8,49–52]. In recent years, the back analysis method has been widely applied in geotechnical engineering especially in the underground works [53–56]. Various related analytical and numerical techniques have been developed [57–60]. The method is based on the required input physical information, and can be divided into deformation back analysis method, stress back analysis method and coupled back analysis method. The physical information in the coupled back analysis method requires both deformation and stress. The back analysis has been applied to various rock engineering projects, particularly to underground excavation, to verify the support design and opening stability [57–62]. Construction of rock engineering projects usually requires a long duration, from a few months to a few years. The construction of these rock engineering projects will disturb the initial stable state of the rock masses. The various rock mass parameters interact in a dynamic interactive process until the rock mass reaches a new equilibrium state [63–65]. The construction is therefore a dynamic interactive process in time and in space. The success in constructing and managing a rock engineering project not only depends on the eventual state of the project, but also on the interim process and the construction methods adopted [66–68]. Construction of large-scale rock engineering projects is implemented by continual excavation of new working faces. Each newly excavated face interacts dynamically with the existing excavated space in time and in space. This dynamic interactive process of rock engineering constructions is non-inverse and non-linear. Its

Introduction

3

eventual state (or solution) is not unique but changeable with the interim process [1,63–65,69,70]. In other words, the eventual state is strongly dependent on the stress paths or stress histories. This leads to the possibility of the optimisation of construction process. Adopting a proper excavation sequence and installing eﬀective rock reinforcement are engineering measures to stabilise large-scale underground excavations.

1.2.

STRUCTURE OF THE BOOK

This book is divided into eight chapters. Chapter 1 provides an overview of rock mechanics issues in underground excavation in fractured rock masses, on the various topics covered by the main chapters of this book. Chapter 2 addresses physical modelling of jointed rock masses. The observation and discussion are based on extensive laboratory studies on modelled rock masses with discontinuities. The physical modelling includes plane stress and plane strain tests of various jointed rock masses and bridged rock masses, and large-scale physical modelling of rock masses surrounding an excavation, and deformation analysis of excavation opening. Chapter 3 deals with ﬁnite element based numerical modelling of the jointed rock masses together with computational examples. Deformation and strength equivalence formulations as well as treatment of rock joints are discussed. The numerical modelling is compared with physical modelling and examples of engineering application are given. Treatments of thick joints, collinear and multiple discontinuities are also dealt with in Chapter 3. It provides solutions on deformation and strength equivalence treatment for thick discontinuities, and collinear and multiple discontinuities. Chapter 4 focusses on the sensitivity analysis of rock mass parameters. It covers the sensitivity analysis of common rock material and rock joint parameters, including methods of sensitivity analysis, and application of equivalent model for jointed rock mass. Examples on sensitivity analysis of rock mass parameters based on the extent of damage in the surrounding rock mass for a large underground cavern project is illustrated. Weak and soft rocks with rheologic behaviour are often diﬃcult mediums to model and analyse due to their time-dependent nature. Chapter 5 speciﬁcally addresses stability analysis of rheologic rock masses. It outlines rheological mechanical models for rocks and rock masses, visco-elastic and visco-plastic modelling, interaction of rock mass and support, and rheological damage and damage constitutive equations in numerical modelling.

4

Chapter 1

Chapter 6 presents the principle of back analysis and observational methods commonly used in underground excavation stability assessment. The back analysis methods described include elastic back analysis, visco-elastic back analysis, back analysis and optimised methods in transverse isotropic rock masses. The method extends to plane and three-dimensional back analysis of jointed anisotropic rock masses, and stability analysis of excavation stability in such rock masses. Applications of various statistics models in deformation prediction in the stability assessment are discussed. The principle of construction mechanics and excavation optimisation is presented in Chapter 7. It covers the principles of interactive construction mechanics and their applications to optimise cavern construction. It also outlines artiﬁcial intelligence techniques for construction optimisation and problem-solving algorithm in cavern construction optimisation. Engineering applications of artiﬁcial intelligence optimisation method and optimisation of excavation sequence are also presented in this chapter. Chapter 8 speciﬁcally deals with reinforcement mechanism of rock bolts, including the eﬀect of bolts on supporting the rock mass and reinforcement mechanism of rock bolts. Physical and numerical modelling of rock bolts are presented, with varying bolt spacing, length and layout, and with varying rock stress conditions. A scaled engineering model test is illustrated to verify the modelling results. A comprehensive list of literatures on those topics is given at the end of this book, in the reference section.

Chapter 2

Physical Modelling of Jointed Rock Mass Rock structures are generally constructed within comparatively hard rock masses. The rock masses consist of various structural features, such as faults, joints and fractures. They signiﬁcantly inﬂuence the stability of rock masses. In general, faults are treated as locally special fractures [71–73]. The characteristics of strength and deformation of the faults are studied speciﬁcally for the stability analysis. However, for joints and fractures, due to their abundant numbers, such special treatment generally does not apply. Instead they are regarded as basic rock mass elements in a considerable dimension [4–7,74–95]. However, the joints and fractures signiﬁcantly aﬀect the mechanical properties of the rock masses. Generally, distributions of joints are orderly in sets. In physical and numerical modelling, it is common that for a project, based on in situ geological investigation, two to three major joint sets are identiﬁed. The mean density, persistence, orientation and typical distribution patterns are generalised and modelled. Often, these joints are distributed intermittently. Therefore, the study presented here will emphasise the behaviour of rock masses where joints distribute orderly and intermittently. If in situ tests are conducted to study the behaviour of rock mass containing suﬃcient joints, the dimension of the specimen needs to be a few tens of meters. However, these tests are not commonly conducted. Usually, the more practical method is the physical modelling with a reduced scale [78,96–102]. The mechanics of deformation and failure are observed, which forms the basis for further study.

2.1.

MODELLING OF JOINTED ROCK MASSES UNDER PLANE STRESS STATE

2.1.1 Mechanical properties of equivalent materials For physical model tests, it is important to select a modelling material of which the properties are similar to those of the material modelled. Obviously, not all the properties can meet the law of similarity. Usually, the main parameters are made to meet the similarity conditions and secondary parameters are made to meet the conditions approximately [102]. A physical model test is performed to model a large-scale hydropower cavern project. The model is made of sand, barite powder and an organic polymer resin. Blocks are made by those materials through compaction and heating. The mechanical properties of the modelling material and the modelled rock masses are summarised in Table 2.1. 5

6

Chapter 2

Table 2.1. Mechanical properties of the modelling material and the modelled rock mass. Mechanical properties sc (MPa) E (MPa) c (MPa) f ð Þ cj (MPa) fj ð Þ

Modelling material and fracture

Modelled rock material and fracture

Actual rock material and fracture

0.545 68.3 0.79 44.4 0.0094 33.6

218 2.73 104 31.16 44.4 3.76 33.6

212.4 3.0 104 16 56 0.5 36.9

The physical model has the property similarity ratio of 400. The ratio of the rock strength to modelling material strength and the ratio of the rock modulus to the modelling material modulus are both 400. Rock joints in the rock mass are simulated by the joints between blocks. To form intact portion, the blocks are cemented by the polymer resin. The testing results indicate that the physical model worked well in simulating the rock masses of diﬀerent joint systems.

2.1.2 Model tests of jointed rock mass In general, only two or three major joint sets have the governing inﬂuence on the engineering properties of the rock mass [102–106]. In most modelling studies, two representative joint sets are usually taken in the plane modelling. The two joint sets modelled usually intersect approximately orthogonally at an acute angle. Studies on both joint set arrangements were performed and they are presented in this section. Rock masses that consist of two orthogonal joint sets are fairly common, e.g., the rock mass surrounding at Ertan and Xiaolangdi hydroelectric power stations. Physical modelling of the Ertan hydroelectric power cavern is conducted. From the mapped joint data, three common joint distribution arrangements are modelled, to represent a number of possible combinations of the intersection between joint sets, as shown in Figure 2.1. The modelling of mechanical characteristics of jointed rock masses should take into account the eﬀect of joint quantity and joint properties. Studies by Muller and co-workers [107] have indicated that when the ratio of the rock mass dimension (Dm) over the joint spacing (sj) is greater than 10 (i.e., Dm/sj >10), the strength and deformation characteristics of the rock mass model are consistent, and the size eﬀect has little inﬂuence. The model has a size of 50 50 7 cm and consists of blocks of 5 10 7 cm, that satisﬁes the requirement of Dm/sj >10.

Physical Modelling of Jointed Rock Mass

7

Figure 2.1. Patterns of joint set arrangements modelled. (a) intact (no joints rock); (b) two joint sets with the same persistence of 50% forming running-through in one direction; (c) two joint sets with the same persistence of 50% forming T-shapes; (d) two joint sets with persistence of 50% and 100% respectively.

Figure 2.2. Loading and monitoring set-up of the test system.

The loading and monitoring set-up of the test system is shown in Figure 2.2, which consists of a loading frame, loading and monitoring devices. A 0.5 cm thick layer of sand is placed between the ends of the physical model and the rubber pockets to eliminate end friction. Through a series of experiments, the failure mechanism of rock mass, the eﬀect of joint distribution and eﬀect of shearing are studied.

2.1.2.1 Effect of joint distribution patterns. Three model tests are performed to represent rock masses of diﬀerent joint distribution patterns as shown in Figure 2.1.

8

Chapter 2

In Figure 2.1(b), major principal stress is applied at diﬀerent directions. When the major principal stress is perpendicular to the direction of running-through (Figure 2.3(b)), the model has a low deformation module and a high compressive strength, because the joints can be completely closed and the rock masses become intact. In this case, the s1–e1 curve exhibits a concave shape. While for the same joint sets, when the major principal stress is applied parallel to the direction of runningthrough (Figure 2.3(a)), the lateral deformation is high and the tested model is liable to failure by splitting. In this case, the uniaxial compressive strength is lower than the former. The comparison between the two strain–stress curves is shown in Figure 2.4. Results obtained from models shown in Figure 2.3(a) and Figure 2.3(c) are illustrated in Figure 2.5. As expected, because the joints in the model of Figure 2.3(c) are bridged, both longitudinal and transversal deformations are small, the

Figure 2.3. Failure patterns of jointed rock mass.

Figure 2.4. Stress–strain curves of diﬀerent joint distribution patterns shown in Figure 2.3(a) and (b).

Physical Modelling of Jointed Rock Mass

9

Figure 2.5. Stress–strain curves of diﬀerent joint distribution patterns shown in Figure 2.3(a) and (c).

compressive strength is high and the s1–e1 curve is similar to that of intact rock material with no joints. In the model shown in Figure 2.3(b), because the runningthrough joint surface is roughly perpendicular to s1, its failure mostly exhibits the form in which longitudinal tensile-opening takes place at the sharp turning points of the joints, accompanied by local shearing failure. For the rock mass in Figure 2.3(c), T-shaped joints gradually run through each other longitudinally and lateral cracking occurs under compressive stresses. As a result, compound shear failure planes are ﬁnally formed. For the rock mass shown in Figure 2.3(d), the main failure mode is shear failure along joint planes that have been run through completely, accompanied by lateral tensile cracking of the materials. In summary, there are two basic failure patterns of the rock masses: lateral tensile cracking and shear sliding. Due to various combinations of joint set distributions and arrangements, failure mechanisms and development are diﬀerent. In general, secondary tensile cracks occur at the joint tips ﬁrst, then the cracks link up with the adjacent joints, followed by compound lateral tensile cracks and ﬁnally overall shear failure takes place along a plane shear.

2.1.2.2 Strength of typical jointed rock masses. The eﬀect of lateral shear on the rock mass strength is studied with the model shown in Figure 2.3(a), where major principal shear is applied vertically and linear principal shear (s2) is applied horizontally. Table 2.2 and Figure 2.6 give the typical results of tests under diﬀerent s2. They show that the peak strength, s1, of the rock masses increases with the increasing s2. The deformation module also increases but the lateral deformation decreases rapidly.

10

Chapter 2 Table 2.2. Testing results at different lateral pressure. Testing No. 1 2 3 4 5

s2 (MPa)

s1 (MPa)

E (MPa)

v

0 0.05 0.15 0.25 0.34

0.28 0.41 0.59 0.69 0.96

29.75 65.45 61.85 117.3 122.9

0.1 0.1 0.25 0.31 0.43

Figure 2.6. Change of stress and strain at diﬀerent lateral stress for model in Figure 2.3(a).

The modelling results of the jointed rock mass in Figure 2.3(d) show that when the persistence of joint sets increases, the compressive strength of the rock mass decreases. The modelling results indicate that the deformation modules are governed by the persistence of joint sets, as shown in Figure 2.7. Table 2.3 summarises the uniaxial compressive strength (sc) and deformation modules (E) of the four diﬀerent rock masses shown in Figure 2.3. In summary, the variation of joint set pattern strongly aﬀects the strength and deformation characteristics of a rock mass. From the modelling studies, the following qualitative observation is noted for the rock mass represented by Figure 2.3(a), the peak strength and lateral stress are linearly proportional in general,

11

Physical Modelling of Jointed Rock Mass

Figure 2.7. Stress–strain relationship of rock mass model of Figure 2.3(d).

Table 2.3. Strengths and elastic moduli of four models. Model type 2.3 2.3 2.3 2.3

(a) (b) (c) (d)

sc (MPa)

E (MPa)

0.29 0.37 0.35 0.2

29.75 27.38 43.60 7.13

as shown in Figure 2.8, failures in most cases start with the lateral tensile cracking along joints and follow by the formation of a shear plane. Failure occurs ﬁnally along the inclined plane.

2.1.2.3 Anisotropy of rock strength. The variation of the strength of the jointed rock mass in response to the change of the major principal stress direction is studied. Rock mass of joint pattern shown in Figure 2.3(a) is subjected to the ﬁxed s1 and s2 with the direction of the shears. The testing results are presented in Figure 2.9 and Table 2.4. The study shows that the lowest strength of the rock mass occurs when a1 is at 40 50 .

2.1.2.4 Relation between rock mass strength and joint persistence. Studies on the eﬀect of joint persistence on rock mass strength are carried out. The persistence of

12

Chapter 2

Figure 2.8. Relationship between peak strength and lateral stress for rock mass of Figure 2.3(a).

Figure 2.9. Change of rock mass strength with joint direction.

Table 2.4. Change of strength with joint direction. Strength

s1 (MPa) s1/sc

a1 0

10

20

30

40

0.4 0.72

0.37 0.67

0.31 0.57

0.27 0.49

0.24 0.44

Note: sc ¼ 0.55 MPa; s2 ¼ 0.05 MPa.

13

Physical Modelling of Jointed Rock Mass

Figure 2.10. Change of rock mass strength with joint persistence.

Table 2.5. Effect of joint persistence on the rock mass strength. n2 (%) s1 (MPa)

0

20

30

50

80

0.26

0.25

0.22

0.2

0.18

the second joint set (n2) in Figure 2.3(c) is changed, while the loading conditions remain the same (s2 ¼ 0.05 MPa). Modelling study reveals the relationship between the rock mass strength and the joint persistence, as shown in Figure 2.10 and Table 2.5. It can be seen from Figure 2.10 that the strength of the rock mass decreases with the increasing joint persistence. The rate of the strength change is small when joint persistence is below 20%. A rapid strength decrease is observed when the persistence increases from 20% to 30%.

2.1.3 Large dimension plane strain model experiment In many underground rock engineering works, such as tunnels, the surrounding rock masses are in plane strain condition [108–111]. Therefore, the model study on mechanical behaviour of the surrounding rock masses under plane strain condition is of great importance. In plane strain modelling, the model has suﬃcient length to remove the inﬂuence of end friction. The results of plane strain model experiments for jointed rock mass are discussed in the following sections.

14

Chapter 2

Figure 2.11. Four types of jointed rock mass modelled.

2.1.3.1 Material properties and experimental methods. The preparation of the model material is similar to that of the plane stress model. Deformation characteristics of the joint plane are measured. The material has the following indexes: unit weight g ¼ 2.2 104 N/m3, uniaxial compressive strength sc mm ¼ 0.87 MPa, tensile strength st mm ¼ 0.11 MPa, elastic module Emm ¼ 170 MPa, Poisson’s ratio v ¼ 0.22. Joint plane properties are: cohesion cj ¼ 0.01 MPa, friction angle fj ¼ 39 ; shear stiﬀness Ks ¼ 5 MPa/cm, and normal stiﬀness Kn ¼ 75 MPa/cm. Large-size model of 100 50 14 cm is used to eliminate size eﬀect and end eﬀect. The joints in the rock mass are simulated by the contact surfaces of the blocks. The shear strength and deformation stiﬀness of the joints are measured. To simulate the rock bridge, the contact surfaces of the blocks are cemented to the material strength. The shearing strength of the bridged joint is also measured. The loading is applied through a large steel frame under the condition of plane strain. The tests are performed on 20 rock mass models of four diﬀerent joint patterns. These four patterns are shown in Figure 2.11. 2.1.3.2 Testing results. In the tests, the deformations of the model in three directions are measured. Diﬀerent failure stages are judged mainly by the deformation rate and the tendency of the deformation curves. Figure 2.12 and Table 2.6 show the typical testing results. The rock mass strength is calculated from various peak strengths (s1) under diﬀerent lateral pressures (s2) using the Least Square Method, expressed by the overall equivalent cohesion, cmm, and inner friction angle of fmm. Figure 2.13 shows the typical deformation curves of the rock mass of joint distribution pattern of Figure 2.11(D) under diﬀerent lateral pressures. It can be seen from the modelling results that for the four diﬀerent rock masses, their overall strength can be roughly related to the material strength and the joint strength, by the following equations: Rock mass cohesion ¼ (0.2 0.3) (Rock material cohesion þ Rock joint cohesion), Rock mass friction angle ¼ 0.5 (Rock material friction angle þ Rock joint friction angle).

15

Physical Modelling of Jointed Rock Mass

Figure 2.12. Relationship between speciﬁc strengths of rock mass, material and joint, for four rock masses with diﬀerent joint distribution patterns.

Table 2.6. Results from large-size model tests of jointed rock masses. Model type

s1 (MPa)

s2 (MPa)

s3 (MPa)

s0 (MPa)

s0/s1

90-1 90-2 90-3 90-4 90-5

A

0.31 0.62 0.71 1.45 1.28

0 0.1 0.15 0.25 0.31

0.02 0.05 0.06 0.23 0.11

0.19 0.40 0.52 1.25 1.11

0.61 0.65 0.73 0.86 0.87

91-1 91-6 92-4 92-3

B

0.18 0.68 1.42 1.53

0.02 0.10 0.30 0.31

0.06 0.15 0.62 0.59

0.09 0.35 0.95 0.92

0.50 0.51 0.67 0.60

92-2 91-7 91-8 92-1

C

0.11 0.52 0.97 1.44

0 0.08 0.21 0.27

0.01 0.07 0.16 0.39

0.07 0.28 0.51 —

0.55 0.54 0.53 —

91-2 91-5 91-3 91-4

D

0.12 0.44 0.55 0.69

0.01 0.05 0.09 0.14

0.03 0.06 0.12 0.10

0.09 0.26 0.36 0.48

0.73 0.60 0.66 0.70

Model No

cmm (MPa)

fmm ( )

0.05

42

0.04

39

0.02

40

0.03

39

Note: s0 is the stress in the direction of s1 when the volumetric strain rate is zero; s3 is the normal stress applied on the model surfaces to keep the plane strain state.

16

Chapter 2

Figure 2.13. Stress–strain curves of rock mass model of joint distribution shown in Figure 2.11(D).

2.2.

MODEL TEST OF ROCK MASS WITH ROCK BRIDGES

In this section, modelling is performed to study the failure of rock bridges, the relationship between the joint intensity and the strength of the rock, and the strength of the rock mass of the bridged joint. Results of two phases of modelling are presented in the following sections.

2.2.1

Experiment Phase I

In this phase, the model materials are gypsum, diatomite and water mixture that can be conveniently poured to set. Fractures of given persistence are cast. The principal physico-mechanical parameters of the model are given in Table 2.7. In fracture preparation, thin pieces of steel are buried in the specimen when the material is wet and then drawn out after set. The fractures are naturally closed. The specimens are subjected to drying and curing. In tests, the models are loaded by the plane stress, as shown in Figure 2.14. In addition to deformation transducers, special coating is applied at model surface around the fractures to observe crack initiation and propagation at the tips. The persistence of the fractures is at 0, 20%, 40%, 60% and 80% respectively. Seven normal stresses at 0.08, 0.2, 0.3, 0.35, 0.4, 0.5 and 0.6 MPa are applied. The sheared planes of all the specimens are examined after each test. During the testing, normal stress is kept constant, and horizontal shear load is applied gradually until failure occurs. Typical stages of crack initiation and propagation before failure are observed, as shown in Figure 2.15: (a) Small feather-shaped cracks appear around the existing fractures; (b) Tensile cracks appears along the direction of shear load at one tip of the fracture ﬁrst and then appear at the other tip;

Physical Modelling of Jointed Rock Mass

17

Table 2.7. Materials and fracture parameters. Model material

Dry density (g/cm3) Porosity (%) Uniaxial compressive strength (MPa) Compressive modulus (MPa) Poisson’s ratio Tensile strength (MPa) Tensile modulus (MPa) Fiction angle ( ) Cohesion (MPa)

Fracture

Fracture length (mm) Normal stiffness (MPa/cm) Shear stiffness (MPa/cm) Friction angle ( ) Cohesion (MPa)

0.68 68.9 2.05 1.3 103 0.2 0.27 2.6 103 35 0.53 20, 40, 60, 80 100 20 10 0.1

Figure 2.14. Model test arrangement.

(c) A pair of transversal compressive cracks appear along two approximately orthogonal directions between two adjacent tensile cracks; (d) Rotation of lozenge block takes place, in association with shear dilation and small secondary cracks appearing around large cracks; (e) Visible opening of joints takes place; and macro visible shear failure occurs where the block breaks along the lozenge diagonal.

18

Chapter 2

Figure 2.15. Stages of shear failure of the model.

For samples of diﬀerent joint persistence and diﬀerent vertical stresses, the shear failure has three categories [112–119]: (a) For specimens with low joint persistence and medium vertical stress, considerable dilatancy of the specimen occurs. The failure mode is compressive torsional shear failure. (b) For specimens with medium joint persistence and high vertical stress, the deformation and failure of the specimen undergoes the following stages: (i) micro cracks are generated at the fracture tips, (ii) cracks propagate and the secondary compressive torsional cracks occurs, (iii) lozenge blocks are formed, (iv) shear crack appears along the diagonal of the lozenge blocks and connects with tips of the fractures, and (v) the overall failure of the specimen occurs along the existing fractures and the new shear cracks, and the dilatancy takes place within the whole specimen. The failure mode is torsional tensile shear failure. (c) For specimens with high joint persistence and high vertical stress, the deformation and failure of the specimen develop in the following stages: (i) small ‘‘rock bridge’’ is broken due to the vertical stress, (ii) with the increase of the shear stress, overall failure takes place along the existing fractures and the cracks crossing the rock bridges. Although a slight dilatancy can be observed in the shearing process, the specimen as a whole fails basically due to the compression. This is pure shear failure. Results of over 20 specimens are summarised in Figure 2.16. It shows four shear strength envelopes for four groups of joint persistence. The apparent shear strength

Physical Modelling of Jointed Rock Mass

19

Figure 2.16. Change of shear strength with joint persistence.

falls with increasing persistence. When the persistence increases from 0 to 65%, internal friction angle decreases from 35 to about 15 , and the cohesion decreases from over 0.5 MPa to about 0.3 MPa.

2.2.2

Experiment Phase II

In order to have better control of joint persistence in the specimens, a new material mix is used in the second phase of experiments. The properties of the new material mix of the model are close to those of rock. Joint persistence is properly controlled. The material used is the mixture of sand, barite powder, colophony and alcohol. Fractures are created with polythene ﬁlms of diﬀerent lengths. Two types of fractures with diﬀerent shear strength (friction coeﬃcient) are created by two diﬀerent methods: one by a single ﬁlm and another one by two ﬁlms with sandwiched grease. The designed joint persistence is the same as those in the phase I, i.e., 20%, 40%, 60% and 80%. Mechanical tests show that this material has similar properties as some sedimentary rocks and similar dilatancy character before failure. The tensile–compressive strength ratio is about 0.11. The mechanical parameters are summarised in Table 2.8.

20

Chapter 2

Table 2.8. Properties of model material and fracture. Model material

Dry density Uniaxial compressive strength (MPa) Compressive modulus (MPa) Poisson’s ratio Tensile strength (MPa) Friction angle ( ) Cohesion (MPa)

2.21 0.15 1.45 103 1.12 0.122 38.5 0.25

Single fracture

Fracture length (mm) Normal stiffness (MPa/cm) Shear stiffness (MPa/cm) Friction angle ( ) Cohesion (MPa)

20, 40, 60, 80 43 17 30 0.065

Double fracture

Fracture length (mm) Normal stiffness (MPa/cm) Shear stiffness (MPa/cm) Friction angle ( ) Cohesion (MPa)

20, 40, 60, 80 33 11 24.3 0.041

The test equipment, installation and testing methods are the same as that adopted in Phase I. In the tests, failure processes are observed and noted. The crack initiation and propagation are similar to that in Phase I. From test data of over 60 specimens, a series of peak shear strength with corresponding normal stress are obtained by means of the Least Square Method, and are presented in Figures 2.17 and 2.18. The results indicate a systematic decrease of shear strength of the specimen with increasing joint persistence. For specimens with double fracture the decreasing rate is greater. The strength envelopes show relatively good linearity. With increasing joint persistence, the rock mass cohesion and rock mass internal friction coeﬃcient decrease accordingly. The weighted mean method is commonly adopted for predicting the strength of the fractured rock mass. The shear resistance of the shear plane is obtained by summing the shear resistances contributed by the fracture and by the bridges. By assuming the normal stress on the shear plane before failure is uniformly distributed, the shear resistance of the shear plane can be calculated as, t ¼ ½ncj þ ð1 nÞcr þ sn ½nj þ ð1 nÞ r

ð2:1Þ

where n is the joint persistence, cj is the joint cohesion, cr is the cohesion of rock bridge, fj is the internal friction coeﬃcient of joint plane, fr is the internal friction coeﬃcient of rock bridge.

Physical Modelling of Jointed Rock Mass

21

Figure 2.17. Shear strength at diﬀerent normal stress and joint persistence for single fracture model.

Figure 2.18. Shear strength at diﬀerent normal stress and joint persistence for double fracture model.

22

Chapter 2

Figure 2.19. Change of normalised (a) cohesion and (b) friction angle with persistence.

However, the actual distribution of the normal stress on the shear plane before failure is not uniform. The failure mechanism of the fractured rock mass under the shear loading is complex, characterised by multi-stage development of fracturing [117–129]. It is often a combined tensile-shear failure [130–133]. Therefore the shear strength expressed in equation (2.1) diﬀers from the actual strength. The diﬀerences in cohesion and friction angle are analysed by normalising the actual cohesion and friction to that expressed in equation (2.1), and are shown in Figure 2.19. It can be seen that the diﬀerences are little when the persistence is less than 40%. But with the increase of persistence the diﬀerences becomes greater. When the persistence is between 40–80%, the error is about 20%.

2.3.

MODEL TESTS ON STABILITY OF SURROUNDING ROCK OF LARGE-SCALE CAVITY

Numerous model tests on large-scale underground projects have been conducted (e.g., [25,125,134–136]). However, model tests studying the stability of the surrounding jointed rock mass are limited. In this section, a large-scale model test on jointed rock mass stability is performed by modelling a rock cavern project [137]. Site investigation shows that there are three major joint sets: NE and NW sets with steep dip angles, and EW with a gentle dip angle. The surrounding rock mass of the cavern is cut into prisms and polyhedrons by these three joint sets. For simpliﬁcation of the problem, only the NE and EW sets are modelled, as shown in Figure 2.20. The joint sets have cut the surrounding rock mass into cuboids of 8 5 l0 m in size. In the model, these joint distributions and block arrangements are simulated (Figure 2.21).

Physical Modelling of Jointed Rock Mass

23

Figure 2.20. Polyhedron blocks produced by three joint sets cutting through the surrounding rock mass.

Figure 2.21. Potential failure patterns of the opening and sequence of excavation.

24 2.3.1

Chapter 2 Similarity conditions of modelling

A physical simulation model should meet the similarity to the prototype in the following aspects: (i) (ii) (iii) (iv) (v)

Size and conﬁguration of the engineering project, Geometry of the geological structures, Physico-mechanical properties of the surrounding rock mass, The initial stress state, and The construction sequences.

A scale of 1:200 is chosen between the model and the prototype. The rectangular model has a height of 1.7 m and a width of 1.6 m. The model material is made of gypsum, sands of various granular sizes and water. It is modelled by plane stress state condition. An initial horizontal in situ stress of 28.84 MPa and a vertical in situ stress of 13.35 MPa are applied, before the simulation of the excavation. In order to simulate the actual excavation sequence, a ﬁve-stage excavation sequence is adopted from the top to the bottom for the excavation, as shown in Figure 2.21.

2.3.2

Deformation of the surrounding rock mass during excavation

The model is applied with a vertical and a horizontal compressive stress ﬁrst at boundaries to simulate the in situ stresses. The stresses are maintained at constant, while excavations in ﬁve stages are modelled. The deformation and failure are monitored during the excavation.

2.3.2.1 Deformation of the surrounding rock mass during excavation. In the ﬁrst to the third stage of excavation, both vertical and horizontal displacements are not remarkable. From the beginning of the fourth stage through the ﬁfth stage, large inward vertical displacements take place at the crown, with a maximum displacement of about 0.1 cm. The displacements at base are small. Large horizontal displacements are noted on two side walls, with maximum displacement of 0.3 cm, as shown in Figure 2.22. After the ﬁfth stage when the excavation has been completed, the displacement zones extends to 15 m above the crown and 53 m away from the sidewalls, as shown in Figure 2.22. The results show that the horizontal displacement zones are considerably large, about 8 times that of the excavation width.

Physical Modelling of Jointed Rock Mass

25

Figure 2.22. Displacement curve and aﬀected range in rock mass surrounding the excavation.

2.3.2.2 Failure of the surrounding rock mass during excavation. From the ﬁrst to the third stage excavation, there is basically no failure taking place in the rock mass, only small tetrahedronal blocks are loosened at the crown. From the fourth stage to the end of the ﬁfth stage, the tetrahedronal blocks are sliding at the lower walls. In the upper walls, the blocks are falling along the steep dip joint into the opening. The rock mass at crown displays tensile rupture along joint planes and the rock mass at base starts to loose (Figure 2.21). 2.3.2.3 Stability of the surrounding rock mass. Based on the deformation and failure characteristics of the surrounding rock mass, general conclusions on the stability of the rock mass can be drawn. During the excavation from the ﬁrst to the third stage, the surrounding rock mass is basically stable. From beginning of the fourth stage till the completion of the ﬁfth, noticeable instability takes place. The aﬀected displacement zone in the surrounding rock mass is large. The aﬀected displacement zone extends far into the sidewalls (50 m) and through the crown (15 m). The instability is reﬂected in various patterns: tensile rupture in the crown, structural loosening in the bottom; and block sliding and falling on the sidewalls. The instability occurs in the following sequences: ﬁrst in the lower sidewall, followed by the upper sidewall and ﬁnally the rock mass at the top and bottom. It should be noted that the model exaggerates the joint persistence, as 100% persistence is assumed. Simpliﬁcation and approximation are applied to treat the three-dimensional problem by the two-dimensional model. The modelling is based on the law of scaling and the similarity principle, results obtained and observation made on the mechanism are qualitative, but have signiﬁcant applications to research and engineering.

This Page Intentionally Left Blank

Chapter 3

Numerical Modelling of Jointed Rock Mass It is diﬃcult to quantitatively predict the mechanical properties of jointed rock masses due to their complexity, even by expensive, large-scale in situ tests. The parameters obtained from laboratory or ﬁeld tests are usually for joint planes or rock blocks to a very limited scale. In most cases, the scale is limited to a few meters. The overall equivalent properties of rock masses of large size are almost not available through direct measurement. Prediction of jointed rock mass behaviours by numerical modelling is useful to study the eﬀects of loading, the behaviour of rock material, rock joints and rock masses. Therefore, it has wide and promising applications. However, the validity of numerical modelling should be supported by physical simulation and correlation. For jointed rock masses, various types of numerical models have been developed [12,14,15,138–162]. For rocks containing a small number of joints, joint element [25,26] can be adopted to represent the discontinuous planes. Modelling of densely jointed rock masses can be mainly realised by two methods. One is the approach of continuum mechanics, or the equivalence approach, such as material parameter equivalence, energy equivalence, deformation equivalence, composite equivalence, fracture mechanics and damage mechanics [77,83,114,163–167]. The other approach is to take the rock blocks as particles of a discontinuum, to study the mechanical properties of the assembly of these particles, including stress, strain and stability in light of discontinuum theory, and to derive mechanical law for the blocks. Examples of this approach are the rigid block method, discrete element method [15,150,152– 154,168,169], discontinuity deformation analysis method [20,92,156,159]. However, all the methods have their own limitations, for example, incompatibility of joint elements with adjacent continuum elements, the scale eﬀect in equivalence, deﬁnition of damage tensor, equation of damage evolution and discretion error. This chapter attempts to develop a new numerical approach for jointed rock masses and a modelling technique for rapid and accurate prediction of their mechanical behaviour. An equivalent continuum model for jointed rock mass based joint element assembly concept, coupled with damage–fracture mechanics and analytical approach is introduced in the following sections.

27

28 3.1.

3.1.1

Chapter 3 EQUIVALENT CONTINUUM MODEL FOR JOINTED ROCK MASSES

Basic principles

The basic principle of an ‘‘equivalence continuum method’’ is: (a) to have an overall consideration of the inﬂuence of joints on rock mass properties based on equivalence principles; (b) to make the jointed rock mass homogenous and continuous so as to derive a set of constitutive relations and then to obtain mechanical properties of the jointed rock mass by means of numerical analysis. These equivalence models are usually elastic. Elasto-plastic constitutive relations are seldom adopted. As shown in Figure 3.1, in order to model the surrounding rock mass consisting of two joint sets, typical elements should be identiﬁed. The typical elements should be large enough to include the two joint sets and their interaction characteristics (Figure 3.2). It is desired that the size of a typical element should be suﬃciently small compared with the engineering dimension. Therefore numerical analysis of this typical element can lead to understanding of the strength and deformation properties of the typical rock mass containing joints. It further leads to the development of constitutive relation of the jointed rock mass and the analysis and modelling of rock mass stability. The modelling procedure discussed here involves several steps: (a) the typical jointed rock mass is discretised into intact rock elements and rock joint elements; (b) equivalent constitutive relations and strength-deformation properties of the

Figure 3.1. Jointed rock mass surrounding an opening.

Numerical Modelling of Jointed Rock Mass

29

Figure 3.2. Typical element representing the jointed rock masses.

Table 3.1. Principle of the equivalent continuum model. Original rock model

Equivalent continuum model

Geometry mode Deformability

Discontinuous Overall deformation comprises rock deformation and joint deformation

Strength characteristics

Joint strength is dominant when failure takes place along the joint plane, and rock strength is used when failure takes place in the intact rock

Equivalent continuous body Deformation equivalence: deformation equal to that of original jointed model under the same loading Strength equivalence: failure of model takes place as the original jointed model.

jointed rock elements are established; (c) continuum modelling of the typical jointed rock mass element is performed and the overall mechanical properties of the element are obtained. This approach has the ‘generality’ of the equivalence method and the ‘particularity’ of the joint element method. The modelling method has signiﬁcant applications in modelling jointed rock masses. The principle of the equivalent continuum model is illustrated in Table 3.1. The equivalent constitutive relation is established based on deformation and strength. Both approaches are discussed in the following sections.

3.1.2 Deformation equivalence The principle of deformation equivalence approach assumes that the equivalent continuum element and the jointed rock mass element deform exactly the same under the same loading. Based on this principle, the relation of material constants between

30

Chapter 3

the equivalent continuum element and jointed rock mass element is derived. The equivalent continuum element can be treated as an anisotropic medium. Assuming the joint strike is along z axis, then in the plane-stress condition, the anisotropic material has a stress–strain relation of 0

sx

1

0

10

ex

1

c11

c12

c13

Bs C B @ y A ¼ @ c21

c22

CB C c23 [email protected] ey A

c32

c33

txy

c31

ð3:1Þ

gxy

where sx, sy and txy are stresses acting on x, y and xy plane; ex, ey and gxy are strains in x, y and xy direction; cij are elastic constants, called elastic stiﬀnesses. Because cij ¼ cji (i, j ¼ 1, 2, 3), there are six independent elastic constants. Since these six parameters can be determined, elastic ﬁnite element method analysis will not be diﬃcult. However, in situ measurement of these constants is not easy. As a jointed rock mass can be regarded as a composition of isotropic intact rock material and rock joints, the deformation parameters of the equivalent continuum medium can be obtained from the properties of the rock material and the joints, such as the elastic constants of the rock material and rock joints, or joint stiﬀnesses and the joint geometrical parameters (spacing, persistence, orientation, aperture and roughness) [4–7,170–197]. Methods of obtaining the equivalence are discussed in the following sections.

3.1.2.1 Deformation equivalence with no joint dilation. Assume that an elastic medium is generally anisotropic. The elastic constitutive relation can be written in the form of: 0

ex

1

0

1

0

sx

1

s11

s12

s13

Be C B @ y A ¼ @ s21

s22

C B C s23 A @ sy A

s32

s33

gxy

s31

ð3:2Þ

txy

where sij are the elastic constants, called elastic moduli. For a rock mass containing a single joint shown in Figure 3.3, the stress on the joint plane can be obtained from the equilibrium equation: sn ¼ sx sin2 a þ sy cos2 a txy sin 2a t ¼ sy sin a cos a sx sin a cos a þ txy cos 2a

) ð3:3Þ

Numerical Modelling of Jointed Rock Mass

31

Figure 3.3. Mechanical analysis of a jointed rock element.

From the principle of superposition, the deformation components of the rock mass are, 9 sn t dm sin a cos a > = x ¼ s11 sx d þ s12 sy d þ s13 txy d þ ks kn ð3:4Þ sn t > ; ¼ s s l þ s s l þ s t l þ cos a þ sin a dm 21 x 22 y 23 xy y ks kn After substituting equation (3.3) into equation (3.4) and rearranging, dm x

dm y

9 1 1 2 2 > ¼ s11 d þ sin a sin a þ cos a sin a sx > > > kn ks > > > = 1 1 2 2 þ s12 d þ cos a sin a cos a sin a sy > kn ks > > > > > 1 1 ; s13 d sin 2a sin a cos 2a sin a txy > kn ks

ð3:5aÞ

9 1 1 2 2 > ¼ s21 l þ sin a cos a sin a cos a sx > > > kn ks > > > = 1 1 2 2 þ s22 l þ cos a cos a þ sin a cos a sy > kn ks > > > > > 1 1 ; s23 l sin 2a cos a cos 2a cos a txy > kn ks

ð3:5bÞ

32

Chapter 3

The elastic constitutive relation of the equivalent continuum medium is: 0

eex

1

0

se11

B ee C B e @ y A ¼ @ s21 gexy se31

se12

se13

1

0

sx

1

se22

C B C se23 A @ sy A

se32

se33

ð3:6Þ

txy

Under the same loading, the deformation of the equivalent medium is: dex ¼ se11 sx d þ se12 sy d þ se13 txy d

)

dey ¼ se21 sx l þ se22 sy l þ se23 txy l

ð3:7Þ

e m e Following the deformation equivalence principle, i.e. dm x ¼ dx , dy ¼ dy then

se11 se12 se13 se21 se22 se23

9 1 1 2 2 > > sin a þ cos a sin a ¼ s11 þ > > kn d ks d > > > > > > 1 1 > 2 2 > cos a cos a sin a ¼ s12 þ > > kn d ks d > > > > > > 1 1 > cos 2a cos a þ sin 2a sin a > ¼ s13 > = ks d kn d > 1 1 > > sin2 a sin2 a cos a ¼ s21 þ > > > kn l ks l > > > > > 1 1 > 2 2 > > cos a þ sin a cos a ¼ s22 þ > > kn l ks l > > > > > > 1 1 ; cos 2a sin a sin 2a cos a > ¼ s23 þ ks l kn l

ð3:8Þ

From Figure 3.3, l ¼ d tan a and hence se12 ¼ se21 . From the symmetry of the constitutive relation, it is easy to see that se31 ¼ se13 and se32 ¼ se23 . However, s33 is very diﬃcult to derive. To simplify the analysis, it can be assumed that se33 ¼ s33 . Therefore, all six parameters needed for building the constitutive relations are obtained. For rock mass element with two joints, seij of the rock mass element containing one joint can be obtained from equation (3.8). By replacing sij with seij in equation (3.8) and repeating the analysis for the second joint, seij and the constitutive relation of the equivalent medium with two joints can be obtained. As for the case of multiple joints, the method is similar by repeating the above procedure.

Numerical Modelling of Jointed Rock Mass

33

3.1.2.2 Deformation equivalence with joint dilation. Let the joint dilation (normal displacement caused by shearing) be dd, the dilation angle be i, then dd ¼ t/(ks tan i), by considering dilation, equation (3.4) becomes 9 sn t t sin a cos a tan i sin a > > = ks ks kn sn t t > ; dm cos a þ sin a tan i cos a > y ¼ s21 lsx þ s22 lsy þ s23 ltxy þ kn ks ks dm x ¼ s11 dsx þ s12 dsy þ s13 dtxy þ

ð3:9Þ

Let the two terms on the right-hand side of each of the equations (3.9) be: 9 t t t cos a tan i sin a ¼ 0 cos a > > = ks ks ks t t t > ; sin a tan i cos a ¼ 00 sin a > ks ks ks

ð3:10Þ

where k0s ¼ ks =ð1 þ tan i tan aÞ

)

k00s ¼ ks =ð1 tan i cotanaÞ By replacing ks with k0s in the ﬁrst three equations in equation (3.8), and replacing ks with k00s in the last three equations of equation (3.8), then se11 , se12 , se13 , se21 , se22 and se23 can be obtained. Similarly, as an approximation, se33 ¼ s33 , the deformation equivalence formula with joint dilation is then established.

3.1.3

Formula of strength equivalence

The strength of jointed rock mass is governed by the strengths of rock material and of rock joint. The jointed rock mass may undergo two types of failure: the failure of the rock material and the failure of the rock joint.

3.1.3.1 Strength equivalence in the case of a single joint. Assuming that the strength of the intact rock, rock joint and the equivalent rock mass element follow the Mohr–Coulomb criterion and the parameters are (cr, fr), (cj, fj), (ce, fe), respectively. The strength conditions of the rock material element and equivalent

34

Chapter 3

continuum rock mass element are: s1 s3 s1 þ s3 sin jr ¼ cr cos jr 2 2

ð3:11Þ

s1 s3 s1 þ s3 sin je ¼ ce cos je 2 2

ð3:12Þ

and it is evident that ce ¼ cr

) ð3:13Þ

je ¼ jr For jointed rock mass element, let b be the angle between the joint planes and the plane of the major principal stress, as shown in Figure 3.4. When b < bmin or b > bmax, the strength of the jointed rock mass is dominated by the strength of the rock material. The strength of the jointed rock mass element, in this case, is the same as equations (3.13). When bmin b bmax, the failure of jointed rock mass element takes place along the joint plane, the strength is governed by: s1 s3 s1 þ s3 sinð2b jj Þ sin jj ¼ cj cos jj 2 2

Figure 3.4. Jointed rock element containing single joint.

ð3:14Þ

Numerical Modelling of Jointed Rock Mass

35

when bmin b bmax, by loading the jointed rock mass element, the strength envelope (c and f ) can be obtained. If this strength envelope is regarded as the envelope of the equivalent medium, then ce and fe can be obtained. Rewriting equation (3.11) in the following form: sin jj cos jj s1 s3 s1 þ s3 ¼ cj , 2 2 sinð2b jj Þ sinð2b jj Þ and compared to equation (3.12), leading to: 9 sin jj > > je ¼ sin > sinð2b jj Þ = cos jj > > > ce ¼ cj ; sinð2b jj Þ cos je 1

ð3:15Þ

Unlike the intact rock material, when the failure of jointed rock mass is governed by the joint plane, the strength of jointed rock mass is not constant but varies with joint inclination (b).

3.1.3.2 Strength equivalence in the case of two joint planes. Figure 3.5 shows a jointed rock mass element containing two joints. The failure of the rock mass element is dependent on the geometric combination of the two joints, the stress distribution and the shear strength of the joints [187–207]. Failure usually takes place

Figure 3.5. Jointed rock element containing two joints.

36

Chapter 3

along one of the joint planes. The equivalent strength is therefore dependent on the strength of that joint plane where failure occurs. Let the strength criterion of the joint plane be written in another form: s1 ¼

2cj þ 2 tan jj s3 þ s3 ð1 tan jj cotanbÞ sin 2b

ð3:16Þ

Let b1 and b2 be the inclination of the joint plane 1 and the joint plane 2, and both b1 and b2 meet the condition of bmin b bmax. In addition, let 1 , ð1 tan jj cotanb1 Þ sin 2b1 1 : A2 ¼ ð1 tan jj cotanb2 Þ sin 2b2 A1 ¼

When A1 > A2, joint 1 will be the dominant plane governing the strength of the rock mass. Similarly when A2 > A1, joint 2 will be the dominant plane governing the strength. It is easy to obtain the equivalent strength following the method similar to equation (3.15). For a rock mass element containing three joints, the parameters of equivalent rock mass strength can be determined in a similar way. Hoek and Brown [194] pointed out that the strength and deformation properties of a rock mass element containing four joint sets or more can be considered to be isotropic. 3.1.3.3 Analysis of tensile failure. When a jointed rock mass element is subjected to tension, its tensile strength, to a large extent, depends on the tensile strength of joints. Since the rock joint is normally considered having zero tensile strength, the tensile strength of the jointed rock mass is usually taken as zero [186,194–196]. Therefore, non-tension analysis should be adopted in this circumstance. 3.1.4 Treatment of elements with non-persistent joint In a jointed rock mass, the joints are often not persistent throughout the joint planes [175,178,197]. Therefore, relevant modiﬁcation should be made on the fundamental formula of deformation and strength equivalence. For simplicity, let us deﬁne the joint projection length ratios in the x- and y-directions, respectively as: Rx ¼

Ljx Lx

and

Ry ¼

Ljy , Ly

ð3:17Þ

where Ljx and Ljy are the joint projection lengths in the x- and y-directions. Lx and Ly are the element sizes in the x- and y-directions. It is obvious that deformation of

Numerical Modelling of Jointed Rock Mass

37

elements is positively correlative to Rx and Ry. Thus, the following modiﬁcation should be made for equation (3.8): when 0 a 45 , the terms containing Kn and Ks are multiplied by Ry and Rx; when 45 < a 90 , the terms with Kn and Ks are multiplied by Rx and Ry. The modiﬁed formula is veriﬁed in terms of joint length, joint inclination and scale eﬀect. First, deformation d1 (or strain e1) in the direction of s1 can be calculated using ﬁnite element method (e.g., Goodman’s joint element). With d1 as the equivalent deformation, elastic constant s11 ¼ e1/s1 can be obtained, and D11 is deﬁned as sr11 =s11 . Alternatively, s11 and D11 can be derived from the modiﬁed equation (3.8) according to the joint distribution. The values obtained from FEM modelling and from equation (3.8) can be compared. The relative error of the value of D11 can be estimated by treating the FEM results as the exact solutions.

3.1.4.1 Variation of joint length. Assuming that the jointed rock element is square in shape at the XOY plane with a size of 5 5 cm, joint inclination is at 45 , Young’s modulus of material is 68 MPa and Poisson’s ratio of material is 0.25. The joint has a normal stiﬀness of 75.0 MPa/cm and a shear stiﬀness of 5.0 MPa/cm. Six cases, as illustrated in Figure 3.6, are modelled and discussed. The modelling results are summarised in Table 3.2.

3.1.4.2 Variation of joint inclination. As shown in Figure 3.7, three models are studied to examine the eﬀects of joint orientation. In the models, a joint with a length

Figure 3.6. Veriﬁcation of models with diﬀerent joint persistence.

38

Chapter 3

Table 3.2. Influence of joint length distribution on elastic constant D11 ðD11 ¼ sr11 =s11 Þ: Case

Rx

Ry

sr11 =s11 (Analytical)

sr11 =s11 (FEM)

Relative error (%)

a b c d e f

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

1.00 0.83 0.71 0.62 0.55 0.49

1.00 0.94 0.83 0.72 0.62 0.49

0 12 14 14 11 0

Figure 3.7. Veriﬁcation models with diﬀerent joint inclination.

Table 3.3. Influence of joint angle on elastic constant D11 ðD11 ¼ sr11 =s11 Þ: Case

Rx

Ry

sr11 =s11 (Analytical)

sr11 =s11 (FEM)

Relative error (%)

a b c

0.2 0.2 0.2

0.0 0.2 0.3

0.96 0.83 0.81

0.90 0.88 0.93

7 6 13

of 1.0 cm at the centre of the element dips at 0, 45 and 71.56 degrees, respectively. Results obtained from the three models are given in Table 3.3.

3.1.4.3 Scale effect of element. Scale eﬀects are studied using the jointed element models shown in Figure 3.6(d), with the joint dip angle at 45 , Rx and Ry at 0.6. Six diﬀerent model sizes are considered, and the results are presented in Table 3.4. The results show that when the jointed rock element size is greater than 10 cm, the error decreases to 5% or less. In discretisation of the jointed rock mass in engineering practice, the size of the ﬁnite elements is generally much greater than 10 cm. Therefore, the treatment of the deformation equivalence using modiﬁed equation (3.8) is reasonably accurate and easy to be realised in modelling.

39

Numerical Modelling of Jointed Rock Mass Table 3.4. Scale effect on elastic constant D11 ðD11 ¼ sr11 =s11 Þ(a ¼ 45 , Rx ¼ Ry ¼ 0.6). Joint length (cm) 5 10 20 30 40 50

3.1.5

sr11 =s11 (Analytical)

sr11 =s11 (FEM)

Relative error (%)

0.62 0.77 0.87 0.90 0.93 0.94

0.72 0.81 0.88 0.91 0.93 0.94

14 5 1 1 0 0

Verification of the numerical model by physical modelling

The overall mechanical properties of jointed rock masses can be reﬂected when the proposed jointed rock elements are incorporated as the basic elements into the FEM analysis. Veriﬁcations have been performed through correlation with physical modelling [8,9]. The size of the physical model is 50 50 7 cm, containing two sets of orthogonal joints with the same spacing and length of 5 cm. Figure 3.8 shows three types of joint arrangements and the FEM meshes. Due to the complexity of the physical model test, comparisons are only made on strength and deformation properties under the uniaxial stress state for all the three types of joint patterns (Figure 3.8a, b and c) and the properties under the biaxial stress state for one joint pattern (Figure 3.8a). The comparisons are shown in Figures 3.9 and 3.10. In Figure 3.9, e1 and e2 are the strains in the direction of s1 and s2 under uniaxial loading, i.e., s2 ¼ 0. In Figure 3.10, e1 and e2 are the strains under biaxial loading, i.e., when s1 ¼ s2. Figures 3.9 and 3.10 indicate that the stress–strain relations obtained from the physical and the numerical modelling under the uniaxial stress state match well. In the biaxial stress state, the major principal stress–strain relations, peak strengths and strains match well between the physical and the numerical modelling results. But the major principal stress–lateral strain relations dismatch. The reason may be that the boundary friction in physical modelling leads to smaller measured deformations than the expected values.

3.1.6

Examples of engineering applications

3.1.6.1 Prediction of strength of jointed rock mass. The above method is used to model the strength of the rock mass at the Ertan hydropower cavern project in China. The rock masses at the Ertan project site have three major joint sets, two of which dip steeply and the other dips gently. The two orthogonal main joint sets (Figure 3.11) are considered in the modelling.

40

Chapter 3

Figure 3.8. Models of diﬀerent joint arrangement and FEM mesh.

Figure 3.9. Comparison of s–e relations of the physical and the numerical models under uniaxial loading.

The measured length of the joints is between 4 and 7 m, and is taken as 5 m in average in the modelling, with a joint spacing of 1 m, based on the geological investigation. The joint persistence is 50% and 30%, respectively, for the two joint sets. The model dimension is 17.8 17.8 m, which satisﬁes the requirement that each boundary of the rock mass model contains about 10 joints, in order to represent

Numerical Modelling of Jointed Rock Mass

41

Figure 3.10. Comparison of s–e relations of the physical and the numerical models under biaxial loading.

the rock mass adequately. The distribution of joints and the FEM meshes are shown in Figure 3.11. There are 625 elements, 242 of which are jointed rock elements. The number of nodes is 676. The joint arrangement belongs to the type shown in Figure 3.8a. The mechanical parameters of rock and joints are obtained from the laboratory tests as given in Table 3.5. In the study, only strength equivalence is considered. FEM analysis involves three types of elements: intact rock element, and two types of jointed rock elements, as shown in Figure 3.11. The intact rock element has a size of 71.2 71.2 cm. The equivalence of strength property is automatically treated by the program. Loading simulated by the numerical modelling gives the yield values of s1 with respect to diﬀerent lateral pressures s2. Strength parameters of the rock masses are

42

Chapter 3

Figure 3.11. Distribution of joints and meshing of jointed rock.

Table 3.5. Parameters of rock material and rock joints. Material parameters Intact rock Joint plane

E (MPa)

n

c (MPa)

f ( )

35,000 –

0.30 –

14.58 0.5

65.20 36.89

obtained through back analysis, using the Drucker–Prager criterion [208,209]. Results are summarised in Figure 3.12, showing the strength envelopes of the rock masses.

3.1.6.2 Stability analysis of jointed rock masses. Numerical modelling is performed to study the stability of the jointed rock mass surrounding a large opening 400 m below the ground surface. The surrounding rock mass is granite. The separation between the powerhouse and the transformer chamber, the cavern dimension and the excavation sequence are shown in Figure 3.13. During the excavation, deformation measurement with extensometers was taken on several sections of the caverns. The distribution of two major joint sets and three faults is shown in Figure 3.14.

Numerical Modelling of Jointed Rock Mass

43

Figure 3.12. Rock mass strength envelope obtained from the modelling.

Figure 3.13. Layout of the powerhouse, the transformer chamber and excavation sequence.

Deformation properties of the jointed rock mass and the constitutive relations are obtained by the deformation equivalence method. In modelling the jointed rock element is generally assumed anisotropic and the joints have zero aperture and are evenly spaced across the model. The persistence of the joints is taken as 50% based on site investigation. Constitutive relation of the rock mass with two joint sets is derived with the method stated in Section 3.1.2. Mechanical parameters are

44

Chapter 3

listed in Table 3.6. The in situ stresses are sx ¼ 10.19 MPa, sy ¼ 8.51 MPa, and txy ¼ 1.12 MPa. The size of model was 525 320 m, containing 654 elements and 630 nodes. The initial stresses were obtained from back analysis of the displacement data of the fourth excavation stage of the powerhouse and the data of the second excavation stage of the transformer chamber. Using the above mechanical parameters and stress conditions, non-linear FEM modelling has been performed on the stability of the caverns. The Drucker–Prager criterion is adopted. The extent of failure zone around the caverns and displacements are obtained as shown in Figure 3.14. Table 3.7 gives the comparison of the displacements obtained by modelling of the jointed rock mass and by elastic analysis of the rock without considering joints.

Figure 3.14. Over-stressed areas of the surrounding rock mass.

Table 3.6. Mechanical parameters of the rock material and rock joints. Materials Intact rock Rock joint

E (MPA)

n

c (MPa)

f ( )

s1(MPa)

Kn (MPa/cm)

Ks (MPA/cm)

37,000 –

0.20 –

22.27 0.5

48.1 35

5.0 0.0

– 600,000

– 7500

Table 3.7. Predicted displacement convergence values (mm). Measurement Modelling of jointed rock mass Elastic analysis of non-jointed rock mass

1

2

3

4

5

6

7

8

9

11.4 9.7

18.5 15.5

19.5 16.3

6.0 5.0

6.9 5.8

7.3 6.1

10.8 9.0

7.4 6.3

6.9 5.8

45

Numerical Modelling of Jointed Rock Mass 3.2.

EQUIVALENT ANALYSIS FOR ROCK MASSES CONTAINING THICK JOINTS

In this section an equivalent continuous model is proposed for rock mass consisting of thick joint sets. The method is based on strength analysis with the same principle discussed in Section 3.1.

3.2.1

Equivalent deformation principle and method

When a rock mass is cut by two joint sets in direction a1 and a2 respectively, and the rock material is homogeneous, then there are two parameters describing the rock material (Er and sr) and four parameters describing each joint set (Ej, sj, b and a). E is the elastic modulus, s is the strength, b is the ratio of joint aperture to rock mass width and a is the orientation of joint set. Only the basic formulae for plane condition are given here, which are also illustrated in Figure 3.34. In the process, the eﬀect of one joint set is taken into account ﬁrst to produce an ‘equivalent medium’. The other joint set is subsequently taken into consideration to derive the equivalent continuous medium of the rock mass containing two joint sets. As shown in Figure 3.34, considering the ﬁrst joint set, the joint stress–strain relation is as follows: fsgj ¼ ½ sxj

syj

txyj T

fegj ¼ ½ exj

eyj

gxyj T

ð3:18Þ

The stress–strain relation of the rock material is fsgR ¼ ½ sxR

syR

txyR T

fegR ¼ ½ exR

eyR

gxyR T

ð3:19Þ

If the stress–strain state of the above combined joints and material is replaced by the stress–strain state of the jointed rock mass as shown in Figure 3.15, namely fsg1 ¼ ½ sx1

sy1

txy1 T

feg1 ¼ ½ ex1

ey1

gxy1 T

ð3:20Þ

then they are related by the following equations: 3.2.1.1 Equilibrium condition. 9 sx1 ¼ ð1 b1 ÞsxR þ b1 sxj > = sy1 ¼ syR ¼ syj > ; sxy1 ¼ txyR ¼ txyj

ð3:21Þ

46

Chapter 3

Figure 3.15. Representation of rock joints and rock material.

3.2.1.2 Displacement compatibility condition. ex1 ¼ exR ¼ exj ey1 ¼ ð1 b1 ÞeyR þ b1 eyj gxy1 ¼ ð1 b1 ÞgxyR þ b1 gxyR

9 > = > ;

ð3:22Þ

3.2.1.3 Physical equations. Suppose that both the rock material and joint material are homogeneous, isotropic and elastic, then for joint 8 > < sxj ¼ aj1 exj þ bj1 eyj syj ¼ bj1 exj þ cj1 eyj ð3:23Þ > : txyj ¼ dj1 gxyj for rock material: 8 s ¼ aR exR þ bR eyR > < xR syR ¼ bR exR þ cR eyR > : txyR ¼ dR gxyR

ð3:24Þ

47

Numerical Modelling of Jointed Rock Mass

where aR, bR, cR, dR are deﬁned by Er and sr, the elastic constants of rock material. aj1, bj1, cj1, dj1 are deﬁned by Ej1 and mj1, the elastic constants of joint material. In plane strain condition, 8 a ¼ c ¼ Eð1 mÞ=ð1 þ mÞð1 2mÞ > > < b ¼ Em=ð1 þ mÞð1 2mÞ > > : d ¼ E=2ð1 þ mÞ

ð3:25Þ

Based on further derivation, the relation between {s}1 and {e}1 in X1OY1 coordinate system can be obtained fsg1 ¼ ½D0j ½H1 1 feg1 ¼ ½D1 feg1

ð3:26Þ

where 2

1 6 bR bj1 6 b1 6 cj1 ½H1 ¼ 6 6 4 0

0 1 b1 þ b1

3

0 cR cj1

0

0 1 b1 þ b 1

7 7 7 7 7 dR 5

ð3:27Þ

dj1

2

bj1 bj1 6 ð1 b1 ÞaR þ b1 aj1 þ b1 cj1 ðbR bj1 Þ ð1 b1 ÞbR þ b1 cj1 cR 6 ½D0j ¼ 6 6 cR bR 4 0 0

3 0 7 7 7 ð3:28Þ 0 7 5 dR

In the X2OY2 coordinate system, the elastic matrix of the jointed rock mass is ½D1 ¼ ½T1 ½D1 ½TT1

ð3:29Þ

where 2

cos2 ða1 a2 Þ

sin2 ða1 a2 Þ

6 2 ½T1 ¼ 6 cos2 ða1 a2 Þ 4 sin ða1 a2 Þ 1 1 2 sin 2ða1 a2 Þ 2 sin 2ða1 a2 Þ

sin2 ða1 a2 Þ

3

7 sin2 ða1 a2 Þ 7 5 cos 2ða1 a2 Þ

ð3:30Þ

It is followed by the second joint set subsequently taken into consideration. Similarly, the elastic matrices of the bi-directional jointed rock mass in XOY

48

Chapter 3

coordinate system are given as: T ½D2 ¼ ½T2 ½Dij ½H1 2 ½T2

ð3:31Þ

where 2

cos2 a2

6 2 ½T2 ¼ 6 4 sin a2 1 2 sin 2a2

sin2 a2 cos2 a2 12 sin 2a2

2

1 6 d 1 bj2 6 b 21 6 2 c ½H2 ¼ 6 j2 6 0 4 d31 b2 dj2

sin2 a2

7 sin2 a2 7 5 cos 2a2

0 1 b2 þ b2 b2

0 d0 b2 23 cj2

0 d22 cj2

0 d32

dj2

3

1 b2 þ b2

ð3:32Þ

3 7 7 7 7 7 0 5 d33

ð3:33Þ

dj2

2

bj2 0 bj2 0 0 0 6 ð1 b2 Þd11 þ b2 aj2 þ b2 cj2 ðd21 bj2 Þ ð1 b2 Þd12 þ b2 cj2 d22 6 ½Dij ¼ 6 0 0 6 d21 d22 4 0 d31

ð1

0 d32

3 bj2 0 þ b2 d23 7 cj2 7 7 0 7 d23 5

ð3:34Þ

0 b2 Þd13

0 d33

and dij0 (i, j ¼ 1, 2, . . . , 6) are elements of [D]1 as expressed in equation (3.29). Generalised in the spatial condition, the general matrix for anisotropic elasticity is 32 d11 d12 d16 sx 6 sy 76 d21 d22 d26 76 6 .. 6 s 76 ... . 6 z 76 .. 76 .. 6 6 txy 76 . . 76 . 6 .. 4 tyz 54 . . . d61 d62 d66 tzx 2

32

3 ex 7 6 ey 7 76 7 76 e 7 76 z 7 76 7 76 gxy 7 76 7 54 gyz 5 gzx

ð3:35Þ

49

Numerical Modelling of Jointed Rock Mass

Suppose that [D]i1 is the elastic matrix of rock mass in the ith local coordinate system OXiYiZi of the rock mass cut by the (i1)-th group of joint, the elements of [D]i1 are the coeﬃcients (k, l ¼ 1, 2, . . . , 6) in the above expression. Similarly, suppose that the rock mass is cut by the ith joint set and the joints are parallel to plane OXiYi, then 2

1 0

0 1

0 0

0 0 0 0 i1 i1 d34 d35 bi bi cij cij 1 0 d i1 d i1 bi 54 1 bi þ bi 55 dij dij

6 6 6 d i1 b d i1 bij d i1 ij 6 bi 32 1 bi þ bi 33 6 bi 31 6 cij cij cij 6 0 0 0 ½Hi ¼ 6 6 i1 6 d i1 d i1 6 b d51 bi 52 bi 53 i 6 dij dij dij 6 6 i1 i1 4 d d d i1 d i1 bi 61 bi 62 bi 63 bi 64 dij dij dij dij 2

½DTi1j

bij i1 i1 ð1 bi Þd11 þ bi dij þ bi ðd31 bij Þ 6 cij 6 6 6 ð1 b Þd i1 þ b d þ b bij ðd i1 b Þ 6 ij i 12 i ij i cij 32 6 6 6 bij i1 i1 6 þ bi d33 ð1 bi Þd13 6 cij 6 ¼6 b 6 ij i1 i1 6 ð1 bi Þd14 þ bi d34 6 c ij 6 6 bij i1 6 i1 ð1 bi Þd15 þ bi d35 6 cij 6 6 4 b ij i1 i1 ð1 bi Þd16 þ bi d36 cij

bi

i1 d65 dij

0 0 i1 d36 bi cij 0 d i1 bi 56 dij 1 bi þ bi

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 i1 5 d66 dij ð3:36Þ

bij i1 ðd bij Þ cij 31 bij i1 i1 ð1 bi Þd22 þ bi dij þ bi ðd32 bij Þ cij bij i1 i1 ð1 bi Þd23 þ bi d33 cij b ij i1 i1 ð1 bi Þd24 þ bi d34 cij bij i1 i1 ð1 bi Þd25 þ bi d35 cij b ij i1 i1 ð1 bi Þd26 þ bi d36 cij i1 ð1 bi Þd21 þ bi dij þ bi

i1 d31

i1 ð1 bi Þd41

i1 d51

i1 d61

i1 d32

ð1

i1 bi Þd42

i1 d52

i1 d62

i1 d33

i1 ð1 bi Þd43

i1 d53

i1 d34

ð1

i1 bi Þd44

i1 d54

i1 d35

i1 ð1 bi Þd45

i1 d55

i1 d36

i1 ð1 bi Þd46

i1 d56

3

7 7 7 7 7 7 7 i1 7 d63 7 7 7 7 i1 7 d64 7 7 7 i1 7 d65 7 7 5 i1 d66

ð3:37Þ

50

Chapter 3

Having been cut by the ith joint set, the rock mass has the elastic matrix ½Di ¼ ½Di1j ½H1 i

ð3:38Þ

The direction cosines between system OXiþ1Yiþ1Ziþ1 and OXiYiZi are as follows: Original coordinates

New coordinates

Yi

Xi

Zi

direction cosines li1 li2 li3

Xiþ1 Yiþ1 Ziþ1

mi1 mi2 mi3

ni1 ni2 ni3

In the (i þ 1)-th local system OXiþ1Yiþ1Ziþ1, the elastic matrix of the jointed rock mass is ½Di ¼ ½Ti ½Di ½TTi 2

2 li1

6 6 l2 6 i2 6 6 2 6 li3 ½Ti ¼ 6 6 6 li1 li2 6 6 6 li2 li3 4 li3 li1

ð3:39Þ

m2i1

n2i1

2li1 mi1

2mi1 ni1

m2i2

n2i2

2li2 mi2

2mi2 ni2

m2i3

n2i3

2li3 mi3

2mi3 ni3

mi1 mi2

ni1 ni2

li1 mi2 þ li2 mi1

mi1 ni2 þ mi2 ni1

mi2 mi3

ni2 ni3

li2 mi3 þ li3 mi2

mi2 ni3 þ mi3 ni2

mi3 mi1

ni3 ni1

li3 mi1 þ li1 mi3

mi3 ni1 þ mi1 ni3

2ni1 li1

3

7 7 7 7 7 2ni3 li3 7 7 7 ni1 li2 þ ni2 li1 7 7 7 ni2 li3 þ ni3 li2 7 5 ni3 li1 þ ni1 ni3 2ni2 li2

ð3:40Þ i ¼ 1, 2, . . . , 6.

3.2.2

Basic principle of strength equivalence

Suppose that the strengths of the rock material, the joint and the equivalent continuum body all follow the Mohr–Coloumb criterion, the strengths are cr, fr , cj, fi , ce, fe , respectively. The strength of jointed rock mass element consists of the rock

Numerical Modelling of Jointed Rock Mass

51

material strength and the joint strength. Adopting the equivalent strength method described in Section 3.1, the strength condition is s1 s2 s1 þ s2 sin je ¼ Ce cos je 2 2

ð3:41Þ

With diﬀerent angles of the joint plane to the major principal stress plane b, ce and fe are diﬀerent: when b < bmin or b > bmax, Ce ¼ CR

je ¼ jR

ð3:42Þ

when bmin b bmax 9 cos jj 1 > = Ce ¼ Cj sinð2b jj Þ cos je > ; je ¼ arcsinðsin jj = sinð2b jj ÞÞ

ð3:43Þ

If the rock mass contains more than two joint sets, the related results can be obtained by superposition. But it should be noted that the smaller value of c, f should be adopted if two values exist. For diﬀerent stress states, if all the rock material and rock joint parameters are known, the strength of the rock mass can be derived by the above procedure. The equivalent rock mass strength parameters ce and fe can be determined on the basis of Drucker–Prager criterion as shown before. With two stress states (a and b) known, the equivalent rock mass strength parameters (ce and fe ) can be determined through the Drucker–Prager criterion. pﬃﬃﬃﬃﬃ aI1a þ J2a ¼ K qﬃﬃﬃﬃﬃ ð3:44Þ aI1b þ J2b ¼ K From equation (3.44), we have

a¼

qﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ J2b J2a I1a I1b

Because pﬃﬃﬃ 3 sin j a ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ , 3 3 þ sin2 j

ð3:45Þ

52

Chapter 3

the expression below is obtained,

3a je ¼ j ¼ arcsin pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 3a2

ð3:46Þ

From pﬃﬃﬃ 3C cos j K ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 þ sin2 j we have qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 þ sin2 j K pﬃﬃﬃ Ce ¼ C ¼ 3 cos j

ð3:47Þ

The values of ce and fe for the rock mass can be obtained from equations (3.46) and (3.47).

3.2.3

Engineering applications

The above equivalent analysis of jointed rock masses is applied to model a large underground cavern complex of a hydropower project, as shown in Figure 3.16. The main work includes the selection of separations between caverns, the excavation sequences and the support schemes. Two-dimensional non-linear analysis and three-dimensional anisotropic analysis are conducted. In the FEM modelling, continuous rock mass is simulated with the commonly used isoparametric elements. Dominant faults are simulated with the isoparametric joint elements. The fractured rock mass containing joint sets is simulated with equivalent medium. In two-dimensional modelling, primarily 4-node quadrilateral elements and 3-node triangle elements are used. The number of elements is 661, and the number of nodes is 603. Six schemes for cavern separation optimisation, three schemes for excavation sequence optimisation and two schemes for shotcrete– rockbolt support optimisation are modelled. In three-dimensional modelling, 8-node brick-shaped elements and 6-node pentahedron elements are commonly used. The number of elements is 1094, and the node is 1368. Two schemes of cavern separation optimisation and two schemes of excavation and support optimisation are modelled. A large number of modelling cases have been conducted. Some important conclusions are outlined here. The maximum displacements in plane elastic analysis

Numerical Modelling of Jointed Rock Mass

53

Figure 3.16. Systematic view of cavern complex of a hydropower station.

and in non-linear analysis occur at diﬀerent locations. The former appears at the middle of the left wall in the powerhouse (Figure 3.16). In non-linear analysis the yield plastic zones between the main powerhouse and the transformer room is joined together, and the maximum displacement appears at the middle of the right wall of the powerhouse with a magnitude of about 6.7 cm. The maximum compressive stresses also appear at the vault and the ﬂoor. The maximal compressive stress at the left vault is nearly 47.0 MPa. With the increase of spacing between the caverns, the areas of yield zones signiﬁcantly decrease, and eventually the yield zones become isolated. The cavern separation has signiﬁcant inﬂuence on the area of yield zone. Non-linear analysis is conducted for three schemes of excavation sequence: Scheme 1: I ! II ! III ! IV ! V

ðfive excavation stepsÞ

! III ! Scheme 2: I ! II ! !V ! IV !

ðfour excavation stepsÞ

! II ! ! III ! Scheme 3: I ! ! ! IV ! !V !

ðthree excavation stepsÞ

Modelling results indicate that the maximum displacement is decreased by 32% in Scheme 1, by 35% in Scheme 2 and by 12% in Scheme 3. As for yield zone, Scheme 1 has the least yield zone compared with Schemes 2 and 3. Through comprehensive consideration, Scheme 1 is recommended for the actual construction.

54

Chapter 3

In the non-linear modelling, the eﬀects of two diﬀerent bolt schemes are compared. New concept and method are developed for modelling the bolt eﬀects. From the model tests, the results show that the main eﬀect of the bolts is to increase the strength and the stiﬀness of the surrounding rock mass. Its eﬀect to produce support reaction is much less signiﬁcant. The results show that the peak strength and modulus of the bolted rock mass increase approximately by 10–20%. However, the residual strength of the bolted rock mass increases considerably as compared with unbolted rock mass. Under the uniaxial compression, the residual strength increases by 60–80%, and is inﬂuenced by the bolt spacing and the stiﬀness ratio between the bolt and the rock mass. The study on bolt is discussed in detail in Chapter 8.

3.3.

STRENGTH CHARACTERISTICS OF FRACTURED ROCK MASS UNDER COMPRESSIVE SHEAR STRESS

The natural rock masses consist of faults, joints, fractures and other discontinuities. Engineering projects often generate compressive or shearing stress on the rock mass [186,194–196,210–218]. It is important to understand the interaction of the rock discontinuities and the strength and failure mechanisms of the rock masses under compressive and shear stresses. Joints and fractures in a rock mass generally distribute intermittently and the strength of the joint and the bridge area in the rock mass diﬀer greatly in diﬀerent rock masses [205,206]. Researchers usually pay close attention to the overall shear strength of a jointed rock mass and to the secondary crack initiation. Many researchers suggested that the compressive strength of collinear fractures and rock bridges can be determined by means of the weighted mean method [219–221]. Others believed that the diﬀerent deformation characteristics of the joint and the bridge in a rock mass should be taken into consideration to assess the rock mass strength. Because of diﬀerent stress distributions in the rock bridge and on the joint surfaces, the friction on the joint surfaces must be multiplied by a mobilising factor [199–200]. Tests conducted on gypsum show that the strength properties of a rock mass with multi-rowed intermittent fractures aligning along the same direction [205]. Brown [206] studied the strength of jointed rock mass by means of fracture mechanics. Horii and Nemat-Nasser [219] gave analytical solution and stress intensity factor of the secondary crack trajectory, and discussed the formation of a failure plane. Reyes and Einstein [220], through tests and damage analyses, studied the stress distribution and failure mode of a rock bridge between two fractures in en-echelon crack arrays. Analytical models for fracturing mechanisms of the rock bridge between adjacent fractures in a bilateral compressive stress ﬁeld, based upon the phenomena observed

Numerical Modelling of Jointed Rock Mass

55

in the experimental studies are proposed and discussed in this section. The models can be used to predict the overall shear strengths of the rock mass. The analytical solutions have been veriﬁed through experiments.

3.3.1 Strength of rock mass containing collinear cracks 3.3.1.1 Fracture propagating at tight cracks. Stress concentration phenomena will take place at crack tips under loading. Although the average stress of the crack is generally less than the threshold stress of crack propagating, the stresses at the tips are by far higher than the threshold stress. As a result, secondary cracks will occur near the tips. The initiating point of secondary cracks can be determined from the fracture toughness of the rock material. For collinear cracks (Figure 3.17) loaded by compound stresses of compression and shear, stress intensity factor at the crack tips can be derived as follows: kI ¼ 0

ð3:48Þ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pa kII ¼ ðt s tan jj cj Þ 2b tan 4b

ð3:49Þ

Where compressive stress is deﬁned as positive and tensile stress as negative; cj, fj are the cohesion and the friction angle of the crack, respectively. The condition for the crack to propagate is KII KIIc where KIIc is type II fracture toughness of the crack. The crack propagation does not indicate that the stress reaches the rock strength. The strength is reﬂected by the stress that makes the cracks join together to form a failure plane. The crack initiation and the propagating path become very important

Figure 3.17. Collinear cracks under compressive shear stress.

56

Chapter 3

for estimating the strength. As shown in Figure 3.18, during the test, rupture initiates at crack tips, it tends to propagate gradually along the direction of maximum principal stress, s1. When a crack propagates to a certain extent and meets the adjacent crack, rock failure occurs. Analysis on the above-observed phenomenon has shown that stress intensity factors at the tip (point A) of a crack under A A compressive and shear stresses are kA I ¼ 0 and kII 6¼ 0. When jkII j kIIc , the crack rupture initiates at point A with an initial rupture angle of approximately 70 (the angle between initial rupture line and line of the crack). However, with the increment of external loading, the direction of the secondary crack gradually turns to the direction of s1. The secondary crack path with direction of turning is schematically described in Figure 3.19. The cracking starts along A B. When the crack propagates to point B, the shear stress between B C exceeds the maximum shear strength of the rock material, resulting in shearing failure on plane BC to form crack B C. Therefore, the fracture ABC becomes the ﬁnal fracture plane.

3.3.1.2 Determination of shearing strength of crack body. At present, the commonly used method to analyse the strength of a jointed rock mass takes into consideration the joint persistence Z, the shear strength parameters of the crack, cf and ff, the shear parameter of rock bridge material, cr and fr [221]. f ¼ Z fj þ ð1 ZÞfr

)

c ¼ Zcj þ ð1 ZÞcr

Figure 3.18. Fracture through bridged rock materials.

ð3:50Þ

Numerical Modelling of Jointed Rock Mass

57

Figure 3.19. Stress analysis on the fracture plane.

t ¼ Zðsfj þ cj Þ þ ð1 ZÞðsfr þ cr Þ

ð3:51Þ

Taking the stress diﬀerence between rock bridge and crack plane into account, the following equation is also used. t ¼ Z½ð1 bÞsfj þ cj þ ð1 ZÞ½bsfr þ cr

ð3:52Þ

where b is a factor concerning the normal stress distribution on failure planes and is to be determined through experiments or numerical modelling. As shown in Figure 3.10, DA and CE are the existing crack surfaces and AB is the secondary crack. Shear strength can be expressed as: sn ¼ s1 cos2 a þ s2 sin2 a

ð3:53Þ

tj ¼ sn Hðsn Þ fj þ cj

ð3:54Þ

then

where ( HðxÞ ¼

1

x>0

0

x0

ð3:55Þ

58

Chapter 3

The equilibrium equations of the crack element (Figure 3.19) are Fx ¼ 0

Fy ¼ 0

ð3:56Þ

Consequently, 2bs2 sin a þ 2aðtj cos a sn sin aÞ þ 2cðtBC cos y sBC n sin yÞ ¼ 0

ð3:57Þ

2bs1 cos a þ 2aðtj sin a þ sn cos aÞ þ 2cðtBC sin y sBC n cos yÞ ¼ 0

ð3:58Þ

where c ¼ (b a)(cos a)/(cos y) Rewriting (3.57) and (3.58), then tBC sBC n tan y ¼

aðsn tan a tj Þ bs2 tan a ba

ð3:59Þ

bs1 aðtj tan a þ sn Þ ba

ð3:60Þ

tBC tan y þ sBC n ¼

Solving simultaneously, equations (3.59) and (3.60) gives sBC n ¼

B0 A0 tan y 1 þ tan2 y

ð3:61Þ

tBC ¼

A0 þ B0 tan y 1 þ tan2 y

ð3:62Þ

aðsn tan a tj Þ bs2 tan a ba

ð3:63Þ

bs1 aðtj tan a þ sn Þ ba

ð3:64Þ

where A0 ¼

B0 ¼ Let

BC F ¼ jtBC j sBC n Hðsn Þ fr cr

Hence, when F 0, shear failure occurs at plane BC and a failure plane is formed. At this moment, @[email protected] ¼ 0, thus @tBC @sBC n HðsBC ¼0 n Þfr @y @y

ð3:65Þ

Numerical Modelling of Jointed Rock Mass

59

based on @tBC B0 ð1 tan2 yÞ 2A0 tan y ¼ @y ð1 þ tan2 yÞ2 cos2 y

ð3:66Þ

@sBC A0 ðtan2 y 1Þ 2B0 tan y ¼ @y ð1 þ tan2 yÞ2 cos2 y

ð3:67Þ

Substituting equations (3.66) and (3.67) into (3.65) gives 1 A0 B0 fr HðsBC r Þ y ¼ arcctg 2 B0 þ A0 fr HðsBC n Þ

ð3:68Þ

3.3.1.3 Verification through model tests. The modelling material made in the physical model is a mixture of gypsum and ceyssatile, which has the following mechanical properties: uniaxial tensile strength ¼ 0.27 MPa, compressive elastic modulus ¼ 1.3 103 MPa, Poisson’s ratio ¼ 0.20, internal friction angle ¼ 35 , cohesion ¼ 0.53 MPa. The cracks are closed and have the following properties: length ¼ 20, 40, 60, 80 mm, internal friction angle ¼ 10 and cohesion ¼ 0.1 MPa. Layout of the direct shear test is shown in Figure 2.14. The rock mass model has crack persistence ranging from 0 to 62%. In the test, the normal stress on shear plane is kept constant and the lateral loads are gradually increased till the model fails. The test results are analysed by least square method, giving shear strength curves as shown in Figure 3.20. It can be seen from the Figure 3.20 that the experimental results agree well with the analytical solution. The comparison is summarised in Table 3.8. The above comparison indicates that the strength criterion proposed reasonably represents the properties of the rock mass containing collinear cracks.

3.3.2

Strength of rock mass containing multiple cracks

Joints, fractures and cracks in a rock mass may be distributed parallel but not necessarily co-linear, as shown in Figure 3.21. The failure of such rock mass is indicated by dotted lines. Because the crack is in compressive shear stress state, the stress intensity factor at the tip is KI ¼ 0 and KII 6¼ 0. Hence under the compressive shear state the condition of the crack initiation should be KII KIIc.

60

Chapter 3

Figure 3.20. Comparison of strength envelopes between measured and calculated results.

As shown in Figure 3.23, crack initiates at fracture tips, and the secondary cracks BC and DE are formed at the tips B and D. The secondary cracks expand in the direction of maximum compressive stress, and at a certain stage, the cracks join each other to form CD. The rock mass containing multiple cracks then fails. With understanding the failure mechanism after multi-fractured rock mass, the following assumptions are used to develop a strength criterion: (i) (ii)

The secondary crack propagates in the direction of the maximum compressive stress, and along a straight line, The failure of multi-fractured rock mass is due to shear.

As shown in Figure 3.23, BC and ED are the secondary crack surfaces. Stresses can be expressed as: sn ¼ s1 cos2 a þ s2 sin2 a tj ¼ sn Hðs0 Þ fj þ cj For the element in Figure 3.23, the force equilibrium condition is Fx ¼ 0 Fy ¼ 0

Numerical Modelling of Jointed Rock Mass

61

Table 3.8. Comparison between measured and calculated results. Persistence (%)

0 0 0 9 9 13 13 13 20 42 43 43 44 45 46 46 50 60 60 62 62 62

Normal stress (0.1 MPa)

3 4 5 0.8 4 3 4 6 5 5 0.8 3.5 3 3.5 4 6 5 0.8 2 3 4 6

Shear strength (0.1 MPa) Measured

Calculated

7.43 8.56 8.99 5.76 7.86 6.43 6.80 8.06 7.49 5.91 4.39 5.43 5.36 5.03 5.63 6.10 5.35 3.52 3.65 4.34 4.14 4.95

7.4 8.1 8.8 5.44 7.53 6.64 7.27 8.54 7.42 5.9 3.84 5.12 4.82 4.99 5.16 6.08 5.34 3.03 3.50 3.76 4.14 4.89

Figure 3.21. Rock mass with a group of parallel cracks.

Theoretical error (%)

0.4 5.4 2.1 5.56 4.2 3.3 6.9 6.00 1 0.17 12.5 5.7 10 0.8 8.3 0.3 0.18 14 4.1 13.36 0 1.2

62

Chapter 3

Figure 3.22. Enechelon cracking trajectory.

Figure 3.23. Force analysis.

Namely, 2cs2 sin a þ 2aðtj cos a sn sin aÞ þ 2mðtCD cos y sCD n sin yÞ ¼ 0 2cs2 cos a þ 2aðtj sin a þ sn cos aÞ þ 2mðtCD sin y sCD n cos yÞ ¼ 0 where m ¼ ðc aÞ

cos a cos y

) ð3:69Þ

Numerical Modelling of Jointed Rock Mass

63

Arrangement of equation (3.69) leads to tCD sCD n tan y ¼

aðsn tan a tj Þ cs2 tan a ca

ð3:70Þ

cs1 aðtj tan a þ sn Þ ca

ð3:71Þ

tCD tan y þ sCD n ¼

Solution of equations (3.70) and (3.71) gives sCD n ¼

B0 A0 tan y 1 þ tan2 y

ð3:72Þ

tCD ¼

A0 þ B0 tan y 1 þ tan2 y

ð3:73Þ

in which A0 ¼

aðsn tan a tj Þ cs2 tan a ca

B0 ¼

cs1 aðtj tan a þ sn Þ ca

It follows from Figure 3.23 by geometrical analysis that y ¼ arctan ctana 4d tan a

4L 1 3c 2a sin a

ð3:74Þ

According to the above analysis, the failure criterion for multi-fractured rock mass under the action of compressive shear stress is F 0

ð3:75Þ

where CD F ¼ tCD sCD n Hðsn Þ fr cr

tCD ¼

A0 þ B0 tan y 1 þ tan2 y

sCD n ¼

B0 A0 tan y 1 þ tan2 y

ð3:76Þ

64

Chapter 3 A0 ¼

aðsn tan a tj Þ cs2 tan a ca

ð3:77Þ

csj aðtj tan a þ sn Þ ca

ð3:78Þ

B0 ¼

4L 1 y ¼ arctan c tan a þ 4d tan a 3c 2a sin a

ð3:79Þ

For uniaxial loading; s1 6¼ 0 L¼

s2 ¼ 0 6:703T 2 2 pK1c

ð3:80Þ

For biaxial loading; s1 6¼ 0 s2 6¼ 0 2qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 32 2 1 4 K1C þ 11:614s2 K1C 5 L¼ p 2:234s2

ð3:81Þ

T ¼ 2aFn cos a

ð3:82Þ

Fn ¼ jðs1 s2 Þ sin a cos aj fj Hðsn Þsn cj ( 1 x>0 HðxÞ ¼ 0 x0

ð3:83Þ

Figure 3.24. Diagram of the model test.

ð3:84Þ

Numerical Modelling of Jointed Rock Mass

Figure 3.25. Comparison of predicted values with the tested values when s2 ¼ 0.

Figure 3.26. Comparison of predicted values with the tested values when s2 ¼ 0.2 MPa.

65

66

Chapter 3

In addition, when L 2d/cos a, connection takes place among the propagated cracks, and at the same time, the rock mass strength reaches the peak. The present criterion for multi-fractured rock mass is veriﬁed by physical model tests [8]. Figure 3.24 shows the schematic layout of the physical model tests. The dimension of the existing fracture are: a ¼ 4.8 cm, b ¼ 8.0 cm, d ¼ 1.0 cm. The model material is gypsum and its main physical parameters are: uniaxial compressive strength ¼ 1.84 MPa, uniaxial tensile strength ¼ 0.38 MPa, modulus of elasticity ¼ 0.238 104 MPa, Poisson’s ratio ¼ 0.20, internal friction angle ¼ 45 , cohesion ¼ 0.38 MPa and the fracture toughness of type I K1C ¼ 74.06 N/cm3/2. The main physical parameters of the fracture are: the friction coeﬃcient ¼ 0.365 and the cohesion ¼ 0. The criterion is also implemented with numerical modelling code. A computer program using the new strength criterion is written in FORTRAN and was used to simulate the physical model tests. Comparisons of the results obtained from the numerical modelling and the physical modeling are shown in Figures 3.25 and 3.26. From the comparisons, it can be seen that the numerical modelling based on the new strength criterion generally agree well with the physical test results.

Chapter 4

Sensitivity Analysis of Rock Mass Parameters At present, a variety of numerical methods are available for stability analysis of rock masses. Among them, the most widely used are the ﬁnite element method (FEM) [138–140,222–232], the boundary element method (BEM) [14,233–241], the discrete element method (DEM) [15,148–154,242–246] and coupled methods [16,247–253]. The rationality and reliability of the results from those methods depend, to a great extent, upon the appropriate selection of computational model and mechanical and mathematical parameters. Once the computational model is determined, the key to success hinges on the rational selection of the computing parameters. There are many factors and parameters that aﬀect the rock mass stability. One has to identify the order of importance of all the parameters [1–3]. In terms of computation, the limited resource may be the primary restriction. This section presents some speciﬁc assessments on the eﬀects of various parameters by means of the sensitivity analysis in systematic approaches.

4.1.

SENSITIVITY ANALYSIS OF COMMONLY USED PARAMETERS

4.1.1 Method of sensitivity analysis Sensitivity analysis method is the method used to analyse the stability of a system [254]. Given a system whose character, P, is governed mainly by n factors of a ¼ {a1, a2 , . . . , an}, P ¼ f {a1, a2 , . . . , an}. Under a reference state of a ¼ fa1 , a2 , . . . , an g, the character is described by P*. The sensitivity analysis is to, let the above individual factors vary within respective possible range and then analyse both tendency and extent to which the character of the system, P, departs from the base state due to the variation of the factors. The ﬁrst step of sensitivity analysis is to establish the system model, i.e., the functional relation between the system character and the factors, P ¼ f {a1, a2 , . . . , an}. This relation should be, if possible, described by analytical expression. In the case of a complex system, it can be expressed by numerical method or through presentation of graphic chart. It is a key step of the eﬀective analysis on parameter sensitivity to establish a model that reﬂects the system with the reality as fully as possible. The basic parameter set should be given after the system model has been established. The basic parameter set should be established to reﬂect the subjects considered. For example, when the sensitivity of the stability of an underground project to the variation of rock mass parameters is to be studied, the suggested rock 67

68

Chapter 4

mass parameters of the in situ rock mass can be used as the basic parameter set. Once the basic parameter set is determined, the sensitivity analysis can be performed on each parameter. When analysing the eﬀect of ak on the characteristic of the system, P, the parameter ak in the set is varied within a possible range while the remaining parameters are kept constant. In this case, the system’s character displays the following relation: P ¼ f ða1 , . . . , ak1 , ak , akþ1 , . . . , an Þ ¼ jk ðak Þ

ð4:1Þ

The character curve of P–ak is plotted from equation (4.1), which roughly describes the sensitivity of P to the disturbance from ak. For example, the rapid change of the curve around ak1 shows a high sensitivity of P to ak, i.e., a slight change of ak will cause a great change of P. On the contrary, the curve is gently around ak2, then the system character of P is less sensitive to the parameter of ak, i.e., P varies slightly with a large change of ak. In other words, when the parameter is near ak1, it is a parameter of high sensitivity; when it is near ak2, then is of low sensitivity (Figure 4.1). The above analysis only gives the sensitivity behaviour of the system character P to a single factor. The character of a real system is generally governed by many factors of diﬀerent physical quantities and units. Therefore, it is diﬃcult to compare the sensitivity of the various factors by the above method. To solve this problem, dimensionless treatment and analysis can be applied. In dimensionless analysis, the sensitivity function and the sensitivity factor are deﬁned in dimensionless terms. The ratio of the relative error (dp), of the system character P (dp ¼ |P|/P) to the relative error of parameter ak (dak ¼ |ak|/ak) is deﬁned as the sensitivity function, Sk(ak), of the parameter ak. P ak jPj . jak j k ¼ 1, 2, . . . , n ð4:2Þ Sk ðaK Þ ¼ P ak ak P When |ak|/ak is small, the function of Sk(ak) can be expressed approximately as djkðak Þ ak Sk ðak Þ ¼ dak P

k ¼ 1, 2, . . . , n

ð4:3Þ

From Equation (4.3), the sensitivity function curve of ak can be obtained, which is shown in Figure 4.2. Given ak ¼ ak , the sensitivity factor Sk of the parameter ak is obtained as djkðak Þ a Sk ¼ Sk ðak Þ ¼ ak ¼ ak k k ¼ 1, 2, . . . , n ð4:4Þ dak P

Sensitivity Analysis of Rock Mass Parameters

69

Figure 4.1. Characteristic curve of system P–ak.

Figure 4.2. Curve of sensitivity function Sk–ak.

where Sk and k ¼ 1, 2, . . . , n are a group of non-negative dimensionless real numbers. The higher Sk is, the more sensitive P is to ak. Based on comparison between diﬀerent Sk values, one can give synthetic assessment on the sensitivity of various factors. When jk (ak) is a sectional function, its derivative may be discontinuous at section boundary point of ak0, which makes Equation (4.3) fail to give the sensitivity of S 0k at ak0. In this case, one of the following methods can be used: (i)

Choose the higher value of the left and the right limits Sk(ak) at ak ¼ ak0 as the sensitivity Sk0 , i.e., Sk0 ¼ maxfSk ðak Þ, Sk ðak þÞg

(ii) (iii)

Smooth the function of jk (ak) using cubic spline function ﬁtting method or other techniques to eliminate the discontinuous point of the derivative; and Set a common relative error e for every parameter, i.e., let jak j=ak ¼ e, k ¼ 1, 2, . . . , n and then calculate S 0k using equation (4.2).

70

Chapter 4

4.1.2 Sensitivity analysis of stability of underground works Examples of using the sensitivity analysis method described in the previous section are given in this section. It analyses various factors that aﬀect the stability of a rock cavern complex of a hydropower project. The project comprises main powerhouse, transformer house and tailrace surge chamber, and the sectional layout is shown in Figure 4.3. The powerhouse measures 64.4 m in height and 27 m in width. The deformations of the side wall and the crown are of interest to the stability analysis. The project is located in the region of high in situ stress ﬁeld where horizontal stresses are greater than the vertical stresses. Due to the high overlying depth above the chamber complex, the in situ stress ﬁeld is assumed to be uniform and the directions of its two principal stresses are oriented horizontally and vertically at sx ¼ 13.3 MPa and sy ¼ 9.5 MPa respectively.

4.1.2.1 Computational model. A two-dimensional non-linear FEM programme has been used to model and analyse the stability of the underground structures. The following simpliﬁcations and assumptions have been made: (i)

(ii)

The surrounding rock mass is homogeneous and continuous with the eﬀect of faults neglected, but the joint eﬀect is considered using the equivalent elastic module, Ee, from the in situ measurements. The initial in situ stress is uniformly distributed within the computational domain and the two principal stresses act in horizontal and vertical directions.

Figure 4.3. Sectional diagrammatic sketch of Laxiwa hydroelectric power station.

71

Sensitivity Analysis of Rock Mass Parameters

The excavation is simulated by unloading process and the eﬀect of the excavation of the tailrace surge chamber on the weakening of the rock is treated with the stiﬀness-reduction method. The system character, namely, the stability of the underground structure complex is reﬂected by the maximum horizontal deformation of the left sidewall of the main powerhouse (or by other indexes, e.g., damage areas). The parameters used for the sensitivity analysis include elastic module (E), Poisson’s ratio (n), cohesion (c), internal friction angle (f), horizontal and vertical principal stresses (sx and sy). For the above exercise, the basic parameter set is given in Table 4.1.

4.1.2.2 Analysing the results. All parameters have been analysed one by one using the method stated earlier. The procedure to analyse the horizontal stress sx and the elastic module of E is described below. From experiences, the possible varying range of those two parameters are determined. E is in the range of 1.0–4.5 104 MPa, and sx is in the range of 4.75–19.95 MPa. The values of E and sx are adjusted step by step to calculate the maximum horizontal displacement, u, of the left sidewall of the main powerhouse. Curves representing u–E and u–sx are plotted from the computing results, as shown in Figures 4.4 and 4.5. The function relations of u–E and u–sx have been obtained from the curves and can be expressed as: u ¼ jE ðEÞ ¼ 8:70=j

ð4:5Þ

u ¼ jsx ðsx Þ ¼ 0:225sx 0:2425

ð4:6Þ

From the equation (4.3) we have two sensitivity functions of SE ðEÞ and Ssx ðsx Þ: SE ðEÞ ¼

8:70

1 Eu

Ssx ðsx Þ ¼ 0:225

ð4:7Þ

sx 0:225sx ¼ u 0:225sx 0:2425

ð4:8Þ

Table 4.1. Basic parameter set. E (104 MPa) 3.2

n

c (MPa)

f ( )

sx (MPa)

sy (MPa)

0.21

25

48

13.3

9.5

72

Chapter 4

Figure 4.4. u–E curve.

Figure 4.5. u–sx curve

Sensitivity Analysis of Rock Mass Parameters

73

The corresponding sensitivity curves of SE –E and Ssx–sx are shown in Figures 4.6 and 4.7 respectively. As shown in Figures 4.6 and 4.7, the sensitivity function of SE ðEÞ 1 means that the sensitivity factor, SE , is constantly equal to 1, and is not inﬂuenced by the basic value of the elastic module E. Ssx ðsx Þ is a decreasing function, and the sensitivity is high when sx is low and decreases with increasing sx. The limit of Ssx is 1. By substituting sx ¼ 1.33 MPa into Equation (4.8), the sensitivity factor, Ss x Ss x , of the parameter sx is 1.088. Similar analysis can be performed on other parameters. Table 4.2 summarises the sensitivity factor of other parameters.

Figure 4.6. SE –E curve.

Figure 4.7. Ssx–sx curve.

Table 4.2. Sensitivity factor of various parameters. SE

Sm

Sc

Sj

Ss x

Ss y

1.0

0.077

0.039

0.020

1.088

0.077

74

Chapter 4

Table 4.2 indicates that when the horizontal displacement of the sidewall of the powerhouse is used to judge the stability of the cavern, then the most sensitive factor that aﬀects the stability is the horizontal stress. It has sensitivity as high as 1.088. In other words, if there exists an error of 15% in sx, the relative error (du) of u is 1.088 15% ¼ 16.32%. The internal friction angle f is the least sensitive factor (Sj ¼ 0.02%). An error of 15% in f only results in a 0.3% (0.2 15%) error of u. As seen from the analysis, the horizontal stress and the elastic module have high sensitivity factors and should be treated with great care. It should be noted that the above conclusion is drawn from a given set of basic parameters that are directly involved in a speciﬁc engineering project. The conclusion diﬀers from problem to problem and project to project. As shown in Figure 4.7, the sensitivity factor, Ss x , of sx varies, depending upon the selection of the basic value of sx . In addition, the sensitivity of some parameters is aﬀected by the interaction of other parameters. For example when a structure is in a perfect elastic state, cohesion and friction have no eﬀect on the deformation of the structure. But when the structure is in an elasto-plastic state, and cohesion and friction aﬀect the deformability, such eﬀect becomes more remarkable with increasing plastic area. The sensitivity of c and f is therefore dependent not only on their basic value but also on the values of sx and sy. Further analysis of such interaction between parameters requires more rock mechanics knowledge of the problem. One can use the Rock Engineering System [1–3] or the Grey System [254] to study the interaction between the parameters. For such cases, when sensitivity analysis is performed on a parameter that has active interaction with others, it is desirable to make the latter varying within their possible ranges.

4.1.3 Application to optimisation of test schemes In order to analyse the stability of the underground rock structures, the mechanical and engineering parameters of the rock mass must be known. Accurate rock mass properties can only be obtained from large in situ tests. Such tests are seldom carried out as they are very expensive and time consuming. Sensitivity analysis of parameters can be applied for the optimisation of testing schemes. Test scheme optimisation is to obtain the rock properties that meet the engineering requirement with the least amount of work for large in situ tests. Studies of sensitivities of all the parameters would identify the parameters of high sensitivities that should be measured during the in situ tests. In the example in Section 4.1.2, the horizontal deformation of the high wall, the horizontal stress and the rock mass module are the key parameters. They must be determined through ﬁeld tests, while the other parameters are less critical and can be obtained through less expensive tests or other methods.

Sensitivity Analysis of Rock Mass Parameters

75

Sensitivity analysis helps to avoid mistakes due to subjective conjecture. Diﬀerent basic parameter sets have diﬀerent orders of sensitivity. For this reason, the sensitivity analysis of parameters should be aimed at speciﬁc engineering problems so as to distinguish key parameters from the rocks. The key parameters may vary from project to project. The amount of ﬁeld tests for rock parameters can be rationalised according to their sensitivity factors. In principle, parameters of higher sensitivity should be subjected to more tests. Selections of appropriate test methods shall be made in accordance with the requirements of the engineering project and based on the sensitivity factors. For example, when there are two options available in measuring the horizontal stress sx: one is cheap but has a relative error of dsx1 ¼ 15%, while another is expensive but has a relative error of dsx2 ¼ 10%. As the sensitivity of Ss x is 1.088, the relative errors of the two methods results in measurement errors of du are 16.32% and 10.88%, respectively. If a relative error du < 15% is required, the second method should be adopted although the ﬁrst one is cheap. If a larger error of du < 20% is allowed, the ﬁrst method can be used. Although the sensitivity analysis provides, in view of accuracy, scientiﬁc basis for the optimisation of testing schemes, the overall optimisation should consider other factors, such as testing cost, duration and availability.

4.2.

ANALYSIS OF THE EFFECT OF JOINT PARAMETERS ON ROCK MASS DEFORMABILITY

In the previous section, the sensitivity analysis is performed by assuming rock masses as isotropic and homogeneous media. However, natural rock masses contain sets of joints, which have inﬂuence on the strength and the deformation characteristics of the rock mass. Chapter 3 has suggested several mechanical models for diﬀerent kinds of jointed rock masses. This section examines the sensitivity of the joint mechanical and geometric parameter on the rock mass stability.

4.2.1

Application of equivalent model for jointed rock mass

The sensitivity analysis of the eﬀect of joints on underground rock structures involves a considerable amount of work. The current study therefore adopts the same analytical model and method described in Section 3.3. The rock mass is assumed to contain two joint sets with diﬀerent orientations (Figure 4.8). The following notations are used in the analysis. For the rock material: Er - elastic module, nr - Poisson’s ratio; and for joint sets (subscripts 1 and 2 for set I

76

Chapter 4

Figure 4.8. Typical rock mass containing two joint sets.

and II respectively): Ej1 and Ej 2 - elastic moduli, nj1, nj 2 - Poisson’s ratio, aj1 and aj 2 - dip angle, Zj1 and Zj 2 - persistence, bj1 and bj2 - speciﬁc width (total joint width percentage in the rock mass). The model is treated under the plane strain condition, and the strike of the joint sets coincides with the z axis, and the joint persistence Cj1 and Cj2 are both equal to 1. Equivalent properties of the rock mass are obtained according to the properties of individual joint sets and the rock material, as discussed in Section 3.3. The stress states of the joint and the rock material are respectively (refer to Section 3.3). j1 j1 r fsgj1 ¼ fs j1 x , s y , t xy g

j1 j1 r fegj1 ¼ fe j1 x , e y , g xy g

ð4:9Þ fsgr ¼ fs rx , s ry , t rxy gr

fegr ¼ fe rx , e ry , g rxy gr

4.2.2 Basic parameter for sensitivity analysis The sensitivity analysis of the joint parameter is conducted for the same case as those in the previous section (Section 4.1). Sensitivity of various parameters to the maximum horizontal displacement of the cavern sidewall and to the maximum settlement of the roof is studied. The project was described in Section 4.1.2 and shown in Figure 4.3. There exists two prevailing joint sets in the region and the elastic constants of the rock material are: elastic module Er ¼ 5.8 104 MPa and Poisson’s ratio nr ¼ 0.21. The parameters of the two joint sets are obtained from the site investigation and are summarised in Table 4.3.

77

Sensitivity Analysis of Rock Mass Parameters Table 4.3. Basic parameter values of joint sets used in sensitivity analysis. Joint set I Joint set II

Ej1 ¼ 0.5 104 MPa Ej1 ¼ 1.4 104 MPa

j1 ¼ 0.23 j1 ¼ 0.23

aj1 ¼ 70 aj2 ¼ 110

Zj1 ¼ 0.6 Zj2 ¼ 0.6

bj1 ¼ 3.210 2 MPa bj2 ¼ 3.2 10 2 MPa

The values of parameter in Table 4.3 are also suggested for the sensitivity analysis. Accordingly, given that each parameter varies within a certain range, the side wall displacement (ux) and the roof settlement (uy) are calculated. The change and trend of ux and uy with the variation of the parameters are examined. A FEM program is used for the analysis.

4.2.3 Computational results 4.2.3.1 Effects of joint elastic moduli on displacement. Figure 4.9 shows the relation curves between the joint elastic modulus (Ej) and the surrounding rock mass displacement (m), for each joint set. ux is the deformation of the high wall and uy is the deformation of the crown. Relative convergence (u/B) and the relative ratio of the joint elastic modulus to the rock mass modulus (Ej/Er) are also presented in Figure 4.9. B is the cavern width in calculating ux, and B is the cavern height in calculating uy. From Figure 4.9, it can be seen that the u–Ej curve can be ﬁtted with the function of Y ¼ A/X þ B. The regression analysis determines the u–Ej relationship for

Figure 4.9. Deformation-joint elastic moduli relations for both joint sets.

78

Chapter 4

the two joints as: For joint set I, ux ¼ 527:6=Ej1 þ 1:5505

ð4:10aÞ

uy ¼ 326:4=Ej1 þ 0:4772

ð4:10bÞ

ux ¼ 675:8=Ej2 þ 1:6098

ð4:10cÞ

uy ¼ 286:2=Ej2 þ 0:5251

ð4:10dÞ

For joint set II,

where ux and uy (cm) stand for the maximum displacement of the sidewall and the maximum vertical deformation of the cavern, respectively. Ej1 and Ej2 (MPa) are the modulus of joint sets I and II, respectively. The following conclusions can be drawn from Figure 4.9 and equation (4.10): (i)

(ii)

(iii)

If the actual elastic moduli of the rock joints are lower than the suggested elastic moduli used in the computation, then the actual deformation of the surrounding rock mass is greater than that computed. For the main powerhouse cavern, the total deformation of the sidewall is greater than that of the roof, for the same change in the joint modulus. However, the relative deformation of the sidewall is less than that of the roof. For example, when the elastic modulus of joint set I changes from Ej1 ¼ 5000 to 2500 MPa (given that the maximum relative error of Ej is 50%), then the absolute values of ux and uy are 0.106 and 0.065 cm, while the corresponding relative changes are 6.37 and 12.04%, respectively. The results show that the displacement of roof is more sensitive than the displacement of sidewall to the change of joint elastic modulus. The same changes of the elastic moduli of the joint sets with diﬀerent initial moduli have diﬀerent eﬀects. Given the same relative errors of 50% of both joint sets whose initial moduli are 5000 and 14,000 MPa, the errors caused are 12.04 and 3.76% at the roof, respectively. It shows that lower elastic moduli has higher sensitivity to the rock mass deformation.

4.2.3.2 Effect of joint Poisson’s ratio on rock mass deformation. Figure 4.10 shows the relation curves between the joint Poisson’s ratio (nj) and the rock mass displacement (u). The Poisson’s ratios of the joint sets (nj1 and nj2) vary between 0.184 and 0.276, and the deformations (ux and uy) remain almost the same. This shows that the rock mass deformation is by far less sensitive to the Poisson’s ratio.

Sensitivity Analysis of Rock Mass Parameters

79

Figure 4.10. Eﬀects of joint Poisson’s ratio on rock mass deformation.

Figure 4.11. Relationship between displacement and dip angle.

Therefore, in analysing the stability of the rock mass, selection of the Poisson’s ratio will not result in signiﬁcant errors.

4.2.3.3 Effect of joint dip angle on rock mass displacement. The eﬀect of joint dip angle to the rock mass deformation is shown in Figure 4.11. From the curves, the following conclusions can be obtained.

80 (i)

(ii)

(iii) (iv)

Chapter 4 The change of rock mass deformation is slightly aﬀected by the joint dip angle varying between 0 and 90 . It suggests that the joint dip angle is a non-sensitive factor aﬀecting the rock mass deformation. The sensitivity of joint dip angle is related to the joint elastic modulus. For the joint set II with a high elastic modulus, the change in the dip angle results in little change of the rock mass deformation. On the other hand, for the joint set I with a low modulus, the change of the dip angle causes increase of the rock mass deformation as shown in Figure 4.11(c). It implies that the sensitivity of the joint dip angle increases with the decreasing joint elastic modulus. The joint dip angle is a sensitive factor when the joint elastic modulus is very low. The vertical settlement of roof, uy, is more sensitive to the change of joint dip angle, as compared with the horizontal deformation of the sidewall. It is noted from Figure 4.11(c) that when a joint set of low modulus and dip angle is between 40 and 60 , the rock mass deformation becomes the greatest, which is most unfavourable to the rock mass stability.

4.2.3.4 Effect of joint persistence on rock mass deformations. Figure 4.12 presents the relation curve of the joint persistence (Zj) and the rock mass displacement (u). The displacement (u) shows the following character as Zj varies: (i)

(ii)

(iii)

When Zj 0.6, ux and uy increase more or less linearly with increasing Zj; when Zj > 0.6, the horizontal displacement, ux, increases acceleratively with increasing Zj, while the vertical displacement, uy, increases at a much lower rate. This indicates that the joints with a persistence greater than 0.6 have higher sensitivity to the sidewall deformation. The sensitivity of joint persistence to the roof displacement, however, is rather low. The sensitivity of the persistence depends on the joint elastic modulus. For example, when Zj ¼ 0.6 and the maximum deviation is 0.2, for joint set I ¼ 5000 MPa, the maximum error of the sidewall displacement is 4.5%; with Ej1 while for joint set II with Ej2 ¼ 14,000 MPa, the error of the sidewall displacement is 1.68%. It can be seen that when the joint elastic modulus is low, the sensitivity of the joint persistence to the rock mass deformation is high. The sensitivity of joint persistence to the sidewall displacement is higher than that to the roof displacement, as shown in Figure 4.12.

4.2.3.5 Effect of joint aperture on rock mass deformations. The relation between the joint aperture and the rock mass deformation is as shown in Figure 4.13.

Sensitivity Analysis of Rock Mass Parameters

81

Figure 4.12. Eﬀects of joint persistence on rock mass displacement.

Figure 4.13. Eﬀect of joint aperture on rock mass deformation.

It leads to the following conclusions: (i) (ii) (iii)

The rock mass deformation increases linearly with the increasing joint aperture. The roof settlement is more sensitive to the change of the joint aperture than the sidewall deformation, as shown in Figure 4.13. The sensitivity of the joint aperture to the surrounding rock mass deformation is also inﬂuenced by the joint elastic modulus. The comparison between Figures 4.13a and 4.13b shows clearly that joint set I, with lower elastic modulus has higher sensitivity factor compared with joint set II.

82

Chapter 4

4.2.3.6 Comparison of sensitivity of different parameters. The previous section discussed the sensitivity of various parameters to the rock mass deformation. In this section, attempt is made to compare the sensitivities of various parameters within the same domain. Since the Poisson’s ratio and the joint dip angle are non-sensitive parameters, only elastic modulus, persistence and aperture of the joint sets are used in the comparison study. The sensitivity of a parameter can be measured by the sensitivity factor Sp [254], deﬁned as the ratio of the relative deviation (dp ¼ jp=p j), by which the system’s character p departs from a certain state of P*, to the relative deviation of the parameter, hence, Sp ðaÞ ¼ p=p =a=a

ð4:11Þ

A higher value of Sp suggests greater eﬀect of the parameter a on the system’s character, P, i.e., the greater sensitivity of a. The relative deviation of a may depend on the problem concerned. In the comparison study below, a relative error of 50% is selected for all parameters for the same underground cavern complex analysis in Figure 4.3. The sensitivity factors of various parameters to the sidewall displacement, the roof settlement and their relative deviations are analysed and summarised in Table 4.4. It can be seen from Table 4.4 for that particular underground project, the sensitivity order of the joint parameters to the cavern sidewall deformation: Zj1 (persistence of joint set I), Ej1 (elastic modulus I), bj1 (joint aperture I), Ej2 (elastic modulus II), Zj2 (persistence II), and bj2 (joint aperture II). Zj1 and Ej1 are the parameters of high sensitivity and the others are of low sensitivity. Similarly, the sensitivity order of the joint parameter to the roof settlement is: Ej1, bj1, Ej2, bj2 and Zj2. Ej1 and bj1 are the parameters of high sensitivity and the rest are of low sensitivity. The synthetic analysis on ux and uy shows that joint set I with a low elastic modulus is the main factor aﬀecting the rock mass deformability. A 50% of combined error of the parameters of this joint set will result in an error of over 10% of the rock mass deformation.

Table 4.4. Sensitivity factors of various parameters to roof and sidewall displacement. Sensitivity factor Sux Suy dSux % dSuy %

Ej1

Zj1

bj1

Ej2

Zj2

bj2

0.127 0.241 6.37 12.03

0.203 0.077 10.15 3.86

0.082 0.118 4.11 5.89

0.058 0.075 2.91 3.75

0.0578 0.013 2.88 0.66

0.036 0.047 1.82 2.34

Sensitivity Analysis of Rock Mass Parameters

83

It should be noted that the above conclusions have been drawn on the basis that each basic parameter varies independently. The interactions between the parameters have not been considered. A more comprehensive study of sensitivity can be conducted by coupling the present method with a parameter interaction study, e.g. the Rock Engineering System approach [1–3]. Nevertheless, the present method is usually suﬃcient to quantify the sensitivity of various parameters.

4.3.

SENSITIVITY ANALYSIS OF ROCK MASS PARAMETERS ON DAMAGE ZONES

This section discusses the sensitivity analysis of the rock material and joint parameters on the magnitude of the damaged zone. The rock material parameters and the rock joint parameters are regarded as dependent variables in the analysis. The stability criterion is based on the magnitude of the damaged zone. The engineering case used is the same cavern complex project as the previous section (Figure 4.3).

4.3.1

Failure criterion for the equivalent jointed rock mass

The principle for analysing the deformation equivalence of the jointed rock mass is similar to that discussed in the previous section. However, each ﬁnite element includes at least one joint. The ﬁnite element may have several possible failure modes or damage patterns including: plastic ﬂow (or shear) of the rock material, tensile or shear failure of joints. The rock mass is considered having been damaged, if any of these phenomena takes place. Drucker–Prager criterion is used to judge if the plastic ﬂow (or shearing) of the intact rock occurs, qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 3 sin I1 þ 3 3 þ sin2 J2 3 3c cos 0

ð4:12Þ

I1 and J2 are the ﬁrst invariant of stress tensor and the second invariant of stress deviator respectively, and c and f are the cohesion and the frictional angle of the rock block. The tensile or shear failure of the joint planes in the rock mass follows the criterion of s 0 jtj stg0j þ Cj0

ð4:13Þ

84

Chapter 4

s is the normal stress on the joint plane (negative sign stands for tension); jtj is the absolute value of the shear stress on the joint plane; c0j and 0j are the cohesion and friction angle of the joint plane of failure. c0j and 0j are calculated through weighted mean method taking account of the cohesion and friction angle of the existing joint and the rock block as well as the joint persistence.

4.3.2 Sensitivity analysis of an underground cavern complex The sensitivity analysis is to study the eﬀect caused by the error in the rock mass parameters on the damaged zones of the underground cavern complex. The underground cavern complex comprises the main power cavern, transforming cavern and tailtrace surge cavern, as shown in Figure 4.3. Details of the project are outlined in the Section 4.1.2. The single-parameter approach is adopted for the sensitivity analysis [255–261]. The method allows each parameter to vary at a time within a possible range, and then derives the corresponding variation of the damage zone in the surrounding rock mass. In order to compare the sensitivities of all the parameters, the dimensionless sensitivity factor is deﬁned as: Ak max A A Ak min SðkÞ max A A

ð4:14Þ

where S(k) is the sensitivity of the parameter k, A* is the area of the damaged zone corresponding to the basic parameter set, Ak max and Ak min are the maximum and minimum areas of the damaged zone within the error domain of the parameter k. The deﬁnition of the sensitivity factor here is not the same as the one deﬁned in the previous section. The sensitivity analysis is aimed at the following parameters: rock material elastic modulus Er, Poisson’s ratio nr, cohesion Cr and internal friction angle fr; rock joint Ej, uj, cj, fj and dip angle aj, persistence Zj and speciﬁc width bj. The basic values and error ranges of the parameters are summarised in Table 4.5.

4.3.3 Result and analysis 4.3.3.1 Effect of parameters on damaged zones. The relationship between parameter’s error and damaged zone area has been obtained through computation. Examples are given in Figure 4.14 showing the relationships between the main parameters (Cr, fr and Ej1) to the damaged zone area. The values of the parameters and the damaged zone area are normalised according to respective basic values.

85

Sensitivity Analysis of Rock Mass Parameters Table 4.5. Basic value and error range of parameters. Parameter Basic value Error range

Parameter

Er (103 MPa)

nr

Cr (MPa) fr ( ) Ej1 (103 MPa)

58.0 46.4 69.6

0.21 0.168 0.252

2.5 1.5 2.5

48.0 28.8 67.2

5.0 1.0 9.0

0.23 0.184 0.276

0.1 0 0.2

Zj1

bj1 (%)

Ej2 (10 MPa)

nj2

cj2 (MPa)

fj2 ( )

aj2 ( )

14.0 2.8 252

0.23 0.184 0.276

0.1 0 0.2

24 19 29

110 90 130

nj1

cj1 (MPa) fj1 ( )

aj1 ( )

24 70 19 29 50 90

Zj2

bj2 (%)

3

Basic value Error range

0.6 0.4 0.8

3.2 1.6 4.8

Figure 4.14. Eﬀects of rock mass parameters on damaged zone area.

0.6 3.2 0.4 1.6 4.8 0.8

86

Chapter 4

It can be seen from Figure 4.14 that the damaged zone area increases linearly with decreasing cr, while it increases abruptly with decreasing fr. Therefore, special attention should be paid to fr with respect to rock stability. The decrease of the joint elastic modulus also results in the increase of the damaged zone area. But its sensitivity is much lower than that of fr and cr.

4.3.3.2 Comparison between sensitivities of various parameters. Based upon the error ranges of parameters listed in Table 4.5, the sensitivity factors of all the parameters have been obtained as deﬁned by equation (4.14). The sensitivity factors are summarised in Table 4.6. A parameter with a sensitivity factor greater than 0.2 (S 0.2) is deﬁned as a highly sensitive parameter. S 0.2 means that 20% of the apparent error in damage zone area is resulted from the parameter error. A parameter with 0.04 < S < 0.2 is a moderately sensitive parameter. A parameter of S < 0.04 is considered non-sensitive parameter. From the results, each parameter can be arranged from high to low sensitivity as follows: Highly sensitive parameters: fr, cr (high to low) Moderately sensitive parameters: Ej1, Zj1, Ej2, and aj1 (high to low) Non-sensitive parameters: bj1, nr, Zj2, bj2, Er, aj2, nj1, cj1, fj1, nj2, cj2, and fj2. The results show that the strength of the rock material is the most critical factor aﬀecting the damaged zones in the surrounding rock mass. Particularly, the internal friction angle, fr, is the most sensitive parameter. On the other hand, the deformation parameter of the rock material has little eﬀect on the damaged zone. The deformation and geometry properties of the joints have certain eﬀects on the size of damaged zone in the surrounding rock mass. The comparison between various parameters of the same joint set shows that the elastic modulus is the most sensitive parameter, the persistence and dip angle have almost the same sensitivity factor. The Poisson’s ratio is non-sensitive. As in the previous sections, the eﬀect of joint parameter with a lower elastic modulus on the damaged zone area is larger than that of the joint set with a higher elastic modulus.

Table 4.6. Sensitivity of various parameters. Rock parameter

S(Er) ¼ 0.014

S(nr) ¼ 0.035

S(cr) ¼ 0.264

S(fr) ¼ 0.837

Joint set I S(Ej1) ¼ 0.101 S(nj1) ¼ 0 S(cj1) ¼ 0 S(fj1) ¼ 0 S(aj1) ¼ 0.047 S(Zj1) ¼ 0.053 S(bj1) ¼ 0.036 Joint set II S(Ej2) ¼ 0.051 S(nj2) ¼ 0 S(cj2) ¼ 0 S(fj2) ¼ 0 S(aj2) ¼ 0.034 S(Zj2) ¼ 0.035 S(bj2) ¼ 0.010

Sensitivity Analysis of Rock Mass Parameters

87

The computation results also show that the joint strength parameters of cj and fj have no eﬀects on the damaged zone, which can be explained from the following two aspects: (i)

(ii)

4.3.4 (i)

(ii)

(iii)

Under the given in situ stresses of sx ¼ 13.3 MPa and sy ¼ 9.5 MPa, the damage of the surrounding rock mass mainly behaves yielding ﬂow and only slight damage takes place along the joint plane. With the joint persistence at 0.6, rock mass strength is governed by rock material properties rather than those of the joint. Therefore, the eﬀect of the change in joint strength parameters on the damaged zone is not being reﬂected.

Summary The strength parameters of rock materials are the main factors aﬀecting the size of the damaged zone. When there is a 20% relative error in fr and cr, they will produce about 83.7% and 26.4% increments in damaged zone areas, respectively. Therefore, careful assessment of the two parameters is important to control damage zone. Strength reinforcement such as systematic rock bolts can be applied to reduce the damaged zone. The elastic modulus of the joint set I also aﬀects the damaged zone area in the surrounding rock mass considerably. The existence of weaker joint set has greater eﬀect on rock mass stability, and vice versa. The eﬀect of the geometric parameters of joint sets on the damaged zone area is relatively low and often can be neglected.

This Page Intentionally Left Blank

Chapter 5

Stability Analysis of Rheologic Rock Mass Rocks and rock masses often exhibit rheologic behaviour, especially weak and soft rocks or highly jointed rock masses. Rheologic behaviour is the time-dependent characteristics of the material deformation and strength [262–275]. For example, the deformation of a rock mass may increase under a constant loading with time elapsing, i.e., creep eﬀect, and the strength may decrease. In the case of underground works, the phenomenon is often found that due to the rheologic behaviour, the loading on support elements gradually increases, leading to the ﬁnal failure of the system [276–286]. In the vicinity of an excavated opening, weak rocks or jointed rock masses can creep and deform visco-elastically [287–301]. Often, stresses in the surrounding rock mass exceed the rock strength, causing the rock mass to become visco-plastic and increasing the visco-plastic composition in the total deformation [302–306]. The understanding of rheologic characteristics is important to the design of underground excavation and support. The reinforcement and support design for the rheologic rock mass must take into consideration the visco-elastic and visco-plastic deformation.

5.1.

RHEOLOGICAL MECHANICAL MODELS FOR ROCKS AND ROCK MASSES

There are two basic approaches to study the rheological phenomena of a rock mass. The ﬁrst approach is from the macro point of view to study synthetical and mechanical behaviour of the rock mass by taking a large volume to represent the whole rock mass containing adequate quantities of discontinuities [17–19,307]. The second is to study the intact rock material and discontinuities individually and then to assemble them together [15,20,21]. Here, emphasis is laid on the ﬁrst approach. There are three basic ideal bodies for common rheological mechanical models: Hooke’s elastic solid, Newton’s viscous liquid and St. Venant’s plastic mass [264,291]. In the rock mechanics, these basic bodies are often used to model a variety of rock masses and to simulate diﬀerent rheological characteristics of the rock masses. Table 5.1 summarises the common rheological models of rock mechanics. It should be noted that all the models are linear. The stress–strain curves of the models describe the relationship between deviator tensors. The equations are mainly applicable to the uniaxial loading state. The equations that describe multi-axial loading state can be derived through the superposition theorem.

89

90

Table 5.1. Typical mechanical model, formula and rock type. Formula

Rheological type

Rock type

Hooke (H)

s ¼ Ee

Elastic

Hard or relatively hard rock

Newton (N) St. Venant (V)

s ¼ 2Z e_ s¼y

Viscous Plastic

Soft rock Soft rock or rock under high

Kelvin (K ¼ H/N)

s ¼ 2Ge þ 2Z e_

Visco-elastic

Most medium to soft rock

Model

Model illustration

confining pressure (elastic post effect)

Maxwell (M ¼ H N) Prandtl (P ¼ H V)

sþ

(PT ¼ H/(H N))

Zm s GM

¼ 2GHe þ Bingham (B ¼ H (N/V))

Burgers (Bu ¼ M K)

Schofield–Scott–Blair (SSB ¼ K (M/V))

Visco-elastic ( plastic)

or rock at depth

Elasto-plastic

Elasto-visco-plastic

Soft rock and soil under high confining pressure

sþ

Visco-elastic ( plastic)

GM Zp þ GK Zp þ GM ZK s _ GM GK Zp ZK 2Zp ZK s€ ¼ y þ 2Zpe_ þ e€ GM GK GK

Elasto-visco-plastic

sþ

Medium strength rock and most sedimentation rock

GM þ GH ZM e_ GM

ZMðGM þ GK Þ þ ZKGM s_ GM GK Z Z Z þ M K s€ ¼ 2ZM e_ þ K e€ GM GK GK

Rock under certain confining pressure

Visco-elasto-viscous

8 < s ¼ZEe when s < y s þ ps_ ¼ y þ 2Zp_e G : when s y

Halite under long-term load

Soft rock such as clayed shale, mudstone

Note: s – deviator component of stress tensor, e – deviator component of strain tensor, G – shearing module and Z – viscosity factor.

Halite under long-term load

Chapter 5

Poynting-Thomson

Z _ ¼ 2Z_e sþ s G s ¼ Ee when s < y s ¼ Eðe epl Þ when s y

Stability Analysis of Rheologic Rock Mass

91

From the testing results on the rheological characteristics of most rock materials, it can be seen that the constitutive relations of rocks and rock masses are all non-linear. This is reﬂected in two aspects: ﬁrstly, the rheological mechanical parameters, such as G, Z, E and n are not constant but the functions of stress level and/or time; and secondly, strain, stress and their rates are also related non-linearly [19,194,308,309]. The universal constitutive relation of rocks and rock masses can be expressed as aðs,tÞðsÞn1 þ bðs,tÞðs_ Þn2 þ cðs,tÞðs€ Þn3 ¼ y þ aðs,tÞðeÞn4 þ bðs,tÞð_eÞn5 þ ðs,tÞð€eÞn6

ð5:1Þ

where n1, n2, . . . , nn are all positive, y is the threshold value of stress at which the rock material enters the plastic state. The above non-linear constitutive relation can be used provided that all parameters concerned are available by suﬃcient testing of rock behaviour, i.e., the functional factors such as a(s,t) and power values nn of stress and strain can be obtained. However, it is diﬃcult to determine so many parameters or variables as the rheologic tests are time consuming. In practice, simpliﬁed linear models are often adopted for engineering applications [308,309]. Equation (5.2), for example, is a common constitutive rheological model adopted in rock mechanics study to simulate a variety of rocks: as þ bs_ þ cs€ ¼ y þ ae þ b_e þ s€

ð5:2Þ

where a, b, c, y, a, b, are specially deﬁned as constants. It can be seen that the mathematical equations of various rheological mechanical models in Table 5.1 are in fact special cases of equation (5.2) or diﬀerent combinations of equation (5.2).

5.2.

VISCO-ELASTIC SURROUNDING ROCK MASS AND SUPPORTING PROBLEM

Most rock masses involved in underground engineering exhibit characteristics of visco-elasticity to diﬀerent extents except some very hard and massive rocks [194,308]. It is therefore of paramount importance to study the behaviour of the visco-elastic rock masses. Nowadays, numerical analyses including ﬁnite element method (FEM) and boundary element method (BEM) have been widely used to study the stress state of the surrounding rock masses. Nevertheless, it is still of great signiﬁcance to use the analytical method as it provides theoretical solutions in limited forms. The ﬁnal expression of this method contributes to direct understanding of the problems under consideration.

92 5.2.1

Chapter 5 General solution for circular visco-elastic media

In this section, the excavated opening, the support lining and the surrounding rock mass are treated as a generalised two-dimensional problem. They are assumed to be homogeneous, isotropic and of ﬁnite deformation. A triple-element model of visco-elasticity (Figure 5.1) is adopted. It is equivalent to the Poynting–Thomson model (see Table 5.1). Assuming that the deformations of each element have their own independent physical equations and the Poisson’s ratio is constant, then the rheological physical equations for the plane problems can be expressed as [8,52]: 9 > > > > > 3n 3n > e þ 2Z e_ Z þ e_ > > > > 1 2n 1 2n = 3n 3n > e þ 2Z e_ X þ e_ > sX þ s_ X ¼ 2G eX þ > 1 2n 1 2n > > > > > > > 3n 3n ; e þ 2Z e_ sY þ s_ Y ¼ 2G 1 2n 1 2n

þ _ ¼ G þ Z_ sZ þ s_ Z ¼ 2G eZ þ

ð5:3Þ

where s_ X , s_ Y , s_ Z and _ are the normal stress rate and shear stress rate along the directions of X, Y, Z; e_ X , e_ Z and _ are the normal strain rate and the shear strain

Figure 5.1. Rheological model of triple-element.

Stability Analysis of Rheologic Rock Mass

93

Figure 5.2. Coordinate system of circular medium.

rate along X, Z; is the relaxation time, G is the elastic shear module (inﬁnitesimal loading rate) and Z is the viscosity of shearing deformation. Writing equation (5.3) in the form of integration equation and substituting it into equilibrium equation and considering geometric equation, one can obtain the Volter integration equation set of the second type with the body force neglected. The equation is integral. The components of the stress in a circular medium derived from the general solution can be expressed in polar coordinates [8,52], as shown in Figure 5.2: r 1 0 0 2 Ur ðtÞ ¼ ð1 2nÞC0 ðtÞ þ b0 ðtÞS þ a01 ðtÞð3 4nÞS ln 2G S 1 3 b1 ðtÞS ð1 4nÞC1 ðtÞq þ d1 ðtÞ Q1 ðjÞ q 1 h X n nþ2 þ Az,n ðnÞan ðtÞS nbn ðtÞS n¼2 i n n2 Qn ðjÞ þ Aw,n ðnÞCn ðtÞq þ ndn ðtÞq ð5:4Þ

W y ðtÞ ¼

r 1 a01 ðtÞS ð3 4nÞS ln þ 1 þ b1 ðtÞS3 þ ð5 4nÞC1 ðtÞq þ d1 ðtÞq1 2G S 1 h dQðjÞ r X þ Bz,n ðnÞan ðtÞSn þ bn ðtÞS nþ2 dj 2G n¼2 i dQ ðjÞ n ð5:5Þ þ Bw,n ðnÞCn qn þ dn ðtÞqn2 dj

94

Chapter 5

2

sr ðtÞ ¼ KT C00 ðtÞ b00 ðtÞS þ ½ð3 2nÞSa01 ðtÞ þ 2b1 ðtÞS3 2C1 ðtÞqQðjÞ Zt 0 0 ½C0 ðt Þ b00 ðt0 ÞS 2 þ ½ð3 2nÞSa01 ðt0 Þ þ 2b1 ðt0 ÞS 3 2C1 ðt0 Þq þ Kð1 KT Þ t0

1 X 0 QðjÞ eKðt tÞ dt0 KT nðn 1Þðn þ 2Þan ðtÞS n nðn þ 1Þbn ðtÞS nþ2 n¼2 n

nðn þ 1Þðn 2Þcn ðtÞq nðn 1ÞdnðtÞq

n2

1 X

Qn ðjÞ Kð1 KT Þ

Z

n¼2

t

nðn 1Þ

t0

ðn þ 2Þan ðt0 ÞS n nðn þ 1Þbn ðt0 ÞS nþ2 nðn þ 1Þðn 2Þcn ðt0 Þqn

0 nðn 1Þdn ðt0 Þqn2 Qn ðjÞeKðt tÞ dt0

ð5:6Þ

sy ðtÞ ¼ KT C00 ðtÞ b00 ðtÞS2 ð1 2nÞa01 ðtÞS þ 2b1 ðtÞS 3 þ 6C1 ðtÞq QðjÞ Zt 0 0

C0 ðt Þ þ b00 ðt0 ÞS2 ð1 2nÞa01 ðt0 ÞS þ 2b1 ðt0 ÞS 3 þ 6C1 ðt0 Þq þ Kð1 KT Þ t0

1 X 0 QðjÞ eKðt tÞ dt0 þ KT nðn 1Þðn þ 2Þan ðtÞS n nðn þ 1Þbn ðtÞSnþ2 n¼2 1 X

nðn þ 1Þðn þ 2ÞCn ðtÞqn nðn 1Þdn ðtÞqn2 Qn ðjÞ þ Kð1 KT Þ n¼2

Z

t

nðn 1Þ

t0

ðn 2Þan ðt0 ÞS n nðn þ 1Þbn ðt0 ÞSnþ2 nðn þ 1Þðn þ 2ÞCn ðt0 Þqn

0 nðn 1Þdn ðt0 Þqn2 Qn ðjÞeKðt tÞ dt0

ð5:7Þ

sy ðtÞ ¼ KTn 2C0 ðtÞ þ ½2a1 ðtÞS 8C1 ðtÞqQðjÞ Zt 0 2C0 ðt0 Þ þ ½2a1 ðt0 ÞS 8C1 ðt0 ÞqQ1 ðjÞeKðt tÞ dt0 KTn þ Kð1 KT Þn t0

1 X

4nðn 1Þan ðtÞS n þ 4nðn þ 1ÞCn ðtÞqn Qn ðjÞ Kð1 KT Þn

n¼2

1 Z X n¼2

t

0

½4nðn 1Þan ðt0 ÞS n þ 4nðn þ 1ÞCn ðt0 Þqn Qn ðjÞeKðt tÞ dt0 t0

dQðjÞ ðtÞ ¼ KT ½ð12nÞa1 ðtÞS 2b1 ðtÞS 3 þ2C1 ðtÞq dðjÞ Zt dQ1 ðjÞ Kðt0 tÞ 0 e þKð1KT Þ ½ð12nÞaðt0 ÞS 2b1 ðt0 ÞS 3 þ2C1 ðt0 Þq dt dj t0

ð5:8Þ

95

Stability Analysis of Rheologic Rock Mass þKT

1 X

nðn1Þan ðtÞS n ðnþ1Þbn ðtÞSnþ2 þnðnþ1ÞCn ðtÞqn þðn1Þdn ðtÞqn2

n¼2

1 Z t X dQn ðjÞ þKð1KT Þ nðn1Þan ðt0 ÞSn ðnþ1Þbn ðt0 ÞSnþ2 þnðnþ1ÞCn ðt0 Þqn dj t n¼2 0

þnðn1Þdn ðt0 Þqn2

dQn ðjÞ Kðt0 tÞ 0 e dt dj

A,n ðnÞ n½n þ 2ð1 2nÞ Bz,n ðnÞ ¼ n þ 4ð1 nÞ an ðtÞ ¼ ½a0n ðtÞ, a00n ðtÞ Cn ðtÞ ¼

½Cn0 ðtÞ, C 00n ðtÞ

Qn ðjÞ ¼ ðcos nj, sin njÞ

ð5:9Þ 9 Aw,n ðnÞ ¼ n½n 2ð1 2nÞ > > > > > > Bw,n ðnÞ ¼ n þ 4ð1 nÞ > > = 0 00 bn ðtÞ ¼ ½bn ðtÞ, bn ðtÞ > > > > dn ðtÞ ¼ ½dn0 ðtÞ, d 00n ðtÞ > > > > ; Q1 ðjÞ ¼ ðcos j, sin jÞ

ð5:10Þ

where R0 r q¼ n ¼ 2, 3, 4 R1 r 1 (reciprocal of the relaxation time) K¼ Z T ¼ (time of post-elastic behaviourÞ G S¼

The functions of cos j [with coeﬃcients of a0n ðtÞ, b0n ðtÞ, C 0n ðtÞ, d 0n ðtÞ] and sin j [with b00n ðtÞ, C 00n ðtÞ, d 00n ðtÞ] are determined from the boundary conditions. Applying the previous equation sets describing the components of displacement and stress, one can solve all problems (which can be expressed by the Fourier series) with visco-elastic medium boundary conditions. a00n ðtÞ,

5.2.2

Interaction of visco-elastic surrounding rock mass and elastic lining

Long tunnels at depth can often be treated as plane problems for stability analysis. When the overlying depth (H) is greater than 30 times of tunnel radius, the gravitational ﬁeld can be replaced by the stress state in an inﬁnite plane approximately, as shown in Figure 5.3. The initial vertical stress is Pz ¼ 0 H and horizontal stress is Px ¼ l1 Pz ¼ l1 0 H. H is the overlying depth, is the unit weight and l1 is the lateral pressure coeﬃcient.

96

Chapter 5

Figure 5.3. Plane problem of tunnel stability analysis.

Before excavation, the undisturbed rock mass is generally in an elastic state. According to the theory of elasticity, its initial stress state is 9 1 > sð2Þ ¼ p ½ð1 þ l Þ ð1 l Þ cos 2j z 1 1 > r,0 2 > > > > ð2Þ 1 = sy,0 ¼ 2 pz ½ð1 þ l1 Þ þ ð1 l1 Þ cos 2j > sð2Þ y,0 ¼ l1 pz 0ð2Þ ¼ 12 pz ð1 l1 Þ sin 2j

> > > > > > > ;

ð5:11Þ

and initial displacement components are 9 r > ð1 l1 Þr0 Hð1 cos 2jÞ > > > 4G2 > = r ð2Þ W y,0 ¼ ð1 l1 Þr0 H sin 2j > 4G2 > > > > ð2Þ ; V ¼0 ð2Þ ¼ Ur,0

ð5:12Þ

y,0

If an opening is excavated, the initial stress state around the opening periphery ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ can be described as: (r ¼ a), S ð2Þ 0 ¼ S 0 ðs r,0 , s y,0 , s y,0 , 0 , U r,0 , W y,0 , V y,0 Þ. Such a disturbed stress state does not meet the boundary conditions, and a compensating

Stability Analysis of Rheologic Rock Mass

97

stress state, S(2), is produced in the same area. The new stress state, Scð2Þ , is the summation of the above two stress states, i.e.,

ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ s ,s ,s , ,U ,W ,V s ,s ,s , ,U ,W ,V S ð2Þ ¼ S c r,c y,c y,c c r,c y,c 0 r,0 y,0 y,0 0 r,0 y,0 y,c y,0

ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ þ S ð2Þ sð2Þ ,s ,s , ,U ,W ,V r y r y y y ð5:13Þ Compared with the stress on the lining after excavation, the stress state produced by the self weight is so small that it can be neglected. The initial stress state in the lining, S ð1Þ 0 , can be considered as zero, so

ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ S ð1Þ sð1Þ sð1Þ ,U ð1Þ c r,c ,sy,c ,sy,c , c ,U r,c ,W y,c ,V y,c ¼ S r ,sy ,sy , r ,W y ,V y ð5:14Þ When the overlying depth above the opening is suﬃciently great and the initial in situ stress state is nearly of hydrostatic stress, Px ¼ Py ¼ Pz ¼ 0H, then from equations (5.11) and (5.12) we have 9 ð2Þ ð2Þ > sð2Þ r,0 ¼ st,0 ¼ sy,0 ¼ P > > = ð2Þ ð5:15Þ 0 ¼ 0 > > > ð2Þ ð2Þ ; U ð2Þ r,0 ¼ W t,0 ¼ V y,0 ¼ 0 The stress state after excavation can be derived accordingly from equations (5.4), (5.5), (5.6), (5.7), (5.8) and (5.9). In consideration of one-dimensional axial symmetry of the stress state, all terms relating to the polar angle j are equal to zero, then Z th h i i 0 2 0 0 0 0 2 k2 ðt0 tÞ 0 sð2Þ þ K ðtÞ ¼ K T C ðtÞ b ðtÞS ð1 K T Þ C ðt Þ b ðt ÞS dt 2 2 0,2 2 2 2 r 0,2 2 0,2 0, 2 2 e h

0

i

0 0 2 sð2Þ y ðtÞ ¼ K2 T2 C 0,2 ðtÞ þ b0,2 ðtÞS 2 þ K2 ð1 K2 T2 Þ 0 sð2Þ y ðtÞ ¼ K2 T2 C 0,2 ðtÞ þ K2 ð1 K2 T2 Þ

Z

t 0

Z th 0

i 0 C 00,2 ðt0 Þþb00, 2 ðt0 ÞS 22 ek2 ðt tÞ dt0

0

C 00,2 ðt0 Þek2 ðt tÞ dt 0

ð2Þ ðtÞ ¼ 0 U ð2Þ r ðtÞ ¼

i 1 h r2 ð1 2nÞC 00, 2 ðtÞ þ b 00,2 ðtÞS 22 2G2

W ð2Þ y ðtÞ ¼ 0 V ð2Þ y ¼0

ð5:16Þ

98

Chapter 5

where K2, T2, G2 are the mechanical parameters of the rheological characteristics of the surrounding rock mass; S2 ¼ a/r, r is the radial distance to any point in the rock mass from the centre of the circular tunnel. Because the second stress state after excavation is the sum of the initial and the compensating stress state (equation (5.13)), then Z th h i i 0 0 0 0 0 0 2 k2 ðt0 tÞ 0 sð2Þ þK ¼K T C ðtÞb ðtÞS ð1K T Þ C ðt Þb ðt ÞS dt P 2 2 2 2 2 2 r,c 0,2 0,2 0,2 0,2 2 e 0 Z th h i i 0 0 2 0 0 0 0 2 k2 ðt0 tÞ 0 ðtÞ¼K T C ðtÞþb ðtÞS ð1K T Þ C ðt Þþb ðt ÞS dt P þK sð2Þ 2 2 2 2 2 0,2 0,2 2 0,2 0,2 2 e y,c 0 Zt 0 0 sð2Þ ðtÞ¼K T C ðtÞþK ð1K T Þ C 00,2 ðt0 Þek2 ðt tÞ dt0 P 2 2 3 3 3 y,c 0,2 0

ð2Þ c ðtÞ¼0 U ð2Þ r,c ðtÞ¼

i 1 h r2 ð12nÞC 00,2 ðtÞþb00,2 ðtÞS 22 2G2

W ð2Þ y,c ðtÞ¼0 ð5:17Þ The stress state in the lining which is assumed to be ideally elastic can be derived from equation (5.16). Since y1 and T1 ¼ Z1 =G1 have the same order of magnitude for the common solid materials, and y1 ! 0, T 1 ! 0 for an elastic body, then K1 T1 ¼ ðT1 =y1 Þ ! 1, i.e., K1 T1 ¼ 1

) ð5:18Þ

K1 ð1 K1 T1 Þ ¼ 0 In this case, equation (5.16) expressing the stress state in the lining is 0 0 2 sð1Þ r ðtÞ ¼ C 0,1 ðtÞ b0,1 ðtÞS 1 0 0 2 sð1Þ y ðtÞ ¼ C 0,1 ðtÞ þ b0,1 ðtÞS 1 0 sð1Þ y ðtÞ ¼ 2n1 C 0,1 ðtÞ

ð1Þ ¼ 0

9 > > > > > > > > > > > > > > =

> > > h i> > 1 0 0 2 > > ðtÞ ¼ ð1 2nÞC ðtÞb ðtÞS U ð1Þ 1 > 0,1 0,1 1 > r > 2G1 > > > > ; ð1Þ ð1Þ W y ðtÞ ¼ V y ðtÞ ¼ 0

ð5:19Þ

Stability Analysis of Rheologic Rock Mass

99

where n1 and G1 are the mechanical parameters of the lining; S 1 ¼ a0 =r1 , r1 is the radial distance to a given point in the lining from the centre of the circular tunnel and a0 is the inner radius of the lining. There are four coeﬃcients in equations (5.17) and (5.19) to be determined; 0 C 0, 2 ðtÞ, b00, 2 ðtÞ, C 00,1 ðtÞ and b00,1 ðtÞ, which are the functions of time. They can be obtained according to the following boundary conditions: 9 ð2Þ ð2Þ sð2Þ r,c ¼ sr,0 þ sr ¼ P > > = ð1Þ ðiiÞ r1 ¼ a sr ¼ 0 > > ; ðiiiÞ r1 ¼ r2 ¼ a ðiÞ

r2 ! 1

ð5:20Þ

ð2Þ ð2Þ sð1Þ r ¼ sr þ sr,0 ð2Þ ð2Þ U ð1Þ r ¼ U r U r,ðt¼0Þ

where the last term is the boundary condition of displacement, implicating that once the lining is completed, the elastic displacement at the periphery of the opening is relieved. The term of U ð2Þ rðt¼0Þ can be easily obtained using the method of elasticity theory. The above four functional coeﬃcients can be obtained by simultaneously solving the simultaneous equation set (5.17), (5.19) and (5.20) that consists of algebraic and integration equations. And ﬁnally, the stress states in the surrounding rock mass and the lining can be derived as follows: for the surrounding rock mass, 9 a2 > 0 b2 t > ¼ P 1 2 ½1 a ð1 e Þ > > > r2 > > > = 2 a ¼ P 1 þ 2 ½1 a0 ð1 eb2 t Þ > > > r2 > > > > > ; ð2Þ ¼ P c ¼ 0

sð2Þ r,c sð2Þ y,c sð2Þ y,c where

a 1 a ¼ 1 a þ 2G2 T2 K2 0

¼ 2G1

a2 a20 1 2 a a0 þ ð1 2n1 Þa2

ð5:21Þ

100

Chapter 5

and for the lining, 9 a20 > 0 b2 t > sð1Þ ¼ a Pð1 e Þ 1 > r,c 1 > r21 > > > = 2 a0 ð1Þ 0 b2 t sy,c ¼ a1 Pð1 e Þ 1þ 2 > > r1 > > > > > ; ð1Þ 0 b2 t sy,c ¼ 2n1 a1 Pð1 e Þ

ð5:22Þ

where

a01

ðT2 K2 1Þ ð1 2n1 Þ þ X02 ¼ T2 K2 ½ð1 2n1 Þ þ X02 þ 1 X02 G

G¼

G1 G2

X0 ¼

a0 a

K2 ð1 2n1 Þ þ X02 þ 1 X02 G

b2 ¼ K2 T2 ð1 2n1 Þ þ X02 þ 1 X02 G It clearly shows that the ﬁnal stress states in the rock mass and the lining depend upon the following parameters: (i) the initial stress state, P (ii) thickness of the lining, X0 ¼ a a0 (iii) elastic shearing moduli of the lining and the rock mass, G ¼ G1 =G2 (iv) relation of post-elastic behaviour time to relaxation time in the rock mass, T2 K2 ¼ T2 =y2 (v) the Poisson’s ratio of the lining, n1. The value of b2 in the equation for stress states only aﬀects the rate of transition from the initial state to the ﬁnal state but does not aﬀect the magnitude of the ﬁnal state. The parameters that are related to b2 include n, X0, G, T2K2 and T2.

5.2.3

Interaction of rock mass and lining of different visco-elastic media

When the surrounding rock mass and the lining are composed of visco-elastic media, but diﬀerent rheological parameters, the rheological mechanical stress state of the rock mass can still be expressed by equation (5.17). The stress state in lining can be

101

Stability Analysis of Rheologic Rock Mass

expressed by the following rheological mechanical equations, Z th 9 h i i ð1Þ 0 0 2 0 0 0 0 2 k1 ðt0 tÞ 0 > sr ðtÞ ¼ K1 T1 C 0,1 ðtÞ b0,1 ðtÞS 1 þ K1 ð1 K1 T1 Þ C 0,1 ðt Þ b0,1 ðt ÞS 1 e dt > > > > 0 > > Z > h i i th > ð1Þ 0 0 2 0 0 0 0 2 k1 ðt0 tÞ 0 > sy ðtÞ ¼ K1 T1 C 0,1 ðtÞ þ b0,1 ðtÞS 1 þ K1 ð1 K1 T1 Þ C 0,1 ðt Þ þ b0,1 ðt ÞS 1 e dt > > > > 0 = Zt ð1Þ 0 0 0 k1 ðt0 tÞ 0 > sy ðtÞ ¼ 21 K1 T1 C 0,1 ðtÞ þ K1 ð1 K1 T1 Þ C 0,1 ðt Þ e dt > > > 0 > h i > > r 1 > ð1Þ 0 0 2 > U r ðtÞ ¼ ð1 2n1 ÞC 0,1 ðtÞ þ b0,1 ðtÞS 1 > > > 2G1 > > ; ð1Þ ð1Þ ð1Þ W y ðtÞ ¼ V y ðtÞ ¼ 0 ðtÞ ¼ 0 ð5:23Þ By assuming that the boundary condition is similar to that given in last section, several functional coeﬃcients in the equations of the rock mass and the lining can be determined. The four boundary equations independent to each other are: (i)

when r2 ! 1 ð2Þ sð2Þ r,0 þ sr ¼ P Rt 0 P þ K2 T2 ½C 00, 2 ðtÞ þ K2 ð1 K2 T2 Þ 0 C 00, 2 ðt0 Þek2 ðt tÞ dt0 ¼ P

(ii)

) ð5:24aÞ

when r1 ¼ a0 sð1Þ r ¼0 Rt 0 K1 T1 ½C 00,1 ðtÞ b00,1 ðtÞ þ K1 ð1 K1 T1 Þ 0 ½C 00,1 ðt 0 Þ b 00,1 ðt0 Þ ek1 ðt tÞ dt0 ¼ 0

ð5:24bÞ (iii)

when r1 ¼ r2 ¼ a ð2Þ ð2Þ sð1Þ r ¼ sr þ sr,0 Z th h i i 0 K1 T1 C 00,1 ðtÞ b00,1 ðtÞX02 þ K1 ð1 K1 T1 Þ C 00,1 ðt0 Þ b00,1 ðt0 ÞX02 ek1 ðt tÞ dt0 0 Zt 0 ¼ K2 T2 ½C 00, 2 ðtÞ b00, 2 ðtÞ þ K2 ð1 K2 T2 Þ ½C 00, 2 ðt0 Þ b00, 2 ðt0 Þek2 ðt tÞ dt0 P 0

(iv)

When r1 ¼ r2 ¼ a ð2Þ ð2Þ U ð1Þ r ¼ U r U r,ðt¼0Þ i a h a2 0 aP 0 0 2 ð1 2n1 ÞC 0,1 ðtÞ þ b0,1 ðtÞX0 ¼ b ðtÞ 2G1 2G2 T2 K2 2G2 0, 2

102

Chapter 5

To obtain the solutions to these four simultaneous equations, it is necessary to solve the following second-order homogeneous diﬀerential equations, ab€00,1 ðtÞ þ bb_00,1 ðtÞ þ cb00,1 ðtÞ ¼ d ð5:25Þ where

a ¼ K1 T1 1 X02 G þ K2 T2 ð1 2n1 Þ þ X02

b ¼ K1 1 X02 Gð1 þ K 2 T2 Þ þ K2 ð1 2n1 Þ þ X02 ðK1 T2 þ 1Þ

c ¼ K1 K2 1 X02 G þ ð1 2n1 Þ þ X02

K1 Gð1 T2 K2 Þ P T2 There are three possible solutions to the integration of equation (5.25), depending on the diﬀerent values of the discriminant of ¼ b2 4ac, i.e., > 0, < 0 and ¼ 0. The main results with the derivation omitted are given below: When 6¼ 0, the stress state of the surrounding rock mass 9

> a2 0 0 r1 t 00 r2 t ð2Þ 0 k2 t > sr,c ¼ P 1 2 a2 b2 e b2 e r2 e > > > r2 > = 2

a ð5:26Þ ð2Þ 0 r1 t 00 r2 t 0 0 k2 t sy,c ¼ P 1 þ 2 a2 b2 e b2 e r2 e > > > r2 > > > ; ð2Þ ð2Þ sy,c ¼ P c ¼ 0 d¼

where a02

1 X02 G þ ð1 2n1 Þ þ X02 T2 K2

¼ T2 K2 1 X02 G þ ð1 2n1 Þ þ X02

K2 ð1 þ T2 r1 Þ b02 ¼ C 1 ð1 2n1 Þ þ X02 K2 þ r1

K2 ð1 þ T2 r2 Þ b2 ¼ C 2 ð1 2n1 Þ þ X02 K2 þ r2

C1 C 2 K2 a02 r2 ¼ ðK2 T2 1Þ ð1 2n1 Þ þ X02 þ K2 þ r1 K2 þ r2 K1 ðT2 K2 1Þr2 T2 K2 1

þ C 1 ¼ T2 ðr2 r1 Þc T2 ðr2 r1 Þ K1 T1 1 X 2 G þ K2 T2 ð1 2n1 Þ þ X 2

9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > =

> > > > > > > > > > > 0 0 > > > > > K ðT K 1Þ r T K 1 > 1 2 2 1 2 2 > >

C2 ¼ > 2 2 T2 ðr2 r1 Þc T2 ðr2 r1 Þ K1 T1 1 X0 G þ K2 T2 ð1 2n1 Þ þ X0 > > > > ) > pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > 2 > r1 b b 4ac > > ¼ > ; 2a r2 ð5:27Þ

Stability Analysis of Rheologic Rock Mass

103

and the stress state of the lining n o9 a20 0 r1 t 00 r2 t 0 0 k1 t > > > ¼ 1 2 GP a1 b1 e b1 e r1 e > > r1 > > > > > > n o 2 > a ð1Þ 0 r1 t 00 r2 t = 0 0 k1 t > 0 sy,c ¼ 1 þ 2 GP a1 b1 e b1 e r1 e r1 > > > n o > > 0 r1 t 00 r2 t ð1Þ 0 0 k1 t > > sy,c ¼ 2n1 GP a1 b1 e b1 e r1 e > > > > > > ; 00 0 ¼ 0

ð5:28Þ

9 T2 K2 1 > >

> > > T2 K2 1 X02 G þ ð1 2n1 Þ þ X02 > > > > > > > c1 K1 1 þ T1 r1 > 0 > > b1 ¼ > = K1 þ r1 > > c2 K1 1 þ T1 r2 > > > b001 ¼ > > > K1 þ r2 > > > > > > c c 1 2 0 0 > K 1 a1 > r1 ¼ ðK1 T1 1Þ þ ; K1 þ r1 K1 þ r2

ð5:29Þ

sð1Þ r,c

where

a01 ¼

When ¼ 0, the derivation procedure is the same. For the rock mass, 9

> a2 0 0 r 00 t 00 0 k2 t r0 t > sð2Þ ¼ P 1 a b e b ðtÞe r e > r,c 2 2 2 > > r22 2 > > > = i 2h a ð2Þ 0 r 00 t 00 r0t 0 0 k2 t sy,c ¼ P 1 þ 2 a2 b2 e b2 ðtÞe r2 e > > > r2 > > > > > ; ð2Þ ð2Þ sy,c ¼ P c ¼ 0

ð5:30Þ

104

Chapter 5

where

a02 b02 b002

r 00 c 001 c 002

1 X02 G þ ð1 2n1 Þ þ X02 T2 K2

¼ T2 K2 1 X02 G þ ð1 2n1 Þ þ X02

c00 K2 ð1 þ T2 r Þ þ c002 K2 ðK2 T2 1Þ ¼ ð1 2n1 Þ þ X02 1 ðK2 þ r Þ2 ( 00

00

2 c2 K2 ðT2 r þ 1Þ 0 ¼ ð1 2n1 Þ þ X0 r2 ðK2 T2 1Þ ð1 2n1 Þ þ X02 : 00 K2 þ r 00 00 00 K2 ½c1 ðK2 þ r Þ c 2 0 a2 ðK2 þ r 00 Þ2 b ¼ 2a T2 K2 1

¼ K2 T2 ð1 2n1 Þ þ X02 þ 1 X02 G ðT2 K2 1Þr 00

¼ K2 T2 1 X02 G þ ð1 2n1 Þ þ X02 T2 K2 1

T2 K2 T2 ð1 2n1 Þ þ X02 þ K1 T1 1 X02 G

ð5:31Þ

and for the lining, 9 0 > a20 0 r 00 t 00 r 00 t 0 k1 t > > sð1Þ ¼ 1 GP a b e b e r e > r,c 1 1 1 1 > r21 > > > > 2 = 0 a0 00 ð1Þ 0 r 00 t 00 r t 0 k1 t sy,c ¼ 1 þ 2 GP a1 b1 e b 1 e r1 e r1 > > > 00 00 > ð1Þ > sy,c ¼ 2n1 GP a01 b01 er t b 001 er t r01 ek1 t > > > > ; ð1Þ c ¼ 0

ð5:32Þ

9 T2 K2 1 > > > > T2 K2 ½ 1 X02 G þ ð1 2n1 Þ þ X02 > > > > 00 00 c1 K1 ð1 þ T1 r ÞðK1 þ r Þ þ c2 K1 ðK1 T1 1Þ > > 0 > > b1 ¼ = 2 00 ðK1 þ r Þ > c2 K1 ðT1 r 00 þ 1Þ > > b 001 ¼ > > > K1 þ r 00 > > > 00 > K1 ½c1 ðK1 þ r Þc2 > 0 0 > r1 ¼ ðK1 T1 1Þ a ; 1 2 ðK1 þ r 00 Þ

ð5:33Þ

where a01 ¼

Stability Analysis of Rheologic Rock Mass

105

In some limiting cases, the above equations can be simpliﬁed. When t ! 1, for the rock mass ( > 0 and ¼ 0),

sð2Þ r,c sð2Þ y,c

¼

a02 P

¼

a02 P

sð2Þ y,c ¼ P

a2 1 2 r2 1þ

a2 r2

9 > > > > > > =

ð2Þ c ¼ 0

ð5:34Þ

> > > > > > ;

and, for the lining ( > 0 and ¼ 0),

sð1Þ r,c sð1Þ y,c

0 sð1Þ y,c ¼ 2n1 a1 GP

5.2.4

9 > > > > > > =

a20 ¼ 1 2 r1 2 a ¼ a01 G 1 þ 20 r1 a01 G

ð1Þ c ¼0

> > > > > > ;

ð5:35Þ

Two-dimensional stress state in surrounding visco-elastic rock mass

In most cases, the in situ stress state of rock mass is anisotropic. The magnitude of the vertical stress generally equals to the overburden stress, i.e., directly proportional to the overlying depth. On the other hand, the horizontal stresses vary, which can approximately be derived using the elasticity theory adopting lateral pressure coeﬃcient of l ¼ n/(1n) (n is the Poisson’s ratio of the rock mass). Generally, it is greater than the vertical stress, i.e., l > 1 [310–313]. Therefore it is of great signiﬁcance to study the stress state in surrounding visco-elastic rock masses for l 6¼ 1. From the general equations (5.4)(5.9) for visco-elastic rock mass given in Section 5.2, it can be seen in principle that the equations can be used to solve any boundary problems of multi-layered circular media. Of course, provided that the lining is circular in shape and composed of a visco-elastic medium while the surrounding rock mass is composed of another visco-elastic medium, such twodimensional problem can be solved, even for initial in situ stress with l 6¼ 1. The solving procedure is tedious. For this reason, this section only deals with a simple case of two-dimensional problems, i.e., the variation of the stress state in the rock mass with an unlined opening.

106

Chapter 5

In this case, the terms with subscript 1 in equations (5.4)–(5.9) are zero, i.e., the ﬁrst circular layer does not exist. The basic equations are given as follows: For stress components, 9 > > > > > 0 > > > > > Zt > >

> 2 4 0 2 0 4 0 > KT 8a2 ðtÞS 6b2 ðtÞS 2d2 ðtÞ þKð1KT Þ 8a2 ðt ÞS 6b2 ðt ÞS 2d2 ðt Þ > > > 0 > > > > > > > 0 > kðt tÞ 0 > > e dt cos2j > > > > > > > Zt > >

> 0 > 0 0 2 0 0 0 0 2 kðt tÞ 0 > c0 ðt Þþb0 ðt ÞS e dt s0 ¼KT ½c0 ðtÞþb0 ðtÞS þKð1KT Þ > > > 0 > = > > KT ½6b2 ðtÞS4 24c2 ðtÞq2 2d2 ðtÞþKð1KT Þ > > > > > > > > Zt > > 0 > 0 4 0 2 0 kðt tÞ 0 > > ½6b2 ðt ÞS 24c2 ðt Þq 2d2 ðt Þe dt cos 2j > > > 0 > > > > > > > sy ¼nðsr,c þsy,c Þ > > > > > >

> 2 4 2 0 > ¼2 KT 2a2 ðtÞS 3b2 ðtÞS þ6c2 ðtÞq þd2 ðtÞq > > > > > > Zt > >

0 > 0 2 0 4 0 2 0 0 kðt tÞ 0 > ; þKð1KT Þ 2a2 ðt ÞS 3b2 ðt ÞS þ6c2 ðt Þq þd2 ðt Þq e dt sin 2j sr ¼KT ½c00 ðtÞb00 ðtÞS2 þKð1KT Þ

Z

t

0 c00 ðt0 Þb00 ðt0 ÞS 2 ekðt tÞ dt0

0

ð5:36Þ For displacement components, Ur¼

r ð1 2n0 Þc00 ðtÞ þ b00 ðtÞS 2 þ ½8ð1 n0 Þa2 ðtÞS 2 2b2 ðtÞS 4 2G þ8n0 c2 ðtÞq2 þ d2 ðtÞq cos2j

9 > > > > > > > > > > > > =

r > 4ð1 2n0 Þa2 ðtÞS2 þ 2b2 ðtÞS 4 þ 4ð3 2n0 Þc2 ðtÞq2 þ 2d2 ðtÞq0 sin2j > > > 2G > > > > > > 0

> n n 0 > 2 2 ; y 2c0 ðtÞ 8a2 ðtÞS þ 24c2 ðtÞq cos2j Vy ¼ 2Gn ð5:37Þ

Wy ¼

107

Stability Analysis of Rheologic Rock Mass For strain components, @U r @r 1 ð1 2n0 Þc00 ðtÞ b00 ðtÞS 2 8ð1 n0 Þa2 ðtÞS2 6b2 ðtÞS 4 ¼ 2G

24n0 c2 ðtÞq2 2d2 ðtÞq0 cos 2j

er ¼

1 @Wt Ur þ r @j r 1 ð1 2n0 Þc00 ðtÞ þ b00 ðtÞS 2 þ 8n0 a2 ðtÞS2 6b2 ðtÞS 4 ¼ 2G

24ð1 n0 Þc2 ðtÞq2 2d2 ðtÞq0 cos 2j

ey ¼

9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > =

> > > > > > > > e ¼ er þ ey > > > > >

> 1 0 0 0 2 0 2 > 2ð1 2n Þc0 ðtÞ 8ð1 2n Þa2 ðtÞS 24ð1 2n Þc2 ðtÞq cos 2j > ¼ > > 2G > > > > > @W t W t 1 @U r > > þ 2r ¼ > > r @j @r r > > > > > 1 > 2 4 2 ; 8a2 ðtÞS 12b2 ðtÞS þ 24c2 ðtÞq þ 4d2 ðtÞ sin 2j ¼ 2G

ð5:38Þ

The above equations indicate that the undetermined coeﬃcients of c00 ðtÞ, b00 ðtÞ only exist in the terms with n ¼ 0 (without trigonometric functional terms), whereas those of a2(t), b2(t), c2(t), d2(t) only exist in the terms with n ¼ 2. Therefore, these undetermined coeﬃcients can be divided into two groups with respect to boundary equations, i.e., a group of n ¼ 0 and a group of n ¼ 2, which can be conveniently solved. The equations for solving these coeﬃcients for the following boundary conditions are given below: (a)

When r ¼ a, sr,c ¼ sr þ sr,0 ¼ 0 ¼ þ0 ¼ 0

(b)

sr,c ðn ¼ 0Þ ¼ 0 sr,c ðn ¼ 2Þ ¼ 0

ðn ¼ 2Þ

When r! 1, sr,c ¼ sr þ sr,0 ¼ sr,0 c ¼ þ0

ðn ¼ 2Þ

sr,c ðn ¼ 0Þ ¼ sr,0 sr,c ðn ¼ 2Þ ¼ sr,0

ðn ¼ 0Þ ðn ¼ 2Þ

ð5:39Þ

108

Chapter 5

Hence, six simultaneous equations can be given according to equation (5.36) and the above boundary conditions: 9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > ðsr,c , n ¼ 2Þ > > > Zt > > > 0 0 0 > > KT ½8a2 ðtÞ 6b2 ðtÞ 2d2 ðtÞ Kð1 KT Þ ½8a2 ðt Þ 6b2 ðt Þ 2d2 ðt Þ > > > 0 > > > > 1 0 > kðt tÞ 0 > e dt þ pz ð1 l1 Þ ¼ 0 > > 2 > > > > > ð c ,n ¼ 2Þ > > > Zt > > 1 > 0 0 kðt0 tÞ 0 > dt pz ð1 l1 Þ ¼ 0 > 2KT ½2a2 ðtÞ 3b2 ðtÞ 2Kð1 KT Þ ½2a2 ðt Þ 3b2 ðt Þe = 2

when r ¼ R0 ¼ a, R0 a r a q¼ ¼ !0 S¼ ¼ ¼1 R1 R1 ! 1 r a ðsr,c ,n ¼ 0Þ Zt 0

0 0

0 1 0 KT c0 ðtÞ b0 ðtÞ þ Kð1 KT Þ c0 ðt Þ b00 ðt0 Þ ekðt tÞ dt0 pz ð1 þ l1 Þ ¼ 0 2 0

0

when r ¼ R1 ! 1 R0 a !0 S¼ ¼ r R1 ! 1 ðsr,c ,n ¼ 0Þ KTc00 ðtÞ þ Kð1 KT Þ

q¼ Z

t 0

ðsr,c , n ¼ 2Þ

r R1 ¼ ¼1 R1 R1

1 1 0 c00 ðt0 Þekðt tÞ dt0 pz ð1 þ l1 Þ ¼ pz ðHl1 Þ 2 2

Z

t

2KTd2 ðtÞ þ 2Kð1 KT Þ 0

1 1 0 d2 ðt0 Þekðt tÞ dt0 þ pz ð1 l1 Þ ¼ pz ð1 l1 Þ 2 2

ð c ,n ¼ 2Þ

Z

2KT ½6c2 ðtÞ þ d2 ðtÞ 2Kð1 KT Þ 0

1 ¼ pz ð1 l1 Þ 2

t

1 0 ½6c2 ðt0 Þ þ d2 ðt0 Þekðt tÞ dt0 pz ð1 l1 Þ 2

> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > ; ð5:40Þ

The coeﬃcients in the form of a function can be completely determined from these six simultaneous equations. The explicit formulae of the coeﬃcients can be obtained by converting the above integral equations into diﬀerential equations in accordance with the initial conditions. Then each component of the stress state is divided into

Stability Analysis of Rheologic Rock Mass

109

two parts: static and rheologic. The latter is related to deformation rate. The two expressions for each stress component are shown below, the superscripts of s and r stand for static and rheologic respectively: for radial stress, sr,c ¼ ssr,c þsrr,c

9 > > > > > > > =

2 2 1 a a4 t=T a t=T 1ð1e Þ 2 þ pz ð1l1 Þ 1ð1e Þ 4 2 3 4 cos2j r r r 2 2 > > > > 2 2 4 > > 1 a 1 a a r t=T t=T > ; sr,c ¼ pz ð1þl1 Þe p cos2j ð1l Þe 4 3 z 1 2 2 4 2 r 2 r r ð5:41Þ

1 ssr,c ¼ pz ð1þl1 Þ

for tangential stress, 9 1 a2 1 a4 > ssy,c ¼ pz ð1 þ l1 Þ 1 þ ð1 et=T Þ 2 pz ð1 l1 Þ 1 þ 3ð1 et=T Þ 4 cos 2j > > = 2 2 r r 1 a2 3 a4 sry,c ¼ pz ð1 þ l1 Þet=T 2 pz ð1 l1 Þet=T 4 cos 2j 2 2 r r

> > > ; ð5:42Þ

for shearing stress 9 2 1 a a4 > t=T > ¼ pz ð1 l1 Þ 1 þ ð1 e Þ 2 2 3 4 sin 2j > > = 2 r r 2 > > 1 a a4 > r t=T > 2 2 3 4 sin 2j c ¼ pz ð1 l1 Þe ; 2 r r sc

ð5:43Þ

The displacement components after excavation surrounding the opening are: 9 Pz r KT 1 t=T a2 Pz r > > ð1 þ l1 Þ 1 e ð1 l þ Þ > 1 > > 4G KT r2 4G > > > > > = 2 4 KT 1 t=T a a 0 e 1 4ð1 n Þ 2 4 cos 2j > KT r r > > > > > > 2 4 > Pz r KT 1 t=T a a > 0 ; ð1 l1 Þ 1 e W#¼ 2ð1 2n Þ 2 þ 4 sin 2j > 4G KT r r Ur ¼

ð5:44Þ

110

Chapter 5

Figure 5.4. Variation of stress components in surrounding rock mass with time at roof (j = 90 ) and sidewall (j = 0 ).

Figure 5.5. Variation of radial displacement of the opening periphery with time at roof and sidewall.

Stability Analysis of Rheologic Rock Mass

111

Figure 5.4 plots the variations of the tangential stress with time at two points of the circular opening (j ¼ 0 and 90 ), given l1 ¼ 0.5. The curves for the static and the dynamic components are plotted separately. Figure 5.5 shows the variation of radial displacement with time at the same points under the same condition as in Figure 5.4, given KT ¼ 1.25.

5.3.

INTERACTION BETWEEN THE VISCO-ELASTIC–PLASTIC SURROUNDING ROCK AND LINING

This section studies the interaction between the visco-elastic rock mass within plastic zones and lining.

5.3.1

Stress state in plastic zones of rock mass

The occurrence and conﬁguration of plastic zones in the surrounding rock mass are often non-symmetrical and they depend on the rock mass characteristics and in situ stress. However, in order to give an explicit equation using analytical method, some assumptions are made to simplify the solutions. As a typical condition, in situ stress ﬁeld is isometric, and the surrounding rock mass is homogeneous, isotropic, viscoelastic and continuous. Such a problem can be treated as a symmetrical one. If the plastic zone is assumed incompressible, then the expression for strain state, when using polar coordinates, becomes very simple: ey þ er ¼ 0 du er ¼ dr

9 =

ð5:45Þ

u ey ¼ ; r

Rearrangement and integration of the above two equations yield: u¼

AðtÞ r

ey ¼

AðtÞ r2

er ¼

AðtÞ r2

ð5:46Þ

Given that the physical equations of the medium within the plastic zone are sy s ¼ 2Mey sr s ¼ 2Mer where s is the average stress.

) ð5:47Þ

112

Chapter 5

Based upon experimental data available and Mogi’s criterion [128], the plastic condition of surrounding rock mass can be expressed as OCT ¼ f ðs1 þ s3 þ as2 Þ Because of symmetry, 1 sm ¼ ðs1 þ s3 þ s2 Þ ¼ P ¼ const 3 then s1 þ s3 þ as2 ¼ const ðs1 s2 Þ2 þ ðs2 s3 Þ2 þ ðs3 s1 Þ2 ¼ Kp2 ¼ const

ð5:48Þ

Also for a symmetrical problem s1 ¼ sy

s3 ¼ sr

s2 ¼ sy

By substituting equation (5.47) into equation (5.48), we have Kp M ¼ pﬃﬃﬃ 2 6e y

ð5:49Þ

By substituting equation (5.49) into equation (5.47), it gives 9 Kp > sy ¼ s pﬃﬃﬃ > > 6= Kp > > sr ¼ s þ pﬃﬃﬃ > ; 6

ð5:50Þ

2Kp sy sr ¼ pﬃﬃﬃ 6

ð5:51Þ

Therefore,

The equilibrium equation of the plastic zone is r

dsr þ sr sy ¼ 0 dr

ð5:52Þ

Stability Analysis of Rheologic Rock Mass

113

The load applied to the lining at the opening periphery (r ¼ a) is sð1Þ r¼a ðtÞ. By substituting equation (5.51) into (5.52) and through rearrangement and integration, the stress components in this area are: 9 2Kp r ð1Þ > > p ﬃﬃ ﬃ þ s sð2Þ ðtÞ ¼ ðtÞ ln > r¼a r = a 6 > 2Kp r > ð1Þ ; p ﬃﬃ ﬃ ðtÞ ¼ ðtÞ > sð2Þ þ sr¼a 1 þ ln y a 6

5.3.2

ð5:53Þ

Interaction between surrounding rock mass and lining

The area beyond the plastic zone in the surrounding rock mass is still in visco-elastic state. The stress ﬁeld can be determined using the method stated in previous sections, i.e., the new stress ﬁeld (S ð3Þ c ) after excavation is the summation of the initial stress ﬁeld (S ð3Þ ) and the compensating stress (S ð3Þ ) ﬁeld: 0 ð3Þ ð3Þ S ð3Þ c ¼ S0 þS

where the initial stress ﬁeld is the same as in equation (5.23). The compensating stress ﬁeld can be simpliﬁed because of axial symmetry. The global components of the stress state are 9 Z th h i i > 0 0 2 0 0 0 0 2 k3 ðt0 tÞ 0 > sð3Þ þK e ðtÞ¼K T C ðtÞb ðtÞS ð1K T Þ C ðt Þb ðt ÞS dt P > 3 3 3 3 3 r,c 0,3 0,3 3 0,3 0,3 3 > > t0 > > > Z th > h i i > 0 > ð3Þ 0 0 2 0 0 0 0 2 k3 ðt tÞ 0 C 0,3 ðt Þþb0,3 ðt ÞS 3 e dt P > sy,c ðtÞ¼K3 T3 C 0,3 ðtÞþb0,3 ðtÞS 3 þK3 ð1K3 T3 Þ > > > > t0 > > Zt > = ð3Þ 0 0 0 k3 ðt0 tÞ 0 sy,c ðtÞ¼K3 T3 C 0,3 þK3 ð1K3 T3 Þ C 0,3 ðt Þe dt P > t0 > > > > > ð3Þ ðtÞ¼0 > c > > > h i > 1 > ð3Þ 0 0 2 > r3 ð12nÞC 0,3 ðtÞþb0,3 ðtÞS 3 U r,c ðtÞ¼ > > > 2G3 > > > ; ð3Þ W y,c ðtÞ¼0 ð5:54Þ where K3, T3, G3 are the mechanical parameters in the visco-elastic area of the surrounding rocks (as illustrated in Figure 5.6); S 3 ¼ R=r3 , R is the radius of the plastic zone and r3 is the radial coordinate of the point under consideration.

114

Chapter 5

Figure 5.6. Division of diﬀerent damage zones around an opening.

Figure 5.7. Synthetic rheological mechanical model for plastic zone, visco-elastic area and elastic lining, where G – bKp; b – constant related to the frictional coeﬃcient of rock block; A – visco-elastic area; B – plastic zone; C – lining.

The lining material is assumed to be a Hooke’s elastic body, its basic equation is the same as equation (5.23). The synthetic mechanical model for the plastic zone, visco-elastic area and elastic lining can be described approximately using the diagram shown in Figure 5.7. Each coeﬃcient in equation (5.54) can be determined using the following boundary and physical conditions. (a)

At an inﬁnite point, i.e., r3 ! 1 sð3Þ r,c ¼ P

(b)

On the inner boundary of the lining, i.e., r1 ¼ a0, sð1Þ r¼a0 ¼ 0

Stability Analysis of Rheologic Rock Mass (c)

Along the interface between the lining and the rock mass, i.e., r1 ¼ r2 ¼ a, and provided that the lining is applied with the moment of t ¼ t1, then ð2Þ ð2Þ U ð1Þ r ¼ U r U rðt¼t1 Þ

(d)

115

ð5:55Þ

Along the interface of visco-elastic and plastic zones, i.e., r2 ¼ r3 ¼ R,

For plastic condition 2 ð3Þ sð3Þ y sr ¼ pﬃﬃﬃ Kp 6 For condition of continuous displacements ð3Þ U ð2Þ r ¼ Ur

Based upon the above boundary conditions, the corresponding equations can be given. By assuming that the time for applying lining is t ¼ t1, the excavating time is t0 ¼ 0 and solving integral and algebraic equations in connection with equations (5.54), (5.23), (5.45) and (5.55), functions of C 00,3 ðtÞ, C 00,1 ðtÞ, b00,1 ðtÞ, b00,3 ðtÞ and A(t) can be determined. They ﬁnally give the stress components: For the lining 9 Kp ðdK3 T3 1Þ t1 =T3 GR2 t=T3 2 >

pﬃﬃﬃ ¼ 2 ðe e Þð1 S 1 Þ > > > > a ð1 2nÞ þ X02 6K 3 T 3 > > = 2 Kp ðdK3 T3 1Þ t1 =T3 GR ð1Þ t=T 2 3

p ﬃﬃ ﬃ sy ðtÞ ¼ 2 ðe e Þð1 þ S Þ 1 2 > a ð1 2nÞ þ X0 6K 3 T 3 > > > 2 > Kp ðdK3 T3 1Þ t1 =T3 2n1 GR > t=T ð1Þ > 3

; p ﬃﬃ ﬃ ðe e Þ sy ðtÞ ¼ 2 2 a ð1 2nÞ þ X0 6K 3 T 3

sð1Þ r ðtÞ

ð5:56Þ

where G ¼ G1/G3 is the stress state in the visco-elastic zone, when r 5 R, 9 2 Zt > KTKp K 1 R > p t=T3 > p ﬃﬃ ﬃ p ﬃﬃ ﬃ > ð1 KT Þe Kð1 KT Þ ¼ 1 þ > ð3Þ 2 KT r 6 6> sy,c ðtÞ > 0 = 2 1 R 0 0 > ð1 KT Þet =T3 2 eKðt tÞ dt0 P 1þ > > > KT r > > > ; ð3Þ s ðtÞ ¼ P sð3Þ r,c ðtÞ

y,c

)

ð5:57Þ

116

Chapter 5

For the plastic zone 9 2 2 > GR 1 X 2K K ðK T 1Þ r p p 3 3 > 0 >

pﬃﬃﬃ sð2Þ ðet1 =T3 et=T3 Þ > r ðtÞ ¼ pﬃﬃﬃ ln 2 = 2 a ð1 2nÞ þ X a 6 6K3 T3 0 > GR2 1 X02 2Kp K ðK T 1Þ t1 =T3 r >

ppﬃﬃ3ﬃ 3 > ðe et=T3 Þ > sð2Þ ; y ðtÞ ¼ pﬃﬃﬃ ð1 þ ln Þ 2 2 a a ð1 2nÞ þ X0 6K3 T3 6 ð5:58Þ ð2Þ Given sð3Þ r,c ðtÞ ¼ sr,c ðtÞ, then the boundary, R, between the visco-elastic and plastic zones can be obtained from equations (5.53) and (5.57),

pﬃﬃﬃ 6 ð3Þ ð2Þ R ¼ a exp s ðtÞ sr,c ðtÞ 2Kp r,c

ð5:59Þ

It can be seen from the above equations that the loading on the lining is a function of (a) the plastic zone size (R) at the time when the lining is applied (t1), (b) the ratio of the shearing moduli of the lining and the rock mass (G), and, (c) other related physical, mechanical and geometrical parameters of the lining and the surrounding rock mass.

5.4.

STRESS STATE IN VISCO-ELASTIC–VISCO-PLASTIC SURROUNDING ROCK MASSES

The preceding section discussed the stress states in the surrounding rock mass and the lining when there exist plastic zones. The plastic zone is time-dependent, in other words, the plastic zone is visco-plastic, whereas the rock masses at further distance still behaves visco-elastically [320–323]. This section discusses the interaction between the surrounding rock mass and the lining under such condition. Figure 5.8 shows the mechanical model of the whole system. The mechanical equations for

Figure 5.8. Mechanical model for the surrounding rock and the lining.

117

Stability Analysis of Rheologic Rock Mass

the visco-elastic rock mass and the lining zone are the same as that given in the preceding section; and the stress in the visco-plastic zone consists of two parts, i.e., 0

sð2Þ ¼ sð2Þ þ sð2Þ

00

ð5:60Þ

0

where the ﬁrst part, sð2Þ is the stress component of the Maxwell medium, and the 00 second part, sð2Þ is the stress component of the St. Venant medium. The physical equation for the Maxwell medium is

e_ y e_ ¼

ð2Þ sð2Þ s y sy

9 _ ð2Þ s_ > s_ ð2Þ > y s y > > > þ = 2G

2Z2 2 ð2Þ _ ð2Þ s_ ð2Þ s s_ sr sð2Þ r r s r e_ r e_ ¼ þ 2Z2 2G2

> > > > > ;

ð5:61Þ

in which s is the average stress, s_ is the average stress rate, e_ is the average strain rate. The general solutions to the diﬀerential equation of (5.61) can be obtained. The assumptions similar to the last section are made and the visco-plastic zone is approximately regarded as an incompressible medium. By applying the solutions of equations (5.45) and (5.46), equation (5.61) becomes: u¼

AðtÞ r

ey ¼

AðtÞ r2

er ¼

AðtÞ r2

ð5:62aÞ

u_ ¼

A_ ðtÞ r

e_ y ¼

A_ ðtÞ r2

e_ r ¼

A_ ðtÞ r2

ð5:62bÞ

hence

From equation (5.61), e_ y e_ r ¼

0 1 ð2Þ0 1 ð2Þ0 ð2Þ0 _ _ sy sð2Þ s þ s r r 2Z2 2G2 y

ð5:63aÞ

Substituting (5.62a) into (5.63a), then 1 @ðsy sr Þ sy sr 2 @AðtÞ þ ¼ 2 2G2 @t r @t 2Z2

ð5:63bÞ

118

Chapter 5

Examining the visco-elastic zone and its boundary with the visco-plastic zone, the condition of continuous displacements in the boundary leads to ð3Þ U ð2Þ r¼R ¼ U r¼R

ð5:64Þ

and the plastic condition (see equation (5.51)) leads to ð3Þ sð3Þ y sr ¼ dKp

ð5:65Þ

A solution can be obtained by using equations (5.16), (5.62), (5.63) and (5.65) for stress and displacement in the visco-elastic zone, i.e.,

AðtÞ ¼

R2 dKp 4G3

1

1 et=T3 1 K3 T3

ð5:66Þ

By substituting equation (5.66) into equation (5.63a) and solving the corresponding diﬀerential equation, we have sy sr ¼

M t=T3 e þ c3 eðG2 =Z2 Þt r2

ð5:67Þ

where

M¼

R2 G2 dKp 1 1 1 K3 T3 G3 T3 ðG2 =Z2 Þ ð1=T3 Þ

ð5:68Þ

In equation (5.67), c3 is an undetermined constant. In consideration of t ¼ 0, the stress state in the surrounding rock mass after excavation starts transiting from elastic state to visco-plastic state, i.e.,

sy srjt ¼ 0 ¼ 2P

a2 r2

ð5:69aÞ

In substitution of this equation into equation (5.67), we have c3 ¼ 1=r2 ð2Pa2 þ MÞ. Again, upon substitution of the so-obtained c3 into equation (5.67), we have sy sr ¼

1 Mðet=T3 et=T2 Þ 2Pa2 et=T2 r2

ð5:69bÞ

Stability Analysis of Rheologic Rock Mass

119

It can be obtained according to the equilibrium condition of the medium, sy sr ¼ r

@sr @r

ð5:69cÞ

Solving equations (5.69b) and (5.69c), by (5.69c) separating variables and integrating, we have the solution in the form of 9 d > > sr ¼ BðtÞ þ c 2 TðtÞ > = r > d > sy ¼ BðtÞ þ c þ 2 TðtÞ > ; r

ð5:70Þ

Subtraction of these two equations gives sy sr ¼

2dTðtÞ r2

By substituting this equation into equation (5.69), we obtain TðtÞ ¼

1 t=T3 M e et=T2 2Pa2 et=T2 2d

ð5:71Þ

According to the condition of continuous displacements at the boundary between the lining and the rock mass, we have ð2Þ ð2Þ U ð1Þ r¼a ðtÞ ¼ U r¼a ðtÞ U r¼a ðt1 Þ

ð5:72Þ

where t1 is the time of applying the lining. Making use of equations (5.62), (5.66) and (5.64) and substituting them into equation (5.72), we have sð1Þ r¼a ðtÞ ¼

t=T3 1 X02 dKp G2 R2 1 2 et1 =T3 e 1 2 K3 T3 2 G3 a ð1 2n1 þ X0 Þ

In the case of unlined opening, i.e., t t1, sð2Þ r¼a ðtÞ BðtÞ ¼

d ¼ bðtÞ þ c 2 TðtÞ ¼ 0 a

1 c t=T3 et=T2 2Pa2 et=T2 M e 2 2a 2d

ð5:73Þ

120

Chapter 5

Substituting the expressions for B(t) and T(t) gives sð2Þ r ðtÞ

)

sð2Þ y ðtÞ

1 c t=T3 2 M e et=T2 2Pa2 et=T2 ¼ 2 2a 2r

ðt t1 Þ

ð5:74Þ

while in the case of lined opening, i.e., t t1, sð2Þ r¼a ðtÞ

d ¼ bðtÞ þ c 2 TðtÞ ¼ sð1Þ r¼a ðtÞ a

Substitution of the results from equations (5.73) and (5.71) into the above equation gives t=T3 1 X02 dKp G2 R2 1 BðtÞ ¼ e 1 et1 =T3 K3 T3 2 G3 a2 ð1 2n1 þ X02 Þ

1 c t=T3 þ M e et=T2 2Pa2 et=T2 2a2 2d Substituting this equation and equation (5.71) into equation (5.70) gives sð2Þ r ðtÞ sð2Þ y ðtÞ

)

t=T3 1 X02 dKp G2 R2 1 ¼ e 1 et1 =T3 K3 T3 2 G3 a2 ð1 2n1 þ X02 Þ

1 1 t=T3 þ M e et=T2 2Pa2 et=T2 for t t1 2 2 2a 2r

ð5:75Þ

According to the plastic condition (5.65) and the boundary condition, sð3Þ r ! P for r ! 1, we have at the boundary of r2 ¼ r3 ¼ R 9 dKp > = 2 dKp > ; sð3Þ y ð0Þ ¼ P 2

sð3Þ r ð0Þ ¼ P þ

Let the stress sðtÞ r remain continuous on the boundary of r ¼ R, then when t t1,

dKp 1 1 t=T3 et=T2 2Pa2 et=T2 ¼ P þ M e 2 2 2a 2R 2

ð5:76Þ

Stability Analysis of Rheologic Rock Mass

121

and when t t1,

dKp G2 R2 1 X02 1 1 t=T3 t=T2 2 t=T2 e M e 2Pa e þ 2a2 2R2 2 G3 a2 ð1 2n1 þ X02 Þ t=T3 dKp 1 1 e : ð5:77Þ et1 =T3 ¼ P þ K3 T3 2

Equations (5.76) and (5.77) describe the change of the radius (R) of the viscoplastic zone, with time elapsing in the form of an explicit function. And equations (5.74), (5.75), (5.76) and (5.77) comprise the complete solution to the stress state of the visco-plastic zone. For example, given that d ¼ 0.8, Kp ¼ 1.5 P, a ¼ 2 m, lining thickness ¼ 40 cm, n ¼ 0.25, T2 ¼ T3 ¼ 12 days, K2 ¼ K3 ¼ 1/8 days and G2/G3 ¼ 0.5, then for t ¼ 0, R ¼ 2.31 m and for t ! 1, R ¼ 2.623 m (as shown in Figure 5.9). It is easy to prove that the boundary conditions are met for t ¼ 0 and t ! 1 under limit condition. Given t ! 1 in equation (5.75), we have the stress state in the viscoplastic zone in inﬁnite time after applying lining: 9 sð2Þ r ðtÞ =

1 X02 dKp G2 R2 1 et1 =T3 2 1 ¼ 2 ; ð2Þ K 2 G a T þ X 1 2n 3 3 3 1 0 s ðtÞ

ð5:78Þ

y

Figure 5.9. Stress distribution in various zones in the surrounding rock mass under limit condition.

122

Chapter 5

It can be seen from equation (5.78) that the rock mass in the visco-plastic zone is unable to bear long-term shearing loads because its physical property is close to ﬂuids. Finally, the two main stress components in the whole region tend to balance and the ﬁnal loading on the lining is the same as that given in the above equation, i.e., ð2Þ sð1Þ r¼a ðt ! 1Þ ¼ sr ðt ! 1Þ

It can be seen from above that for this type of visco-elastic and visco-plastic rock mass, the ﬁnal loading on the lining is related not only to the physical and geometrical parameters of the visco-elastic zone in the rock mass and the lining but also to the time when the lining is applied (t1) and the magnitude of the radius of the visco-plastic zone. It is diﬀerent from the previously stated viscoelastic-plastic media. In the present case the ﬁnal loading on the lining is related to the ratio of the shearing deformation moduli G2/G3 of the visco-plastic and visco-elastic zones, but not related to the shearing modulus G1 of the lining itself. Additionally, it also can be seen that to reach a mechanical equilibrium state, the visco-plastic zone extends beyond the lining into the rock mass with the radius reaching R.

5.5.

RHEOLOGICAL ANALYSIS WITH DILATION AND SOFTENING OF THE ROCK MASS

The previous sections assume that the plastic medium is incompressible. However, in most cases there exists a broken-swelling eﬀect after plastic zone produced by excavation. The purpose of this section is to study the stress state accompanied by dilation eﬀect and meanwhile to consider the softening phenomenon and its eﬀect in the post-failure region in the rock mass [324–329]. For this reason, a more complete rheological mechanical model is proposed.

5.5.1

Mechanical model of surrounding rock mass

The assumption of continuum mechanics is still adopted. The media involved is considered to be isotropic, and the initial stress ﬁeld is assumed to be uniform, so axisymmetrical analysis is allowable. Assume that the rock mass in the far region after excavation is in elastic state while the rock mass in the near region is in the visco-plastic state in which certain residual strength still exists. The lining is assumed to be an elastic medium, as shown in Figure 5.10.

Stability Analysis of Rheologic Rock Mass

123

Figure 5.10. Schematic division of zones.

On the boundary between visco-elastic and visco-plastic zones, the stress state is assumed to meet the Mohr–Coulumb criterion, i.e., sy sr ¼ ðsy þ sr Þ sin j þ 2c cos j

ð5:79Þ

The stress components in the visco-elastic zone are the same as in equation (5.54). It is known from equation (5.54) that when r ! 1, Ur(t) ¼ 0; then c0(t) ¼ 0 and the following equation is valid: 1 ðsy þ sr Þ ¼ P 2 By substitution of this equation into Mohr–Coulumb criterion, gives sy sr ¼ 2P sin j þ 2c cos j ¼ Kp

ð5:80Þ

Supposing the minimum strength in the residual strength region is d times (d < 1) Kp, i.e., equal to dKp, then dKp ¼ 2dðP sin j c cos jÞ

ð5:81Þ

By substituting equation (5.80) into equation (5.54), we can obtain the undetermined functional coeﬃcient and then obtain the radial and tangential stresses 9 R22 > = ðP sin j c cos jÞ r2 2 R > sy ¼ P 22 ðP sin j c cos jÞ ; r sr ¼ P þ

ð5:82Þ

124

Chapter 5

and the radial displacement R22 1 t=T3 ðP sin j c cos jÞ 1 e 1 ur ¼ K3 T3 4Gr

ð5:83Þ

5.5.2 Visco-plastic model considering dilation and softening 5.5.2.1 Physical model. The rock mass in the post-failure region displays viscoplastic softening behaviour when the peak strength is reached and the dilation phenomenon takes place simultaneously [325–327]. For this reason, the diﬀerential equation for the Maxwell medium that describes the visco-plastic behaviour is adopted and applied to describe the softening and dilation characteristics, it is expressed as follows: ðsy sr Þ þ

Z 2nG ðs_ y s_ r Þ ¼ 2Zð_ey e_ r Þ þ ðey þ er Þ 2Gðey er Þ þ D G 1 2n

ð5:84Þ

This equation not only reﬂects the softening and relaxation characteristics with visco-plastic property of the rock mass (the third term on the right-hand side of the equation), but also guarantees the stress diﬀerence in the rock mass.

5.5.2.2 Geometric equation. Assuming that the problem under consideration meets the small deformation concept, we have, for an axisymmetrical problem, er ¼

@u @r

ey ¼

u r

If the dilation phenomenon takes place in the visco-plastic zone and the radial strain is l/r times of the tangential strain, i.e., dilation occurs near to the opening periphery, then l er ¼ ey r @u l u u ¼ ¼ l 2 @r r r r Integrate u with respect to r1, then 9 u ¼ AðtÞel=r > > > > = l l=r er ¼ 2 e AðtÞ r > > > 1 l=r > ey ¼ e AðtÞ ; r

ð5:85Þ

Stability Analysis of Rheologic Rock Mass

125

and strain rate 9 l l=r _ > = A ðtÞ e r2 > 1 e_ y ¼ el=r A_ ðtÞ ; r e_ r ¼

5.5.2.3 Equilibrium equation. r

5.5.3

ð5:86Þ

For an axisymmetrical problem, we have

9 @sr > > ¼ sy sr = @r Z 1 > ; sr ¼ ðsy sr Þdr þ f ðtÞ > r

ð5:87Þ

Stress components in each zone

5.5.3.1 Visco-plastic zone. By substitution of equations (5.85) and (5.86) into equation (5.84) and further interpretation, we have s_ y s_ r þ

G ðsy sr Þ ¼ A A_ ðtÞ B AðtÞ þ C Z

where 9 1 l > > 1 þ el=r A ¼ 2G > > r r > > > = 2 2G 1 2n l l l=r 1 e B ¼ þ 1þ 1 2n r r > Z r > > > > > G > ; C¼ D Z

ð5:88Þ

Equation (5.88) is the inhomogeneous partial diﬀerential equation of ðsy sr Þ, with the following solution Z G Z AðtÞeðG=ZÞt dt þ A eðG=ZÞt AðtÞ þ C eðG=ZÞt þ D ðrÞ sy sr ¼ eðG=ZÞt A þ B Z G ð5:89Þ

126

Chapter 5

By substituting equation (5.89) into equilibrium equation (5.87) and integrating it with respect to r, we have 4G2 4n 2 3n 1 l=r sr ¼ þ e e ðG=ZÞt l r Zð1 2nÞ Z Z 2 1 l=r Z 1 ðG=ZÞt AðtÞe dt 2G þ e AðtÞ þ C ln r þ DðrÞ dr þ f ðtÞ l r G r

ð5:90Þ

It is easy to determine D ðrÞ in equation (5.90), if analysing the transient response. The equation of D ðrÞ ¼ D is given by Zhu et al. (1988). Considering this result in equation (5.90), we obtain Z t G 0 sy sr ¼ eðG=ZÞt A þ B Aðt0 ÞeðG=ZÞt dt0 þ A eðG=ZÞt AðtÞ þ D eðG=ZÞt 1 Z 0 ð5:91Þ Substituting equation (5.91) into equilibrium equation (5.87) and integrating the result with respect to r gives Zt 4G2 4n 2 3n 1 l=r ðG=ZÞt 2 1 0 þ e þe sr ¼ eðG=ZÞt Aðt0 ÞeðG=ZÞt dt0 AðtÞ 2G þ el=r l r l r Zð1 2nÞ 0 ðG=ZÞt þD e 1 lnr þ f ðtÞ ð5:92Þ

Applying the condition sr,nt ¼ sr,np on the boundary between R2 visco-elastic and visco-plastic zones and making equality between equations (5.82) and (5.92) results in ðG=ZÞt

4G2 4n 2 3n 1 l=R2 þ e l R2 Zð1 2nÞ

f ðtÞ ¼ p þ ð p sin j c cos jÞ þ e Zt 2 1 l=R2 ðG=ZÞt 0 ðG=ZÞt0 0 ðG=ZÞt e Aðt Þe dt 2G þ e AðtÞ Dðe 1Þ ln R2 l R2 0 ð5:93Þ By substituting equation (5.93) into equation (5.92), we can derive the explicit formula of sr. Applying equations (5.91), (5.92) and (5.93) gives Z t G 0 Aðt0 ÞeðG=ZÞt dt0 þ A eðG=ZÞt AðtÞ þ D eðG=ZÞt 1 þ sr sy ¼ eðG=ZÞt A þ B Z 0 ð5:94Þ

Stability Analysis of Rheologic Rock Mass

127

5.5.3.2 Residual strength zone. Because the residual strength is related to the conﬁning pressure, the strength is higher at greater depth [328–334]. Owing to the small size of this zone, it can be assumed that the residual strength and r have a linear relationship. By examining equations (5.80) and (5.81), the physical equation to describe this zone can be approximately expressed as sy sr ¼ c0 r þ dKp

ð5:95Þ

Substituting equation (5.95) into equation (5.87) and integrating the result, we have sr ¼ c0 r þ dKp ln r þ jðtÞ

9 =

sy ¼ 2c0 r þ ð1 þ ln rÞdKp þ jðtÞ ;

ð5:96Þ

In the case of unlined opening, r ¼ a, sr ¼ 0, then, from equation (5.96), 1 c0 ¼ ½dKp ln r þ jðtÞ a

ð5:97Þ

The term of j(t) in the above equation can be determined from the condition that the radial stresses sr along the boundary R1 between the visco-plastic zone and the residual strength zone are equal, i.e., the equation (5.92) is equal to equation (5.96). The stress components in the visco-elastic zone are expressed by equations (5.82) and (5.83) respectively.

5.5.4

Stress state without lining

5.5.4.1 Visco-plastic zone, R1 < r < R2. In the previous equations, there is a function A(t) that is undetermined. It can be derived from the condition of continuous displacements on the boundary between the visco-elastic and the viscoplastic zones (R2). By letting unp ¼ une, then we have its expression from equations (5.83) and (5.89),

AðtÞ ¼

R2 l=R2 1 e et=T3 1 ðP sin j C cos jÞ 1 K3 T3 4G

ð5:98Þ

128

Chapter 5

By substituting equation (5.98) into equations (5.92) and (5.94) and integrating the result, we have the expression of the stress state in the visco-plastic zone, GR2 1ð1=K3 T3 Þ l=R2 sr ¼ PþðpsinjccosjÞþe ðpsinjccosjÞ e G=ðZ1=T3 Þ Zð12nÞ Z 4n2 3n1 l=R2 4n2 3n1 l=r eððG=ZÞð1=T3 ÞÞt 1 ðeðG=ZÞt 1Þ e þ þ e G l R2 l r R2 1 2 1 l=R2 2 1 eðG=ZÞt el=R2 ðpsinjccosjÞ 1 þ þ el=r et=T3 1 e K3 T3 l R2 l r 2 DðeðG=ZÞt 1ÞðlnR2 lnrÞ ð5:99Þ ðG=ZÞt

G R2 l=R2 e sy ¼ e A þB ð p sin j c cos jÞ Z 4G Z ðG=ZÞt 1 ð1=K3 T3 Þ ððG=ZÞð1=T3 ÞÞt e e 1 1 ðG=ZÞ ð1=T3 Þ G ð5:100Þ 1 ðG=ZÞt R2 l=R2 t=T3 e e þA e ð p sin j c cos jÞ 1 1 K3 T3 4G þD eðg=ZÞt 1 þ sr ðG=ZÞt

5.5.4.2 Residual strength zone, a < r < R1. As stated previously, the explicit formula j(t) can be derived from equations (5.96), (5.97) and (5.99) by applying the condition of equal sr on the boundary between two adjacent zones (R1):

GR2 el=R2 ðpsin j ccosjÞ jðtÞ ¼ P þ ðp sin j c cos jÞ e Zð1 2nÞ Z ðG=ZÞt 1 ð1=K3 T3 Þ ððG=ZÞð1=T3 ÞÞt e 1 e 1 ðG=ZÞ ð1=T3 Þ G 4n 2 3n 1 l=R2 4n 2 3n 1 l=R1 1 þ þ e e R2 l R2 l R1 2 1 2 1 l=R2 þ eðt=T3 Þ 1 e eðG=ZÞt eðl=R2 Þ ðp sin j ccosjÞ 1 K3 T3 l R2 2 1 l=R1 R1 R1 1 DðeG=Zt 1Þðln R2 ln R1 Þ dKp ðlnR1 lnaÞ þ e l R1 a a ðG=ZÞt

ð5:101Þ

Stability Analysis of Rheologic Rock Mass

129

The stress components in this zone are,

9 r r > sr ¼ d Kp ln r ln a þ jðtÞ 1 = a a 2r 2r > ; sy ¼ 1 þ ln r ln a dKp þ jðtÞ 1 a a

ð5:102Þ

5.5.4.3 Determination of boundary R2. Since the rates of displacement components on the boundary between the visco-elastic and visco-plastic zones are equal, we can determine the value of R2 by letting _ ð3Þ u_ ð2Þ r ¼ u r , and derive the corresponding displacement components in equations (5.83) and (5.85) with respect to t and let the results be equal, then R2 ¼ l

ð5:103Þ

Although this equation is extremely simple, its physical meaning is rationally clear. The assumption of er ¼ l=rey in the geometric equation in the previous sections, when r ¼ R2, er ¼ ey, which is the starting point where the volumetric deformation in the visco-plastic zone begins to dilate in volume. Therefore, the result of R2 ¼ l coincides with the assumption made initially. 5.5.4.4 Determination of boundary R1. Similarly, the stress diﬀerences between two sides of the boundary are equal when r ¼ R1, R1 can be determined. For the residual strength zone, we have, from equation (5.102)

r r ð5:104Þ sy sr ¼ dKp 1 ln a jðtÞ a a By comparing the stress diﬀerence of (sysr)vp in the visco-plastic zone shown in equation (5.104) to equations (5.91), (5.93) and (5.98), we have R1 R1 dKp 1 lna jðtÞ a a 4G 1 l=R1 n l l e 1 ¼ eðG=ZÞt þ 1þ Z R1 1 2n R1 R1 Z R2 l=R2 1 ð1=K3 T3 Þ ððG=ZÞð1=T3 ÞÞt e e ðpsinj ccosjÞ 1 eðG=ZÞt 1 ðG=ZÞ ð1=T3 Þ G 4 1 R2 l 1 elðR2 R1 Þ=R1 R2 ðpsinj ccosjÞ 1 et=T3 1 þ 1þ 2 R1 R1 K3 T3 ðG=ZÞt ðG=ZÞt e 1 ð5:105Þ þ De

130

Chapter 5

By substituting the values of R2 and j(t) into equation (5.105), we can obtain the value of R1 using trial and error method. In the limit case, we can derive the following equation for t ¼ 0 R1 R1 1 ln a p þ ð p sin j c cos jÞ R2 el=R2 ð p sin j c cos jÞ dKp 1 2 a a 1 2 1 l=R2 2 1 l=R1 R1 R1 þ þ ln a 1 e e dKp ln R1 K3 T3 l R2 l R1 a a R2 l 1 elðR2 R1 Þ=R1 R2 ð p sin j c cos jÞ ð5:106Þ 1þ R1 K3 T3 2R1 when t ! 1 R1 R1 R2 dKp 1 ln a p þ ð p sin j c cos jÞ þ el=R2 ð p sin j c cos jÞ a a 1 2n 4n 2 3n 1 l=R2 4n 2 3n 1 l=R1 þ þ e e l R2 l R1 R2 l=R2 e ð p sin j c cos jÞ 2 2 1 2 1 l=R1 þ þ el=R2 e þ Dðln R2 ln R1 Þ l R2 l R1 R1 R1 ln a 1 dKp ln R1 a a R2 lðR2 R1 Þ=R1 R2 n l l 1 ¼ þ 1þ e ð p sin j c cos jÞ 1 2n R1 R1 R1 R2 lðR2 R1 Þ=R1 R2 l e 1þ ð p sin j cos jÞ þ D ð5:107Þ R1 2R1

5.5.5

Stress state with lining

Assume that circular lining of elastic media is applied soon after the excavation, i.e., t ¼ 0, the stress state in this case can be expressed as [8,52]: 9 a20 > sr ðtÞ ¼ cðtÞ bðtÞ 2 > > r = > a2 > > sy ðtÞ ¼ cðtÞ þ bðtÞ 20 ; r

ð5:108Þ

131

Stability Analysis of Rheologic Rock Mass

If the inner surface of the lining is free from loading, i.e., sr ¼ 0 when r ¼ a0, then, cðtÞ ¼ bðtÞ So equation (5.108) becomes, 9 a20 > > sr ðtÞ ¼ cðtÞ 1 2 > r = a2 > > sy ðtÞ ¼ cðtÞ 1 þ 20 > ; r

ð5:109Þ

Again, sr and sy at the boundary (r ¼ a) of two adjacent zones are equal, and the values of stress diﬀerences (sy sr) are also equal at r ¼ R1. The following undetermined parameter or function can be determined when solving the simultaneous equations (5.102), (5.109), (5.91), (5.95) and (5.98), 2G 1 n1 l l c0 ¼ eðG=ZÞt el=R1 þ 1þ 1 Z R1 R1 R1 1 2n R2 l=R2 1 ð1=K3 T3 Þ ððG=ZÞð1=T3 ÞÞt Z ðG=ZÞt e ðe ðpsinj ccosjÞ 1Þ ðe 1Þ ðG=ZÞ ð1=T3 Þ G 2 R2 l 1þ þ elðR2 R1 Þ=R1 R2 R1 2R1

t=T3 ðG=ZÞt ðG=ZÞt R1 ðpsinj ccosjÞ ð1 ð1=K3 T3 ÞÞe 1 þ De ðe 1Þ dKp ð5:110Þ 9 > > > =

a2 0 ðc a þ dKp Þ 2a20 2 2 > a 3 a 1 > 0 > jðtÞ ¼ þ dK ac ln a ; p 2 2 2a0 2 2a0 2 cðtÞ ¼

ð5:111Þ

The term of f(t) in the stress component in the visco-plastic zone can be determined using the condition of equal sr at r ¼ R1 of the two adjacent zones, R2 G 4n 2 3n 1 ðpsinj ccosjÞ þ f ðtÞ ¼ c R1 þ dKp lnR1 þ jðtÞ þ elðR2 R1 Þ=R1 R2 Zð1 2nÞ l R1

Z ðG=ZÞt 1 ð1=K3 T3 Þ ððG=ZÞð1=T3 ÞÞt R2 ðG=ZÞt ðe 1 ðe 1Þe elðR2 R1 Þ=R1 R2 ðG=ZÞ ð1=T3 Þ G 2 2 1 1 1 et=T3 1 ðpsinj ccosjÞ þ ð5:112Þ l R1 K3 T3 0

132

Chapter 5

The stress components in the visco-plastic zone, residual strength zone and the lining can be obtained from the function or parameter determined using equations (5.110), (5.111) and (5.112). The locations of R1 and R2 are then determined according to the boundary condition that the stress or displacement components are equal at r ¼ R2 of two adjacent zones. For simpliﬁcation, only the stress component in the lining for r!1 is given by the equation below, 2 3 sr a a 1 R2 n l l lðR2 R1 Þ=R1 R2 1 ðpsinj ccosjÞe ¼ 20 þ 1 þ 1 2n R1 R1 r 2a20 R1 R1 sy 2 R2 l a þ D dKp þ 2 dKp ð5:113Þ elðR2 R1 Þ=R1 R2 ðpsinj ccosjÞ 1þ R1 2 2a0

5.6.

EFFECT OF BOLT REINFORCEMENT IN VISCO-ELASTIC ROCK MASS

Bolt and shotcrete support is to reinforce surrounding rock masses, and to control deformation. The technique eﬀectively controls the rock deformation, block loosening and cave-in. It reinforces the complete rock mass system by adjusting the stress distribution in the rock mass, and by mobilising the self-supporting ability of the surrounding rock mass [66,194–196,330–339]. The mechanism of bolting technique has not yet been thoroughly understood, although the technology has been well developed and advanced in recent years [180,218,340–381]. The purpose of this section is to examine the interaction between the rock mass and the bolts, to study the use of bolt reinforcement and its eﬀect on the stress state of the rock mass.

5.6.1

Stress state in different zones

Similar to the previous cases, the rock mass surrounding an opening in far region is regarded as a visco-elastic medium, whereas the rock mass in the bolted region is considered to be another visco-elastic medium having diﬀerent parameters, as shown in Figure 5.11. The assumptions on the continuum mechanics adopted in the previous sections are still used, and the initial stress state is assumed to be uniform and the surrounding rock mass is assumed to be isotropic and homogeneous. The visco-elastic stress state in the far region is the same as in equation (5.54). Assume that bolts are applied when t ¼ t1, the bolts are (Ra) in length and create a bolted circular ring. The parameters of T, K, G in the bolted region have basically

Stability Analysis of Rheologic Rock Mass

133

Figure 5.11. Schematic representation of surrounding rock mass zones.

the same value as those in the visco-elastic zone, the bolts only restrain the displacement of the bolted rock mass. According to the principle of superposition, the components of stress and displacements in the bolted region should meet the following conditions: sð2Þ r

¼

K3 T3 ½c2 ðtÞ b2 ðtÞS 22 þ K3 ð1 K3 T3 Þ

Z

t

t1

0 ½c2 ðt0 Þ b2 ðt0 ÞS 22 eK3 ðt tÞ dt0 P 1 S 22 ð5:114Þ

2 sð2Þ y ¼ K3 T3 ½c2 ðtÞ þ b2 ðtÞS 2 þ K3 ð1 K3 T3 Þ

U ð2Þ r

Z

t

t1

0 ½c2 ðt0 Þ þ b2 ðt0 ÞS 22 eK3 ðt tÞ dt0 P 1 þ S 22

ð5:115Þ 1 1 1 ¼ r½ð1 2nÞc2 ðtÞ þ b2 ðtÞS 22 þ PrS 22 1 ð5:116Þ et=T3 1 2G3 2G3 K3 T3

where S2 ¼ a/r and S3 ¼ R/r. It should also satisfy the boundary condition below. (i)

When r ! 1, because U ð3Þ r ¼ 0, we have, from equation (5.54) c3 ðtÞ ¼ 0

(ii)

ð5:117Þ

When r ¼ a, the stress condition is ð2Þ ð2Þ sð2Þ r ¼ aU þ s0 ¼ aU r¼a U r¼R þ s0

ð5:118Þ

134

Chapter 5

where a is a constant related with the length, cross-sectional area of rock bolt, and bolting density and mechanical property of the bolts, i.e.,

a¼

E1 B1 ðR aÞB2

ð5:119Þ

In the above equation, E1 is the elastic modulus of the bolts; B1 is the sum of the cross-sectional areas of all bolts in unit length of the opening; B2 is the total area of the opening face in unit length; s0 is the equivalent normal stress converted from the ð2Þ axial stress in the bolt; jU ð2Þ r¼a U r¼R j is the displacement diﬀerence in the bolted circular ring after bolts installed at t t1. This boundary condition reﬂects the action of the bolt, i.e., if relative displacements take place in the bolted region after bolting, they will be restrained by bolts. In addition, the bolt’s bearing force in the surrounding rock mass is proportional to the displacement. Substituting equation (5.116) into equation (5.118), we have sð2Þ r¼a ¼

a ½ð1 2nÞða RÞc2 ðtÞ þ að1 X2 Þb2 ðtÞ 2G3 1 þ Pa 1 ð1 X2 Þðet=T3 et1 =T3 Þ þ s0 K3 T3

ð5:120Þ

where X2 ¼ a/R Substituting r ¼ a into equation (5.114) gives

sð2Þ r¼a ¼ K3 T3 ½c2 ðtÞ b2 ðtÞ þ K3 ð1 K3 T3 Þ

Z

t

0

½c2 ðt0 Þ b2 ðt0 ÞeK3 ðt tÞ dt0 t1

From equations (5.120) and (5.121), we have Z

t

0

½c2 ðt0 Þ b2 ðt0 ÞeK3 ðt tÞ dt0

K3 T3 ½c2 ðtÞ b2 ðtÞ þ K3 ð1 K3 T3 Þ t1

a ¼ ½ð1 2nÞ ða RÞc2 ðtÞ þ að1 X2 Þb2 ðtÞ 2G3 1 þ Pa 1 ð1 X2 Þðet=T3 et1 =T3 Þ þ s0 K3 T3

ð5:121Þ

Stability Analysis of Rheologic Rock Mass

135

Multiplying the two sides of the above equation by eK3 t , deriving the result with respect to t and eliminating the term of eK3 t , we have

a a _ ð1 2nÞða RÞ K3 T3 c_2 ðtÞ þ ð1 X2 Þ þ K3 T3 Þb2 ðtÞ 2G3 2G3 a a ð1 2nÞða RÞ 1 c2 ðtÞ þ K3 að1 X2 Þ þ 1 b2 ðtÞ þ K3 2G3 2G3 aPa 1 1 t=T3 t1 =T3 ð1 X2 Þ K3 e e K3 s0 ¼ 1 K3 2G3 K3 T3 T3

(iii)

ð5:122Þ

When r ¼ R, the displacement condition is ð3Þ U ð2Þ r ¼ Ur

According to equations (5.54), (5.116) and (5.117), we have b3 ðtÞ ¼ ð1 2nÞc2 ðtÞ þ X22 b2 ðtÞ (iv)

ð5:123Þ

When r ¼ R, the stress condition is ð2Þ ð2Þ sð3Þ r¼a ¼ sr¼R X2 sr¼R

The last term in this equation reﬂects the action of the bolts. Substituting equations (5.54), (5.114) and (5.117) into the above equation and interpreting the results, we have K3 T3 ½ðX2 1Þc_2 ðtÞ þ X2 ðX2 1Þb_2 ðtÞ b_3 ðtÞ þ K3 T3 ½ðX2 1Þc2 ðtÞ þ X2 ðX2 1Þb2 ðtÞ b3 ðtÞ ¼ 0 By solving this diﬀerential equation and substituting the result into equation (5.123), we have

2Rðn 1Þ b2 ðtÞ ¼ 1 þ c2 ðtÞ þ C 1 et=T3 a where C1 is an undetermined constant.

ð5:124Þ

136

Chapter 5

Substituting equation (5.124) back into equation (5.122) gives

a 2R K3 T3 ðn 1Þ c_2 ðtÞ ð1 X2 Þða þ 4nR 3RÞ þ 2G3 a a 2R ðn 1Þ c2 ðtÞ þ K3 ð1 X2 Þ ða þ 4nR 3RÞ þ 2G3 a aPa 1 1 t=T3 1 ð1 X2 Þ K3 et=T3 K3 e ¼ 2G3 K3 T3 T3 aa 1 K3 s0 þ ð1 X2 Þ K3 C 1 et=T3 2G3 T3

then solving this equation gives a PaðK3 T3 1Þ C 1 et=T3 et=T3 3R a 4nR K3 T3 ða þ 4nR 3RÞ ð1 X2 ÞaPK3 a2 1 2K3 G3 a et1 =T3 þ 1 s0 K3 T3 N2 N2

c2 ðtÞ ¼ C 2 eðN2 =N1 Þt þ

ð5:125Þ

where N1 ¼ aað1 X2 Þða þ 4nR 3RÞ þ 4G3 K3 T3 ðn 1ÞR

) ð5:126Þ

N2 ¼ K3 aað1 X2 Þða þ 4nR 3RÞ þ 4G3 RK3 ðn 1Þ and C1, C2 are undetermined constants. For equation (5.114), when t ¼ t1, r ¼ a and r ¼ R, we have respectively c2 ðt1 Þ b2 ðt1 Þ ¼ 0

and

c2 ðt1 Þ X22 b2 ðt1 Þ ¼ 0

From this relation, we have c2 ðt1 Þ ¼ b2 ðt1 Þ ¼ 0 If we substitute this result back into equations (5.124) and (5.125), then the undetermined constants of C1 and C2 can be obtained: C 1 ¼ 0 PaðK3 T3 1Þ aPK3 a2 ð1 X2 Þ 1 C2 ¼ 1 K3 T3 ða þ 4nR 3RÞ N2 K3 T3 2K G as 3 3 0 ðN2 =N1 Þt1 e eððN2 =N1 Þð1=T3 ÞÞt þ N2

9 > > > > = > > > > ;

ð5:127Þ

Stability Analysis of Rheologic Rock Mass

137

Substituting equation (5.127) back into equations (5.124) and (5.125) gives PaðK3 T3 1Þ aPK3 a2 ð1 X2 Þ 1 et=T3 et=T3 þ 1 c2 ðtÞ ¼ C 2 eðN2 =N1 Þt K3 T3 ða þ 4nR 3RÞ N2 K3 T3 2K3 G3 a s0 ð5:128Þ N2 2Rðn 1Þ PðK3 T3 1Þða þ 2nR 2RÞ t=T3 e b2 ðtÞ ¼ 1 þ C 2 eðN2 =N1 Þt a K3 T3 ða þ nR 3RÞ aPK3 að1 X2 Þða þ 2nR 2RÞ 1 2K3 G3 ða þ 2nR 2RÞ þ 1 s0 et1 =T3 N2 K3 T3 N2 ð5:129Þ Furthermore, by substituting equations (5.128) and (5.129) back into equations (5.114), (5.115) and (5.116) and integrating the results, we have the expressions for the stress and strain components in the bolted region: N1 T3 N2 2S 22 Rðn 1Þ N1 2 C 2 eðN2 =N1 Þt sð2Þ ¼ K 1 S K3 ð1 K3 T3 Þ 3 r 2 a K3 N1 N2 K3 N1 N2 2S 22 Rðn 1Þ N2 PðK3 T3 1Þ 1 S 22 t1 K 3 t c2 exp K3 a a þ 4nR 3R N1 1 aPK 3 að1 X2 Þ ½a S 22 ða þ 2nR 2RÞ exp K3 t1 K 3 t þ T3 N2 1 aPK að1 X 3 2 Þð1 K3 T3 Þ ½a S 22 ða þ 2nR 2RÞ 1 et1 =T3 K3 T3 N2 1 1 2K3 G3 s0 2 exp K3 t þ K3 t1 ½a S 2 ða þ 2nR 2RÞ 1 K3 T3 T3 N2 2 2 ½a S 2 ða þ 2nR 2RÞ½1 ð1 K3 T3 Þ exp½K3 ðt1 tÞ P 1 S 2 ð5:130Þ N1 T3 N2 2S 22 Rðn 1Þ N1 ð2Þ 2 K3 1 þ S 2 þ K3 ð1 K3 T3 Þ sy ¼ C 2 eðN2 =N1 Þt a K3 N1 N2 K3 N1 N2 2S 22 Rðn 1Þ N2 PðK3 T3 1Þ 1 þ S 22 þ C 2 exp K3 t1 K 3 t a a þ 4nR 3R N1 1 aPK3 að1 X2 Þ 2 t1 K 3 t þ ½a þ S 2 ða þ 2nR 2RÞ exp K3 T3 N2 1 aPK3 að1 X2 Þð1 K3 T3 Þ ½a þ S 22 ða þ 2nR 2RÞ 1 et1 =T3 K3 T3 N2 1 1 2K3 G3 s0 ½a þ S 22 ða þ 2nR 2RÞ 1 exp K3 t1 K3 t K3 T3 T3 N2 ½a þ S 22 ða þ 2nR 2RÞf1 ð1 K3 T3 Þ exp½K3 ðt1 tÞg Pð1 þ S 22 Þ ð5:131Þ

138

Chapter 5 2S 22 Rðn 1Þ PðK3 T3 1Þ C 2 eðN2 =N1 Þt 1 2n þ S 22 þ a K3 T3 ða þ 4nR 3RÞ aPK að1 X2 Þ 3 ½að1 2nÞ þ S 22 ða þ 2nR 2RÞet=T3 þ N2 1 2K3 G3 s0 2 t1 =T3 ½að1 2nÞ þ S 2 ða þ 2nR 2RÞ 1 e K3 T3 N2 1 1 ½að1 2nÞ þ S 22 ða þ 2nR 2RÞ þ PrS 22 1 et=T3 1 2G3 K3 T3

U ð2Þ r ¼

r 2G3

ð5:132Þ By substituting equations (5.128), (5.129) back into equation (5.123), we can obtain the explicit formula of b3(t). By further substituting b3(t) and equation (5.117) back into equation (5.54) and integrating the results, we have the expression for the stress and strain components in the visco-elastic zone as following: N1 T3 N2 sð3Þ K3 S 23 r ¼ K3 N1 N2

2X 2 Rðn1Þ 12nþX22 þ 2 a

C 2 eðN2 =N1 Þt

N1 2X 2 Rðn1Þ C2 K3 S 23 ð1K3 T3 Þ 12nþX22 þ 2 a N1 K3 N2 N2 PðK3 T3 1Þ 2 S ½að12nÞþX22 ðaþ2nR2RÞ t1 K3 t þ exp K3 aþ4nR3R 3 N1 1 aPK3 að1X2 Þ 2 exp K3 t1 K3 t S 3 ½að12nÞþX22 ðaþ2nR2RÞ T3 N2 1 aPK3 að1X2 Þ 2 S 3 ½að12nÞþX22 ðaþ2nR2RÞ 1 et1 =T3 þ K3 T3 N2 1 1 2K3 G3 s0 2 S 3 ½að12nÞþX22 ðaþ2nR2RÞ 1 exp K3 t1 K3 t þ K3 T3 T3 N2 f1ð1K3 T3 Þexp½K3 ðt1 tÞgP 1S 22 ð5:133Þ þ

2 2 2X2 Rðn1Þ 12nþX2 þ C 2 eðN2 =N1 Þt a 2 N1 2 2 2X2 Rðn1Þ K3 S 3 ð1K3 T3 Þ 12nþX2 þ C2 a K3 N1 N2 N2 PðK3 T3 1Þ 2 S ½að12nÞþX2 ðaþ2nR2RÞ t1 K3 t exp K3 aþ4nR3R 3 N1 1 aPK3 að1X2 Þ 2 t1 K3 t þ S 3 ½að12nÞþX22 ðaþ2nR2RÞ exp K3 T3 N2

N1 T3 N2 K3 S 23 sð3Þ y ¼ K3 N1 N2

Stability Analysis of Rheologic Rock Mass 139 1 aPK3 að1 X2 Þ 2 et1 =T3 1 S 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ K3 T3 N2 1 1 exp K3 t1 K 3 t 1 K3 T3 T3 2K3 G3 s0 2 S 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ N2 i h 1 ð1 K3 T3 ÞeK3 ðt1 tÞ Pð1 þ S 22 Þ ð5:134Þ 2X 2 Rðn 1Þ PðK3 T3 1Þ 1 2n þ X22 þ 2 C 2 eðN2 =N1 Þt a K3 T3 ða þ 4nR 3RÞ aPK að1 X2 Þ 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞet=T3 þ N2 1 2K3 G3 s0 2 t1 =T3 e ½að1 2nÞ þ X2 ða þ 2nR 2RÞ 1 K3 T3 N2 1 1 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ þ PrS 22 1 et=T3 1 2G3 K3 T3

U ð3Þ r ¼

rS 23 2G3

ð5:135Þ By allowing t ! 1, the expression of stress and strain components can be obtained for the diﬀerent zones. (i)

In the bolted region

sð2Þ r

sð2Þ y

aPK3 að1 X2 Þ 1 2 et1 =T3 ¼ ½a S 2 ða þ 2nR 2RÞ 1 N2 K3 T3 2K3 G3 s0 ½a S 22 ða þ 2nR 2RÞ P 1 S 22 N2

aPK3 að1 X2 Þ 1 2 ¼ ½a þ S 2 ða þ 2nR 2RÞ 1 et1 =T3 N2 K3 T3 2K3 G3 s0 ½a þ S 22 ða þ 2nR 2RÞ Pð1 þ S 22 Þ N2

aPK3 að1X2 Þ U ð2Þ r½að12nÞþS 22 ðaþ2nR2RÞ r ¼ 2G3 N2

ð5:136Þ

1 1 et1 =T3 K3 T3

K3 s0 1 r½að12nÞþS 22 ðaþ2nR2RÞ PrS 22 2G3 N2

ð5:137Þ

140

Chapter 5

(ii)

In the visco-elastic zone

aPK3 að1 X2 Þ 2 1 2 ¼ S 3 ½að1 2nÞ þ X2 ða þ 2nR 2RÞ 1 et1 =T3 N2 K3 T3 2K3 G3 s0 2 þ S 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ P 1 S 22 N2 aPK3 að1 X2 Þ 2 1 2 ¼ S ½að1 2nÞ þ X ða þ 2nR 2RÞ 1 sð3Þ et1 =T3 3 2 y N2 K3 T3 2K3 G3 s0 2 S 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ Pð1 þ S 22 Þ ð5:138Þ N2 aPK3 að1 X2 Þ 2 1 2 U ð3Þ et1 =T3 ¼ rS ½að1 2nÞ þ X ða þ 2nR 2RÞ 1 r 3 2 2G3 N2 K3 T3 K3 s0 2 1 rS 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ PrS 22 ð5:139Þ 2G3 N2 sð3Þ r

It can be seen clearly from equation (5.139) that the expression of stress or displacement comprises three terms, each having distinct physical meaning. The ﬁrst term is the eﬀect of the time (t1) of bolt installation on the surrounding rock mass. It indicates that the earlier the installation the better to improve the stress state in the rock mass, i.e., reducing t1 as fully as possible. The second term reﬂects the eﬀect of prestress of the bolts (s0). The higher s0 gives the more eﬀective improvement of the rock mass. The third term reﬂects the eﬀect of the initial in situ stress, P.

5.6.2

Discussion and application

Figure 5.12 shows stress distribution in the surrounding rock mass reinforced with bolts. According to equations (5.126) and (5.119), we have N2 ¼ K3 aað1 X2 Þða þ 4nR 3RÞ þ 4G3 RK3 ðn 1Þ

E1 B1 a K3 a 3 4n 4G3 RK3 ð1 nÞ ¼ R B2 Because B2 B1 , 4G3 RK3 ð1 nÞ >> ðE1 B1 =B2 ÞK3 að3 4n ða=RÞÞ: The above equation can be approximately reduced to N2 ¼ 4G3 RK3 ð1 nÞ

ð5:140Þ

141

Stability Analysis of Rheologic Rock Mass

Figure 5.12. Stress distribution in the surrounding rock mass after bolting.

Substituting equations (5.119) and (5.140) into equations (5.136) and (5.137) gives the approximate solution to the stress and strain in the bolted zones:

E1 B1 P a ha a i 1 þ S 22 2 2n 1 et1 =T3 4B2 G3 ð1 nÞ R R R K3 T3

s0 h a a i þ S 22 2 2n þ P 1 S 22 R 2ð1 nÞ R

sð2Þ r ¼

sð2Þ y

E1 B1 P a h 2 a ai 1 S 2 2n et1 =T3 1 ¼ 4B2 G3 ð1 nÞ R 2 R R K3 T3

s0 h a a i S 22 2 2n þ P 1 S 22 R 2ð1 nÞ R

ð5:141Þ

ð5:142Þ

r E1 B1 P a ha a i 1 ð1 2nÞ S 22 2 2n et1 =T3 1 2G3 4B2 G3 ð1 nÞ R R R K3 T3

s0 h a a i ð1 2nÞ S 22 2 2n þ PS 22 ð5:143Þ R 2ð1 nÞ R

U ð2Þ r ¼

From the plastic condition of equation (5.51) given in Section 5.3, we have 2 sy sr ¼ pﬃﬃﬃ dKp 6

ð5:144Þ

142

Chapter 5

Considering equations (5.141) and (5.142), we have sð2Þ y

sð2Þ r

E1 B1 PS 22 a a 1 2 2n 1 et1 =T3 ¼ R K3 T3 2B2 G3 ð1 nÞ R

s0 a S 22 2 2n 2PS 22 þ R ð1 nÞ

ð5:145Þ

Therefore, in order to prevent the opening wall (S2 ¼ 1) from becoming plastic state after an inﬁnite long time, we may apply the following criterion according to equations (5.144) and (5.145): E1 B1 P a a 1 s0 a 2 2 2n 2 2n et1 =T3 þ 1 2P pﬃﬃﬃ dKp 2B2 G3 ð1 nÞ R R K3 T3 R ð1 nÞ 6 ð5:146Þ To meet the condition in equation (5.146), it is necessary to enlarge the ﬁrst and the second terms. To enlarge the second term, however, it needs to increase both prestress and length of the bolt, i.e., s0 and R. However, there is a limit to increase the prestress of bolts, because a bolt of a certain cross-sectional area can bear only a limited prestress corresponding to the tensile strength. Therefore, eﬀorts should be made to increase the ﬁrst term as far as possible. There are several ways to increase this term, for instance, install bolts as early as possible (decreasing t1), or enlarge the bolt cross-section area (i.e. increase B1) and increase the bolting density (i.e. decrease B2), or change the bolt length. The ﬁrst two measures are more eﬀective. When the prestress (s0) is not taken into consideration or the bolt is not prestressed, equation (5.145) becomes sð2Þ y

sð2Þ r

E1 B1 P a a 1 2 2n 1 et1 =T3 2P ¼ 2B2 G3 ð1 nÞ R R K3 T3

ð5:147Þ

This equation takes the status of sidewalls of the opening into account, i.e., S2 ¼ 1. ð2Þ The relationship between sð2Þ y sr and a=R is expressed in diagram shown in Figure 5.13. Given that E1 ¼ 2.1 105 MPa, B1 ¼ 1 cm2, B2 ¼ 50 cm2, G3 ¼ 4 103 MPa, ¼ 0.25, T3 ¼ 12d, K2 ¼ 1/8d, t1 ¼ 0, then from the ﬁgure, D1 ¼ (0.01 2)P, D2 ¼ (0.006 2)P, D3 ¼ (0.004 2)P. ð2Þ It can be seen from equation (5.147) or from Figure 5.13 that sð2Þ y sr has the minimum value when a=R ¼ 1 n. The above analysis leads to the conclusion that in order to avoid the surrounding rock mass damage due to stress concentration or to make the stress diﬀerence in

Stability Analysis of Rheologic Rock Mass

143

Figure 5.13. Eﬀect of relative bolt length on stress diﬀerence of surrounding rock mass.

opening walls reach the lowest level: (a) the bolts must be installed as early as possible, (b) the prestress in the bolts must be suﬃciently large, (c) the bolts must have suﬃcient length and cross-sectional area, (d) bolting should be installed with suﬃcient density. The optimal bolt length is n=ð1 nÞ times opening radius, i.e., a=R ¼ 1 n, when non-prestressed bolts are used. If bolts are installed late, the ﬁrst term in equation (5.145) becomes basically non-functional. The second term in the equation shows that the higher R, the better, i.e., long bolts should be selected, up to the optimal length. If failure is determined by the magnitude of displacements, the conditions of which the displacement of the opening wall becomes in minimum should be examined. From equation (5.143) and substituting S2 ¼ 1, r ¼ a into equation (5.143), we can obtain U

ð2Þ r¼a

a E1 B1 P a a 1 a t1 =T3 1 1 P e ¼ þ ðs0 Þ 1 2G3 2B2 G3 R R K3 T3 R

If the prestress s0 ¼ 0, then the above equation becomes, U ð2Þ r¼a ¼

a E1 B1 P a a 1 1 1 et1 =T3 P 2G3 2B2 G3 R R K3 T3

Under the conditions given in this section, ½Dn is proven to be symmetrical and to have the expression similar to that of matrix [D]. In the case of ! ¼ 0, ½Dn becomes

144

Chapter 5

Figure 5.14. Eﬀect of relative bolt length on displacement of surrounding rock mass.

an elastic matrix and for ! ¼ constant with no damage development occurring, the above proposed algorithm constitutes the constant stiﬀness algorithm. In Figure 5.14, D4 ¼ ð0:0025 1Þ ðaP=2G3 Þ and D5 ¼ ð0:0018 1Þ ðaP=2G3 Þ. Figure 5.14 indicates that U ð2Þ r reaches the minimum value when a=R ¼ 1=2, i.e., when bolt length equals to the opening radius, opening wall has minimum displacement. Similarly, in order to minimise the displacement of opening walls, bolts should be installed as early as possible and the prestress of bolts should be increased, in addition to enlarge bolt cross-sectional area and to increase bolting density. In the case of non-prestressed bolts, the optimum length of bolt should have the ratio of a=R ¼ 1=2. If bolts are installed late, the same conclusion can be drawn from equation (5.147) that long bolts should be used. In summary, the optimum time for installing bolts and the magnitude of prestress applied in the bolts can be determined quantitatively using the method presented in this section. A clear conclusion can be drawn that the most rational bolt length (R a) for visco-elastic surrounding rock mass should be a=ð1 nÞ, where a is the opening radius, R is the radius of the visco-elastic zone, n is the Possion’s ratio of the rock mass.

5.7.

RHEOLOGICAL DAMAGE ANALYSIS OF THE ROCK MASS STABILITY

Previous sections have studied the problems of the stress state in surrounding rock masses and the interaction of the rock mass and support using various rheological models. The present section combines damage mechanics and rheological mechanics

Stability Analysis of Rheologic Rock Mass

145

to study the stability of rock mass. When excavation is in soft rock or at great depth, the rock mass deforms. A great portion of the deformation is caused by the loosening due to relatively high stresses as compared to the strength of rock around the opening. It produces cracks in the surrounding rock mass, and results in remarkable dilatancy phenomena reﬂected by obvious volume increment in the rock mass [39,206,382–389]. As a consequence, the conventional analysis methods are not adequate to study the rock mass exhibiting the above behaviour. The concept and method of damage mechanics combining with rheological mechanics are applied to analyse the rock mass in this section. 5.7.1

Damage evolution equation

The basic concepts and assumptions of damage mechanics have been discussed in Chapter 3. This section adopts the damage equation by Desai [291,292,382] based upon the isotropic damage concept to express the eﬀective stress and apparent stress, i.e., sij ¼ ð1 !Þsij þ

! dij skk 3

ð5:148Þ

where ! is the damage variable. Considering that the rupture failure planes have residual strength, a speciﬁc maximum damage value !p less than 1 of the rock mass can be assumed. The value of !p depends upon the type of the rock. Most rocks have brittle characteristics. Under loading, microcracks initiate and propagate and a failure plane forms. The failure can be caused by tensile, shear or combined tensile and shear. The rock dilates before the ﬁnal failure as cracks are growing in number and extending gradually. The dilatancy phenomenon is the most important warning sign of rock failure. There are three main types of damage evolutions for brittle material such as rocks: (i) (ii) (iii)

Evolution and development of damages are governed mainly by strains, especially by the tensile strain. Damage evolution and development are governed mainly by stresses, especially by the deviator stress or the major principal stress. Such evolution is governed by energy or relief rate of energy.

Tan [39] studied the mechanism and criterion of rock dilatancy phenomenon and suggested the condition of dilatancy generation, soct k2 1 f3

ð5:149Þ

146

Chapter 5

where f3 ¼ k mp is the strength envelope curve (or surface) for a critical surface, which is roughly corresponding to the long-term envelope (surface); p is the mean spherical stress; k and m are the strength index; k2 is a real number greater than unity that is determined from test, and soct is the second variant of deviator stress tensor. The damage evolution equation adopted in the present section that follows the assumption of isotropic damage expressed by equation (5.149) is, soct ! ¼ K1 ð1 !Þ F 1 f3

ð5:150Þ

and, 8 soct k2 > > < 1 soct f3 ¼ F > f3 > :0

soct >1 f3 soct when 1 f3

when

in which tensile stress is taken as positive. The above damage evolution equation is associated to both increments in number and extension of the cracks and the dilatancy of the material.

5.7.2

Viscoelastic–viscoplastic-damage constitutive equation and FEM method

5.7.2.1 Constitutive equation. The common constitutive equation of non-damaged viscous materials has the following increment expression, fdsg ¼ ½Dðfdeg fdeV gÞ

ð5:151Þ

where [D] is the elastic matrix; fdsg,fdeg and fdeV g are the apparent stress increment, overall strain increment and viscous strain increment in the form of column matrix respectively. Based on the concept of equivalent strain, i.e., the eﬀect of damage on the strain behaviour is reﬂected only by the equivalent stress. This means that the constitutive relation of a damaged material can use equation (5.151) by replacing the stress term with the equivalent stress. Hence, we have the following constitutive equation for viscoelastic–viscoplastic-damage rock masses: fds g ¼ ½Dðfdeg fdeV gÞ

ð5:152Þ

Stability Analysis of Rheologic Rock Mass

147

Figure 5.15. A rheological model (ss is the plastic limit).

A typical rheological model with visco-elastic and visco-plastic characteristic is shown in Figure 5.15. In the case of one-dimensional problem, the constitutive equation of the model can be divided into two stages according to the stress level, sþ

Z2 E1 E2 E1 Z2 e_ s_ ¼ eþ E1 þ E2 E1 þ E2 E1 þ E2

for s ss

ð5:153Þ

and s ss þ

Z3 Z2 þ Z3 Z Z Z Z s_ þ 2 3 s€ ¼ Z2 e_ 2 3 e€ þ E1 E2 E1 E2 E2

for s > ss

ð5:154Þ

in the case of creep, the equation becomes e¼

s s þ 1 eðE2 =Z2 Þt E1 E2

for s ss

ð5:155Þ

and e¼

s ss s s þ 1 eðE2 =Z2 Þt þ t for s > ss E1 E2 Z3

ð5:156Þ

and in the case of creep damage, s s þ 1 eðE2 =Z2 Þt for s ss E1 E2

ð5:157Þ

s ss s s þ 1 eðE2 =Z2 Þt þ t for s > ss E1 E2 Z3

ð5:158Þ

e¼ and e¼

148

Chapter 5

5.7.2.2 FEM method. According to the virtual work principle, we have the following equilibrium equation, Z

½BT fsgn dV ¼ ff gn

ð5:159Þ

V

where [B] is the strain matrix; fsgn , ff gn are the stress increment and the exterior R R loading with the time interval of tn; ff gn ¼ S t ½NT ftgn dS þ v ½NT fpgn dV; [N] is the matrix of shape functions; {t}n is the surface force exerted on the boundary of St within tn; and {p}n is the body force within tn. Integration on equation (5.148) gives 1 dsij ¼ ð1 !Þdsij di !j dskk S ij d! 3

ð5:160Þ

where S ij is the eﬀective deviator stress. Equation (5.160) can be written in the matrix form, fdsg ¼ ½C ! fds g d!fS g

ð5:161Þ

where [C!] is the matrix related to the damage variable !, called damage matrix, for a three-dimensional problem, it is expressed as 1!þ! 3 ½C ! ¼

! 3

! 1!þ 3 symmetry

! 3 ! 3

! 1!þ 3

0

0

0

0

0

0

0

0

0

ð1 !Þ

0 ð1 !Þ

0

ð1 !Þ ð5:162Þ

Writing the diﬀerential expression of equation (5.186) in the increment form gives fsgn ¼ ½C ! n fs gn !n fS gn

ð5:163Þ

Equation (5.163) shows that within the time interval of tn, the apparent stress increment is correlated to the eﬀective stress state, the damage state at the current moment, as well as the increment of damage and the increment of eﬀective stress within tn.

Stability Analysis of Rheologic Rock Mass

149

Within tn, from equation (5.152), we have fs gn ¼ ½Dðfegn feV gn Þ

ð5:164Þ

Substituting equations (5.163) and (5.164) into the equilibrium equation (5.159), we have the equilibrium equation that describes the damage eﬀect, Z

Z

½BT ½C ! n ½D½BdVfugn ¼ ff gn þ V

!n ½BT fS gn dVþ V

Z

½BT ½C ! n ½Df!V gn dV V

ð5:165Þ By letting ½D n ¼ ½C ! n ½D

ð5:166Þ

fS ! gn ¼ !n fS gn

ð5:167Þ

we can rewrite equation (5.165) as ½kfugn ¼ ff gn þ fF gn þ fF 0 gn

ð5:168Þ

where Z

½BT ½Dn ½B dV

½k ¼

ð5:169Þ

V

is the stiﬀness matrix. Z

fF gn ¼ V

½BT fS ! gn dV

ð5:170Þ

is the additional force caused by damage evolution. The following equation fF 0 gn ¼

Z

½BT ½Dn feV gn dV

ð5:171Þ

V

is the additional force caused by the viscous stress increment. Under the conditions given in this section, ½Dn is proven to be symmetrical and to have the expression similar to that of matrix [D]. In the case of ! ¼ 0, ½Dn becomes

150

Chapter 5

an elastic matrix and for ! ¼ constant with no damage development occurring, the above proposed algorithm constitutes the constant stiﬀness algorithm. 5.7.3

Application to stability analysis of an underground opening

Analysis of underground excavation project using damaged rheological model is conducted and compared with non-damage analysis. The underground project model described is a metal mine located in a very complicated geological environment where the old metamorphic rock mass has experienced many times of tectonic movement. In addition to the faults of various sizes, the joints and fractures are densely distributed in the mining area. The surrounding rock mass displays remarkable rheological characteristics. In order to study the surrounding rock mass stability and to rationalise the supporting scheme, a special testing gallery has been excavated, with a variety of monitoring instrumentation. Large-size in situ triaxial rheological tests have been conducted in the gallery to study the rheological behaviour. The gallery has overlying depth of 500 m and a span of 3 m with arch roof. The testing gallery has been supported using initial shotcrete-bolting that follow the advance of working face closely and secondary shotcrete-steel net supporting. The grouted bolts are 1.5–1.8 m in length and the total thickness of the shotcreting layer is about 25 cm. Two sections for multi-point borehole extensometer were installed in the testing segment, each measuring 10 m in depth with six monitoring points. The observed deformation curve shows obvious 3-stage rheological deformability. Owing to the uniformly cracked or fractured structure of the surrounding rock mass, it can be considered approximately as a quasi-continuous and homogeneous rheological medium. The rock mass stability around the opening is studied using the viscoelastic– viscoplastic-damage method discussed in Section 5.7.2, incorporated in a FEM program. The results from the analysis are compared with the in situ measurements to check the eﬀectiveness of the model and the method. To make comparison between the methods, calculations have also been carried out using the rheological model with no damage evolution. 5.7.3.1 Decomposition of the rheological deformation. According to the model (Figure 5.15), the overall deformation comprises three parts. (a) Transient elastic deformation

It can be obtained according to Hooke’s law

fee g ¼

1 ½Afsg E1

ð5:172Þ

Stability Analysis of Rheologic Rock Mass

151

where E1 is the transient elastic module; [A] is the constant matrix related to Poisson’s ratio. In the case of plane strain, 2

1 n2

½A ¼ 4

nð1 þ nÞ 1 n2

symmetry

(b) Visco-elastic deformation. time interval tn,

feve gn ¼

3 0 5 0 2ð1 þ nÞ

ð5:173Þ

It is calculated from the following equation, for a

1 ½Afsgn ð1 eðE2 =Z2 Þtn Þ ð1 eðE2 =Z2 Þtn Þfeve gn E2

ð5:174Þ

where E2 is the delay elastic modulus; Z2 is the visco-elastic coeﬃcient; [A] is the constant matrix related to Poisson’s ratio. (c) Visco-plastic deformation. When the material is proven having entered the yield state according to the criterion of viscoplastic yield, this part of deformation can be calculated by the following methods.

(i) Visco-plastic yield criterion. It adopts Mohr–Coulomb yield criterion, whose expression under a complex stress state is 1 1 F ¼ sin jI1 þ cos y pﬃﬃﬃ sin y sin j ðJ2 Þ1=2 C cos j ¼ 0 3 3

ð5:175Þ

where I1 ¼ sij , is the ﬁrst variant of stress tensor; J2 ¼ 12 sij sij , is the second variant of deviator stress tensor; y is Lode parameter, pﬃﬃﬃ 3 3 J3 ; sin 3y ¼ 2 ðJ2 Þ3=2

1 J3 ¼ sij sjk s, 3

is the third variant of deviator stress tensor. (ii) Visco-plastic stress increment. It can be expressed by:

f_evp g ¼

@Q 1 fðF Þ Z3 @fsg

ð5:176Þ

152

Chapter 5

where Z3 is the visco-plastic ﬂowing coeﬃcient; and

fðFÞ ¼

fðF Þ 0

ðF > 0Þ ðF 0Þ

fðFÞ has the form of fðFÞ ¼ F/F0 where F0 is a reference value. The yield function value of F is dimensionless, and F0 ¼ C cos f. Q is the plastic potential function. Q 6¼ F stands for irrelevant ﬂowing, and Q ¼ F stands for relevant ﬂowing; the relevant ﬂowing law is used here. From Equation (5.20), the visco-plastic strain increment that occurs in the time interval of tn ¼ tnþ1 tn can be calculated using the following equation, fevp gn ¼ tn ½ð1 Þf_evp gn þ f_evp gnþ1

ð5:177Þ

When ¼ 0, we have Euler’s time integration method and the strain increment is determined by the strain rate at the current moment, tn; when ¼ 1, we have the complete implicit integration method and the strain increment is determined by the strain rate at the end of the time interval; when ¼ 1=2, The implicit trapezoid method can be adopted. The above equation is for strain increment of non-damaged material. For damaged material, the equation is similar, except that the eﬀective stress term is used instead of the apparent stress term, based on the eﬀective strain concept.

5.7.3.2 Determination of Model’s parameters. The Model’s parameters used for analysis are obtained through ﬁeld triaxial compressive rheological tests on the rock mass. Each parameter used in the damage evolution equation is obtained by ﬁtting the elastic module of specimens from multi-loading-stage creep tests, i.e., !¼1

E E0

ð5:178Þ

where E0 is the elastic modulus of the tested body with no damage evolution; the failure damage value is calculated using equation (5.178) from the elastic modulus. The values of cohesion (C) and friction angle (j) are determined from the stressvolumetric strain curve at maximum volumetric strain. At this stress level, the swelling deformation initiates. It can be seen from equation (5.150) that the criterion for damage development is to judge if dilatancy of the rock mass occurs. Therefore, the method also reveals that the yield condition of the rock mass is the damage evolution condition. The property parameters are listed in Table 5.2 and the initial in situ stress ﬁeld has the vertical and horizontal components of sx ¼ 20.0 and

153

Stability Analysis of Rheologic Rock Mass Table 5.2. Material’s properties. Material

Parameters E2 E1 (MPa) (MPa)

Rock mass 7500 Shotcrete layer 22000

6500

0.3 0.18

¼ 1/Z3 C Z2 (MPah) [l/(MPah)] (MPa) f( ) 174400

4

0.45 10

1.0

K1

K2 3

33.0 5.87 10

!p

4.82 0.40

Figure 5.16. Layout and section view of displacement monitoring in the test gallery.

sy ¼ 14.0 MPa, respectively. The shotcrete concrete layer is treated as a linear elastic medium.

5.7.3.3 Comparison between calculated and in situ measured results. The layout of monitoring points in the testing gallery in the mining under consideration is shown as in Figure 5.16. The measured results obtained from extensometer observation have been compared with the calculated results.

154

Chapter 5

Figures 5.17–5.20 compare the calculated results and the measured results. From these results, we can see that the displacement of the gallery periphery calculated by taking the damage evolution of the rock mass into account is somewhat higher than the result from conventional viscous calculation. With respect to the attenuation law of the displacement rate at the periphery and to the distribution of displacements in the depth of the surrounding rock mass, the computed results by damage model are better coincident with the measured one. On the distribution of displacements along depths, the diﬀerence between the conventional calculation and the damage computation lies in the plastic zone. In the plastic zone around the gallery, the

Figure 5.17. Displacement–time curve of the gallery roof.

Figure 5.18. Displacement–time curve of the sidewall.

Stability Analysis of Rheologic Rock Mass

155

Figure 5.19. Calculated displacement–time curve of the sidewall with damage evolution of rock mass.

Figure 5.20. Displacement distribution with depth on the sidewalls.

displacement gradient obtained from the damage analysis is greater than that from the conventional analysis. The conventional value from the damage analysis is closer to the ﬁeld measurement. It shows that the visco-elasto-plastic calculation with rock mass damage evolution can describe more realistically the loosening failure characteristics in the rheological process of the surrounding rock mass than the conventional analytical method.

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Chapter 6

Back Analysis and Observational Methods In recent years, the back analysis method has been widely applied in geotechnical engineering especially in the underground works. Since Peck [390] proposed this method in various related analytical methods and numerical methods have been developed [53–58,391–400]. As a relatively new branch of mechanics, rock mechanics has been developed by adopting traditional analytical methods of mechanics based on continuum mechanics. The typical mathematical and mechanical analytical model is somewhat to ﬁnd the stress ﬁeld and deformation ﬁeld in material by applying the known external loads (or internal loads) with the known material constitutive relationship, geometrical shape and mechanical parameters [53–56]. However, back analysis is a reverse procedure, which is to solve the external load or partial material parameters, based on the known deformation and stresses at limited points and the partially known material parameters. In the stability analysis of the geotechnical engineering problems, it is often necessary to know the in situ stress ﬁeld, material mechanical parameters and even the mechanical model, by utilising the monitored physical information such as deformation, strain, stress and pressure during constructions. Such a methodology is called back analysis method. The back analysis method, based on the required input physical information, can be divided into deformation back analysis method, stress back analysis method and coupled back analysis method [53]. The physical information in the coupled back analysis method requires both deformation and stress. The back analysis has been applied to various geotechnical engineering projects, particularly to the rock engineering projects. The reason is due to the fact that the rock masses are very complicated and inhomogeneous. Excavation may disturb the rock mass to diﬀerent extents. Therefore, the construction process is not a close system behaviour, and is aﬀected by the environment and at the same time aﬀects the environment [63–70,401,402]. In underground excavation and stability analysis, we face two problems. Firstly, the information available is generally ‘grey’ and uncertain. It is therefore very diﬃcult to obtain explicitly the exact solution by using the conventional mathematical and mechanical analytical methods. Secondly, since the excavation is a process of forming an underground opening, the exchange of energy would occur between the rock mass and its surrounding environment.

157

158

Chapter 6

In other words, the rock mass would absorb some external energy and at the same time, release some energy to the external environment. This would result in some energy concentration and energy relaxation zones. Generally, the energy concentration and relaxation zones are obtained by using conventional continuum-based mechanical methods and they do not account for the energy dissipation. In fact, actual rock masses do not obey the continuity assumption, because some cracks and fractures are induced due to stress concentration and blast eﬀect during the construction. The inhomogeneity and relative slips at fracture interfaces cause energy dissipation, e.g., thermal energy. It is often diﬃcult to obtain exact results by using the continuum-based analytical method to such a discontinuous medium. Therefore, the analysis of rock excavation must combine both structure analysis and behaviour analysis, since the information monitored on-site is related to the rock behaviour during the excavation. The information can supplement the results obtained by using the conventional analytical method. The deformation monitoring and back analysis is to identify the ‘grey’ problems by using the information of rock mass behaviour. The rock engineering problems usually involve several to several thousand meters. It is very diﬃcult to determine representative or average external loads (in situ stresses in most cases) and material parameters (such as the Young’s modulus, the Poisson’s ratio, cohesion and friction coeﬃcient), by testing rock samples. Large-scale in situ tests are generally very expensive, and often conducted at very few numbers. Hence, it is diﬃcult to ensure the result as a representative one reﬂecting the whole project based on in situ tests. On the other hand, the information used for back analysis is monitored directly from the actual site. As compared with the in situ large-scale mechanical tests, the deformation monitoring and back analysis have following features and advantages: (a) Monitored deformation is the average response of a rock mass in a large scale, from several to several tens of meters, sometimes several hundreds of meters. (b) It corresponds the actual engineering response to the excavation. (c) It gives large quantity of information, as each monitoring location produces diﬀerent information. (d) It can be correlated with laboratory tests to generate the correlation relationship. (e) It can monitor the response of rock outside the blast inﬂuence zone. (f) The deformation monitoring is easy to perform. (g) Monitoring is cost-eﬀective and can be widely used in projects of various sizes. With reliable monitoring data obtained from construction site, it is possible to back analyse the external load and mechanical parameters, by using proper physical material model and the analytical method.

Back Analysis and Observational Methods 6.1.

6.1.1

159

ELASTIC BACK ANALYSIS AND STRESS DISTRIBUTION ANALYSIS

Elastic back analysis

Elastic deformation back analysis method can be classiﬁed into three types: force method, mathematical regression method and graphic method [53–56]. The force back analysis method can be analytical or a numerical method. The analytical method is usually used when the excavation shape is simple and analytical solution can be found. The numerical method, ﬁrstly proposed by Sakurai et al. [53,54], is to assume the deformation at a point in a direction being sum of the individual deformations produced by individual loads at the same point in the same direction. Corresponding equation set can then be created. The number of equations or the number of monitored deformations should be equal to or more than the number of variables. The variables can then be solved by using damping least square method. For homogeneous media, the deformation modulus can also be solved. This is termed the inverse approach of the back analysis method.

6.1.1.1. Basic formulation.

The basic formula can be expressed as Z fP 0 g ¼

½BT s0 dV

ð6:1Þ

v

where {s0} is the in situ stress; {P0} is the nodal load acting on the excavation face; V is the excavated volume; [B] is a matrix relating to the element geometry. The relationship of nodal load {P} and nodal deformation {U} is fPg ¼ ½KfUg

ð6:2Þ

where [K ] is the stiﬀness matrix. Using ER and EL to represent the modulus of rock and lining, respectively, we have ½K ¼ E

R

K

R

EL L

þ R K ¼ E R ½K E

ð6:3Þ

where [KR] and [KL] denote the stiﬀness matrixes when ER ¼ 1 and EL ¼ 1, respectively. The nodal displacement {U} in equation (6.2) can be divided into two parts: displacements at monitoring points and displacements at other points. They are: fU m g ¼

1 ½A s0 ¼ ½Afs 0 g R E

ð6:4Þ

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Chapter 6

where [A] is the compliance matrix which is a function of the Poisson’s ratio and the co-ordinates of monitoring points, s 0 is the normalised in situ stress expressed as n oT s0 ¼ s0x =E s0y =E t0xy =e

ð6:5Þ

Replacing the absolute displacements in equation (6.4) with relative monitored displacements, then fU m g ¼ ½T½Afs 0 g ¼ ½A fs 0 g or

1 fs 0 g ¼ ½A T ½A ½A T fU m g

ð6:6Þ

6.1.1.2. Deformation monitoring and back analysis. An example is given in the following sections to illustrate deformation monitoring and back analysis technique. (i) Deformation monitoring of surrounding rock mass. A large underground hydropower complex is located in a high in situ stress zone in southwest China. Before the construction, a rectangular test tunnel of 2.5 m wide, 5 m high and 30 m long was excavated for deformation monitoring and stability analysis. The layout of testing tunnel is shown in Figure 6.1. The tunnel is parallel to the exploratory tunnel. The distance between the two tunnels is 15 m. Three monitoring sections are set in the test tunnel. Smooth blasting is used around the monitoring sections. Partial monitoring results obtained from a section are illustrated and discussed in this example. Three multiple-point borehole extensometers (MPBX) including two bar electrical transducer and one wire electrical transducer were installed at the roof of the section. Two MPBXs of steel wire electrical transducer were installed at sidewalls. The MPBXs are of 10–13 m in length, for roof and sidewalls, respectively. The monitoring layout is shown in Figure 6.2. The end points are located in the undisturbed zone and have no deformation. In addition, several monitoring holes of 5 m length are set at the location close to the sections for ultrasonic measurements and convergence monitoring. Figure 6.2 shows the monitored ﬁnal displacements at diﬀerent points of the MPBXs. Figure 6.3 shows the displacement-time curves obtained from the MPBX in the left sidewall. From Figure 6.2, it can be seen that the maximum displacement reaches about 1.3 mm on the sidewalls, but less than 0.3 mm at the roof. This may be

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Figure 6.1. Layout plan of the monitoring in experimental tunnel.

Figure 6.2. Displacement measurements by multiple-point borehole extensometers (Section 1 of Figure 6.1).

caused by the high horizontal in situ stress, and high walls of rectangle shape. In addition, the longitudinal ultrasonic wave velocity measured from the boreholes before and after the construction does not change apparently. It indicates that the surrounding rock has not been damaged or disturbed. The rock is at perfectly elastic state. From Figure 6.3, it can be seen that the curve tends to be ﬂat when the excavation approaches Section II (v/v = 1).

162

Chapter 6

Figure 6.3. Changes of displacement with excavation.

The displacement monitored at Section I is the deformation after the 1m excavation from Section I towards Section II. Three-dimensional ﬁnite element modelling is carried out to simulate the total displacement and displacements of diﬀerent excavation stages. The total displacements at left and right walls are computed to be 2.54 mm and 2.45 mm, respectively. (ii) Back analysis and results. The rock mass located around the monitoring sections of the underground complex is a seinite. The rock mass is fractured and the fractures are generally closed. The rock mass is under high in situ stresses. The rock mass is assumed to be uniform, homogeneous and linearly elastic. The in situ stress in rock is assumed to be uniform. Therefore, the linear elastic theory can be used to perform back analysis. If the Young’s modulus and the Poisson’s ratio are known, it is not diﬃcult to back analyse the in situ stress. By inputting the monitored displacements and adopting the least square method for solving equation (6.6), the in situ stress can be computed. Taking E ¼ 40,000 MPa and n ¼0.15 and treating it as a plane strain problem, the maximum principal in situ stresses are computed as s1 ¼ 23.7 MPa and s2 ¼ 18.7 MPa. a1 is at 31.5 to the horizontal plane (Figure 6.4). Figure 6.4 shows comparison of monitored displacement with the back analysis result at left wall. From the ﬁgure, it can be seen that the back analysis agrees well with the monitoring. It indicates that back analysis using linear elastic theory is applicable in this particular case. It is also found that the in situ stress back analysed based on the monitored displacements is very close to in situ stress measured. In other words, the 2-D rock deformation back analysis incorporating rock mechanics parameters E and n and in situ stresses measured on-site, has veriﬁed that the monitored data are reliable.

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Figure 6.4. Monitored and back-analysed displacements at left wall.

6.1.2 Back analysis of in situ stress distribution In situ stress is generally a necessary input parameter in the analysis of rock stability. However, for a large underground complex such as the hydroelectric caverns, the in situ stress cannot be treated to be uniform, as the rock covers very large area (often more than 300 m). The in situ stress ﬁeld changes with depth and distance. A stress function interpolation method [8,52,137] is adopted. Its principle is to use the least square method to obtain the in situ stress ﬁeld satisfying stress equilibrium and deformation condition, based on-site monitoring data and boundary conditions. The following describes the method for the in situ stress ﬁeld of the same hydroelectric complex as in Section 6.1.1.

6.1.2.1. Computational zone and monitored in situ stress. The computational zone of the underground complex is from the ground surface to a depth of 800 m and about 200 m radius from the boundary of the opening in horizontal direction. Six sets of measured in situ stresses are taken [8,52].

6.1.2.2. Determination of stress function. The equilibrium equation at a point in rock is a set of inhomogeneous linear diﬀerential equations. The solution of the equation set is the sum of its corresponding general solution of the homogenous equation set and speciﬁc solution of the inhomogeneous equation set. The speciﬁc solution is taken as sx ¼ sy¼ txy ¼ tyx¼sz¼ 0, txz ¼ gx, and the general solution of the homogeneous equation set is expressed by the stress functions f1 (x, y, z), f2 (x, y, z) and f3 (x, y, z).

164

Chapter 6

Based on equilibrium equation and stress function, following relations can be derived. 9 @2 j 3 > > > txy ¼ > @[email protected] > > > > = 2 @ j1 tyz ¼ @[email protected] > > > > > @2 j 2 > > > ; tzx ¼ @[email protected]

@2 j3 @2 j2 sx ¼ þ 2 @y2 @z sy ¼

@2 j1 @2 j3 þ 2 @z2 @x

sz ¼

@2 j2 @2 j1 þ 2 @x2 @y

ð6:7Þ

By considering the ﬂuctuation and deviation of the measurement data of in situ stresses, stress functions are assumed to be polynomial of power 4 and the corresponding stress components to be polynomial of power 2. The general expression of stress function j1(x, y, z) is: j1 ðx, y, zÞ ¼ a1 x2 þ a2 y2 þ a3 z2 þ a4 xy þ a5 yz þ a6 zx þ a7 x3 þ a8 y3 þ a9 z3 þ a10 xyz þ a11 xy2 þ a12 x2 y þ a13 yz2 þ a14 y2 z þ a15 zx2 þ a16 z2 x þ a17 x4 þ a18 y4 þ a19 z4 þ a20 xyz2 þ a21 xy2 z þ a22 x2 yz þ a23 xy3 þ a24 x3 y þ a25 yz3 þ a26 y3 z þ a27 zx3 þ a28 z3 x þ a29 x2 z2 þ a30 y2 z2 þ a31 z2 x2

ð6:8Þ

where ai is the extrapolation parameter to be determined. The stress components can be expressed as sx ¼ b0 þ b1 x þ b2 y þ b3 z þ b4 x2 þ b5 y2 þ b6 z2 þ b7 xy þ b8 yz þ b9 zx

ð6:9Þ

with b0 b2 b4 b6 b8

¼ 2ða002 þ a03 Þ ¼ 2ð3a008 þ a016 ¼ 2ða0026 þ a027 Þ ¼ 2ða0028 þ a019 Þ ¼ 6ða0025 þ a023 Þ

b1 b3 b5 b7 b9

¼ 2ða0013 þ a015 Þ ¼ 2ða0014 þ 3a09 Þ ¼ 2ð6a0018 þ a028 Þ ¼ 2ð3a0020 þ a031 Þ ¼ 2ða0030 þ 3a021 Þ

where a0 and a00 are the corresponding coeﬃcients of j2 and j3. The expressions of stress functions j2 and j3 and other stress components are similar to equations (6.8) and (6.9) and are not listed here.

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By substituting equation (6.7) into six deformation equations (other components are ignored here), we have ð1 þ nÞr2 sx þ

@2 ¼0 @x2

ð6:10Þ

where ¼ sx þ sy þ sz ; n is the Poisson’s ratio. The six extrapolation functions can be eliminated. By considering the relationship between stress functions and stress components above, each stress function can eliminate some extrapolation coeﬃcients, and the amount of independent extrapolation coeﬃcients to be determined becomes 55. The corresponding boundary conditions must be determined to obtain the extrapolation functions by using least squares method. They include the stress measurements at the ground surface and at selected locations. The normal and tangent stresses at ground surface with outward normal cosine directions of l, m and n are equal to zero, which can be expressed as 9 X0 ¼ sx l þ txy m þ txz n ¼ 0 = Y0 ¼ tyx l þ sy m þ tyz n ¼ 0 ; Z0 ¼ tzx l þ txy m þ sz n ¼ 0

ð6:11Þ

Substituting the stress components with those extrapolation function coeﬃcients as listed in equation (6.9) into equation (6.11), all the boundary conditions can be expressed by extrapolation function coeﬃcients. Therefore, each stress measurement point below ground has six equations and each measurement at ground surface point has three equations. These equations have a general form as A1 Q1 ðxi Þ þ A2 Q2 ðxi Þ þ þ An Qn ðxi Þ ¼ Bi

ð6:12Þ

where An is the extrapolation coeﬃcient to be determined (n ¼ 55); xi is the co-ordinate (x, y, z) of the point; Qi is the exponent function of co-ordinate ( j ¼ 1, 2, . . ., n); Bi is the stress measurement values at the measurement points or zero values at ground surface points; i ¼ 1, 2, . . . , n, m ¼ 6 I þ 3 K, I is number of stress measurement points, K is the number of ground surface points. Based on the above equations, all the coeﬃcients of stress functions can be solved by using least squares method to obtain the co-ordinates of the measurement points. The in situ stress distribution in the whole zone can thus be obtained. The stress distribution computed satisﬁes force equilibrium conditions, continuous deformation conditions and minimum value in the squares error with actual measurements

166

Chapter 6

Figure 6.5. Comparison of stresses measured and computed.

and the given boundary conditions. Figure 6.5 shows the comparison of stresses between the actual measurements and the computed results using the above method. It can be seen that in general they agree well with each other. The results obtained by this method were compared with the results obtained by multiple-variable regression method. It is found that both methods provide similar results, and the result obtained by this method generally agree better with the actual measurements. This method is simpler and has the advantage in computer running time and the preparation work. This method also eliminates the human error in determining stress ﬁeld. Figure 6.6 shows the stress distribution at a section in the middle of the underground complex.

6.2.

6.2.1

VISCO-ELASTIC BACK ANALYSIS AND ITS ENGINEERING APPLICATIONS

Method of site deformation monitoring and its application results

This section introduces the back analysis method for visco-elastic media using a case study on a shallow tunnel.

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Figure 6.6. Stress distribution at a section in the middle of the underground complex.

Figure 6.7. Tunnel section and multiple-stage excavation procedure.

A large railway tunnel is located in a loess layer. The tunnel is a double-lane heavy vehicle tunnel of 11.5 m high, 12 m wide and with 12 m thick overburden. The tunnel is constructed by the New Austrian Tunnelling Method (NATM) and has an experimental section to conduct monitoring studies. Monitored vertical deformation at the crown due to excavation is presented here to illustrate the application of back analysis. Figure 6.7 shows the tunnel section and multiple-stage excavation procedure.

168

Chapter 6

Figure 6.8. Ground condition and instrumentation layout.

Figure 6.8 shows the installation of instrumentation. It can be seen that the multiple-point borehole extensometers (MPBX) are installed in the boreholes before the excavation to monitor the absolute deformation in the soil during the tunnel construction. Relationship between ground settlement and MPBX monitored displacement are shown in Figure 6.9. The monitoring results and analysis of MPBX 3 with a period of 65 days during the excavation of the upper part of the tunnel are shown in Figure 6.10. The largest deformation of 15.29 mm is at the end point (point 1). The deformation can be divided into three stages: (a) Initial deformation stage (compression deformation stage) in which the deformation occurs during the tunnel excavation from the beginning to the monitoring section, with a period of 24 days. The maximum compression deformation is 1.91 mm and displacement rate is 0.08 mm/day. All the six points at diﬀerent depths have negative deformations. Initially the monitoring data has small ﬂuctuation in the ﬁrst 10 days. Then the negative deformation increases and the curves tend to be close together, indicating that deformations occurs mainly due to the ground settlement. (b) Sharp deformation stage. As the tunnel excavation passes through the monitoring section, the deformations become positive immediately and increase sharply for 10 days before reaching a maximum of 15.19 mm. The average displacement rate is 1.72 mm/day.

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Figure 6.9. Notation of negative displacement in pre-installed extensometers.

Figure 6.10. Monitored displacement in borehole 3 with a period of 65 days during the excavation.

(c) Stable deformation stage. When the tunnel excavation is 14 m away from the monitoring sections, the deformations increase slightly (displacement rate was 0.06 mm/day) for 1 month and then become stable. However, the displacements at two deeper points (points 1 and 2) decrease in the ﬁrst several days, indicating that there may be saturated sand-lens at these locations. While the lower part is excavated completely, the displacement increased again about 1–5 mm and the average displacement rate is 0.14 mm/day.

170 6.2.2

Chapter 6 Visco-elastic back analysis

6.2.2.1. Computation method. As noted in the observation, the deformation in ground above the tunnel develops with the excavation of the tunnel. Therefore, theoretically it is a 3-D problem. To save the computation time, it is often simpliﬁed as a virtual 3-D problem or a plane problem, but uses the change of a virtual support force to simulate the unloading due to excavation. Because the monitoring data were obtained from the excavation of the upper part of the tunnel, analysis is performed only for the upper part of the excavation. Finite element elastic analysis is conducted. Figure 6.11 shows the generated mesh. Because of the symmetry, only half of the domain is taken as the computational model with 144 elements (4-node isotropic element) and 170 nodes. The in situ stress is obtained from gravity of the overburden soil. The top of the tunnel is 12 m below the ground surface. The elastic solution of the stress boundary condition is

uðx, y, zÞ ¼

f ðx, y, zÞ P E

ð6:13Þ

where u(x, y, z) is the displacement at a point in the 3-D space; f (x, y, z) is the displacement induced by unit load at unit elastic modulus which can be solved by the

Figure 6.11. Generated mesh in modelling.

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ﬁnite element analysis; P is the excavation load as a function of time. 1 P ¼ Pðx,y,zÞ 0:5 arctan ðt t0 2Þ

ð6:14Þ

6.2.2.2. Visco-elastic analysis results. Using three-unit visco-elastic model shown in Figure 6.12 and assuming the Poisson’s ratio as a constant, the visco-elastic displacement solution can be expressed as follows by transforming equation (6.14) with visco-elastic responding principle: Uðt, x, y, zÞ ¼ f ðx, y, zÞMðtÞ

ð6:15Þ

where, MðtÞ ¼

1 1 1 E0 arctan ðt t0 2Þ 1 þ eE2 = ðtt0 Þ E1 2 E2

f(x, y) can be computed using 2-D FEM. Subsequently back analysis regression can be performed for the monitored data by using equation (6.15). Figure 6.13 shows the

Figure 6.12. The three-unit visco-elastic model.

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Chapter 6

Figure 6.13. Regression curves of measurements at points 1 and 4.

regression curves of measurement points 1 and 4. The monitored data are the absolute displacements combining the ground surface settlements and relative displacements. It can be seen that the regression data agree well with the monitored data after the excavation face passed the monitoring sections. Nevertheless, the overall visco-elastic parameters of the soil can also be obtained from the regression of the latter part of the curves. The analysis gives E1 ¼ 15.2 MPa and E2 ¼ 45.8 MPa, and are seemingly reliable compared with the empirical values.

6.3.

BACK ANALYSIS AND OPTIMISED METHODS IN TRANSVERSE ISOTROPIC ROCK

The layered rocks such as sedimentary formation are often encountered, so it is very important to investigate the back analysis method for such media.

6.3.1

Basic formulae of transverse isotropic mechanics

In continuum mechanics, if the media satisﬁes the assumptions of uniformity, continuity, small deformation and linear elasticity, the constitutive equations can be expressed as sij ¼ Dijkl "kl

ð6:16Þ

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Considering the symmetry of the strain tensor and stress tensor, the elastic parameters can be reduced from 81 to 36. The matrix form is fsg ¼ ½Df"g

ð6:17Þ

Due to the symmetry of the elastic matrix, the elastic independent parameters can be reduced to 21. This can be used in general for anisotropic material that has three perpendicular elastic symmetric planes. If the axes are located in the elastic symmetric planes, the general Hooke’s law can be expressed in the form of matrix as 9 38 8 9 2d "x > d12 d13 sx > > 11 > > > > > 6 d21 d22 d23 > sy > > "y > > 7> 0 > > > > > > > > 7 6 = 6 < < = 7 " d d d sz z 31 32 33 7 6 ¼ ð6:18Þ 7> xy > 6 txy > d66 > > > > 7> 6 > > > > > 4 > txz > > xz > > 5> 0 d55 > ; : > > : ; tyz yz d44 containing only nine independent elastic parameters. The stress–strain relationships in the components of the anisotropic media are 9 sx ¼ d11 "x þ d12 "y þ d13 "z > > > sy ¼ d21 "x þ d22 "y þ d23 "z > > > = sz ¼ d31 "x þ d32 "y þ d33 "z ð6:19Þ txy ¼ d66 xy > > > > txz ¼ d55 xz > > ; tyz ¼ d44 yz Assuming that there is a plane parallel to xoz plane at each point in the elastic body as shown in Figure 6.14, the elastic properties in any direction are equivalent. The body is called orthotropic medium. The general Hooke’s law has a form of: 9 38 8 9 2d "x > d12 d13 sx > > 11 > > > > > 6 > > > 7> "y > d21 d22 d23 0 > > > > sy > > > > > 7 6 < < = 6 = 7 " d d d sz z 31 32 33 7 6 ¼ ð6:20Þ 1 7 6 txy > xy > ðd11 d12 Þ > > > > > > 2 6 7 >t > > > 4 > > 5> xz > 0 d44 > > > : xz > ; > > : ; tyz yz d44 In this case, only ﬁve independent elastic parameters are unknowns. Therefore, equation (6.17) can be modiﬁed to be f"g ¼ ½Afsg where [A] is the compliance matrix.

ð6:21Þ

174

Chapter 6

Figure 6.14. Transverse isotropic elastic body.

As the xz plane is an elastic isotropic plane, thus Ex ¼ Ez ¼ Es

Ey ¼ En

nxz ¼ ns nxy ¼ nyz ¼ nn Gxz ¼ Gs ¼ 0:5 Es =ð1 þ ns Þ Gxy ¼ Gyz ¼ Gsn where [A] becomes 2

1 6 Es 6 6 nn 6 6 En 6 n 6 s 6 6 Es ½A ¼ 6 6 6 6 6 6 6 6 4

nn En 1 En nn En

0

ns Es nn En 1 Es

3

0

1 Gsn

1 Gsn

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 2ð1 þ ns Þ 5 Es

ð6:22Þ

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If the main axes of the medium are not the same as the coordinate axes, then [A] needs to be transformed in coordinate and new compliance matrix is ½A0 ¼ ½T½A½TT

ð6:23Þ

where [T ] is the coordinate transforming matrix. For plane strain problems, [A] can be further simpliﬁed as 2

1 n2s nnn ð1 þ ns Þ 6 1 6 nnn ð1 þ ns Þ nð1 nn2n Þ ½A ¼ 4 Es 0 0

3 0 0 7 7 Es 5 Gsn

ð6:24Þ

The stiﬀness matrix is 2

nð1 nn2n Þ 6 E n 6 nnn ð1 þ ns Þ ½D ¼ ½A1 ¼ m4 0

nnn ð1 þ ns Þ 1 nn2n 0

3 0 0 7 7 mGsn 5 En

ð6:25Þ

where n ¼ Es/En, m ¼ (1þns)(1 ns 2nn2n ) Let us discuss a special and useful case. Assuming the layer plane orientation is parallel to the tunnel direction (z axis), the excavation face is xoy plane, x axis is horizontal and is parallel or oblique to the layer plane, the analysis can be conducted using equations (6.24) and (6.25). If there is a joint set parallel to the layer plane and with the same mechanical properties, and shear stiﬀness is Ks, normal stiﬀness is Kn and joint spacing is b, the elastic parameters in equations (6.24) and (6.25) can be determined by using the following equation: 9 Es ¼ E,ns ¼ n > > > > > 1 > > En ¼ > > > 1=Es þ 1=Kn b = 1 > > Gsn ¼ Gn ¼ > 2ðð1 þ ns Þ=Es Þ þ 1=Ks b > > > > > > n > ; nn ¼ En Es

ð6:26Þ

176

Chapter 6

6.3.2 Optimisation analysis method As stated earlier, the inverse approach of back analysis method can be used in a number of cases with simple conditions. In most cases with complicated conditions, only the direct approach of back analysis method can be used. This method is actually a regression method. The most important issue of this direct approach method is the appropriate understanding of the subject to be back analysed. It depends extensively on the experience and knowledge. For example, in underground excavation analysis, the mechanical model (elastic isotropic or anisotropic, viscoelastic or elasto-plastic models), mechanical analysis method or computational program have to be determined. When the mechanical model and analytical method are determined, the next step is to rationalise the regression computation to make it scientiﬁc and fast. This is the optimisation of the direct back analysis to be discussed in this section. The objective of optimisation, based on monitored convergence displacement, is to obtain the best sets of material parameters and stress parameters. Assuming the computational displacement is ui(X ), monitored convergence displacement is u0i (i ¼ 1, 2, . . . , n), the objective function is deﬁned as FðXÞ ¼

n X

ui ðXÞ u0i

2

ð6:27Þ

i1

where X ¼ ½sx , sy , txy , E, n, c, f where sx, sy and txy are three components of in situ stress, E is the elastic modulus, n is the Poisson’s ratio, c is the cohesion and f is the friction angle. The optimisation method is to make the objective function approach gradually to minimum in the direct approach. Generally, giving one set of initial parameters X0 and their allowable ranges, the optimisation program searches automatically one set of parameters X to make the objective function satisfying the given accuracy requirement. There are many optimisation methods with or without restraints available [49–51]. Brief outlines of the common optimisation methods are presented below. (a) Pure shape speeding method: Pure shape acceleration method was developed from the pure shape method. The pure shape method is to compare the function values at (n þ 1) peak points of an initial pure shape in n-D space, replacing the points of maximum function value, with a new pure shape, and approaching gradually to the points of minimum value through iterations. (b) Composite shape acceleration method: This method is used to solve optimisation problems of multiple variables (normally not more than 20) with constraints of

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unequal equations. The optimisation procedure is to take 2n peak points from non-linear constraints in n-D space to form a pure shape. It is then to compare function values at each peak point one by one, replace the worst point with a new point to improve the objective function and satisfy constraint conditions, and approach gradually to the optimised point. (c) Mixing penalty function method: This is also termed as sequential unconstrained minimisation technique (SUMT). It is to add constraint function with equal equations and unequal equations in penalty, respectively, to the objective function to form a new objective function with no constraint (penalty function). So that it converts the original minimisation of objective function with constraints into a new minimisation of penalty function with no constraints. (d) New Powell method: This is an advanced optimisation method to solve the minimisation of complex functions. The iterative procedure is to solve the minimisation of the objective function along a series of conjugate directions starting from the initial point (X0). The forming of the conjugate directions uses only the objective function values at the iteration points. Therefore, it is a direct method. This method is useful when the objective function is a non-convex function. In the case of multiple minimum value points within a certain range, this method can ﬁnd the optimal point. For complicated non-linear problems, the objective function is often non-convex function, and the minimum point is not unique. To obtain the optimal solution, the optimal searches in one dimension and multiple dimensions are added to seek minimum points of objective function along conjugate directions. Meanwhile, the step should be decreased in the area with sharp slope of objective function. After obtaining the minimum point, the searches will be performed for its two sides to obtain second and third class minimum points. Comparing them with each other gives the optimal solution. An optimisation generally has three loops to execute diﬀerent functions: optimisation methods, model types (linear, non-linear or isotropic models) and selection of diﬀerent back analysis parameters section. Figure 6.15 illustrates a typical program ﬂow.

6.3.3

Examples of engineering applications

The following example is to illustrate the application of the optimisation method in back analysis, for a large hydroelectric project. The power complex is located underground in sedimentary rocks. The surrounding rocks are mainly siltstone interlayered with clay slate. Finite element displacement back analysis coupled with an optimal program was used to examine the displacement distribution, in situ stresses and rock parameters.

178

Chapter 6

Figure 6.15. A typical program ﬂow.

As shown in Figure 6.16, Zones 1 and 3 are the siltstone and Zones 2 and 4 are the clay slate. The siltstone has the Young’s modulus of 28 GPa and 35 GPa in directions perpendicular to and parallel to the rock layer, and Poisson’s ratio of 0.25. The clay slate has the Young’s modulus of 21 GPa and 27 GPa in directions perpendicular to and parallel to the rock layer, and Poisson’s ratio of 0.27. The monitored convergence curves are shown in Figure 6.17. The tunnel was constructed by using one-stage excavation. The in situ stresses are uniformly distributed in the surrounding rock masses. This example is treated as a plane strain problem. The ﬁnite element mesh is generated into four-node four-side isotropic parameter elements with 217 elements and 252 nodes. The rock masses are assumed to be isotropic. The modelling assumes that the Young’s modulus between the measured modulus in the two directions, and the Poisson’s ratio does not change. The back analysis results, obtained using four optimisation methods, are summarised in Table 6.1. The analysis results of in situ stresses obtained by the Powell method are sx ¼ 9.27 MPa, sy ¼ 4.80 MPa. The measured stresses are sx ¼ 9.39 MPa,

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Figure 6.16. Cross section of the tunnel and monitoring holes.

Figure 6.17. Monitored displacement histories.

sy ¼ 4.75 MPa. The results from back analysis with optimisation are very close to the actual measured values.

6.3.4 Discussions The discussions below are based on the experience in using the four optimisation methods to a variety of engineering problems involving visco-elastic and

180

Chapter 6

Table 6.1. Results for different analysis methods. Method

Pure shape Composite shape Mixing penalty Powell method

No of iteration steps

15 50 15 43

Target function (105)

3.2 2.5 2.2 3.6

Back analysis results (MPa) sx

sy

E1

E2

9.04 9.00 9.04 9.27

4.74 4.80 4.74 4.80

21829.6 21700.0 21721.6 21700.0

30294.0 30560.0 30224.5 30800.0

elasto-plastic back analysis. (a) Pure shape acceleration method: It has a fast convergence and a high accuracy, especially for elastic, isotropic problems with more than two parameters. But initial value of back analysis parameters must be properly determined in advance. (b) Composite shape acceleration: Similar to the pure shape acceleration method, it has a fast convergence and a high accuracy, but has constraint functions. Giving constraint functions and an allowable value range of parameters, the program is able to perform back analysis automatically under the constraint conditions. (c) Mixing penalty function method: It has a constraint optimisation function after the Powell method. It does not require to set properly initial values of back analysis parameters in advance and can ﬁnd most optimal solutions satisfying constraint conditions automatically. It converges faster than the Powell method because of the 1-D search method feature. (d) Powell method: It converges slower than that of pure shape and composite shape, but is applicable when the objective functions are complex non-convex functions. It can ﬁnd the most optimal point when minimum points are not unique in a certain range, especially in the cases of non-linear problems with less than three parameters. In summary, for the back analysis of general elastic, anisotropic problems, the pure shape acceleration method and composite shape acceleration method are the best choices since they have a fast convergence and a high accuracy. However, for complex elasto-plastic problems, the Powell method or the mixing penalty function method should be used to conduct optimisation search, and then use either pure shape or composite shape methods to perform global optimisation back analysis to obtain the ﬁnal results. So the unique result can be obtained with fast convergence.

Back Analysis and Observational Methods

181

The criteria to select optimisation methods with or without constraint conditions are: (a) For the cases when the parameters cannot be determined in range, but can be estimated as a possible value, the optimal back analysis approach with no constraint conditions can be used. (b) However, when the parameters have constraints (equal or unequal constraints), the optimal back analysis approach with constraints will be a better alternative. 6.4.

BACK ANALYSIS OF JOINTED ROCK MASS AND STABILITY PREDICTION

The sections earlier introduced the back analyses of elastic, visco-elastic, anisotropic and elasto-plastic problems, but does not attempt the back analysis of jointed rock mass. This section discusses the back analysis of jointed rock mass. It is illustrated by an engineering example of a pumped storage hydroelectric complex.

6.4.1

Description of the project and monitoring

6.4.1.1. Description of the project. The large-scale pumped storage hydroelectric facilities include main powerhouse, transformer house, ventilation and transport tunnels located underground 200–400 m in depth. The surrounding rock mass is mainly a medium-grained granite. There are mainly two joint sets at the crown of the powerhouse, with dip direction of 020–030 and 300–320. The joint spacing is usually 1–2 m. The average persistence is about 50%. Two cases are considered in the analysis. One is to treat the rock mass as an isotropic medium. Another is to consider the joint sets and treat the rock mass as an approximately perpendicularly anisotropic medium. During construction, displacements are monitored at three sections of ventilation tunnel and transport tunnel. The positions of measurement Sections I and II in a ventilation tunnel and the layout of MPBX monitoring is shown in Figure 6.18.

6.4.1.2. Data processing and modification. Due to the ﬂuctuation of the monitored data, data were processed. In addition, as most instruments are installed after the face of excavation, the data do not include the displacements occurring before the installation. The lost displacements are determined using 3-D ﬁnite element analysis for all the points and corresponding modiﬁcation coeﬃcients are derived. Therefore, the ﬁnal displacement based on the 2-D back analysis should be d ¼ dm þ dl ¼ dm,

182

Chapter 6

Figure 6.18. Monitoring layout in ventilation tunnel.

where dm is monitored displacements, dl is lost displacement, is modiﬁcation coeﬃcient which is determined from 3-D ﬁnite element analysis. The instruments (multiple-point extensometers or convergence meters) were installed behind the excavation face about 1 m in the two measurement tunnels. Therefore, the displacement measurements reﬂect partial displacement during the tunnel construction. Three-dimensional ﬁnite analysis is to obtain the modiﬁcation coeﬃcient and determine total displacement at each measurement point. The overburden thickness of the tunnels is 405 m, so the in situ stresses are calculated as sx ¼ 6:89 MPa sy ¼ 9:16 MPa sz ¼ 13:1 MPa

txy ¼ 0:49 MPa tyz ¼ 0:65 MPa tzx ¼ 0:37 MPa

The Young’s modulus is taken as 3 104 MPa, and the Poisson’s ratio as 0.20. Three-dimensional ﬁnite element modelling is carried out for each monitoring point to estimate the total displacement. Details of the modelling can be found in [8,9]. Based on the computed total displacement, the monitored displacement will be modiﬁed. It was found that the modiﬁcation coeﬃcient between the monitored and the computed displacements was 1.27 (average). The monitored displacements and modiﬁed displacements are presented in Table 6.2.

6.4.2

Back analysis using pure shape acceleration method

The approach of ordinary back analysis method is to: (a) create constitutive model of the rock mass, (b) assume an initial value of parameter to be back analysed,

183

Back Analysis and Observational Methods Table 6.2. The monitored and modified displacements. Measurement lines

MP-4 MP-3 MP-2 MP-1

Remark

Measurement points (mm) 1

2

3

4

0.29 0.35 0.21 0.25 0.25 0.30 0.45 0.58

0.34 0.50 0.33 0.39 0.39 0.52 0.60 0.77

0.54 0.82 0.38 0.49 0.41 0.62 0.77 0.99

0.62 0.99 0.40 0.55 – – 0.85 1.09

monitored modified monitored modified monitored modified monitored modified

(c) conduct numerical modelling, (d) solve displacements at measurement points, and (e) compare the computed displacement with monitored displacements. The ﬁnal parameter obtained from the back analysis is the parameter giving minimum diﬀerence between the computed displacement and the monitored displacement. Normally the diﬀerence between the computed displacement and the monitored displacements is expressed by the error objective functions as shown in equation (6.27). The pure shape acceleration method is used here to conduct optimisation analysis, because it is an eﬀective method to make the error objective function be minimum.

6.4.2.1. Computational procedure. Displacement analysis procedure consists of optimisation analysis and positive modelling. The optimisation analysis is discussed earlier and its program ﬂow is listed in [403]. The ordinary positive modelling program is modiﬁed from a commercial program, capable to model anisotropic material. The modelling program ﬂow is given in [404]. The whole program ﬂow is presented in Figure 6.19, to explain the concept of back analysis.

6.4.2.2. Computational results. The back analysis considers two mechanic models. One is to assume rock as uniform and isotropic medium with no joint. The material parameters are E, n, c and f. The other is to treat rock as an anisotropic media with one joint set. The parameters are E, Kn and Ks to be obtained by back analysis using corresponding constitutive relationship. (a) Access tunnel. In case of isotropic rock, two sets of input displacements are taken in back analysis. Upon optimisation back analysis, the relevant parameters are

184

Chapter 6

Figure 6.19. The complete cycle of program ﬂow.

obtained as listed below: E ¼ 3:70 104 MPa s1 ¼ 12:87 MPa

v ¼ 0:15 s3 ¼ 5:47 MPa

F ¼ 0:6 103 a ¼ 50

In case of anisotropic rock, in situ stresses are assumed constant, the back analysis is conducted by inputting in situ stress and corresponding displacement set. The empirical formulae Ks ¼ 1/5 1/10Kn (Belytschko et al. 1984), is taken as a constraint condition. The back analysis results are list below: E ¼ 6:419 104 MPa F ¼ 0:59 103

Kn ¼ 7:25 105 MPa=cm Ks ¼ 0:91 105 MPa=cm

(b) Ventilation tunnel In case of isotropic rock, the back analysis uses modiﬁed displacements at measurement holes MP-4, MP-2. Assuming the Poisson’s ratio n ¼ 0.15, optimised back analysis gives: E ¼ 3:77 104 MPa F ¼ 0:1 102

Back Analysis and Observational Methods

185

s1 ¼ 10:02 MPa s3 ¼ 6:78 MPa a ¼ 25 In case of anisotropic rock, taking the above in situ stress as input data, the back analysis results are: E ¼ 5:91 104 MPa F ¼ 0:2 102 Kn ¼ 48:65 104 MPa=cm Ks ¼ 6:08 104 MPa=cm The measured displacements and back analysis results for the access tunnel and the ventilation tunnel are compared as shown in Figures 6.20 and 6.21. It can be seen that the results agree well with each other. Inputting the above back analysis results for each case into the ordinary ﬁnite element modelling, the results are compared with the modiﬁed displacement measurements as listed in Tables 6.3 and 6.4. It can be seen that the results obtained from back analysis by combining the access tunnel and the ventilation tunnel are better. For each MPBX, the correlation of measured and back-analysed results is better at greater depth than that at shallower depth. This may be because the rock at shallower depth is greatly disturbed by blasting vibration.

6.4.3

Stability prediction of powerhouse and transformer chamber

The powerhouse was constructed by using multiple-stage excavation. In the ﬁnite element modelling, however, only the ﬁrst and last stages are involved. In this section, only the latter is discussed. Two rock mechanical models are considered in the modelling: uniform linear elastic rock with no joint and anisotropic rock mass with joints. In adopting with joints, the joint inﬂuences are estimated by equivalent method discussed in Chapter 3. Two joint sets are considered and equivalent deformation modulus and equivalent strength parameters are obtained. Three key faults are involved in each modelling. The damage distribution is shown in Figure 6.22. The mesh of the computational model has 596 elements and 589 nodes. The in situ stresses of ventilation tunnel are obtained by using ﬁnite element modelling as: sx ¼ 9.42 MPa, sy ¼ 7.369 MPa, txy ¼ 1.245 MPa.

186

Chapter 6

Figure 6.20. Comparison of monitored and back-analysed displacement of the access tunnel.

Material parameters of rock are: E ¼ 3.7 104 MPa; n ¼ 0.20; c ¼ 22.27 MPa; f ¼ 48.1o; st ¼ 5.0 MPa. Joint stiﬀness are estimated from back analysis, and joint strength parameters are estimated from a similar project. The joint parameters are: Kn ¼ 6.0 105 MPa/cm; Ks ¼ 7.5 103 MPa/cm; c ¼ 0.5 MPa; f ¼ 35 .

Back Analysis and Observational Methods

187

Figure 6.21. Comparison of monitored and back-analysed displacements for the ventilation tunnel.

Faults material parameters are: E ¼ 9.0 103 MPa; n ¼ 0.25; c ¼ 0.4 MPa; f ¼ 30 ; st ¼ 1.0 MPa. The layouts of convergence measurements in the powerhouse and transformer chamber are shown in Figure 6.23. The displacement prediction results are listed in

188

Chapter 6

Table 6.3. Comparison of measured and computed results in access tunnel. Measurement holes

MJ-1

MJ-2

MJ-3

MJ-4

Remark

Measurement points (mm) 1

2

3

4

0.28 0.38 0.22 0.25 0.13 0.08 0.20 0.34 0.02 0.03 0.21 0.11 0.08 0.35 0.26 0.25

0.56 0.64 0.47 0.46 0.35 0.25 0.39 0.79 0.12 0.08 0.60 0.21 0.11 0.62 0.51 0.47

0.65 0.72 0.64 0.61 0.63 0.54 0.56 1.29 0.16 0.18 0.97 0.28 0.74 0.74 0.71 0.68

0.73 0.76 0.77 0.78 0.76 0.93 0.75 1.82 0.45 0.29 1.34 0.31 0.84 0.75 0.91 0.84

monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2)

Table 6.4. Comparison of measured and computed results in ventilation tunnel. Measurement holes

MP-4

MP-3

MP-2

MP-predrilled hole

Remark

Measurement points (mm) 1

2

3

4

0.35 0.45 0.38 0.36 0.25 0.07 0.15 0.13 0.30 0.10 0.23 0.20 0.58 0.12 0.10 0.09

0.50 0.75 0.63 0.60 0.39 0.28 0.60 0.50 0.52 0.39 0.57 0.53 0.77 0.34 0.30 0.29

0.82 0.95 0.95 0.92 0.49 0.64 1.08 0.87 0.62 0.50 0.63 0.59 0.99 0.77 0.69 0.67

0.99 1.02 1.13 1.10 0.55 0.79 1.07 1.02 – – – – 1.09 0.98 1.06 1.04

monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2)

Back Analysis and Observational Methods

189

Figure 6.22. Distribution of damage zones.

Figure 6.23. Layout of convergence measurements.

Table 6.5. From the table, it can be seen that the convergence in the case of considering the joint inﬂuence is about 10% greater than that in the case of no joint. The displacements on tunnel walls and rock failure zone by equivalent model are shown in Figure 6.22. It can be seen that in addition to fault zone failure, shear failure occurs at top and bottom of the tunnel, and tensile failure on the sidewalls.

190

Chapter 6

Table 6.5. Predicted displacement of convergence (mm). Measurement lines Equivalent approach Linear elastic

6.5.

1

2

3

4

5

6

7

8

9

11.4 9.7

18.5 15.1

19.5 16.3

6.0 5.0

6.9 5.8

7.3 6.1

10.8 9.0

7.4 6.3

6.9 5.8

THREE-DIMENSIONAL BACK ANALYSIS OF ANISOTROPIC ROCK

The example using 3-D back analysis of anisotropic rock mass is shown in this section, in a large-scale hydroelectric power cavern complex. The hydroelectrical power is 70 m high and 30 m wide and constructed underground at the downstream of the right side of a river. The overburden depth is 30–107 m. Two sides of the river are symmetric with average slope of 45–50 . The project is very complicated with the following problems. (a) Poor engineering geology conditions: The rocks encountered are mainly killas and chorismite with intrusion of other types of rock which are heavily jointed. Breaking of rock in the tunnel occurs easily due to the excavations. The rocks are divided into three types. There are several faults intersecting the powerhouse that may induce instability to the tunnel walls. (b) High horizontal in situ stress: The site investigation shows that the horizontal in situ stress is more than twice of the vertical in situ stress. (c) Faults: There are a number of faults intersecting the complex and in the vicinity. Experimental and numerical analyses show that the faults intersecting the powerhouse cavern may be slipped during excavation. In summary, this project is constructed under high in situ stresses and complicated ground conditions with non-uniformity, anisotropy and discontinuities. Therefore, displacement monitoring and back analysis are performed to understand the rock mass behaviour and in situ stresses [8,52].

6.5.1

Displacement monitoring in trial tunnel and results

6.5.1.1. Set-up of displacement monitoring. In order to understand the behaviour of underground rock masses during excavation, a pilot tunnel of 1/6 of the cavern size was excavated at the cavern position. The pilot tunnel is 5 m wide, 9.5 m high and 54 m long. The pilot tunnel crosses all the three rock types, and three sections are set up for displacement monitoring. Seven boreholes for MPBX installation are drilled in Section I and II separately, and four are drilled in Section III. Each section has a

Back Analysis and Observational Methods

191

Figure 6.24. Setting up of MPBX in Section II.

pre-embedded MPBX to monitor whole deformation process. The pre-embedded MPBX is drilled from the surge chamber to the wall of the pilot tunnel as shown in Figure 6.24. Each monitoring section has 5–7 measurement points on the wall to form more than six convergence monitoring lines to correlate with results obtained by extensometers. In addition, some tests are conducted in the pilot tunnel including measurements of shotcrete and rock bolt stresses, rock bolt pulling test and seismic measurement of rock disturbance zone. 6.5.1.2. Monitoring results. The displacement monitoring results are given here with a period of 250 days until all the displacements become stable. It shows: (a) Displacement curve tends to be stable in 60–80 days. A typical displacement set is shown in Figure 6.24. A convergence measurement set is shown in Figure 6.25. (b) The smallest deformation occurs in Section II due to the good rock quality, while the largest displacement is observed in Section III due to the weak rock. Deformation in Section I is remarkable due to the faults. However, in overall, all deformations are quite uniform and not extensive. (c) The disturbance zone of surrounding rock is about 1–1.5 times of the tunnel size.

6.5.2

Back analysis

Three-dimensional back analysis is conducted by considering anisotropic rock, complex ground condition and that the in situ stress axis does not coincide with the

192

Chapter 6

Figure 6.25. A typical displacement set.

Figure 6.26. A convergence measurement set.

tunnel axis. Only the monitored displacements at middle measurement points are used in the back analysis, because it is found that the monitored data at two ends of MPBXs have relatively large error and low reliability. The procedure of back analysis modelling is similar to that outlined in Figure 6.19 of Section 6.4.2. The comparison of modelled results and monitored data for Section II are summarised in Table 6.6. The back-analysed material parameters of the anisotropic rock mass and in situ stresses are presented in Tables 6.7 and 6.8. The results indicate that the backanalysed results are lower than the results obtained from laboratory tests. The results also reﬂect the scale eﬀects. In general, large-size rock masses have lower strength and modulus. The back-analysed in situ stresses agree well with those measured, as shown in Table 6.7. From the above results, it can be seen that satisfactory results are obtained from 3-D non-linear regression based on monitoring displacements. Figure 6.27 shows modelled displacement curves compared with monitored

193

Back Analysis and Observational Methods Table 6.6. Comparison of calculated and monitored results in Section II. Measurement lines Relative displacement

Calculated Monitored Calculated Monitored Calculated Monitored Calculated Monitored Calculated Monitored

u1 u0 u2 u0 u3 u0 u4 u0 u5 u0

II-1

II-2

II-3

II-4

II-5

II-6

II-7

0.18 0.19 0.44 0.46 0.68 0.64 1.13 1.13 2.46 2.78

0.14 0.18 0.36 0.40 0.76 0.67 1.21 1.23 2.66 3.14

0.10 0.56 0.58 0.86 0.90 1.22 1.53 1.48 2.98 3.27

0.15 0.15 0.76 0.70 1.09 1.03 1.67 1.57 2.73 2.50

0.15 0.16 0.68 0.37 0.49 0.67 2.03 2.06 2.90 2.99

0.24 0.17 0.67 1.59 1.64 1.78 2.36 2.25 3.57 3.21

0.23 – 0.74 – 1.58 – 2.34 – 3.36 –

Table 6.7. Mechanical parameters of rock mass obtained by back analysis. Classification of rock mass A B1 B2 C f27 f25

E||( 103 MPa)

E?( 103 MPa)

u||

u?

17.0 13.5 11.0 8.5 1.0 2.0

11.5 9.0 7.4 4.3 – –

0.27 0.28 0.28 0.29 0.3 0.3

0.27 0.28 0.28 0.29 – –

Table 6.8. Comparison of calculated and measured in situ stresses. In situ stress

sx

sy

sz

txy

tyz

ca

Measured Calculated

5.9 5.4

7.2 6.7

7.7 8.0

0.3 0.9

0.2 0.4

0.6 0.5

Figure 6.27. Modelled displacement curves compared with monitored displacement curves in two typical MPBXs.

194

Chapter 6

displacement curves obtained from multiple-point extensometers for two typical MPBXs. Again, good matches are observed. 6.6.

THREE-DIMENSIONAL BACK ANALYSIS OF JOINTED ROCK MASS AND STABILITY ANALYSIS

A 2-D back analysis of jointed rock mass for a pumped storage hydroelectric power station was conducted in Section 6.4. The analysis was based on the monitoring information of two branch tunnels, before the excavation of the powerhouse cavern. This section is to illustrate, for the same project, the 3-D back analysis based on monitoring of the powerhouse cavern. Brief concept of the method will be introduced and some main results will be discussed in this section. The theory has been discussed in the previous sections. 6.6.1

Mechanic model

Similar to the example in Section 6.4, the equivalent model of jointed rock is also used in this 3-D back analysis case. For the joint persistence, the 3-D back analysis uses area equivalence, while 2-D uses line equivalence. The material parameters of rock and joints are obtained from approximate weighted average based on the joint arrangement. The joint persistence is taken as 80% from site investigation results. The material parameters of rock element used in the 3-D modelling are E, n, cr, fr and st, where st is tensile strength of rock. The parameters used for joints are Kn, Ks, cj, fj and aj, where aj is the joint dip angle. Equivalently transferring jointed rock mass into quasi-continuous rock mass, new stiﬀness matrix [D] can be obtained. It is a 6 6 full matrix for 3-D (3 3 matrix for 2-D). The new matrix will replace elastic matrix in the ﬁnite element modelling for the rock mass.

6.6.2

Summary of site monitoring data

Five monitoring sections are set up in the powerhouse cavern, while two monitoring sections are set in the transformer chamber cavern. Altogether, a total number of 36 monitoring MPBXs and 144 monitoring points are installed in those sections. Upon the completion of powerhouse and transformer chamber constructions, it is found that only data at 90 monitoring points in 28 MPBXs give eﬀective data. After data sorting, the data from 15 points in 3 MPBXs have clean trends and are considered reliable, and these data are used in the 2-D and 3-D back analysis.

195

Back Analysis and Observational Methods

Figure 6.28 shows the multiple-stage excavation of the powerhouse and the transformer chamber and the layout of instruments. The instrument details and monitoring data in powerhouse are shown in Tables 6.9 and 6.10. Data in Table 6.9 are used for 2-D analysis and data in Table 6.10 are used for 3-D analysis. From the monitoring data it can be seen that the displacements are small with a maximum value of 6 mm close to the right side of powerhouse cavern. In Table 6.10, the n horizontal lines refer to (n 1) stage excavation. The vertical lines across the horizontal lines represent excavation face position and

Figure 6.28. Excavation stage of the caverns and layout of instrumentation.

Table 6.9. Measured displacement by MPBX (mm). 1

2

3

4

Original status

M1-3 M1-5

– –

2.73 –

1.12 2.44

3.11 4.12

M1-7 M1-9

0.14 0.17

1.01 –

1.04 0.90

3.51 1.72

M2-2 M2-6

1.31 3.57

1.47 –

– 5.25

2.28 6.05

After excavation of Stage IV 1.5 m away from excavation face in Stage I

M5-1 M5-2

– –

0.11 0.72

0.30 2.44

0.60 5.05

After excavation of Stage II

After excavation of Stage II 5 m away from excavation face in Stage I After excavation of Stage I After excavation of Stage V

196

Chapter 6

Table 6.10. Instrumentation set-up and measured displacement. Arrangement

Point

Depth (m)

Measured value (mm)

M13 5

8

2.44

M14

5

23

4.12

M21 6

1

3.57

M23 6

8

5.25

M24 6

23

6.05

CM11 2

1

0.09

CM12 2

3

0.63

CM14 2

19

0.85

4

CM14 3

19

0.71

5

CM11 4

1

0.42

CM13 4

8

1.14

CM14

4

19

2.41

6

CM22 2

3

35

7

CM23 3

8

0.34

8

CM21 4

1

0.91

CM23

8

1.72

M11 7

1

0.14

M13 7

3

1.01

M13 7

8

1.04

M14

7

23

3.51

M12 3

3

2.73

M13 3

8

1.12

M14 3

23

3.11

M52 1

3

0.11

M53 1

8

0.30

M54 1

23

0.60

1

2

3

9

10

11

4

Location

1 5m

1 2

1.5m

2

1 1m

1

2 2

1 1

5 5

(continued)

197

Back Analysis and Observational Methods Table 6.10. Continued. Arrangement

12

13

14

15

Point

Depth (m)

Measured value (mm)

Location

M52 2

3

0.72

M53 M54

2

8

2.44

2

23

5.05

CM21 5

1

0.09

CM23 5

6

0.34

CM24 5

12

1.05

2

M21 2

1

1.31

2

M23 M24

2

3

1.47

2

12

2.28

M11 5

1

0.17

M13 M14

5

6

0.90

5

12

1.72

2

2 1 1

that non-crossing the horizontal lines mean that the excavation stage has been completed. 6.6.3 Finite element back analysis of underground powerhouse complex Two-dimensional modelling uses 630 nodes and 635 elements, while 3-D modelling uses 4304 nodes and 3994 elements. Rock material parameters adopted in the 2-D back analysis are given in Section 6.4. To reduce the mesh preparation work, all the computations use the same mesh arrangement as shown in Figure 6.29. However, mesh relating to the excavation will be changed to simulate the multiple-stage excavation process. Upon the back analysis, the ﬁnal rock material parameters are obtained as following: E ¼ 3:7 104 MPa n ¼ 0:24 c ¼ 1:29 MPa f ¼ 41 st ¼ 3:5 MPa

198

Chapter 6

Figure 6.29. 3-D FEM model mesh.

Joint parameters are: Kn ¼ 6:0 105 MPa Kt ¼ 7:5 104 MPa c ¼ 0:5 MPa f ¼ 35 In situ stresses are: sx ¼ 8:95 MPa sy ¼ 12:84 MPa sz ¼ 6:89 MPa txy ¼ 0:58 MPa tyz ¼ 0:38 MPa tzx ¼ 0:41 MPa Comparing above results with results obtained in Section 6.4, there is no much diﬀerence, except the rock material strength parameters.

Back Analysis and Observational Methods

199

The results from the 2-D back analysis and from the monitoring in the powerhouse and the transformer chamber are compared in Figures 6.30 and 6.31. The 3-D back analysis results and the monitoring data in the powerhouse are compared in Figure 6.32.

Figure 6.30. 2-D back analysis and monitoring in the powerhouse.

Figure 6.31. 2-D back analysis and monitoring in the transformer chamber.

200

Chapter 6

Figure 6.32. 3-D back analysis and monitoring data in the powerhouse.

Figure 6.33. Displacement distribution and failure zone in the rock mass around the cavern.

6.6.4

Stability of powerhouse and transformer chamber

Non-linear analysis of powerhouse and transformer chamber using above back-analysed parameters provides the displacement distribution in the surrounding rock mass. The displacement distribution and failure zone in the rock mass around the cavern at a particular section are shown in Figure 6.33. The displacements at roof are smaller than those at the walls and the bottom of the powerhouse. The upward displacement at the lower corner of powerhouse is the largest. Shear failure occurs locally. The above results are obtained in

Back Analysis and Observational Methods

201

the case of no supports. The rock mass can be stabilised by reinforcement and support. From the above analysis results, it can be concluded that the mechanics models, especially the equivalent continuous model for the jointed rock mass proposed, are applicable and validated. In addition, the rock mechanic parameters from the back analysis using those analytic models have given good results. Therefore, the analytical methods are further veriﬁed.

6.7.

APPLICATIONS OF STATISTICS MODEL IN DEFORMATION PREDICTION

In underground constructions, monitoring of rock mass deformation becomes more and more important and is widely applied. Especially when the observational construction method is adopted in tunnelling. The deformation monitoring provides the basic information for back analysis. The information obtained from deformation monitoring can be used in at least three aspects. Firstly, it is used to predict the safety during construction. The deformation monitoring can reﬂect the total deformation and deformation rate. Secondly, ordinary conditions including in situ stress and material parameters can be back analysed. Lastly, the monitored deformation can be used to predict the rock behaviour in subsequent construction [53–56,398,405,406]. This section is to discuss the last application. There are several methods to predict rock parameters and behaviour in subsequent construction, for example, the mathematical analytical method. In usual analytical method, the rock model with material parameters and in situ stresses must be determined in advance and provided to the model as input data. In back analysis method, the in situ stresses and material parameters are obtained from back analysis with monitored data and computational work. However, the statistic method is aimed at saving computational work. The method relies on the observed rock behaviour and monitored data, develops the inter-relation between the properties and behaviour, and predicts the behaviour of the rock [407–413]. This method can accordingly avoid inﬂuences of many mechanical uncertainties and provides results in short time and with high accuracy. There are many methods to be used for data regression in statistics approach, such as Laglongi, power sliding, spline, regression model, time series analysis and grey system theory [408,410,412,414–417]. In this section, a method combining non-linear regression method and modiﬁed grey system model is introduced through an example to predict displacement in a project.

202 6.7.1

Chapter 6 Non-linear regression model

As a statistical model, the regression model requires to make judgement on the data distribution and to assume a model in advance based on the user’s experience and skills. Then the parameters of the given model are recognised by least square estimation of error square sum function. Finally, statistic checking is conducted and prediction is made. Presently, linear regression model is well developed. For general non-linear regression model, the least square estimation of parameters is often carried out by using numerical modelling [24,30,33]. It is transformed to general linear model by mathematical transformation under certain conditions, and then obtain least square estimation of parameters. From the features of rock deformation curves, the regression model is assumed as hyperbolic function

u¼

t A þ Bt

ð6:28Þ

where {ti} denotes time sequence, {ui}, (i ¼ 1, 2, . . . , n) is the measured displacements. Let y ¼ t/u, then, Y ¼ A þ Bt

ð6:29Þ

So it is transformed to be a linear regression model. Setting n 1X t¼ ti n i¼1

y¼

Ltt ¼

n 1X yi n i¼1

n X

ðti tÞ2

9 > > > > > > > > > > > > > > > > > > > > > > > > =

> > > > > > > n > X > 2 > > Lyy ¼ ðyi yÞ > > > > i¼1 > > > > > n > X > > Lyt ¼ ðti tÞðyi yÞ > ; i¼1

i¼1

ð6:30Þ

Back Analysis and Observational Methods

203

It is known that B ¼ Lyt =Ltt A ¼ y Bt The linear relationship of variable y and t can be expressed by using relation coeﬃcient, Lyt r ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ltt Lyy Obviously |r| 1. When |r| ¼ 1, it becomes fully linearly related. When |r| ¼ 0, it becomes fully linearly unrelated. The nearer |r| to 1, the more linearly it is related. Maximum displacement is given as: u1 ¼ lim

t!1 A

t 1 ¼ þ Bt B

ð6:31Þ

It is the predicted ﬁnal displacement. The initial displacement rate is given as: u0t!0 ¼ 1=A From the above two formulae, the physical meaning of A and B can be recognised.

6.7.2

Grey system theory model

Because all the grey models GM have the same basic conditions and principles, the basic grey model GM(1,1) is used here for analysis [414,415]. The grey theory assumes ð0Þ ð0Þ discretised source data series as uð0Þ ¼ fuð0Þ 1 , u2 , . . . , un g, by conducting one accumulað1Þ ð1Þ tion forming treatment (AGO), generates a forming series uð1Þ ¼ fuð1Þ 1 , u2 , . . . , un g, and then creates a one-stage deviation equation, GM(1,1) is created as duð1Þ þ auð1Þ ¼ b dt

ð6:32Þ

b at b ð0Þ ¼ u1 e þ a a

ð6:33Þ

The solution of GM(1,1) is: u^ ð1Þ tþ1

204

Chapter 6

Equation (6.33) is the grey prediction formulae. Deviating uð1Þ tþ1 , we can get the prediction formulae of source data as:

ð0Þ at u^ ð0Þ tþ1 ¼ au1 þ b e

ð6:34Þ

GM(1,1) requires equal time step. However, monitoring data do not satisfy this requirement. Local internal insertion or smooth treatment can be used to form equal time step. The GM(1,1) model of equal time step is subsequently improved, to make it applicable for solving problems of non-equal time step. The improved model is applied to predict single pile capacity and showed that it was eﬀective. The key feature of non-equal time step model is to replace grey deviation by the diﬀerence form: dxð1Þ xð1Þ ðk þ 1Þ xð1Þ ðkÞ xð0Þ ðk þ 1Þ ¼ 0 ¼ ð1Þ t ðk þ 1Þ t0 ðkÞ t ðk þ 1Þ dt

ð6:35Þ

The problem now is how to take the background value of dx(1)/dt. It was noted that this value does not change signiﬁcantly when x changes from x(1)(k) to x(1)(kþ1) if t(0)(kþ1)t(0)(k) is suﬃciently small. Therefore, we have zð1Þ ðk þ 1Þ ¼ 0:5xð1Þ ðk þ 1Þ þ 0:5xð1Þ ðkÞ as the background value of dx(1)/dt between t(0)(k) and t(0)(k þ 1). The background value of dx(1)/dt can further be improved based on the features of rock deformation curves that change sharply in the initial stage. Usually u–t curves is up convex, the background value of dx(1)/dt can be taken as:

zð1Þ ðk þ 1Þ ¼ xð1Þ ðkÞ þ xð1Þ ðk þ 1Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ xð1Þ ðkÞxð1Þ ðk þ 1Þ

ð6:36Þ

From equations (6.31), (6.34) and (6.36), we have, xð0Þ ðk þ 1Þ þ atð1Þ ðk þ 1Þzð1Þ ðk þ 1Þ ¼ btð1Þ ðk þ 1Þ

ð6:37Þ

205

Back Analysis and Observational Methods Based on least squares method, we have, 1 a ¼ BT B BT Yn b

ð6:38Þ

where 2 6 6 6 6 6 6 ½B ¼ 6 6 6 6 6 6 4

tð1Þ ð2Þ

0 tð1Þ ð3Þ : : :

0 2

3 7 7 7 7 7 7 7 7 7 7 7 7 5

tð1Þ ðnÞ h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃi xð1Þ ð1Þ þ xð1Þ ð2Þ xð1Þ ð1Þxð1Þ ð2Þ h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃi xð1Þ ð2Þ þ xð1Þ ð3Þ xð1Þ ð2Þxð1Þ ð3Þ

6 6 6 6 6 6 6 : 6 6 6 : 6 4 h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃi xð1Þ ðn 1Þ þ xð1Þ ðnÞ xð1Þ ðn 1Þxð1Þ ðnÞ

1

3

7 7 7 17 7 7 7 7 7 7 7 5 1

3 xð0Þ ð2Þ 6 xð0Þ ð3Þ 7 7 6 6 : 7 7 6 Yn ¼ 6 7 6 : 7 4 : 5 xð0Þ ðnÞ 2

The measured displacements in underground excavation consist of time series {ti} and corresponding displacement series {ui}. The displacement prediction formulae is obtained from the monitored data as b b u€ kþ1 ¼ u1 eaðt1 tkþ1 Þ þ a a

ð6:39Þ

206

Chapter 6

where 2 6 6 6 ½B ¼ 6 6 6 4

3 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðu1 þ u2 u1 u2 Þ 1 7 6 ðu2 þ u3 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ u2 u3 Þ 17 7 7 6 7 7 6 : 7 76 7 7 6 : 7 7 6 5 5 4 : p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðun1 þ un un1 un Þ 1 tð1Þ ðnÞ

tð1Þ ð2Þ

0

tð1Þ ð3Þ : : : 0

3

2

3 u2 u1 6 u3 u2 7 7 6 7 6 : 7 6 Yn ¼ 6 7 : 7 6 5 4 : un un1 2

The stability of the model is often examined by backward error ratio c and small error ratio p [111]. Error is given as: "ðiÞ ¼ uðiÞ u^ ðiÞ i ¼ 1, 2, . . . , n Mean error is given as: " ¼

n 1X "ðiÞ n i¼1

Variance is given as: s21 ¼

n 1X ½"ðiÞ " n i¼1

Mean of source data is calculated by: u ¼

n 1X uðiÞ n i¼1

Variance of source data is calculated by: s22 ¼

n 1X ½uðiÞ u n i¼1

207

Back Analysis and Observational Methods Table 6.11. Classification of accuracy grade. Accuracy c p

Excellent

Good

Fair

Bad

0.95

0.80

0.70

0.65 0.70

Posterior error ratio is estimated by: c¼

s1 s2

Probability of small error is estimated by: p ¼ p "ðiÞ " < 0:6745s2 A small value of c represents a good model. A small value of c generally leads to a large s2 and a small s1. Small s1 represents small error deviation, and small c means small diﬀerence between the computational data and the actual data. A large value of p represents a good grey model, and means more points with small diﬀerence between the error and the mean of error (< 0.6745s2). The accuracy can be divided into several grades based on c and p value as shown in Table 6.11. If the c and p values are within the allowable range, the model can be used to make prediction. Otherwise, the model needs to be modiﬁed until the accuracy is in a satisfactory range.

6.7.3

Engineering application

The grey model is applied to a trial tunnel of a hydroelectric power station. The layout of extensometers is shown in Figure 6.34. Some regression displacement curves and monitoring curves are shown in Figure 6.35. The monitoring displacements and prediction results in later days are listed in Table 6.12. From Figure 6.35 and Table 6.12, it can be seen that hyperbolic model and grey model all agree well with monitoring data. The ﬁnal monitoring displacements are between the two predictions. Therefore, the use of both models is suggested.

6.7.4

Discussion

The grey model can be created only when the source series is a smooth discrete function. It requires the source series determined or has a determinable trend.

208

Chapter 6

Figure 6.34. Layout of extensometers.

Figure 6.35. Regression displacement curves and monitoring curves.

However, the feature of the grey model is to treat source series and to generate forming series, and to describe x(1) by using the grey amplitude and indirectly describe x(0). Therefore, in practice, the grey model does not require x(0) to be a smooth discrete function, but allows x(0) to have certain ﬂexibility. This extends

209

Back Analysis and Observational Methods Table 6.12. Comparison of measured and predicted displacements. Time (day)

I-3-1 M

140 149 167 173 Infinity

5.28 5.31 5.33 5.30

I-3-2

HM GM 5.29 5.30 5.32 5.32 5.49

5.21 5.21 5.21 5.21 5.21

M 5.46 5.47 5.49 5.46

II-1-1 94 103 118 128 Infinity

2.29 2.32 2.31 2.32

2.26 2.29 2.33 2.36 2.68

I-3-3

HM GM 5.38 5.39 5.42 5.43 5.66

5.26 5.27 5.28 5.28 5.29

M 5.16 5.18 5.21 5.14

II-1-2 2.14 2.22 2.32 2.37 2.77

2.57 2.59 2.59 2.62

2.58 2.60 2.62 2.64 2.83

I-3-4

HM GM 5.07 5.10 5.13 5.14 5.47

4.92 4.93 4.95 4.96 4.98

M 5.10 5.11 5.13 5.03

II-1-3 2.49 2.50 2.51 2.52 2.52

1.44 1.46 1.43 1.41

1.46 1.47 1.48 1.50 1.62

I-3-5

HM GM 4.81 4.84 4.87 4.88 5.15

4.94 4.98 5.05 5.09 5.21

M 4.17 4.16 4.18 4.15

II-1-4 1.50 1.51 1.52 1.53 1.54

2.15 2.14 2.13 2.10

2.41 2.15 2.16 2.17 2.26

HM GM 4.04 4.05 4.08 4.09 4.31

4.15 4.19 4.28 4.31 4.60

II-1-5 2.19 2.19 2.20 2.20 2.21

1.66 1.65 1.63 1.61

1.56 1.57 1.59 1.60 1.71

1.57 1.59 1.62 1.63 1.68

Note: (1) M – Measured; HM – Hyperbolic Model; GM – Grey Model; (2) I-3-1 refers to measurement point 1 in Section I MPBX 3, similarly for other points.

the application range of the grey model. The behaviour of rock displacement in underground projects makes the grey model more suitable and applicable. However, all models are created based on the existing monitoring information. If geological conditions of the excavation are changed, the models need to be re-created or otherwise the expected prediction accuracy will become low. Generally the grey model is applicable in predicting ﬁnal displacements in underground engineering. However, hyperbolic model is applicable for describing monitoring displacements that follows a simple function. It is not suitable for other situations.

This Page Intentionally Left Blank

Chapter 7

Construction Mechanics and Optimisation of Excavation Schemes Construction of rock engineering projects, especially large-scale underground excavations in rock masses, usually requires a long period to complete, i.e., a few months to a few years. To the extreme, a mining project may take tens or hundreds of years. The construction of these rock engineering projects will disturb the initial stable state of the rock masses. After that, various rock mass parameters interact in a dynamic interactive process until the rock mass reaches a new stable state [1–3]. The construction is therefore a dynamic interactive process in time and in space. The success in constructing and managing a rock project not only depends on the eventual state of the project, but also on the interim process and the construction methods adopted [63–65,69,70]. Construction of large-scale rock projects is implemented by continual excavation of new working faces. Each newly excavated face interacts dynamically with the existing excavated space in time and in space [63,64,66,69]. Rock mass is a geological medium, which is generally discontinuous, inhomogeneous and anisotropic. The rock mass has uncertain and variable parameters, which are further changed by the engineering activities. It is very important to consider the eﬀects of engineering activities, especially under complex geological conditions such as high rock stress, weakness zones, faults, joints, and ground water [69,180,348,350,406]. For any large-scale rock engineering project, the eﬀects of engineering activities must be taken into account for the design and construction. From the viewpoint of mechanics, the dynamic interactive process of rock engineering constructions is non-inverse and non-linear. Its eventual state (or solution) is not unique but changeable with the interim process [1–3]. In other words, the eventual state is strongly dependent on the stress paths or stress histories. This leads to the necessity of the optimisation of construction process. It is generally impossible to complete the construction of any large-scale projects just at one stage of full face excavation, especially for large underground caverns. In practice, they are constructed step by step following a certain sequence of excavation [67,68]. The excavation scheme is decided according to the layout of access tunnels, types of tunnelling machines and characteristics of rock masses. In the series of sequential construction, each step of construction corresponds to a certain type of temporary cavern geometry, i.e., diﬀerent sequences of construction correspond to the diﬀerent temporary loading conditions. During and after the 211

212

Chapter 7

construction, rock mass parameters are aﬀected by the continually variable cavern geometry and loading conditions. Typical rock mass parameters are rock stress, rock damage and circumferential displacement around the cavern. The disturbance to the surrounding rock mass and the redistribution of rock stress are primarily due to excavation activities. Therefore, excavation sequences and associated methods have widely crucial implication on the deformation and stability of underground excavation [418–437]. Many engineering measures have been proposed to stabilise the large-scale underground excavations. Of them, the most eﬀective measure is to adopt a proper excavation sequence and to install eﬀective rock reinforcement [418–426]. This forms the principle of the construction mechanics. The initiation and development of the construction mechanics are presented and discussed in various literatures [8,9,52,63,68,404,431]. It was proposed that the excavation steps should be reduced to a minimum number, because the rock mass is of very poor strength against cyclic loads and movements [68,404]. Some researchers studied the eﬀects of excavation sequences on the rock stress distribution at diﬀerent points of surrounding rock mass [68]. Some suggested that the excavated geometry should match the initial distribution and orientation of ﬁeld stresses [194,212,432]. Zhu et al. [52] conducted comparative studies on multiple schemes of diﬀerent excavation sequences, and obtained an optimum scheme for improving the surrounding rock mass. Yu and Yu [433] presented a new concept of rock memory and emphasised the importance of analyses on the functions of surrounding rock masses. The basic principles and applications of construction mechanics are brieﬂy discussed in the following sections.

7.1.

7.1.1

BASIC PRINCIPLES OF INTERACTIVE CONSTRUCTION MECHANICS

Basic principles

It has been commonly recognised that the schemes and sequences of rock excavations and supports have signiﬁcant eﬀects on the stabilisation and cost of underground rock engineering projects. The approach of the New Austrian Tunnelling Method (NATM) actually reﬂects the signiﬁcant eﬀects imposed by excavation activities. NATM emphasises the eﬀects of excavation methods and attends to the economical aspects of construction [438]. It does not, however, examine the mechanisms of construction from a wider viewpoint of philosophy. It is necessary to examine a new concept in rock mechanics and engineering – interactive construction mechanics of rock engineering. The concept of interactive construction mechanics applies the rock mechanics theory to rock engineering practice, by examining the interactive mechanics in the

Construction Mechanics and Optimisation of Excavation Schemes

213

excavation and support process [8,9,434]. The principles of the interactive construction mechanics are outlined in detail in the following paragraphs. (a) Rock engineering construction under complex rock mass conditions is a complete open system. This open system is aﬀected by the uncertainty in natural geological factors. As a result, analyses of rock mass stability and estimation of construction cost become a systematic work. In order to take the full consideration of the eﬀects of various factors, it is required to study not only the natural factors (e.g., geological conditions, initial stress and mechanical properties of rock mass), but also human factors (excavation sequence, excavation method and geometry). The main idea of this principle is to view a rock engineering project as an open and interactive system rather than a close and static system. The interactive viewpoint emphasises both the natural factors and the human activities. For example, optimisation methods may be adopted in advance at the design stage to minimise the potential problems. At construction stage, pre-measures may be taken to reduce the possibility of anticipated problems. Optimisation analyses should be conducted in advance. (b) Rock mass stability and economic aspects during and after construction are not only related to the eventual states, but also to the excavation sequences and methods. This is because the boundaries of excavated rocks change in time and in space. From the mechanics point of view, construction is a non-linear process that is related to both the eventual states and stress paths as well as stress histories. The sequences of large cavern construction under complex geological conditions can signiﬁcantly inﬂuence the safety and economy of the caverns during excavation and operation. The cavern excavation is actually a process of loading and unloading the rock masses at diﬀerent time and positions. As the surrounding rock mass is a non-linear mechanical medium, the diﬀerent excavation sequences imply diﬀerent stress paths and histories imposed on the rock mass. Hence, it produces very diﬀerent damage to the rock mass. The diﬀerence in damage produced by the various sequences of excavation may be very signiﬁcant. (c) The approach should be adopted before the construction to determine the optimum excavation schemes for ﬁnal decision-making. In the optimisation analyses, rock supports should be considered as an important factor. Optimisation analyses on the excavation sequences should be taken after the cavern design but before the actual construction, in view of the mechanics of excavation process. (d) The design and control of construction process have signiﬁcant eﬀects on the rock engineering projects. It is important to understand the interaction and the response of surrounding rock masses, to adopt proper excavation methods and

214

Chapter 7

rock supporting measures, and to minimise the potential instability. It emphasises the control on the construction process. Careful considerations must be taken to stabilise the rock masses by choosing very suitable construction technologies and methods, such as charge weight of blasting, instant rock bolting and shotcreting, and groundwater drainage. That is, the rock mass damages resulting from the excavating and supporting activities should be minimised. (e) The optimum construction scheme should be continually modiﬁed during the construction based on observation and monitoring of the surrounding rock mass. The information on rock mass conditions shall be updated during the construction, so that the existing scheme can be continually assessed and improved. The actual rock mass conditions may be diﬀerent from expected conditions during excavation due to the high uncertainty in natural geological environment. The traditional site investigation techniques can only access to a very small quantity of rock mass. Hence, in situ monitoring can be used to verify and modify the predictions of the rock mass. Furthermore, the new information obtained in monitoring can be used for back analyses, or statistical analyses. Accordingly, modiﬁcations can be made to the originally proposed mechanical model of rock masses, geological and mechanical parameters, and eventually to the rock structure design and rock supporting measures. The modiﬁcations should be continually conducted during the excavation. (f) Site investigation, design, construction and research shall be fully combined within an integrated system. It is unwise to strictly follow a speciﬁc scheme and to ignore the changes in rock mass conditions. Instead, modiﬁcation of the existing scheme should be continually carried out according to the current conditions. This principle emphasises the policy guarantee for the implementation of the above principles. It is because the construction of large-scale complex rock projects requires interactive management and close cooperation between geologists, design engineers, contractors, researchers, clients and site engineers. One of the diﬀerences between rock engineering projects and other civil engineering projects is that rock engineering projects require continual updating of the site data and modiﬁcations to the initially proposed design schemes. The excavation methods and schemes should have the ﬂexibility for timely updating and modiﬁcations, which need to be reﬂected in the design codes and the construction speciﬁcations.

7.1.2

Engineering applications

Applications of the construction mechanics to an engineering project are illustrated in this section to demonstrate the necessity and importance of optimisation analyses on excavation sequences.

Construction Mechanics and Optimisation of Excavation Schemes

215

7.1.2.1 Description of the project. The hydroelectric power project is a large underground cavern complex, consisting of a main powerhouse cavern, a transforming cavern and a tailrace cavern. The powerhouse cavern is 31.2 m wide and 72.6 m high, the transformer chamber is 17.4 m wide and 36.0 m high, and the surge chamber is 21.7 m wide and 76.9 m high. Bolts and shotcrete are needed for the rock reinforcements. The rock type is syenite. The in situ horizontal ﬁeld stress at the main cavern is about 25 MPa. The cavern complex is 200 m deep below ground [52]. The initial in situ stress distribution is analysed by the method described in Chapter 6. The plane strain model is used to analyse the changes in stress ﬁelds disturbed by diﬀerent excavation sequences. The ﬁnite element mesh is composed of 815 quadrilateral elements and a few triangular elements. The central part of the mesh around the caverns is shown in Figure 7.1. An improved two-dimensional non-linear ﬁnite element program, is used in the computation. The mechanical properties of the rock masses are summarised in Table 7.1. Figure 7.2 shows the constitutive model of the rock mass. The strength criterion shown in Figure 7.3 is mainly based on the

Figure 7.1. FEM meshes of the surrounding rock masses.

Table 7.1. Mechanical properties of the rock mass. Mechanical properties Young’s modulus E (MPa) Poisson’s ratio m Internal frictional angle j ( ) Cohesion C (MPa) Tensile strength (MPa)

Elastic properties

Residual properties

3.5 104 0.2 60 5.0 1.7

54 1.25 0

216

Chapter 7

Figure 7.2. Constitutive model of rock masses.

Figure 7.3. Strength criterion.

Prager–Drucker criterion, which is further modiﬁed by increasing the cohesion to a reasonable value. The distributions of bolts and pre-stressed cables are shown in Figure 7.4 [52]. The contributions of rock bolts to rock mass stiﬀness are treated by equivalent area and stiﬀness in the computational model, while the contributions to rock mass strength are treated by increasing the cohesion (internal frictional angle remains unchanged). These treatment methods adopted have been discussed in Section 3.3 of Chapter 3.

7.1.2.2 Computational implementation and results for different excavation sequences. A total number of 11 computational schemes are compared to examine the inﬂuences of diﬀerent excavation sequences on the rock mass stability. This includes the schemes of one-, ﬁve- or six-stage excavation. The detailed explanations

Construction Mechanics and Optimisation of Excavation Schemes

217

Figure 7.4. Installation scheme of rock bolts and cables.

to the diﬀerent schemes are shown in Table 7.2 and Figure 7.5. The computational results on some of the schemes are compared and discussed in the following sections: (i) Non-linear one-stage excavation As an extreme case, the scheme of non-linear one-stage excavation is ﬁrstly considered. The stress in the surrounding rock mass increases, and becomes greater than the strength. The over-stressed area continually expands into the surrounding ﬁelds. The damage zone is larger than that obtained from elastic analysis. A large plastic zone connecting the three caverns eventually forms, as shown in Figure 7.6. (ii) Non-linear five-stage excavation (5–I and 5–II) The excavation sequence is given in Table 7.2. The computational results indicate that the damage area is signiﬁcantly reduced in comparison with the one-stage excavation scheme. The diﬀerence in displacements at cavern corners also decreases. However, the damage area around each cavern is still connected, forming a continuous area. After the rock bolts are taken into account in the computation, the results indicate that the damage areas around the left two caverns are disconnected, but those around the right two caverns are still inter-connected, as shown in Figure 7.7. This implies that the results are not perfect. (iii) Non-linear six-stage excavation (6–I, 6–II and 6–III) This is the suggested excavation scheme by the design engineers. It is divided into three sequences of excavation: 6–I, 6–II and 6–III (Table 7.2). The eventual stress state for Scheme 6–I is shown in Figure 7.8. Although the total damage area is smaller than that of onestage scheme, the damage areas between the three caverns are still connected. In this six-stage scheme, the damage area is in fact slightly larger than that of the ﬁve-stage scheme. This implies that more excavation stages do not always lead to less damage

218

Table 7.2. Schedule of different excavation schemes and steps for three caverns. No 1

No 2

No 3

No 4

No 5

No 6

Note Elastic one-step excavation

Elasto-plastic one-step excavation

(5–III) (6–I)

A,B,C,D,E, F,G,H,I,J, K,L,M,N,P A,I A

B,F,J I

G B,F,J

H,K,L,M C,D,E

C,D,E,N,P G,K,L,M,N,P

5 6 7 8

(6–II) (5–III) (5–II) (5–II)

A,I A,I A,I A,I

B,F,J B,J B,F,J B,F,J

G C,K G G

H,K,L,M D,F,L H,K,L,M H,K,L,M

N,P E,G,M,N N,P C,D,E,N,P

9

(5–II)

A,I

B,F,J

G

H,K,L,M

C,D,E,N,P

10

(6–III)

A,I

B,J

C,K

D,F,L

E,G,M,N

H,P

11

(6–III)

A,I

B,J

C,K

D,F,L

E,G,M,N

H,P

(1–I)

2

(1–II) (1–III)

3 4

Note: assume that tunnels are completed at the first step of excavation.

H C,D,E H,P C,D,E

The originally proposed scheme

No tunnels Having rock bolts (only short bolts render support to rock mass) Having condensed rock bolts (both short and long bolts render support, but they are not pre-stressed) Having condensed rock bolts and pre-stressed rock cables Having condensed rock bolts and pre-stressed rock cables; subject to additional load induced by crane beams

9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > ;

Chapter 7

A,B,C,D,E, F,G,H,I,J, K,L,M,N,P

1

Elasto-plastic

Construction Mechanics and Optimisation of Excavation Schemes

219

Figure 7.5. Diﬀerent schemes for multiple-stage excavation.

Figure 7.6. Damage zones and displacements of one-stage excavation without support (non-linear analysis).

Figure 7.7. Damage zones and displacements after ﬁve-stage excavation with anchoring of rock bolts (non-linear analysis).

220

Chapter 7

Figure 7.8. Damage zones and displacement of last stage of six-stage excavation without anchoring (non-linear analysis).

Figure 7.9. Damage zones of Scheme 6-III, (a) at excavation stage 2, (b) at excavation stage 4, (c) at excavation stage 6, (d) at excavation stage 6 with reduced bolt spacing.

of the surrounding rock mass. Scheme 6–III is suggested based on the analysis and experience. The two caverns at the left and right sides are ﬁrst excavated, and the cavern at the centre is excavated later. Figure 7.9 illustrates the damage area and displacement distribution at Stages 2, 4 and 6 in the excavation process. It can be seen that the eﬀects are very signiﬁcant in isolating the damage areas between the three caverns, reducing the damage zones around each cavern (within 20 m thickness) and reducing the displacements at the cavern perimeter (half the displacement in the previous case). Figure 7.9d indicates that the rock mass stability

Construction Mechanics and Optimisation of Excavation Schemes

221

is further improved after reducing the spacing of rock bolts and cables. The damage areas are therefore signiﬁcantly reduced such that many of them become isolated individual small areas.

7.1.2.3 Discussions. (a) The results from numerical analyses show that diﬀerent excavation sequences produce very diﬀerent eﬀects on the rock mass stability. The one-stage excavation may generate large damage areas and displacements at the cavern perimeter. This implies that the stress history plays an important role, and the optimisation of excavation sequences is necessary. (b) In the situation of multiple cavern excavations in rock mass with high horizontal stress, in order to minimise the damage zone, excavations should be conducted at diﬀerent times. In concurrent excavations, the excavations should be kept at far distance to reduce the interaction between the excavations. (c) The excavated volume at each stage of the excavation should be limited to a reasonable amount in order to reduce the area of over-stressing. In addition, rock support measures such as rock bolt and shotcrete should be applied instantly. All of these have positive eﬀects on the rock mass stability. (d) At the connections between caverns and tunnels, three-dimensional numerical analyses should be conducted, because the two-dimensional analyses may not give quantitative results with the simpliﬁed model. (e) Since the above method of excavation optimisation cannot consider the energy loss in rock mass, the damage areas are generally over-estimated. So they should be normally used as the reference values for comparisons between diﬀerent excavation schemes. Back analysis from monitoring can overcome the shortcoming of the above method.

7.2.

APPLICATIONS OF INTERACTIVE PROGRAMMING IN OPTIMISATION OF CAVERN CONSTRUCTION

The concepts and principles of the interactive construction mechanics of rock engineering, together with the applications to an engineering project have been presented in the previous section. However, the method used has its limitations. For example, the schemes for optimisation analyses are selected based on engineering experiences. It is possible that the optimal scheme obtained with this method is a local optimal scheme within the limited number of proposals rather than a global optimal scheme. In this section, the interactive programming principle and associated methods are discussed and applied to construction optimisation.

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7.2.1 Principles of interactive programming The method of interactive programming, originally proposed by Bellman [439] is a speciﬁc technique solving the optimisation problems. Generally, optimisation is the process of ﬁnding the best solution to the problems from a group of possible solutions according to the given requirements. Correspondingly, the interactive programming is a multistage sequential decision-making process of ﬁnding the best solution. An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the ﬁrst decision. This is known as Bellman’s Principle of Optimality [439]. The main advantages of the interactive programming method are: (a) An optimisation problem of multiple (n) dimensions can be changed into n problems of one dimension, which may be solved in a sequential manner. This solution-searching procedure cannot be implemented with the traditional optimisation methods. (b) The global maximum or minimum value can be directly determined. By contrast, the traditional optimisation methods possibly give the local maximum or minimum value, which leads to the diﬃculty of judging the globality or locality of the solutions obtained. In fact, the interactive programming method is not an ‘algorithm’ but an approach to break and reconstruct the problems so that a suitable optimisation method can be applied. The following technical terms and statements are commonly used in the interactive programming method in describing the optimisation problems. A physical system can always be taken as a hierarchical system in a certain sequence. The interactive programming method divides the optimisation problem into a series of stages, which may represent, for example, time or space. At each stage, the state of system can be depicted by a relatively small group of variables. These variables are termed as state variables or state vectors (or simply state). Single or multiple decisions need to be made in any state of system at each stage. These decisions may depend on either the stage or the state, or both, while the history during which the system comes to the current state or stage is not important. In other words, the decision-making is only based on the current stage or state. After a decision is made, a beneﬁt is obtained correspondingly. At the same time the system changes to the next stage. The beneﬁt is governed by a known single-value function of a certain given state. Similarly, the state of the system after alteration is governed by a single-value function of the current state altered. The eventual objective of hierarchical development of the system stages is to ﬁnd the maximum or minimum values of the function of state variables. The key elements related to the interactive programming are stage, state, decision, alteration and beneﬁt.

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(a) Stage: The concept of stage is required in representing the sequence of decisionmaking. The function of state is to give the number of the sequences involved in the system developing process. Many problems seemingly without stage nature can still be analysed in a stage manner. For example, during the process of excavation, several excavation phases may be combined into one excavation stage. The optimisation problems of multiple variables can be treated as a process of multiple decision-making according to a certain sequence. During this process, each step in the sequence can be represented by a stage, which depends on the current state. (b) State: State space can be represented as a non-nil combination 1 . An element l 2 . This element is called as state, which is the description of a variable (or a group of variables). The state space is composed of all the state variables. For example, during the process of excavation, the state variables describe the current step in excavation sequences within the excavation system. The corresponding state space is composed of all the possible excavation steps. (c) Decision: The system state must include all the information required for determining all possible decisions. Therefore, for each state l 2 , there is a non-nil combination Xl. It is termed as decision combination of l. One of the elements, X3 ðlÞ 2 Xl is a decision or a decision variable. It represents an allowable choice when the system is at the state l. Decision combination Xl is composed of all the possible choices when the system is in the state l. For example, as shown in Figures 7.10 and 7.11, the decision under the state P1T1T2T3 is T4 or P2.

Figure 7.10. Excavation stages of the powerhouse and transformer chamber.

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Figure 7.11. Possible combinations of excavation sequences.

(d) Beneﬁt: Because a problem of dynamic programming is an optimisation problem, there may be an objective function to assess the given decisions. The total beneﬁt is a combination (sum or product) of beneﬁts at all the stages. It is the accumulation of state alteration (from one stage to another). The beneﬁt function is changeable from one stage to another. There is a requirement for proper deﬁnition of the beneﬁt at each stage. For example, for the optimisation of the cavern construction, the beneﬁt function can be deﬁned as the area of rock damage around the caverns.

7.2.2

Applications to the optimisation of cavern construction

A hydropower station is chosen to demonstrate the applications of the interactive programming method. The station mainly consists of a powerhouse cavern (20 m span and 45 m height) and a transformer cavern (18 m span and 28 m height). The design excavation scheme includes six stages (numbered as P1, P2, . . . , P6) for the power house cavern and in four stages (numbered as T1, T2, T3, T4) for the transformer cavern, respectively, as shown in Figure 7.10. To simplify the optimisation problem, some assumptions are made as follows: (a) Caverns are excavated in the sequence from top to bottom within each stage; (b) Each excavation stage is considered as one excavation phase; and

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(c) Stage P1 corresponds to the ﬁrst phase of excavation for the powerhouse cavern, and so on. A total of 168 possible construction schemes can be obtained. However, it is impossible to conduct ﬁnite element computations for all the 168 excavation schemes. In the past study, a limited number of schemes were selected for the ﬁnite element computations, and the computing results are compared to obtain a local optimal scheme. In the present study, a global optimal scheme is determined from n number of possible schemes, and, n ¼ X sNc where Xs is the total number of excavation sequences, and Nc is the number of caverns. For the case in this study, n ¼ 10 2 ¼ 20. That is, only 20 ﬁnite element (FEM) computations are required to obtain the global optimal scheme. Figure 7.11 schematically illustrates some of the possible combinations of excavation sequences. In the FEM computation, rock mass is treated as an isotropic elastic-plastic medium. The mechanical properties are shown in Table 7.3. The in situ ﬁeld stress is calculated based on the back analysis conducted in Section 6.5 of Chapter 6, sx ¼ 9:43 MPa, sy ¼ 7:43 MPa, xy ¼ 1:24 MPa To investigate the eﬀects of in situ stress on selecting the construction sequences, four conditions of stress ﬁeld are assumed by changing the angle a between s1 and horizontal direction. These are: (a) a ¼ 25.2 , (b) a ¼ 0 , (c) a ¼ 90 , and (d) a ¼ 25.2 , as shown in Figure 7.12. The beneﬁt function is deﬁned by the area of rock damaged around the cavern. Figure 7.13 illustrates the optimisation process for Case (a), where the value in the bracket is the beneﬁt. Figure 7.14 shows the distribution of damage zones around the cavern corresponding to the four cases in Figure 7.12. Table 7.4 shows the computational results for the four conditions of ﬁeld stresses of Figure 7.12.

Table 7.3. Mechanical properties of the surrounding rock mass. Parameter Value

E (MPa)

n

c (MPa)

f ( )

Rt (Mpa)

35000

0.22

1.30

50.20

5.0

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Figure 7.12. Diﬀerent distributions of principal stresses.

Figure 7.13. Optimisation process for excavation case (a) in Figure 7.12.

Figure 7.14. Damage zones in surrounding rock masses corresponding to cases in Figure 7.12.

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Table 7.4. Computational results for different stress field in Figure 7.12. Stress field Optimal scheme Damage area (m2) Maximum horizontal relative displacement

(a)

(b)

(c)

(d)

P1. . .P5T1 . . .T4P6 836 1.24

The same as (a) 1119 1.46

P1P2T1T2P3 T3T4P5P6 1370 0.87

The same as (a) 1282 1.44

Table 7.5. The eventual damage area and displacements for different construction schemes. Schemes Damage area (m2) Maximum horizontal relative displacement

1

2

3

4

5

1730 1.90

859 1.21

928 1.36

938 1.23

836 1.24

To compare the results for diﬀerent construction schemes under the same stress ﬁeld (a), the following schemes are ﬁrst selected: Scheme 1: PT (one-step full-face excavation) Scheme 2: P1-P2-P3-P4-P5-P6-T1-T2-T3-T4 Scheme 3: P1-T1-P2-T2-P3-T3-P4-T4-P5-P6 Scheme 4: P1-P2-T1-P3-P4-T2-P5-P6-T3-T4 (actual construction) Scheme 5: P1-P2-P3-P4-P5-T1-T2-T3-T4-P6 (optimal construction) The damage area and the displacement at the cavern edges for the above ﬁve schemes are compared in Table 7.5.

7.2.2.1 Discussions. The interactive programming method is more eﬃcient than the traditional enumerating method in the optimisation of large volume underground excavation. Although it cannot be totally ensured that the obtained optimal scheme is the real global one due to constraints from the assumptions and the specially deﬁned beneﬁt function, the result can at least be taken as the relatively global optimal scheme under the assumed conditions. Improvement is still required in the future. The following conclusions can be drawn from this study: (a) After the excavation steps are determined, the optimal excavation scheme is signiﬁcantly inﬂuenced by the in situ stress magnitude and direction. (b) The optimal scheme searching method used in the present study largely shortens the computation time in comparison with the traditional enumerating method.

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Chapter 7

The required computing time is proportional to the product of the total excavation steps and cavern number. (c) The beneﬁt function needs to be properly pre-determined. For example, it can be the convergent displacement, damage area, or both. On the other hand, because the beneﬁt function adopted in the present study is the damage area at each step of excavation rather than the accumulated damage area after the last excavation step, it may not be ensured to ﬁnd the global optimal scheme. Global optimal scheme can be obtained by combining the beneﬁt functions. (d) In the present study, only stress ﬁeld is taken into consideration. In the cases of presence of major geological structures, further investigation is required. (e) The results in this study are obtained through two-dimensional numerical computation. They provide qualitative basis for three-dimensional FEM problems of cavern construction. In fact, it is very diﬃcult to fully solve the optimisation problems by three-dimensional numerical modelling. The possible approach to the three-dimensional optimisation problems is to compare the computational results for a few representative construction schemes. (f) It can be seen from Table 7.5 that the cavern layout and excavation steps have been arranged in such a way that an optimal construction scheme can be achieved. However, the actual construction may not follow the optimal scheme. The damage area in the actual construction is 12% larger than that in the optimal construction scheme, while the maximum horizontal displacement at sidewalls of the cavern is almost the same. It should be noted that the method in this study provides the optimal scheme for construction sequences in view of rock mass stability of the caverns. Before the method is used, the excavation steps have been pre-determined. Cavern construction optimisation shall be conducted by considering both the rock stability and the economics, based on the site conditions. 7.3.

ARTIFICIAL INTELLIGENCE TECHNIQUES IN CONSTRUCTION OPTIMISATION

In the previous section, the interactive programming theory is applied and engineering case studies are conducted to illustrate the method of cavern construction optimisation. However, there are still two problems to be solved: (a) determining all the possible construction sequences; and (b) creating the input data ﬁles required by the numerical computations according to the selected construction sequences. Traditionally, this work is manually implemented with very low eﬃciency. The artiﬁcial intelligence technique has the capabilities of reasoning, understanding, planning, decision-making and learning [1–3,440–443]. Its applications to cavern

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construction optimisation can help to solve the above problems. In this section, the artiﬁcial intelligence technique is utilised in such aspects as automatic determination of construction sequences, automatic generation of input data ﬁles and automatic decision-making for construction optimisation.

7.3.1 Artificial intelligence language prolog Prolog is a very important language tool in artiﬁcial intelligence programming, and is mainly used to deal with logic problems. Prolog language enables computers to have reasoning capability in problem solving. Since it was produced in 1970s, it has quickly become a popular language of artiﬁcial intelligence. Prolog is similar to natural languages, and is therefore easy to learn and utilise by the users. The computer program written with this language is simple and understandable. The reasoning capability of prolog has lead to wide applications in the ﬁelds of database, logic algorithm, expert system, natural language interpretation, and automatic programming. One of the typical prolog languages is Turbo Prolog. Turbo Prolog has kept the main advantages of the traditional explanation type prolog, with enhanced runningspeed improved functions. In comparison with other computer languages such as Pascal, Turbo Prolog is a more advanced language. The program written with Turbo Prolog is 10 times shorter than that with Pascal for the same problem. This is because Turbo Prolog has an internal pattern-matching mechanism, and a simple but eﬀective method to deal with recurrent process. These two advantages are utilised in this study for construction sequence determination.

7.3.2 Problem solving algorithm in cavern construction optimisation 7.3.2.1 Automatic determination of cavern excavation sequences. Determining cavern excavation sequence is to ﬁnd out all the possible excavation sequences. Suppose that a cavern complex comprises Cavern A and Cavern B. Caverns are divided into diﬀerent excavated faces a1, a2 , . . . , a6 and b1, b2 , . . . , b4, as shown in Figure 7.15. The basic rules for excavation are (Zhu et al. 1992): (a) Excavation face a1 is the ﬁrst step; (b) One face is excavated as one step; (c) Each cavern is excavated from top to bottom. A data table is the basic structure in Turbo Prolog language. It is equivalent to the data group in Pascal language. This table consists of a set of sequential elements. The order of data in the table is an important feature of table. In this context, the

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Figure 7.15. Stepwise excavation of cavern complex.

excavated faces in each excavation step are represented by the elements of the table, and the excavation sequence is represented by the element order in the table. For example, a table L ¼ [a1, b1, a2, b2, a3, b3, a4, b4, a5, a6] represents that: (a) the excavated face at ﬁrst excavation step is a1, (b) the face at the last step is a6, (c) the element sequence in the table is the excavation sequence, and (d) there are a total of 10 excavation steps. For the initially generated table [a1, a2, a3, a4, a5, a6, b1, b2, b3, b4], the element order in the table is quickly determined according to the above three rules. The results are written into the output ﬁle ‘order.out’, which includes the combination of all the possible excavation sequences. To implement the above process of sortation in the table, eleven predicates are deﬁned. It is commonly known that a sentence in computer program is of two-fold characteristics: description and operation. When a computer program is constructed, people always pay attenuation to its operation. However, when the program needs to be veriﬁed and explained, people always concern about its description. When a logic algorithm is constructed, one of the methods for combining the two characteristics is to construct the program by considering its operation at ﬁrst instance, and then translate it into descriptive sentences. A computer program is generally constructed from top to bottom according to the sequential steps of problem solving. In other words, a general problem is ﬁrstly broken into several sub-problems, and then the sub-problems are individually solved. This is a natural process to construct computer programs. Describing the logic in the table order is to ﬁnd the sequential position exchange of the elements in the table. The logic module is ‘sort (Xs, Ys)’, where Ys is the table obtained when all the elements in Xs satisfy the ‘ordered’ condition. sort (Xs, Ys): – permutation (Xs, Ys), ordered (Ys) The statement ‘sort (Xs, Ys)’ has been broken into ‘permutation (Xs, Ys)’ and ‘ordered (Ys)’, which are further explained as below.

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Checking whether a table is arranged in the right order according to given requirements can be implemented with the following sentences: ordered ([-]). ordered (Ys): — is_after (a2, a1, Ys), / *a2 and Ys are after a1 */ is_after (a3, a2, Ys), is_after (a6, a5, Ys), is_after (b1, a1, Ys), is_after (b2, b1, Ys), is_after (b4, b3, Ys), ‘Fact’ indicates that only the tables consisting of one element have been arranged in order, while ‘rule’ indicates that the order must satisfy the three assumptions. Permutation represents a manner of element position exchange in the table. That is, an element is randomly selected, then it is taken as the ﬁrst element in the table, and then the other elements are iteratively exchanged. Its basic fact indicates that a nil table is the only exchange. Permutation ([a1 , . . . , an], L) feeds back the result that L has a total number of n! solutions. Permutation (Xs, [ZjZs ]): —select (Z, Xs, Ys), permutation (Ys, Zs) Permutation ([ ], [ ]) Determining the relative positions of two elements involves three predicates: member, position and is_after. Member is to determine whether an element is involved in the table: member (X, [XjXs ]), member (X, [YjYs ]): —member (X, Ys). The above program means: if X is the ﬁrst part of the table, or X is the element of the last part of the table, then X is an element in the table. Position is to calculate the position order of an element in the table: position (A, position (A, member (A, position (A, N ¼ No þ 1

[A|_ ], 1), [ _|T], N): — T), T, No),

It is speciﬁed that A is placed at the ﬁrst position if A is the ﬁrst element in the table. The meaning of ‘is_after (A2, A1, L)’ is: if the position number of A2 is bigger than that of A1, A2 is after A1 in the table.

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The result of selecting X from [XjXs ] is Xs, or, if the result of selecting X from Ys is Zs, then the result of selecting X from [YjYs ] is [YjZs ], i.e., select (X, [XjXs ], Xs) select (X, [YjYs ], [YjZs ]): — select (X, Ys, Zs) ‘Writelist (L)’ and ‘write10 (list, N)’ mean that a table is exported and each line of the table contains N symbols. ‘Run’ means that the results of order arrangement are saved in the ﬁle ‘order.out’. ‘Make-cut (L): L ¼ L0’ (cut/fail method) means that the program is repeatedly run until L is equal to L0. L0 can be displayed on the screen in advance. In general, the above method can be described as: the initial table of excavation sequence is processed through Predicate ‘sort’, and then a table of order arrangement is produced according to the ‘ordered’ rule. The program Autoorder.PRO for generating the excavation sequence is described as below: ———————— Autoorder.PRO code ¼ 10000 trail ¼ 4000 errolevel ¼ 0 domains A ¼ symbol N ¼ integer ﬁle ¼ myﬁle list ¼ symbol* predicates run ordered (List) member (A, List) make_cut (List) write10 (List) sort (List, List) writelist (List) is_after (A, A, List) position (A, List, N) select (A, List, List) permutation (List, List)

Construction Mechanics and Optimisation of Excavation Schemes clauses position (A, [A|_ ], 1). position (A, [ _|T], N): — member (A, T), position (A, T, NO), N ¼ NO þ 1. is_after (A2, A1, L): — position (A1, L, K1), position (A2, L, K2), K2 > K1. member (Name, [Name|_ ]). member (Name, [ _|Tail]): — member (Name, Tail). select (P, [P|Xs], Xs). select (P, [Y|Ys], [Y|Zs]): — select (P, Ys, Zs). permutation (Xs, [Z|Zs]): — select (Z, Xs, Ys), permutation (Ys, Zs). permutation ([ ], [ ]). writelist (NL): — write10(NL, 0), nl. write10 (TL, 10): — !, write10 (TL, 0). Write10 ([H|T], N): — !, write (H, ‘‘ ’’), N1¼Nþ1, write10 (T, N1). Write10 ([ ], _ ). ordered ([ _ ]). ordered (Ys): — is_after (a2, a1, Ys), is_after (a3, a2, Ys), is_after (a4, a3, Ys), is_after (a5, a4, Ys), is_after (a6, a5, Ys), is_after (b1, a1, Ys), is_after (b2, a1, Ys), is_after (b3, b2, Ys), is_after (b4, b3, Ys),

233

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Chapter 7

sort (Xs, Ys): — permutation (Xs, Ys), ordered (Ys). run: — openwrite (myﬁle, ‘‘order.out’’), sort ([a1, a2, a3, a4, a5, a6, b1, b2, b3, b4], L), writedevice (myﬁle), writelist (L) make_cut (L), !, closeﬁle (myﬁle). Make_cut (L): — L ¼ [a1, b1, b2, b3, b4, a2, a3, a4, a5, a6]. goal run. ———————— It can be seen that the program is simple and easy to understand. Figure 7.16 shows the computational results on all the possible construction sequences (a total of 126 schemes). It only requires about one minute to run the program in a 486 computer. By comparison, the manual method is time-consuming and some schemes may be missed. Figure 7.17 schematically illustrates partial manual results by manual sorting.

7.3.2.2 Automatic generation of data files for finite element computation. The ﬁnite element program used in this study is written by the authors. The structure of data ﬁles required for determining the excavation sequences is shown in Figure 7.18. Each data ﬁle above is divided into two ﬁles: const.dat and excav.dat. The ﬁrst one is the same for each excavation scheme, and the second one is changeable with diﬀerent excavation schemes. The information on multi-stage excavation is saved into data ﬁles named according to the excavated faces. For example, Data ﬁle a1 contains the information on excavating face a1. Data ﬁles a1, a2, a3, a4, a5, a6, b1, b2, b3, b4 are then processed to form a complete ﬁle excav.dat, which represents excavation scheme [a1, a2 , . . . , a6, b1, b2 , . . . , b4]. Two predicates are used in the processing: ‘ﬁle_str (Filename, Text)’ and ‘form_fort.ﬁle (L)’. The ‘ﬁle_str (Filename, Text)’ reads symbols from a data ﬁle and sends them into a variable or a ﬁle. The ‘form_fort ﬁle (L)’ deﬁnes a data ﬁle as the symbol at the top of the table (the ﬁrst excavation step); and then sends the content of data ﬁle to a variable termed as ‘text’. The ‘text’ is ﬁnally written into the data ﬁle ‘excav.dat’. This process continues until

Construction Mechanics and Optimisation of Excavation Schemes a1 a2 a3 a4 a5 a6 b1 b2 b3 b4 a1 a2 b1 b2 b3 a3 a4 a5 b4 a6 a1 a2 a3 a4 a5 b1 b2 b3 a6 b4 a1 a2 b1 b2 b3 a3 a4 b4 a5 a6 a1 a2 a3 a4 b1 a5 b2 a6 b3 b4 a1 b1 a2 a3 a4 a5 a6 b2 b3 b4 a1 a2 a3 a4 b1 b2 a5 a6 b3 b4 a1 b1 a2 a3 a4 a5 b2 b3 b4 a6 a1 a2 a3 a4 b1 b2 b3 a5 a6 b4 a1 b1 a2 a3 a4 b2 a5 b3 b4 a6 a1 a2 a3 b1 a4 a5 a6 b2 b3 b4 a1 b1 a2 a3 a4 b2 b3 b4 a5 a6 a1 a2 a3 b1 a4 a5 b2 b3 b4 a6 a1 b1 a2 a3 b2 a4 a5 b3 b4 a6 a1 a2 a3 b1 a4 b2 a5 b3 b4 a6 a1 b1 a2 a3 b2 b3 b4 a4 a5 a6 a1 a2 a3 b1 a4 b2 b3 b4 a5 a6 a1 b1 a2 a3 b2 b3 a4 b4 a5 a6 a1 a2 a3 b1 b2 a4 a5 b3 b4 a6 a1 b1 a2 b2 a3 a4 a5 b3 a6 b4 a1 a2 a3 b1 b2 a4 b3 b4 a5 a6 a1 b1 a2 b2 a3 a4 b3 a5 b4 a6 a1 a2 a3 b1 b2 b3 a4 b4 a5 a6 a1 b1 a2 b2 a3 b3 a4 a5 b4 a6 a1 a2 b1 a3 a4 a5 b2 a6 b3 b4 a1 b1 a2 b2 b3 a3 a4 a5 a6 b4 a1 a2 b1 a3 a4 b2 a5 a6 b3 b4 a1 b1 a2 b2 b3 a3 b4 a4 a5 a6 a1 a2 b1 a3 a4 b2 b3 a5 a6 b4 a1 b1 b2 a2 a3 a4 a5 b3 a6 b4 a1 a2 b1 a3 b2 a4 a5 a6 b3 b4 a1 b1 b2 a2 a3 a4 b3 a5 b4 a6 a1 a2 b1 a3 b2 a4 b3 a5 a6 b4 a1 b1 b2 a2 a3 b3 a4 a5 b4 a6 a1 a2 b1 a3 b2 b3 a4 a5 a6 b4 a1 b1 b2 a2 b3 a3 a4 a5 a6 b4 a1 a2 b1 a3 b2 b3 b4 a4 a5 a6 a1 b1 b2 a2 b3 a3 b4 a4 a5 a6 a1 a2 b1 b2 a3 a4 a5 b3 b4 a6 a1 b1 b2 b3 a2 a3 a4 a5 b4 a6 a1 a2 b1 b2 a3 a4 b3 b4 a5 a6 a1 b1 b2 b3 a2 b4 a3 a4 a5 a6

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Figure 7.16. Sorting results of cavern construction (126 schemes).

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Figure 7.17. Hand-sorted construction sequences.

Figure 7.18. Structure of data ﬁles.

the bottom of the table (the last excavation step) is reached. The program for automatic generation of data ﬁles is shown below: ———————— Autoform.Pro domains ﬁle¼myﬁle L¼symbol* A¼symbol predicates form_fort_ﬁle (L)

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form_fort_ﬁle (H|T): — ﬁlename¼H, ﬁle_str (Filename, Text), write (Text). run (L): — openwrite (myﬁle, ‘exc.dat’), writedevice (myﬁle), form_fort_ﬁle (L), closeﬁle (myﬁle). ————————

7.3.2.3 Implementation of excavation scheme optimisation. The automatic optimisation of cavern excavation schemes is implemented based on the results of an order arrangement on excavation sequence and data ﬁles generated in the last section. The whole process of excavation optimisation is schematically shown in Figure 7.19. The process can be simply described as: (a) make known the ﬁrst step a1 through reading ﬁle ‘order.out’; (b) make known the second step a2 or b1; (c) form two tables L1 ¼ [a1, a2] and L2 ¼ [a1, b1]; (d) call Auto_form.PRO and the ﬁnite element program; and (e) search for the optimal scheme according to the beneﬁt function. By repeating this process, an optimal scheme is obtained, that is, Lopt ¼ [a1, a2, a3, a4, a5, T1, T2, T3, T4, a6].

Figure 7.19. Optimisation process of cavern complex construction.

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7.3.3 Discussions Earlier in this chapter, a new method for the optimisation of cavern construction schemes is outlined. The interactive programming theory is used for cavern construction optimisation, based on the concepts of interactive construction mechanics associated with rock engineering. The traditional optimisation approach, which is based on the simple comparisons among multiple schemes from ﬁnite element computations, is improved in this method based on multi-stage decisionmaking. To further improve the eﬃciency of optimisation process, the artiﬁcial intelligence Prolog language is utilised. Accordingly, the optimisation is automatically implemented. The computer program is simple and easy to understand. Decision-making in cavern construction optimisation is to select the optimal scheme from all the possible schemes according to the given requirements. After all the possible schemes are evaluated, an optimal scheme is then determined as the ﬁnal option [8,9]. If there is only one index for evaluating the schemes in a system, the system is called single index decision-making system. Optimisation in this system is relatively easy, and can be implemented by comparing the indices of all the possible schemes. However, cavern construction is a complex system with multiple indices for evaluation. In other words, it is a multiple indices decision-making system. In such a system, the optimal scheme cannot be obtained directly from the evaluation results on a single index. The comprehensive evaluation on the multiple indices is required. In the study illustrated in this chapter, damage area around the cavern is taken as the beneﬁt function (evaluation index). This means that the optimisation of cavern construction is still based on a single index. Further improvement on the interactive optimisation method can adopt multiple indices for construction optimisation.

7.4.

ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE OPTIMISATION METHODS

An engineering example is given to demonstrate the eﬀectiveness of applying the construction mechanics and artiﬁcial intelligence methods in cavern construction optimisation. The project is an underground hydroelectrical power station excavated in rocks.

7.4.1 Description of the project The power station is excavated in a Triassic sandstone. The geological structure is uniclines. The dip direction of rock strata is NEE-SE, and dip angle is about 10 . No large fault is found except for bedding fractures. Groundwater level is deep below ground surface, and the permeability of the rock masses is low. The cavern complex

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Table 7.6. Mechanical parameters of rock layers. Layer

T12

T131

T132

T14

T151

T152

T153

Young’s modulus (GPa) Poisson’s ratio Cohesion (MPa) Friction angle ( )

11.5 0.22 1.0 35

12.8 0.21 1.17 35

12.0 0.22 1.0 35

13.5 0.2 1.5 35

9.5 0.24 0.83 35

12.8 0.21 1.25 35

10.5 0.22 1.0 35

is approximately 70–100 m below the ground level. According to the in situ measurement, the horizontal stress is about 3 MPa; and the vertical stress is equal to the overburden stress. Underground cavern complex is housed in fractured mudstones (T131 , T132 and T14 ). Among them, T131 comprises thick layers (about 30 m) of calc-silicon ﬁne-grained quartz sandstone and calc sandstone with small amount of mudstone or siltstone. T132 comprises thick layers (about 30 m) of calc or argillaceous-calc siltstone and ﬁne-grained sandstone. T14 comprises very thick layers (about 60 m) of silicon or calc sandstone containing little amount of mudstone or siltstone. The mechanical properties of each layer of rock formations are presented in Table 7.6.

7.4.2

Layout of cavern group and arrangement of step excavation

The layout of the cavern complex is shown in Figure 7.20, together with the distribution of rock layers. Excavation was designed to be completed in 10 stages. This excavation scheme is used in the study of cavern construction optimisation. In addition, several assumptions are made: (a) C0 is the ﬁrst excavated; (b) Excavation sequence is from top to bottom; (c) Each stage is excavated in one run.

7.4.3

Optimisation of excavation sequence

Two stages of study have been performed. At the ﬁrst stage, the surrounding rock mass is assumed isotropic and the same mechanical properties are used. At the second stage, the rock mass is divided into seven layers of diﬀerent mechanical properties (locally isotropic medium) according to the geological conditions (Figure 7.20 and Table 7.6). The results obtained from the two diﬀerent studies are diﬀerent and outlined in the following sections.

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Figure 7.20. Cavern layout and excavation stage.

7.4.3.1 Rock mass assumed as isotropic medium. The studies are conducted using the method outlined in Section 7.3.2. The analysis is based on two dimensional ﬁnite element modelling. The average depth of cavern complex to ground surface is 85 m. The material parameters for modelling are: ¼ 2610 kg/m3, E ¼ 11000 MPa, n ¼ 0.20, c ¼ 0.3 MPa, f ¼ 33 . Two cases of in situ stress ﬁelds are modelled, one based on the elastic theory and the other based on the in situ stress measurement. The in situ stresses are estimated as: Case I: elastic theory sv ¼ s1 ¼ 2.61 85/10 ¼ 2.22 MPa (vertical) n s1 ¼ 0.56 MPa (horizontal) sh ¼ s3 ¼ 1n Case II: In situ stress measurement sv ¼ s1 ¼ 2.22 MPa, sh ¼ s3 ¼ 3.0 MPa.

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Figure 7.21. Mesh of the computation model.

To simulate the excavation process, the whole cavern area is divided into a mesh of triangle and quadrilateral shapes. The total number of elements is 1326, and the total number of nodes is 1284. The central part of the model is shown in Figure 7.21. The artiﬁcial intelligence language Turbo Prolog 2.0 Version is used (see Section 7.32). The results present 1260 possible excavation sequence schemes. It is impossible and unnecessary to carry out ﬁnite element computation for so many schemes. However, by using the construction mechanics theory, optimum schemes can be quickly sorted out. The searching processes for the optimum scheme with two diﬀerent cases of in situ stress ﬁelds are shown in Figure 7.22. The optimum excavation sequences using optimisation method, are diﬀerent from the original scheme, C0 ! C4 þ W3 ! b1 þ W1 þ C1 ! C2 þ b2 þ W2 ! C3 , where C4 þ W3 means that C4 and W3 are simultaneously excavated (Figure 7.20). Assessment of damage areas around the caverns are conducted, and comparisons between the diﬀerent schemes are shown in Figures 7.23 and 7.24, and Table 7.7. Usually, in a large underground cavern complex, there are many possible excavation schemes. However, by using the construction mechanics together with interactive programming theory and artiﬁcial intelligence method, the optimum scheme can be obtained. Compared with the originally proposed scheme, the damage area from the original scheme is 3–10 times larger than that of the optimum scheme. Although this is obtained with certain assumptions, it however indicates the importance of optimisation of cavern construction scheme.

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Figure 7.22. Optimisation search processes for two cases of in situ stress ﬁelds, (a) Case I, and (b) Case II.

Figure 7.23. Damage distribution of the stress ﬁeld case I, (a) original excavation scheme, and (b) optimised excavation scheme.

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Figure 7.24. Damage distribution of the stress ﬁeld case II, (a) original excavation scheme, and (b) optimised excavation scheme.

Table 7.7. Comparisons between the optimised and original excavation schemes. Scheme

Stress case I Stress case II

Optimised Original Optimised Original

Total damage area (m2)

Vertical convergence of powerhouse (mm)

Horizontal convergence of powerhouse (mm)

513 5139 690 2540

9.64 11.75 4.41 6.67

1.68 0.69 20.88 12.51

7.4.3.2 Rock mass assumed as layered isotropic medium. According to the geological conditions, the rock mass is divided into seven layers. Each layer is treated as an isotropic medium. The density of rock mass in all the layers is 2610 kg/m3. The mechanical properties adopted for each rock layer are the same as those shown in Table 7.6.

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Figure 7.25. Optimisation searching process of construction schemes.

The average depth of cavern complex to ground surface is 85 m. The average rock mass density is 2610 kg/m3. The initial stress ﬁeld is determined from the in situ stress measurement. The horizontal stress obtained from measurement is 3.0 MPa. Therefore, sv ¼ s1 ¼ 2.61 85/10 ¼ 2.22 MPa (vertical) sh ¼ s3 ¼ 3.0 MPa. The ﬁnite element computation is based on plane strain elastic-plastic model. The total number of ﬁnite elements is 1506, and the node number is 1211. The program and the method are the same as those in the previous modelling. 1260 possible schemes are obtained, and subsequently reduced to 9 schemes. If the beneﬁt function is deﬁned by the damage area of surrounding rock mass, the smaller damage area is then taken as the optimum scheme. The optimum scheme searching process is shown in Figure 7.25. The excavation sequence in the original scheme is C0 ! C4 þ W3 ! b2 þ W2 þ C1 ! C2 þ b2 þ W2 ! C3 ; where C4 þ W4 represents that C4 and W3 are simultaneously excavated. The comparisons between the optimised scheme and original scheme are shown in Figure 7.26 and Table 7.8. It can be seen that the damage area of the optimised excavation scheme is reduced by 90% in comparison with the original scheme. A substantial improvement can be achieved by the optimisation. It should be noted that the optimised scheme obtained in this study is diﬀerent from that for rock mass assumed to be isotropic in the previous study. This implies that the optimisation results depend on the rock mass conditions.

7.4.3.3 Summaries. (a) Applications of the interactive construction mechanics theory together with the artiﬁcial intelligence method can optimise the construction schemes of cavern complex scientiﬁcally and automatically, as a result high eﬃciency is achieved.

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Figure 7.26. Damage distribution of (a) original excavation scheme, and (b) optimal excavation scheme.

Table 7.8. Comparisons between the optimised and the original excavation schemes. Scheme

Optimised Original

Total damage area (m2)

Vertical convergence of powerhouse (m)

Horizontal convergence of powerhouse (mm)

329 4533

0.16 1.13

9.6 1.9

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(b) Advantages of the optimisation for construction are signiﬁcant for large-scale rock engineering projects. The damage area of the optimised scheme can sometimes be reduced to 1/10 of that of the unoptimised scheme. (c) The application of the optimisation method of combining interactive construction mechanics theory and artiﬁcial intelligence technique to the heavily fractured rock mass and to the three-dimensional problems needed to be studied.

Chapter 8

Reinforcement Mechanism of Rock Bolts Rock bolts have been used in rock engineering from end of the 19th century [445]. In 1890, reinforced steel was used in reinforcing the rock mass in mines in north Wales. After the 1940s, this support technique was adopted worldwide. The use of shotcrete together with rock bolts further promotes the application of bolt as a main supporting method. Based on the rock bolt technique, a new construction method combining the use of rock bolts and shotcrete gradually formed in Europe in the 1950s and 1960s. This method sets a milestone in the underground construction technology. In recent years, many researchers and engineers have studied the eﬀects of bolts in rock reinforcement. This chapter summarises some of the research ﬁndings.

8.1.

EFFECT OF BOLTS ON SUPPORTING THE ROCK MASS

8.1.1 Effects of rock bolts Two main types of rock bolts are widely employed in the underground excavation in rocks. One is the end-anchored bolts varying from wedge to resin anchoring. The other one is the full-length anchored type varying from fully grouted to split-set bolts. The eﬀects of rock bolts on the surrounding rock are mainly two-fold: providing the reaction force and the reinforcement to the surrounding rock mass. The former prevents deformation occurring in the surrounding rock mass and provides the tensile force to resist the movement of rock blocks. The latter reinforces the rock mass to be an integrated medium and to easily form arches around the excavation [446–477].

8.1.1.1 Reinforcement. The eﬀorts of other support methods such as timber frame, segment lining, and cast concrete lining are mainly external supports. As the rock bolts are inserted into the rock masses, the eﬀects of rock bolts on the rock mass are similar to that of steel bars in reinforced concrete. They increase the integrity and overall strength of the rock masses. The bolts prevent the rock masses from sliding and failing and increase the load-bearing capacity eﬀectively. The rock mass is reinforced by the bolts and becomes an eﬀective self-support medium [446–473].

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8.1.1.2 Post effect of pre-stressing. The deformation modulus of the rock bolt (usually steel) is generally greater than that of the surrounding rock masses. The diﬀerence between the moduli restricts the surrounding rock mass to deform (usually tensile deformation towards the centre of the opening) after excavation. The restriction occurs at the ends of anchored rock bolts and along the full length of grouted rock bolts. This restriction is mobilised once deformation starts to occur and it can be regarded as a complementary pre-stress on the excavation surface after the excavation. Rock bolts signiﬁcantly increase the integrity of the rock mass due to this pre-stressing eﬀect [446–448,474].

8.1.1.3 Prompt prevention. Rock bolts and shotcrete are often applied immediately after excavation. In some cases, rock bolts can also be implemented prior to the excavation, to restrain the relaxation of the surrounding rock and to improve the rock mass strength early. The early application of rock bolts provides prompt prevention of failure.

8.1.1.4 Good match to deformation. Rock bolts possess better match ability to the deformation of rock mass than other support methods. It is often referred as ‘ﬂexible support’. This support method is also very suitable for tunnels in soft rock.

8.1.1.5 Flexibility in construction. The size and distribution of rock bolts can be adjusted according to the geological conditions of the surrounding rock. Bolting and shotcreting can be completed in several steps [475,476]. Additional bolts can be installed at a later stage. The application of bolts has ﬂexibility during the construction.

8.1.2 Reinforcement mechanism of rock bolts The reinforcement mechanisms of rock bolts are very complex. In this section, they are discussed by examining the shear strength and other properties of the bolted rock masses. A series of studies on the reinforcement mechanism of rock bolts have been carried out. Test results indicated that application of rock bolts increases the peak compressive strength of the rock mass by 50100%, compared with that of the rock mass without bolts. For a single opening, the shear strength of the surrounding rock mass can be expressed by the Mohr–Coulomb strength condition, as shown in Figure 8.1.

Reinforcement Mechanism of Rock Bolts

249

The reaction force contributed by rock bolts to the surrounding rock mass (s2) can be estimated from the allowable tensile stresses in the rock bolts. For example, for a tunnel at depth of 500 m with hydro-static in situ stress condition, and assuming the density of rock mass of 2500 kg/m3, the in situ stress ﬁeld can be obtained as: six ¼ siy ¼ siz ¼ 12:5 MPa

ð8:1Þ

The uniaxial compressive strength of the rock material (sc) is assumed as 15.0 MPa, the uniaxial compressive strength of the rock mass strength (smass) is assumed as 5.0 MPa (1/3sc) and the internal friction angle (fmass) is 35 . The cohesion of rock mass can be derived as: cmass ¼ smass ð1 sin fmass Þ=2 cos fmass ¼1:31 Mpa

ð8:2Þ

If the compressive strength of the properly bolted rock mass (c0 mass) is 50% greater than that of unbolted rock mass, then c0 mass ¼ 1.5 cmass ¼ 1.97 MPa. Assuming that the length of bolt is 60 cm and the allowed tensile load is 100 kN, the reaction stress on the rock surface provided by the bolt can be calculated as s2 ¼ 0.28 MPa. For the end-anchored bolts, the reinforcement is applied through a reaction force at the rock wall. The reinforcement eﬀect of a bolt is therefore on the rock wall and the bolt anchor while the other parts of the bolt are not in contact with the rock mass. For full-length grouted or frictional bolts, the reinforcement bolts increase the shear-bearing capacity of the rock masses, in addition to the reaction force at the rock wall generated by the bolts. Assuming that the stresses s1 and s2 on the opening wall reach the rock mass shear strength, the stress condition of the rock wall can be expressed by the Mohr circle s2As1 with centre O. As shown in Figure 8.1, this circle is tangential to the strength envelope BCA at A. The reaction force of rock bolts to the rock wall can be deﬁned as s2 for the end-anchored bolt. In this case the stress circle change to the small Mohr circle s0 2A0 s1 with centre O0 , and the strength envelope is B0 C0 A0 . The new minor principal stress is s20 ¼ s2 þ s2. From Figure 8.1, the shift of the strength envelope, OA ¼ ðc0mass cmass Þ cos f

ð8:3Þ

For the full-length grouted bolts, the shear strength of surrounding rock is enhanced by the bolts, and the shear envelope shifts up to the position of B00 C00 A00 (assuming no change in friction angle). The increased cohesion equals to c00 c0 . The strength

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Figure 8.1. Strength analysis of rock bolt eﬀects.

envelope of full-length grouted bolt is increased by an oﬀset O0 A0 ¼ (c00 c0 ) cosf from the end-anchored bolt. The analysis above is under the assumption that the strength oﬀset is based on the shortest relative distance from centre of the Mohr circle to the envelope. The analysis clearly shows that the full-length grouted bolts achieve high strength storage than the end-anchored bolts. The diﬀerence of the eﬀects of two types of rock bolts can be signiﬁcant, due to the eﬀects of grouting.

8.2.

PHYSICAL MODELLING OF ROCK BOLTS

Rock bolts enhance and reinforce the rock masses. However, the mechanisms of reinforcement, and the deformation and damage of bolts in the surrounding rock mass subjected to change of stress ﬁeld have not been fully understood [449,463,467,477]. This section presents studies on the rock bolt mechanism using physical modelling. When the surrounding rock mass is soft and/or subjected to high in situ stresses, displacement of the tunnel wall can often be as high as 50–100 cm. It is of great interest to have a scientiﬁc approach in selecting the rock bolt to overcome such large deformation problems. At present, the common way of applying bolts is to install bolts perpendicular to the rock surface. The systematic studies are conducted on the installation angle, density and other parameters, and results are of interest to engineers.

8.2.1 Similarity of model materials Rock masses close to the surface are treated as separable blocky materials. The model materials should have similar characteristics as the real rock masses, such as stress–strain relation, properties of dilation and softening. After testing diﬀerent model materials, the selected model material (made of sand and white glue) has elastic modulus of 230 MPa. The bolt is modelled by the bamboo material with

Reinforcement Mechanism of Rock Bolts

251

Figure 8.2. Results of compression tests.

Figure 8.3. Results of tensile tests.

elastic modulus of 100–170 MPa. A series of tests are conducted including uniaxial, biaxial and triaxial compression, tensile test, and cyclic loading test for the model specimens with and without bolts, as shown in Figures 8.2–8.4. From the results, it can be seen that the ratio of modulus to strength of the model material is 178 and that of the rock mass is 200. The ratio of modelling material compressive strengths to rock mass strength is 1/15. From the bulk deformation curves, it also can be seen that the modelling material has the similar dilation property as the modelled rock mass.

8.2.2 Comparison of different bolting methods The eﬀects of diﬀerent bolt materials, bolting methods, bolt densities and installation angles are systematically compared in the tests. The strength of the material

252

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Figure 8.4. Results of cyclic compression tests.

Figure 8.5. Diﬀerent bolt arrangement used in the tests.

modelling the rock mass is 1.0 MPa. Bamboo (E ¼ 10 103 MPa) and plexiglass (E ¼ 3 103 MPa) are used as modelling materials of the bolts. Both full-length grouting and end-anchored bolts are modelled. Diﬀerent densities are used at 8, 10, 12, and 36 bolts per 200 cm2, equivalent to the actual bolt spacing of 1.2 to 0.4 m. The bolting methods in the tests include end-anchored and fully grouted parallel bolts perpendicular to the rock wall, fully grouted perpendicular, vertically and horizontally intersected bolts. The details of various arrangements are summarised in Figure 8.5 and Table 8.1. Figures 8.6–8.8 indicate the various load-deformation curves. From the ﬁgures, it can be seen that high density of bolts can improve the strength of surrounding rock mass. The horizontally obliquely intersected full-length grouted bolts give high strength of the rock mass and small number of bolts. Figure 8.8a illustrates that for the same bolt spacing and length, the rock mass can have the highest

253

Reinforcement Mechanism of Rock Bolts Table 8.1. Various types of bolts and their parameters. Type

No bolt

Normal parallel

Normal parallel

Normal parallel

Vertically oblique

Vertically oblique

Symbols NB NP-E10 NP-F10 NP-F36 VO-F8 VO-F12 – 10 10 36 8 12 Bolt density (per 200 cm2) 90 90 73 68 Inclination to wall – 90 Anchor and grouting – Two ends Full-length Full-length Full-length Full-length Bolt material – Bamboo Bamboo Bamboo Bamboo Bamboo Peak strength (MPa) 1.0 1.5 1.5 3.0 1.54 2.2 Softening behaviour Yes Yes No No Yes No

Horizontally oblique HO-F12 12 75 Full-length Bamboo 2.06 No

Figure 8.6. Axial load – axial deformation of diﬀerent bolt arrangement.

strength and the lowest dilation when the bolts are installed at 22.5 and 67.5 , to the rock wall surface. For the same bolting angle and the same total lengths of bolts, the test results shown in Figure 8.8 suggest that mixed length bolts (total 9 bolts of 5.41 and 10.82 cm long and evenly spaced) provide the best reinforcement results, while short bolts with high density (total 12 bolts of 5.41 cm long and evenly spaced) give the

254

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Figure 8.7. Axial load – volumetric strain for diﬀerent bolt arrangement.

Figure 8.8. Axial load – volumetric deformation for diﬀerent (a) bolting angle, and (b) bolt length.

Reinforcement Mechanism of Rock Bolts

255

worst reinforcement results, as compared with uniformly long bolts (total 6 bolts of 10.82 cm long and evenly spaced).

8.2.3 Analysis of test results From the test results shown in Figures 8.6, 8.7 and 8.8 and Table 8.1, following observations are obtained: (a) The peak uniaxial compressive strength of the rock mass with bolts increases signiﬁcantly (up to 20% from the model tests) compared with that without bolts while the residual strength and the tensile strength increase up to 100%. In biaxial case, the peak compressive strength can increase by 50–100% (Figure 8.6) when the density of bolts is high, rock mass strength can increase by 3 times with little volumetric dilation. (b) Under the plane strain condition, the strength of the models is aﬀected not only by the bolt densities but also by bolt installation angles, types, and shear strength and lateral stiﬀness of bolt. (c) The obliquely intersected bolt distribution enhances the rock mass strength and limits the dilation signiﬁcantly. The best arrangement appears to the bolts at 65 angle to the wall. But the bolts are required to possess a high lateral stiﬀness in this case. (d) Compared to the end-anchored bolts, the improvement of full-length grouted bolts on the peak strength of the surrounding rock mass is not obvious. However, the bulk displacement curves are diﬀerent for the two diﬀerent bolt types. The dilation of the fully grouted bolts starts at a later stage. The post-peak softening phenomenon is not obvious for the fully grouted bolts while it is signiﬁcant for the end-anchored bolts. Therefore, end-anchored bolts have low post-peak strength.

8.3.

NUMERICAL MODELLING OF BOLT

In this section, non-linear ﬁnite element numerical modelling is applied to study eﬀects of rock bolts on the stability of tunnels in soft rocks. Parameters obtained from physical modelling are used as input to the numerical modelling.

8.3.1 Basic parameters of the numerical model A simple circular opening model is used in the parametric study. Two in situ stress conditions are modelled: (a) far ﬁeld stresses s1 ¼ s2 ¼ 20 MPa, and (b) far ﬁeld

256

Chapter 8

Figure 8.9. Element division near the opening.

stresses s1 ¼ 2s2 ¼ 20 MPa. The surrounding rock mass is regarded as an elastoplastic material. The problem is treated as a plain stress one. Based on the symmetry, one quarter of problem is analysed by a ﬁnite element program. A total of 136 elements and 162 nodes are used in the model, as shown in Figure 8.9. Three diﬀerent cases are studied: opening without bolts, opening with bolts that are normal to the wall surface, and opening with obliquely intersected bolts. The radius of the opening is 2 m and the thickness of bolted region equals to the opening radius. The mechanical parameters of surrounding rock mass are adopted from the physical model tests described in the previous section, and is summarised in Table 8.2.

8.3.2 Models with far field stresses s1 ¼ s2 ¼ 20 MPa Stress distributions of models I, II and III when s1 ¼ s2 are presented in Figure 8.10, and the largest displacements of the wall are summarised in Table 8.3. The obliquely intersected bolting gives the smallest displacement, and it is less than half of that of the opening without bolts. Figure 8.11 and Table 8.3 illustrate the range of damage zones in the surrounding rock mass occurring near the wall. The results show that the surrounding rock mass with bolts is generally less damaged than that without bolts. The surrounding rock mass with obliquely intersected bolt system is the least damaged.

257

Reinforcement Mechanism of Rock Bolts Table 8.2. Mechanical parameters of rock mass. Parameter

Model I: No bolts

Model II: Normal bolts

Model III: Obliquely intersected bolts

sc (MPa) st (MPa) E (MPa) (MPa) f (MPa) fr (MPa) sr (MPa) c (MPa) cr (MPa)

26 1/15sc ¼ 1.73 0.4 104 0.25 35 35 200.8 3.58 2.86

1.5sc ¼ 39 1.2st ¼ 2.03 1.15E¼0.58104 0.25 35 35 39 5.36 5.36

2.0sc ¼ 52 1.6st ¼ 2.77 1.5E ¼ 0.75 104 0.25 35 35 52 7.15 7.15

Figure 8.10. Stress distribution of models I, II and III, at roof, when s1 ¼ s2 .

Table 8.3. Largest displacement and damage of models I, II and III when far field s1 ¼ s2. Model

Maximum displacement (cm) Number of damage elements

No bolts

Normal bolts

Oblique intersection bolts

1.80 24

1.11 16

0.84 8

258

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Figure 8.11. Damage around the opening of models I, II and III when s1 ¼ s2 .

8.3.3

Models with far field stresses s1 ¼ 2s2 ¼ 20 MPa

The modelling results for the models I, II and III are presented in Figures 8.12 and 8.13, showing the stress distribution in the roof and sidewall respectively. Figure 8.14 shows the damaged zones near the wall surface for the three models. Again, it can be concluded that obliquely intersected bolt system gives the best results. Table 8.4 shows the maximum displacement on the wall surfaces and the numbers of damaged elements of the three diﬀerent models. The obliquely intersected bolted rock mass sustains less than one-third of the damaged area than that of the unbolted rock mass. The bolt quantity of the obliquely intersected is 20% more than that of normal bolt system. However, the eﬀects of the obliquely intersected bolts on reducing the damaged zones and the displacements of the surrounding rock are signiﬁcant compared with that of normal bolt system. The ﬁnite element modelling uses elastoplastic approach, and post-failure dilation of the rock mass is not considered. By considering the post-failure dilation, the eﬀects of the obliquely intersected bolt system are expected to be more signiﬁcant.

8.4.

SCALED ENGINEERING MODEL TEST

In order to study the reinforcement eﬀects of rock bolts on the surrounding rock mass in an underground excavation, a series of biaxial compression tests on engineering physical model were conducted. The eﬀects of various bolt parameters are compared. The modelling materials and mechanical parameters are selected according to modelling scale to model an excavation in a rock mass. The actual excavation is in a sedimentary rock. The surrounding rock mass is faulted and fractured. The in situ horizontal stresses are 19.5 MPa and 12.8 MPa, respectively. The estimated uniaxial compression strength of the rock mass is about 30 MPa. The elastic modulus and Poisson’s ratio are approximately 1.0 104 MPa and 0.25 respectively. Both stress and geometry scale ratios are 40. The model is 50 50 cm, and the bolts are modelled by bamboo of 2 mm diameter and 40 mm long, to match the stiﬀness ratio. The opening shape is an arched roof with

Reinforcement Mechanism of Rock Bolts

259

Figure 8.12. Stress distribution of models I, II and III, at roof, when s1 ¼ 2s2 .

Figure 8.13. Stress distribution of models I, II and III, at sidewall, when s1 ¼ 2s2 .

vertical walls, as shown in Figure 8.15. The tests are performed under the plain strain condition. Three diﬀerent bolt distributions are simulated: (a) obliquely intersected with bolt at 67.5 to the opening surface, and in the same vertical plane; (b) bolts perpendicular to the opening surface, and (c) no bolts, as shown in

260

Chapter 8

Figure 8.14. Damage around opening of models I, II and III when s1 ¼ 2s2 .

Table 8.4. Largest displacement and damage of models I, II and III when s1 ¼ 2s2 . Model Maximum displacement (cm) Number of damage elements

No bolts

Normal bolts

Oblique intersection bolts

1.45 20

1.23 11

1.03 8

Figure 8.15. Scaled engineering model with dimensions.

Figure 8.16. Bolts are fully grouted in the model tests. The test follows the following procedures: (a) Stresses are applied to the horizontal and vertical directions simultaneously, with vertical stress kept twice that of the horizontal stress. The increment of stress is 0.1 MPa horizontally at each step.

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261

Figure 8.16. Arrangement of (a) no bolt, (b) normal bolts and (c) obliquely intersected bolts.

Figure 8.17. Convergent displacements of the tunnel monitored during the loading.

(b) Loading is stopped when the horizontal and vertical stresses reach 1.0 MPa and 2.0 MPa, respectively. (c) The vertical stress is increased gradually to 4.0 MPa while the horizontal stress is kept at 1.0 MPa. The convergent displacements of the tunnel are monitored during the loading. The monitored displacements are presented in Figure 8.17. It is observed that: (a) The convergent displacement of the rock mass reinforced by the obliquely intersected bolts is generally smaller than that of the rock mass reinforced by normal bolts and the displacement of the roof is reduced by 1640%. The plastic

262

Chapter 8

damage zone occurring in the sidewalls are also smaller for the normal bolting system. (b) Bolting in general, as compared to no bolting, improves the surrounding material properties. The convergent displacement of the opening decreases signiﬁcantly and the plastic damage is greatly reduced by bolting. Normal bolt reduces the displacement of the roof by 1443%. (c) Scaled model tests were also performed on circular opening (Figure 8.15) and similar results were obtained.

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Li S (1991). The time series analysis model for displacement of gallery support. Proc. 4th Nat. Symp. Numer. and Anal. Method in Geotechnique, Beijing, China. Muller L (1978). Remove misconceptions on the New Austrian Tunnelling Method. Tunnels and Tunnelling, October 1978, pp. 29–32. Bellman R (1957). Dynamic Programming. Princeton Univ. Press, Princeton, N.J. Milar DL, Hudson JA (1993). Rock engineering system performance monitoring using neural networks. Artificial Intelligence in the Mineral Sector, Proceedings of an Institution of Mining and Metallurgy Conference, April. Singh VK, Singh D, Singh TN (2001). Prediction of strength properties of some schistose rocks from petrographic properties using artificial neural networks. Int. J. Rock Mech. Min. Sci., 38(2), 269–284. Cai J, Zhao J, Hudson JA, Wu X (1996). Using neural networks in rock engineering systems for cavern performance auditing. Eurock’96 – Int. Symp. on Prediction and Performance in Rock Mech. and Rock Engng., Torino, Italy, pp. 965–972. Kacewicz M (1994). Model-free estimation of fracture apertures with neural networks. Math Geol., 26(8), 985–94. Kim CY, Bae GJ, Hong SW, Park CH, Moon HK, Shin HS (2001). Neural network based rediction of ground surface settlements due to tunnelling. Comput. Geotech., 28(6–7), 517–547. Kovari K (2003). History of the sprayed concrete lining method. Tunnelling and Underground Space Technology, 18, 57–83. Schach R, Garachol K, Heltzen AM (1979). Rock Bolting: A Practical Handbook. Pergamon, Oxford. Stacey TR, Page CH (1986). Practical Handbook for Underground Rock Mechanics. Trans. Tec. Clausthal-Zellerfeld. Stillborg B (1986). Professional Users Handbook for Rock Bolting. Trans Tech Publication, Series on Rocks and Soil Mechanics, Vol. 15. Huang Z, Broch E, Lu M (2002). Cavern roof stability – mechanism of arching and stabilization by rockbolting. Tunnelling and Underground Space Technology, 17(3), pp. 249–261. Beer G, Meek JL (1982). Design curves for roofs and hangingwalls in bedded rock based on voussoir beam and plate solutions. Trans. Instn. Min. Metall., 91, pp. A18–A22. Lee IM, Park JK (2000). Stability analysis of tunnel keyblock: a case study. Tunnelling Underground Space Technology, 15(4), 453–462. Lin D, Fairhurst C (1988). Static analysis of the stability of three-dimensional blocking systems around excavations in rock. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 25, 139–148. Lin D, Fairhurst C, Starfield AM (1987). Geometrical identification of threedimensional rock block systems using topological techniques. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 24(6), 331–338. Gerrard CM, Pande GN (1985). Numerical modelling of reinforced jointed rock masses. Comput. Geotech., 1, 293–318. Bjurstrom S (1979). Shear strength on hard rock joints reinforced by grouted untensioned bolts. Proc. 4th Congr., Int. Soc. Rock Mech., Vol. 2, Balkema, Rotterdam, pp. 1194–1199. Fairhurst C, Singh B (1974). Roof bolting in horizontally laminated rock. Eng. Min. J., February, pp. 80–90.

286 457. 458. 459.

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Index A aperture 30, 43, 45, 80–82 artiﬁcial intelligence technique 246

damage mechanics 27, 144 damaged rheological model 150 damaged zone 83–87, 258 damage tensor 27 deformation back analysis 157 deformation equivalence 27, 29, 30, 32, 33, 38, 43, 83 deformation modulus 8–10 deformation monitoring 158, 160, 166, 201 deformation stiﬀness 14 dilation 17, 33, 122, 124, 250, 251, 253, 255, 258 discontinuity deformation analysis 27 discontinuum theory 27 discrete element method 27, 67 displacement prediction 187, 205 displacement series 205 displacement zone 24, 25 Drucker–Prager criterion 42, 44, 51, 83, 216

4, 228, 229,

B back analysis 2, 4, 42, 44, 157–163, 166, 167, 171–173, 176, 178–187, 190–194, 197, 199–201, 221, 225 2-D back analysis 181, 194, 197, 199 3-D back analysis 190, 194, 199, 200 beneﬁt function 224, 225, 227, 228, 237, 238, 244 biaxial stress 39 block sliding 25 bolt 4, 52, 54, 87, 132–134, 137, 139–144, 150, 191, 214–217, 219–221, 247–262 bolting method 251, 252 boundary element method 67, 91 bridged joint 14, 16 C character curve 68 collinear crack 55, 59 compressive strength 8–10, 14, 17, 19, 20, 54, 66, 248, 249, 251, 255 compressive torsional shear failure 18 construction mechanics 4, 211–214, 221, 238, 241, 244, 246 construction process 3, 157, 211, 213, 214 convergence displacement 176 convergent displacement 261 coupled back analysis 2, 157 crack initiation 16, 20, 54, 55, 59 crack propagation 55 creep 89, 147, 152 crown 24, 25, 70, 77, 167, 181

E elastic back analysis 4, 159 elastic constant 30, 47, 76 elastic shearing modulus 100 elasto-plastic back analysis 180 end eﬀect 14 end-anchored bolt 247, 249, 255 equivalence approach 27 equivalence continuum 28 equivalent continuum model 27–29 excavation scheme 211, 213, 217, 221, 224, 225, 227, 234, 237, 239, 241–245 excavation scheme optimisation 237 excavation sequence 3, 4, 24, 42, 43, 53, 212–217, 221, 223–225, 229, 230, 232, 234, 237, 239, 241, 244 extensometer 42, 150, 153, 160, 161, 168, 169, 182, 191, 194, 207, 208

D damage analysis 144, 150, 155 damage evolution equation 145, 146, 152

287

288 F faults 1, 5, 42, 52, 54, 70, 150, 185, 187, 190, 191, 211 FEM 37–39, 44, 52, 67, 70, 77, 91, 146, 148, 150, 171, 198, 215, 225, 228 ﬁnite element method 30, 37, 67 fracture mechanics 27, 54, 99 fracture toughness 55, 56 fracture 1, 5, 16, 18, 19, 54, 59, 150, 158, 162, 238 frictional bolt 249 full-length grouted bolt 249, 250, 252, 255 G global optimal scheme 221, 225, 227, 228 grey system 74 grey system theory 201, 203 H Hooke’s elastic body 114 horizontal shear load 16 I in situ stress 24, 44, 70, 87, 97, 105, 111, 140, 152, 157, 159–161, 163, 170, 176, 184, 185, 190, 191, 193, 201, 215, 225, 227, 240, 241, 244, 249, 255 in situ test 5, 27, 74, 158 intact rock elements 28 interactive programming 221, 222, 224, 227, 228, 238, 241 internal friction angle 19, 59, 66, 71, 74, 84, 86, 249 J joint dip angle 38, 79, 80, 82, 194 joint distribution 6–8, 14–16, 22, 37 joint elastic modulus 77, 78, 80, 81, 86 joint element 27–29, 37, 52 joint spacing 6, 40, 175, 181 joints 1, 5, 6, 8, 9, 14, 17, 27, 28, 30, 32, 35, 36, 39, 40–46, 49, 54, 59, 75, 78, 80, 83, 86, 150, 185, 194, 211 L lateral deformation 8, 9 lateral tensile crack 9, 11

Index local optimal scheme 221, 225 local shearing failure 9 longitudinal deformation 8 M maximum displacement 52 maximum volumetric strain 152 Maxwell medium 117, 124 modelling material 5, 6, 59, 251, 252, 258 Mohr–Coulomb criterion 33, 50, 123 multiple crack 59, 60 multiple-point borehole extensometer 160, 161, 168 multiple-stage excavation 167, 185, 195, 197, 219 N New Austrian Tunnelling Method 167, 212 non-equal time step 204 non-linear regression 192, 201, 202 normal stress 15, 16, 20–22, 57, 59, 61, 84, 92, 134 O optimisation analysis 176, 183 optimisation process 225, 226, 237, 238 orientation 5, 30, 37, 45, 75, 175, 212 P peak strength 9, 10, 12, 14, 39, 54, 124, 253, 255 persistence 5, 7, 10, 11, 13, 16, 18–22, 25, 30, 37, 40, 43, 56, 59, 61, 76, 80, 82, 84, 86, 87, 181, 194 physical model test 5, 39, 66, 256 plane strain 3, 13–15, 47, 76, 151, 162, 175, 178, 215, 244, 255 plane strain modelling 13 plane stress 3, 5, 14, 16, 24, 30 plastic zone 53, 111, 113–116, 122, 154, 217 post-failure 122, 124, 258 Principle of Optimality 222

289

Index R reinforcement mechanism 4, 247, 248 relative convergence 77 relative error 37–39, 68, 69, 74, 75, 78, 82, 87 residual strength zone 127–129, 132 rheologic behaviour 3, 89 rock bridge 14, 16, 18, 20, 54, 56, 57 rock engineering system 74, 83 rock mass cohesion 14, 20 rock mass dimension 6 rock mass friction angle 14 rock mass strength 9, 11–13, 36, 43, 51, 54, 66, 87, 216, 248, 249, 251, 255 rock reinforcement 3, 212, 215, 247 roughness 30 S scale eﬀect 27, 37–39, 192 secondary tensile crack 9 sensitivity analysis 2, 3, 67, 68, 70, 74–77, 83, 84 sensitivity factor 68, 73, 74, 81, 82, 84, 86 sensitivity function 68, 69, 71, 73 sensitivity order 82 shear dilation 17 shear failure 9, 18, 22, 58, 83, 189, 200 shear failure plane 9 shear sliding 9 shear strength 14, 18, 19, 20, 22, 35, 54–59, 61, 248, 249, 255 shotcrete 52, 132, 150, 153, 191, 215, 221, 247, 248 sidewall 24, 25, 142, 155, 160, 189, 228, 262 similarity 5, 6, 24, 25, 250 similarity ratio 6 size eﬀect 6, 14 softening 122, 124, 250, 253, 255 spacing 4, 6, 30, 39, 40, 53, 175, 181, 220, 221, 252 St. Venant medium 117

stability analysis 3–5, 42, 67, 70, 89, 96, 150, 157, 160, 194 statistic method 201 stiﬀness-reduction method 71 strength equivalence 3, 29, 33, 35, 36, 41, 50 stress back analysis 2, 157 stress concentration 55, 142, 158 stress intensity factor 56 stress-volumetric strain curve 152 structural loosening 25 superposition theorem 89 swelling 152 system character 67, 68, 71 T tensile crack 9, 16, 17 tensile rupture 25 tensile–compressive strength ratio 19 threshold stress 55 time series 201, 205 time series analysis 201 time-dependent characteristics 89 torsional tensile shear failure 18 transversal compressive crack 17 transverse deformation 8 transverse isotropic 4, 172, 174 U unloading process

71

V visco-elastic back analysis 4, 166, 170 visco-elastic deformation 89, 151 visco-elastic zone 115, 116, 118, 122, 123, 126, 127, 129, 133, 138, 140, 144 visco-plastic deformation 89, 151 visco-plastic zone 117, 118, 121–129, 131, 132 Y yield zone

53

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Stability Analysis and Modelling of Underground Excavations in Fractured Rocks

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ELSEVIER GEO-ENGINEERING BOOK SERIES VOLUME 1

Stability Analysis and Modelling of Underground Excavations in Fractured Rocks Weishen Zhu Shandong University, Jinan, China

Jian Zhao Nanyang Technological University, Singapore

Geo-Engineering Book Series Editor

John A. Hudson Imperial College of Science, Technology and Medicine, University of London, UK

2004

Amsterdam – Boston – Heidelberg – London – New York – Oxford Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo

ELSEVIER Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB, UK ß 2004 Elsevier Ltd. All rights reserved. This work is protected under copyright by Elsevier, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-proﬁt educational classroom use. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (þ44) 1865 843830, fax: (þ44) 1865 853333; e-mail: [email protected] You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (þ1) (978) 7508400, fax: (þ1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (þ44) 207 631 5555; fax: (þ44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier’s Science & Technology Rights Department, at the phone, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made. First edition 2004 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for. British Library Cataloguing in Publication Data A catalogue record from the British Library has been applied for.

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Series Preface I am delighted to introduce the new Elsevier Geo-Engineering Book Series. Our objective is to publish high quality books on subjects within the broad Geo-Engineering subject area — e.g. on engineering geology, soil mechanics, rock mechanics, civil/mining/environmental/petroleum engineering, etc. The topics potentially include theory, ground characterization, modelling, laboratory testing, engineering design, construction and case studies. The principles of physics form a common basis for all the subjects, but the way in which this physics is manifested across the wide spectrum of geo-engineering applications provides a rich variety of potential book themes. Accordingly, we anticipate that an exciting series of books will be developed in the years ahead. We welcome proposals for new books. Please send these to me at the email address below. Professor John A Hudson FREng Series Editor [email protected]

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Preface It is well recognised that rock masses are very diﬀerent from common man-made engineering materials. The nature-created geological medium is a discontinuous, inhomogeneous, anisotropic and non-linearly elastic medium. The characteristics of rock masses vary with time, location and direction. Due to the nature of most of underground rock excavations, those excavated structures are usually of a large size and require long-term serviceability. Therefore, the way rock mass characteristics vary with time, location and direction will be reﬂected in the surrounding rock masses of the excavations. Such variation should, ideally, be captured in the modelling and analysis of rock masses. This book attempts to tackle the problems of modelling and analysis of excavations in rock masses, by taking into account discontinuity inﬂuence, time dependence behaviour and construction method. The content of this book is largely based on many years of research work by the senior author and his research group in China, supplemented by the contribution by the junior author and his research groups in Singapore and China. Much of the content of this book has been published in the form of technical papers, while some content is presented for the ﬁrst time. The physical and numerical modelling work described in this book is conducted with several research grants awarded to the senior author, including the China National Natural Science Foundation (NSF) program ‘‘Interaction between geomaterials and hydraulic structures’’ sub-project ‘‘Mechanical characteristics of jointed rock masses and construction mechanics of rock structures’’, other NFC projects (Numbers 59939190, 40272120 and 50229901), as well as the 7-5 National Programs on Science and Technology. Much of the work presented in this book results from the eﬀorts of the research groups headed by the authors. The accomplishment of this book would not have been possible without the great contributions by their colleagues and research students. The authors wish to thank Shiwei Bai, Kejun Wang, Rongming Pan, Xinping Li, Jingnan Xu, Guang Zhang, Ping Wang, Zuoyuan Liang, Bailin Wu, Suhua Li, Rui Ding, Haibin Xu, Haiying Bian, Jungang Cai, Shaogen Chen, Yuhui Zhao, Haibo Li, Yuyong Jiao, Xiaobao Zhao, Hongwei Song and Ashraf Hefny, for their contributions, comments and reviews. The authors also appreciate the encouragement and advice from Elsevier’s editorial team James Sullivan, Vicki Wetherell and Lorna Canderton. Weishen Zhu Jian Zhao

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About the Authors Weishen Zhu is a professor of rock engineering and director of the Geotechnical and Structural Engineering Centre at Shandong University in Jinan, China. Dr Zhu graduated in construction engineering from the Beijing Institute of Mining and Technology in 1956 and obtained a PhD from the Institute of Mining and Metallurgy, Cracow, Poland in 1962. Between 1962 and 2001, Dr Zhu was a research scientist in the Chinese Academy of Science Wuhan Institute of Rock and Soil Mechanics, and he was also a director of the Institute. Dr Zhu is Editor-in-Chief of the Chinese Journal of Rock Mechanics and Engineering, President of the Chinese Society of Underground Engineering and Underground Space, and an editorial board member of the international journal Rock Mechanics and Rock Engineering. Dr Zhu has published 4 books and over 150 technical papers. Dr Zhu’s main research interests are mechanical properties of fractured rock masses, rock reinforcement by bolts, construction mechanics and stability analysis of large rock structures. Dr Zhu has more than 40 years experience in rock mechanics and engineering, and has been involved in numerous rock engineering and underground excavation projects in China. Jian Zhao chairs the underground technology and rock engineering program at Nanyang Technological University of Singapore, and is an adjunct CKSP professor of geotechnical engineering at China University of Mining and Technology. Dr Zhao graduated in civil engineering from the University of Leeds in 1983 and obtained a PhD from Imperial College, London University, in 1987. Dr Zhao is an Editor of the international journal of Tunnelling and Underground Space Technology, editorial board member of the International Journal of Rock Mechanics and Mining Sciences and of 5 other technical journals. Dr Zhao’s main research interests are rock joint properties, rock excavation and support, and rock structure stability under dynamic loads, and has published over 180 technical papers and 3 monographs. Dr Zhao also actively consults on rock engineering and tunnelling projects.

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Contents Series Preface Preface About the Authors

v vii ix

CHAPTER 11 INTRODUCTION

1.1. 1.2.

Rock Mechanics and Underground Excavation Stability Structure of the Book

1 3

CHAPTER 22 PHYSICAL MODELLING OF JOINTED ROCK MASS

2.1.

2.2.

2.3.

Modelling of Jointed Rock Masses Under Plane Stress State 2.1.1 Mechanical Properties of Equivalent Materials 2.1.2 Model Tests of Jointed Rock Mass 2.1.2.1. Effect of Joint Distribution Patterns 2.1.2.2. Strength of Typical Jointed Rock Masses 2.1.2.3. Anisotropy of Rock Strength 2.1.2.4. Relation Between Rock Mass Strength and Joint Persistence 2.1.3 Large Dimension Plane Strain Model Experiment 2.1.3.1. Material Properties and Experimental Methods 2.1.3.2. Testing Results Model Test of Rock Mass with Rock Bridges 2.2.1 Experiment Phase I 2.2.2 Experiment Phase II Model Tests on Stability of Surrounding Rock of Large-scale Cavity 2.3.1 Similarity Conditions of Modelling 2.3.2 Deformation of the Surrounding Rock Mass during Excavation 2.3.2.1. Deformation of the Surrounding Rock Mass during Excavation 2.3.2.2. Failure of the Surrounding Rock Mass during Excavation 2.3.2.3. Stability of the Surrounding Rock Mass

5 5 6 7 9 11 11 13 14 14 16 16 19 22 24 24 24 25 25

xii

Contents

CHAPTER 33 NUMERICAL MODELLING OF JOINTED ROCK MASS

3.1.

3.2.

3.3.

Equivalent Continuum Model for Jointed Rock Masses 3.1.1 Basic Principles 3.1.2 Deformation Equivalence 3.1.2.1. Deformation Equivalence with no Joint Dilation 3.1.2.2. Deformation Equivalence with Joint Dilation 3.1.3 Formula of Strength Equivalence 3.1.3.1. Strength Equivalence in the Case of a Single Joint 3.1.3.2. Strength Equivalence in the Case of Two Joint Planes 3.1.3.3. Analysis of Tensile Failure 3.1.4 Treatment of Elements with Non-persistent Joint 3.1.4.1. Variation of Joint Length 3.1.4.2. Variation of Joint Inclination 3.1.4.3. Scale Effect of Element 3.1.5 Verification of the Numerical Model by Physical Modelling 3.1.6 Examples of Engineering Applications 3.1.6.1. Prediction of Strength of Jointed Rock Mass 3.1.6.2. Stability Analysis of Jointed Rock Masses Equivalent Analysis for Rock Masses Containing Thick Joints 3.2.1 Equivalent Deformation Principle and Method 3.2.1.1. Equilibrium Condition 3.2.1.2. Displacement Compatibility Condition 3.2.1.3. Physical Equations 3.2.2 Basic Principle of Strength Equivalence 3.2.3 Engineering Applications Strength Characteristics of Fractured Rock Mass Under Compressive Shear Stress 3.3.1 Strength of Rock Mass Containing Collinear Cracks 3.3.1.1. Fracture Propagating at Tight Cracks 3.3.1.2. Determination of Shearing Strength of Crack Body 3.3.1.3. Verification Through Model Tests 3.3.2 Strength of Rock Mass Containing Multiple Cracks

28 28 29 30 33 33 33 35 36 36 37 37 38 39 39 39 42 45 45 45 46 46 50 52 54 55 55 56 59 59

CHAPTER 44 SENSITIVITY ANALYSIS OF ROCK MASS PARAMETERS

4.1.

Sensitivity Analysis of Commonly Used Parameters 4.1.1 Method of Sensitivity Analysis 4.1.2 Sensitivity Analysis of Stability of Underground Works

67 67 70

Contents

4.2.

4.3.

4.1.2.1. Computational Model 4.1.2.2. Analysing the Results 4.1.3 Application to Optimisation of Test Schemes Analysis of the Effect of Joint Parameters on Rock Mass Deformability 4.2.1 Application of Equivalent Model for Jointed Rock Mass 4.2.2 Basic Parameter for Sensitivity Analysis 4.2.3 Computational Results 4.2.3.1. Effects of Joint Elastic Moduli on Displacement 4.2.3.2. Effect of Joint Poisson’s Ratio on Rock Mass Deformation 4.2.3.3. Effect of Joint Dip Angle on Rock Mass Displacement 4.2.3.4. Effect of Joint Persistence on Rock Mass Deformations 4.2.3.5. Effect of Joint Aperture on Rock Mass Deformations 4.2.3.6. Comparison of Sensitivity of Different Parameters Sensitivity Analysis of Rock Mass Parameters on Damage Zones 4.3.1 Failure Criterion for the Equivalent Jointed Rock Mass 4.3.2 Sensitivity Analysis of an Underground Cavern Complex 4.3.3 Result and Analysis 4.3.3.1. Effect of Parameters on Damaged Zones 4.3.3.2. Comparison Between Sensitivities of Various Parameters 4.3.4 Summary

xiii 70 71 74 75 75 76 77 77 78 79 80 80 82 83 83 84 84 84 86 87

CHAPTER 55 STABILITY ANALYSIS OF RHEOLOGIC ROCK MASS

5.1. 5.2.

5.3.

5.4.

Rheological Mechanical Models for Rocks and Rock Masses Visco-elastic Surrounding Rock Mass and Supporting Problem 5.2.1 General Solution for Circular Visco-elastic Media 5.2.2 Interaction of Visco-elastic Surrounding Rock Mass and Elastic Lining 5.2.3 Interaction of Rock Mass and Lining of Different Visco-elastic Media 5.2.4 Two-dimensional Stress State in Surrounding Visco-elastic Rock Mass Interaction between the Visco-elastic–Plastic Surrounding Rock and Lining 5.3.1 Stress State in Plastic Zones of Rock Mass 5.3.2 Interaction between Surrounding Rock Mass and Lining Stress State in Visco-elastic–Visco-plastic Surrounding Rock Masses

89 91 92 95 100 105 111 111 113 116

xiv 5.5.

5.6.

5.7.

Contents Rheological Analysis with Dilation and Softening of the Rock Mass 5.5.1 Mechanical Model of Surrounding Rock Mass 5.5.2 Visco-plastic Model Considering Dilation and Softening 5.5.2.1. Physical Model 5.5.2.2. Geometric Equation 5.5.2.3. Equilibrium Equation 5.5.3 Stress Components in Each Zone 5.5.3.1. Visco-plastic Zone 5.5.3.2. Residual Strength Zone 5.5.4 Stress State without Lining 5.5.4.1. Visco-plastic Zone, R1 < r < R2 5.5.4.2. Residual Strength Zone, a < r < R1 5.5.4.3. Determination of Boundary R2 5.5.4.4. Determination of Boundary R1 5.5.5 Stress State with Lining Effect of Bolt Reinforcement in Visco-elastic Rock Mass 5.6.1 Stress State in Different Zones 5.6.2 Discussion and Application Rheological Damage Analysis of the Rock Mass Stability 5.7.1 Damage Evolution Equation 5.7.2 Viscoelastic–Viscoplastic-damage Constitutive Equation and FEM Method 5.7.2.1. Constitutive Equation 5.7.2.2. FEM Method 5.7.3 Application to Stability Analysis of an Underground Opening 5.7.3.1. Decomposition of the Rheological Deformation 5.7.3.2. Determination of Model’s Parameters 5.7.3.3. Comparison between Calculated and In Situ Measured Results

122 122 124 124 124 125 125 125 127 127 127 128 129 129 130 132 132 140 144 145 146 146 148 150 150 152 153

CHAPTER 66 BACK ANALYSIS AND OBSERVATIONAL METHODS

6.1.

Elastic Back Analysis and Stress Distribution Analysis 6.1.1 Elastic Back Analysis 6.1.1.1. Basic Formulation 6.1.1.2. Deformation Monitoring and Back Analysis 6.1.2 Back Analysis of In Situ Stress Distribution

159 159 159 160 163

Contents

6.2.

6.3.

6.4.

6.5.

6.6.

6.7.

6.1.2.1. Computational Zone and Monitored In Situ Stress 6.1.2.2. Determination of Stress Function Visco-elastic Back Analysis and Its Engineering Applications 6.2.1 Method of Site Deformation Monitoring and Its Application Results 6.2.2 Visco-elastic Back Analysis 6.2.2.1. Computation Method 6.2.2.2. Visco-elastic Analysis Results Back Analysis and Optimised Methods in Transverse Isotropic Rock 6.3.1 Basic Formulae of Transverse Isotropic Mechanics 6.3.2 Optimisation Analysis Method 6.3.3 Examples of Engineering Applications 6.3.4 Discussions Back Analysis of Jointed Rock Mass and Stability Prediction 6.4.1 Description of the Project and Monitoring 6.4.1.1. Description of the Project 6.4.1.2. Data Processing and Modification 6.4.2 Back Analysis Using Pure Shape Acceleration Method 6.4.2.1. Computational Procedure 6.4.2.2. Computational Results 6.4.3 Stability Prediction of Powerhouse and Transformer Chamber Three-dimensional Back Analysis of Anisotropic Rock 6.5.1 Displacement Monitoring in Trial Tunnel and Results 6.5.1.1. Set-up of Displacement Monitoring 6.5.1.2. Monitoring Results 6.5.2 Back Analysis Three-dimensional Back Analysis of Jointed Rock Mass and Stability Analysis 6.6.1 Mechanic Model 6.6.2 Summary of Site Monitoring Data 6.6.3 Finite Element Back Analysis of Underground Powerhouse Complex 6.6.4 Stability of Powerhouse and Transformer Chamber Applications of Statistics Model in Deformation Prediction 6.7.1 Non-linear Regression Model 6.7.2 Grey System Theory Model 6.7.3 Engineering Application 6.7.4 Discussion

xv 163 163 166 166 170 170 171 172 172 176 177 179 181 181 181 181 182 183 183 185 190 190 190 191 191 194 194 194 197 200 201 202 203 207 207

xvi

Contents

CHAPTER 77 CONSTRUCTION MECHANICS AND OPTIMISATION OF EXCAVATION SCHEMES

7.1.

7.2.

7.3.

7.4.

Basic Principles of Interactive Construction Mechanics 7.1.1 Basic Principles 7.1.2 Engineering Applications 7.1.2.1. Description of the Project 7.1.2.2. Computational Implementation and Results for Different Excavation Sequences 7.1.2.3. Discussions Applications of Interactive Programming in Optimisation of Cavern Construction 7.2.1 Principles of Interactive Programming 7.2.2 Applications to the Optimisation of Cavern Construction 7.2.2.1. Discussions Artificial Intelligence Techniques in Construction Optimisation 7.3.1 Artificial Intelligence Language Prolog 7.3.2 Problem Solving Algorithm in Cavern Construction Optimisation 7.3.2.1. Automatic Determination of Cavern Excavation Sequences 7.3.2.2. Automatic Generation of Data Files for Finite Element Computation 7.3.2.3. Implementation of Excavation Scheme Optimisation 7.3.3 Discussions Engineering Applications of Artificial Intelligence Optimisation Methods 7.4.1 Description of the Project 7.4.2 Layout of Cavern Group and Arrangement of Step Excavation 7.4.3 Optimisation of Excavation Sequence 7.4.3.1. Rock Mass Assumed as Isotropic Medium 7.4.3.2. Rock Mass Assumed as Layered Isotropic Medium 7.4.3.3. Summaries

212 212 214 215 216 221 221 222 224 227 228 229 229 229 234 237 238 238 238 239 239 240 243 244

CHAPTER 88 REINFORCEMENT MECHANISM OF ROCK BOLTS

8.1.

Effect of Bolts on Supporting the Rock Mass 8.1.1 Effects of Rock Bolts 8.1.1.1. Reinforcement 8.1.1.2. Post Effect of Pre-stressing 8.1.1.3. Prompt Prevention

247 247 247 248 248

Contents

8.2.

8.3.

8.4.

8.1.1.4. Good Match to Deformation 8.1.1.5. Flexibility in Construction 8.1.2 Reinforcement Mechanism of Rock Bolts Physical Modelling of Rock Bolts 8.2.1 Similarity of Model Materials 8.2.2 Comparison of Different Bolting Methods 8.2.3 Analysis of Test Results Numerical Modelling of Bolt 8.3.1 Basic Parameters of the Numerical Model 8.3.2 Models with Far Field Stresses s1 ¼ s2 ¼ 20 MPa 8.3.3 Models with Far Field Stresses s1 ¼ 2s2 ¼ 20 MPa Scaled Engineering Model Test

xvii 248 248 248 250 250 251 255 255 255 256 258 258

REFERENCES

263

SUBJECT INDEX

287

This Page Intentionally Left Blank

Chapter 1

Introduction This book attempts to provide techniques for solving the problems in modelling and analysis of excavations in fractured rock masses, by taking into account the discontinuity inﬂuence, time-dependent behaviour and construction method dependent phenomenon. This chapter provides an overview on the rock mechanics issues related to underground excavation and stability.

1.1.

ROCK MECHANICS AND UNDERGROUND EXCAVATION STABILITY

Rock structures are generally excavated within comparatively competent rock masses. However, rock masses usually consist of various discontinuity features, such as faults, joints and fractures. They signiﬁcantly inﬂuence the stability of rock masses surrounding an excavation opening. The deformation and stability of the surrounding rock masses is controlled by the mechanics of the rock masses subjected to the change of conditions, which is a function of in situ condition before the excavation and disturbance due to the construction activities. The instability of underground works is mainly caused by the redistribution of stresses in the surrounding rock masses due to the excavation activity, due to excessive stress or excessive deformation. The process and result involves the interaction among the rock masses, the boundary conditions, and the engineering activity [1–3]. In order to understand the behaviour of the rock masses surrounding the excavations, it is necessary to understand the basic behaviour of the fractured rock masses subjected to loading and unloading conditions [4–7]. Common methods to study such behaviour are through theoretical analysis when the problems are simple, and through physical and numerical modelling when analytical solutions are not readily available [8–16]. Physical modelling is one of the basic tools to understand the behaviour and mechanism of jointed rock masses subjected to various boundary conditions, primarily loading conditions [17–19]. Whenever possible, physical modelling and testing should be conducted to provide the direct observation and basic understanding of engineering behaviour. It is diﬃcult, often impossible, to quantitatively predict the mechanical properties of jointed rock masses by physical modelling due to their complexity and large scale. Numerical modelling oﬀers wide applications to simulate jointed rock mass

1

2

Chapter 1

behaviours including the eﬀects of loading and time, and the behaviour of rock material, rock joints and rock masses [20–26]. The rationality and reliability of the results from numerical methods depend, to a great extent, upon the appropriate selection of computational model and mechanical and mathematical parameters [27–30]. Once the computational model is determined, the key to success hinges on the rational selection of the computing parameters. There are many factors and parameters that aﬀect the rock mass behaviour and stability. One has to identify the order of importance of all the parameters [31,32]. In terms of computation, the limited resource may be the primary restriction. One of the common methods is the sensitivity analysis of various parameters within a system [33,34]. Rocks and rock masses often exhibit time-dependent behaviour, especially weak and soft rocks or highly fractured rock masses [35–45]. In underground excavation, the time-dependent phenomenon can be found that the loading on support elements gradually increases, leading to the ﬁnal failure of the excavated structure [46–48]. Study of underground excavation in rock masses usually involves various modelling techniques and analysis approaches. Those methods can be physical tests, numerical modelling, observation and back analysis. The ultimate goal is to optimise excavation and support [8,49–52]. In recent years, the back analysis method has been widely applied in geotechnical engineering especially in the underground works [53–56]. Various related analytical and numerical techniques have been developed [57–60]. The method is based on the required input physical information, and can be divided into deformation back analysis method, stress back analysis method and coupled back analysis method. The physical information in the coupled back analysis method requires both deformation and stress. The back analysis has been applied to various rock engineering projects, particularly to underground excavation, to verify the support design and opening stability [57–62]. Construction of rock engineering projects usually requires a long duration, from a few months to a few years. The construction of these rock engineering projects will disturb the initial stable state of the rock masses. The various rock mass parameters interact in a dynamic interactive process until the rock mass reaches a new equilibrium state [63–65]. The construction is therefore a dynamic interactive process in time and in space. The success in constructing and managing a rock engineering project not only depends on the eventual state of the project, but also on the interim process and the construction methods adopted [66–68]. Construction of large-scale rock engineering projects is implemented by continual excavation of new working faces. Each newly excavated face interacts dynamically with the existing excavated space in time and in space. This dynamic interactive process of rock engineering constructions is non-inverse and non-linear. Its

Introduction

3

eventual state (or solution) is not unique but changeable with the interim process [1,63–65,69,70]. In other words, the eventual state is strongly dependent on the stress paths or stress histories. This leads to the possibility of the optimisation of construction process. Adopting a proper excavation sequence and installing eﬀective rock reinforcement are engineering measures to stabilise large-scale underground excavations.

1.2.

STRUCTURE OF THE BOOK

This book is divided into eight chapters. Chapter 1 provides an overview of rock mechanics issues in underground excavation in fractured rock masses, on the various topics covered by the main chapters of this book. Chapter 2 addresses physical modelling of jointed rock masses. The observation and discussion are based on extensive laboratory studies on modelled rock masses with discontinuities. The physical modelling includes plane stress and plane strain tests of various jointed rock masses and bridged rock masses, and large-scale physical modelling of rock masses surrounding an excavation, and deformation analysis of excavation opening. Chapter 3 deals with ﬁnite element based numerical modelling of the jointed rock masses together with computational examples. Deformation and strength equivalence formulations as well as treatment of rock joints are discussed. The numerical modelling is compared with physical modelling and examples of engineering application are given. Treatments of thick joints, collinear and multiple discontinuities are also dealt with in Chapter 3. It provides solutions on deformation and strength equivalence treatment for thick discontinuities, and collinear and multiple discontinuities. Chapter 4 focusses on the sensitivity analysis of rock mass parameters. It covers the sensitivity analysis of common rock material and rock joint parameters, including methods of sensitivity analysis, and application of equivalent model for jointed rock mass. Examples on sensitivity analysis of rock mass parameters based on the extent of damage in the surrounding rock mass for a large underground cavern project is illustrated. Weak and soft rocks with rheologic behaviour are often diﬃcult mediums to model and analyse due to their time-dependent nature. Chapter 5 speciﬁcally addresses stability analysis of rheologic rock masses. It outlines rheological mechanical models for rocks and rock masses, visco-elastic and visco-plastic modelling, interaction of rock mass and support, and rheological damage and damage constitutive equations in numerical modelling.

4

Chapter 1

Chapter 6 presents the principle of back analysis and observational methods commonly used in underground excavation stability assessment. The back analysis methods described include elastic back analysis, visco-elastic back analysis, back analysis and optimised methods in transverse isotropic rock masses. The method extends to plane and three-dimensional back analysis of jointed anisotropic rock masses, and stability analysis of excavation stability in such rock masses. Applications of various statistics models in deformation prediction in the stability assessment are discussed. The principle of construction mechanics and excavation optimisation is presented in Chapter 7. It covers the principles of interactive construction mechanics and their applications to optimise cavern construction. It also outlines artiﬁcial intelligence techniques for construction optimisation and problem-solving algorithm in cavern construction optimisation. Engineering applications of artiﬁcial intelligence optimisation method and optimisation of excavation sequence are also presented in this chapter. Chapter 8 speciﬁcally deals with reinforcement mechanism of rock bolts, including the eﬀect of bolts on supporting the rock mass and reinforcement mechanism of rock bolts. Physical and numerical modelling of rock bolts are presented, with varying bolt spacing, length and layout, and with varying rock stress conditions. A scaled engineering model test is illustrated to verify the modelling results. A comprehensive list of literatures on those topics is given at the end of this book, in the reference section.

Chapter 2

Physical Modelling of Jointed Rock Mass Rock structures are generally constructed within comparatively hard rock masses. The rock masses consist of various structural features, such as faults, joints and fractures. They signiﬁcantly inﬂuence the stability of rock masses. In general, faults are treated as locally special fractures [71–73]. The characteristics of strength and deformation of the faults are studied speciﬁcally for the stability analysis. However, for joints and fractures, due to their abundant numbers, such special treatment generally does not apply. Instead they are regarded as basic rock mass elements in a considerable dimension [4–7,74–95]. However, the joints and fractures signiﬁcantly aﬀect the mechanical properties of the rock masses. Generally, distributions of joints are orderly in sets. In physical and numerical modelling, it is common that for a project, based on in situ geological investigation, two to three major joint sets are identiﬁed. The mean density, persistence, orientation and typical distribution patterns are generalised and modelled. Often, these joints are distributed intermittently. Therefore, the study presented here will emphasise the behaviour of rock masses where joints distribute orderly and intermittently. If in situ tests are conducted to study the behaviour of rock mass containing suﬃcient joints, the dimension of the specimen needs to be a few tens of meters. However, these tests are not commonly conducted. Usually, the more practical method is the physical modelling with a reduced scale [78,96–102]. The mechanics of deformation and failure are observed, which forms the basis for further study.

2.1.

MODELLING OF JOINTED ROCK MASSES UNDER PLANE STRESS STATE

2.1.1 Mechanical properties of equivalent materials For physical model tests, it is important to select a modelling material of which the properties are similar to those of the material modelled. Obviously, not all the properties can meet the law of similarity. Usually, the main parameters are made to meet the similarity conditions and secondary parameters are made to meet the conditions approximately [102]. A physical model test is performed to model a large-scale hydropower cavern project. The model is made of sand, barite powder and an organic polymer resin. Blocks are made by those materials through compaction and heating. The mechanical properties of the modelling material and the modelled rock masses are summarised in Table 2.1. 5

6

Chapter 2

Table 2.1. Mechanical properties of the modelling material and the modelled rock mass. Mechanical properties sc (MPa) E (MPa) c (MPa) f ð Þ cj (MPa) fj ð Þ

Modelling material and fracture

Modelled rock material and fracture

Actual rock material and fracture

0.545 68.3 0.79 44.4 0.0094 33.6

218 2.73 104 31.16 44.4 3.76 33.6

212.4 3.0 104 16 56 0.5 36.9

The physical model has the property similarity ratio of 400. The ratio of the rock strength to modelling material strength and the ratio of the rock modulus to the modelling material modulus are both 400. Rock joints in the rock mass are simulated by the joints between blocks. To form intact portion, the blocks are cemented by the polymer resin. The testing results indicate that the physical model worked well in simulating the rock masses of diﬀerent joint systems.

2.1.2 Model tests of jointed rock mass In general, only two or three major joint sets have the governing inﬂuence on the engineering properties of the rock mass [102–106]. In most modelling studies, two representative joint sets are usually taken in the plane modelling. The two joint sets modelled usually intersect approximately orthogonally at an acute angle. Studies on both joint set arrangements were performed and they are presented in this section. Rock masses that consist of two orthogonal joint sets are fairly common, e.g., the rock mass surrounding at Ertan and Xiaolangdi hydroelectric power stations. Physical modelling of the Ertan hydroelectric power cavern is conducted. From the mapped joint data, three common joint distribution arrangements are modelled, to represent a number of possible combinations of the intersection between joint sets, as shown in Figure 2.1. The modelling of mechanical characteristics of jointed rock masses should take into account the eﬀect of joint quantity and joint properties. Studies by Muller and co-workers [107] have indicated that when the ratio of the rock mass dimension (Dm) over the joint spacing (sj) is greater than 10 (i.e., Dm/sj >10), the strength and deformation characteristics of the rock mass model are consistent, and the size eﬀect has little inﬂuence. The model has a size of 50 50 7 cm and consists of blocks of 5 10 7 cm, that satisﬁes the requirement of Dm/sj >10.

Physical Modelling of Jointed Rock Mass

7

Figure 2.1. Patterns of joint set arrangements modelled. (a) intact (no joints rock); (b) two joint sets with the same persistence of 50% forming running-through in one direction; (c) two joint sets with the same persistence of 50% forming T-shapes; (d) two joint sets with persistence of 50% and 100% respectively.

Figure 2.2. Loading and monitoring set-up of the test system.

The loading and monitoring set-up of the test system is shown in Figure 2.2, which consists of a loading frame, loading and monitoring devices. A 0.5 cm thick layer of sand is placed between the ends of the physical model and the rubber pockets to eliminate end friction. Through a series of experiments, the failure mechanism of rock mass, the eﬀect of joint distribution and eﬀect of shearing are studied.

2.1.2.1 Effect of joint distribution patterns. Three model tests are performed to represent rock masses of diﬀerent joint distribution patterns as shown in Figure 2.1.

8

Chapter 2

In Figure 2.1(b), major principal stress is applied at diﬀerent directions. When the major principal stress is perpendicular to the direction of running-through (Figure 2.3(b)), the model has a low deformation module and a high compressive strength, because the joints can be completely closed and the rock masses become intact. In this case, the s1–e1 curve exhibits a concave shape. While for the same joint sets, when the major principal stress is applied parallel to the direction of runningthrough (Figure 2.3(a)), the lateral deformation is high and the tested model is liable to failure by splitting. In this case, the uniaxial compressive strength is lower than the former. The comparison between the two strain–stress curves is shown in Figure 2.4. Results obtained from models shown in Figure 2.3(a) and Figure 2.3(c) are illustrated in Figure 2.5. As expected, because the joints in the model of Figure 2.3(c) are bridged, both longitudinal and transversal deformations are small, the

Figure 2.3. Failure patterns of jointed rock mass.

Figure 2.4. Stress–strain curves of diﬀerent joint distribution patterns shown in Figure 2.3(a) and (b).

Physical Modelling of Jointed Rock Mass

9

Figure 2.5. Stress–strain curves of diﬀerent joint distribution patterns shown in Figure 2.3(a) and (c).

compressive strength is high and the s1–e1 curve is similar to that of intact rock material with no joints. In the model shown in Figure 2.3(b), because the runningthrough joint surface is roughly perpendicular to s1, its failure mostly exhibits the form in which longitudinal tensile-opening takes place at the sharp turning points of the joints, accompanied by local shearing failure. For the rock mass in Figure 2.3(c), T-shaped joints gradually run through each other longitudinally and lateral cracking occurs under compressive stresses. As a result, compound shear failure planes are ﬁnally formed. For the rock mass shown in Figure 2.3(d), the main failure mode is shear failure along joint planes that have been run through completely, accompanied by lateral tensile cracking of the materials. In summary, there are two basic failure patterns of the rock masses: lateral tensile cracking and shear sliding. Due to various combinations of joint set distributions and arrangements, failure mechanisms and development are diﬀerent. In general, secondary tensile cracks occur at the joint tips ﬁrst, then the cracks link up with the adjacent joints, followed by compound lateral tensile cracks and ﬁnally overall shear failure takes place along a plane shear.

2.1.2.2 Strength of typical jointed rock masses. The eﬀect of lateral shear on the rock mass strength is studied with the model shown in Figure 2.3(a), where major principal shear is applied vertically and linear principal shear (s2) is applied horizontally. Table 2.2 and Figure 2.6 give the typical results of tests under diﬀerent s2. They show that the peak strength, s1, of the rock masses increases with the increasing s2. The deformation module also increases but the lateral deformation decreases rapidly.

10

Chapter 2 Table 2.2. Testing results at different lateral pressure. Testing No. 1 2 3 4 5

s2 (MPa)

s1 (MPa)

E (MPa)

v

0 0.05 0.15 0.25 0.34

0.28 0.41 0.59 0.69 0.96

29.75 65.45 61.85 117.3 122.9

0.1 0.1 0.25 0.31 0.43

Figure 2.6. Change of stress and strain at diﬀerent lateral stress for model in Figure 2.3(a).

The modelling results of the jointed rock mass in Figure 2.3(d) show that when the persistence of joint sets increases, the compressive strength of the rock mass decreases. The modelling results indicate that the deformation modules are governed by the persistence of joint sets, as shown in Figure 2.7. Table 2.3 summarises the uniaxial compressive strength (sc) and deformation modules (E) of the four diﬀerent rock masses shown in Figure 2.3. In summary, the variation of joint set pattern strongly aﬀects the strength and deformation characteristics of a rock mass. From the modelling studies, the following qualitative observation is noted for the rock mass represented by Figure 2.3(a), the peak strength and lateral stress are linearly proportional in general,

11

Physical Modelling of Jointed Rock Mass

Figure 2.7. Stress–strain relationship of rock mass model of Figure 2.3(d).

Table 2.3. Strengths and elastic moduli of four models. Model type 2.3 2.3 2.3 2.3

(a) (b) (c) (d)

sc (MPa)

E (MPa)

0.29 0.37 0.35 0.2

29.75 27.38 43.60 7.13

as shown in Figure 2.8, failures in most cases start with the lateral tensile cracking along joints and follow by the formation of a shear plane. Failure occurs ﬁnally along the inclined plane.

2.1.2.3 Anisotropy of rock strength. The variation of the strength of the jointed rock mass in response to the change of the major principal stress direction is studied. Rock mass of joint pattern shown in Figure 2.3(a) is subjected to the ﬁxed s1 and s2 with the direction of the shears. The testing results are presented in Figure 2.9 and Table 2.4. The study shows that the lowest strength of the rock mass occurs when a1 is at 40 50 .

2.1.2.4 Relation between rock mass strength and joint persistence. Studies on the eﬀect of joint persistence on rock mass strength are carried out. The persistence of

12

Chapter 2

Figure 2.8. Relationship between peak strength and lateral stress for rock mass of Figure 2.3(a).

Figure 2.9. Change of rock mass strength with joint direction.

Table 2.4. Change of strength with joint direction. Strength

s1 (MPa) s1/sc

a1 0

10

20

30

40

0.4 0.72

0.37 0.67

0.31 0.57

0.27 0.49

0.24 0.44

Note: sc ¼ 0.55 MPa; s2 ¼ 0.05 MPa.

13

Physical Modelling of Jointed Rock Mass

Figure 2.10. Change of rock mass strength with joint persistence.

Table 2.5. Effect of joint persistence on the rock mass strength. n2 (%) s1 (MPa)

0

20

30

50

80

0.26

0.25

0.22

0.2

0.18

the second joint set (n2) in Figure 2.3(c) is changed, while the loading conditions remain the same (s2 ¼ 0.05 MPa). Modelling study reveals the relationship between the rock mass strength and the joint persistence, as shown in Figure 2.10 and Table 2.5. It can be seen from Figure 2.10 that the strength of the rock mass decreases with the increasing joint persistence. The rate of the strength change is small when joint persistence is below 20%. A rapid strength decrease is observed when the persistence increases from 20% to 30%.

2.1.3 Large dimension plane strain model experiment In many underground rock engineering works, such as tunnels, the surrounding rock masses are in plane strain condition [108–111]. Therefore, the model study on mechanical behaviour of the surrounding rock masses under plane strain condition is of great importance. In plane strain modelling, the model has suﬃcient length to remove the inﬂuence of end friction. The results of plane strain model experiments for jointed rock mass are discussed in the following sections.

14

Chapter 2

Figure 2.11. Four types of jointed rock mass modelled.

2.1.3.1 Material properties and experimental methods. The preparation of the model material is similar to that of the plane stress model. Deformation characteristics of the joint plane are measured. The material has the following indexes: unit weight g ¼ 2.2 104 N/m3, uniaxial compressive strength sc mm ¼ 0.87 MPa, tensile strength st mm ¼ 0.11 MPa, elastic module Emm ¼ 170 MPa, Poisson’s ratio v ¼ 0.22. Joint plane properties are: cohesion cj ¼ 0.01 MPa, friction angle fj ¼ 39 ; shear stiﬀness Ks ¼ 5 MPa/cm, and normal stiﬀness Kn ¼ 75 MPa/cm. Large-size model of 100 50 14 cm is used to eliminate size eﬀect and end eﬀect. The joints in the rock mass are simulated by the contact surfaces of the blocks. The shear strength and deformation stiﬀness of the joints are measured. To simulate the rock bridge, the contact surfaces of the blocks are cemented to the material strength. The shearing strength of the bridged joint is also measured. The loading is applied through a large steel frame under the condition of plane strain. The tests are performed on 20 rock mass models of four diﬀerent joint patterns. These four patterns are shown in Figure 2.11. 2.1.3.2 Testing results. In the tests, the deformations of the model in three directions are measured. Diﬀerent failure stages are judged mainly by the deformation rate and the tendency of the deformation curves. Figure 2.12 and Table 2.6 show the typical testing results. The rock mass strength is calculated from various peak strengths (s1) under diﬀerent lateral pressures (s2) using the Least Square Method, expressed by the overall equivalent cohesion, cmm, and inner friction angle of fmm. Figure 2.13 shows the typical deformation curves of the rock mass of joint distribution pattern of Figure 2.11(D) under diﬀerent lateral pressures. It can be seen from the modelling results that for the four diﬀerent rock masses, their overall strength can be roughly related to the material strength and the joint strength, by the following equations: Rock mass cohesion ¼ (0.2 0.3) (Rock material cohesion þ Rock joint cohesion), Rock mass friction angle ¼ 0.5 (Rock material friction angle þ Rock joint friction angle).

15

Physical Modelling of Jointed Rock Mass

Figure 2.12. Relationship between speciﬁc strengths of rock mass, material and joint, for four rock masses with diﬀerent joint distribution patterns.

Table 2.6. Results from large-size model tests of jointed rock masses. Model type

s1 (MPa)

s2 (MPa)

s3 (MPa)

s0 (MPa)

s0/s1

90-1 90-2 90-3 90-4 90-5

A

0.31 0.62 0.71 1.45 1.28

0 0.1 0.15 0.25 0.31

0.02 0.05 0.06 0.23 0.11

0.19 0.40 0.52 1.25 1.11

0.61 0.65 0.73 0.86 0.87

91-1 91-6 92-4 92-3

B

0.18 0.68 1.42 1.53

0.02 0.10 0.30 0.31

0.06 0.15 0.62 0.59

0.09 0.35 0.95 0.92

0.50 0.51 0.67 0.60

92-2 91-7 91-8 92-1

C

0.11 0.52 0.97 1.44

0 0.08 0.21 0.27

0.01 0.07 0.16 0.39

0.07 0.28 0.51 —

0.55 0.54 0.53 —

91-2 91-5 91-3 91-4

D

0.12 0.44 0.55 0.69

0.01 0.05 0.09 0.14

0.03 0.06 0.12 0.10

0.09 0.26 0.36 0.48

0.73 0.60 0.66 0.70

Model No

cmm (MPa)

fmm ( )

0.05

42

0.04

39

0.02

40

0.03

39

Note: s0 is the stress in the direction of s1 when the volumetric strain rate is zero; s3 is the normal stress applied on the model surfaces to keep the plane strain state.

16

Chapter 2

Figure 2.13. Stress–strain curves of rock mass model of joint distribution shown in Figure 2.11(D).

2.2.

MODEL TEST OF ROCK MASS WITH ROCK BRIDGES

In this section, modelling is performed to study the failure of rock bridges, the relationship between the joint intensity and the strength of the rock, and the strength of the rock mass of the bridged joint. Results of two phases of modelling are presented in the following sections.

2.2.1

Experiment Phase I

In this phase, the model materials are gypsum, diatomite and water mixture that can be conveniently poured to set. Fractures of given persistence are cast. The principal physico-mechanical parameters of the model are given in Table 2.7. In fracture preparation, thin pieces of steel are buried in the specimen when the material is wet and then drawn out after set. The fractures are naturally closed. The specimens are subjected to drying and curing. In tests, the models are loaded by the plane stress, as shown in Figure 2.14. In addition to deformation transducers, special coating is applied at model surface around the fractures to observe crack initiation and propagation at the tips. The persistence of the fractures is at 0, 20%, 40%, 60% and 80% respectively. Seven normal stresses at 0.08, 0.2, 0.3, 0.35, 0.4, 0.5 and 0.6 MPa are applied. The sheared planes of all the specimens are examined after each test. During the testing, normal stress is kept constant, and horizontal shear load is applied gradually until failure occurs. Typical stages of crack initiation and propagation before failure are observed, as shown in Figure 2.15: (a) Small feather-shaped cracks appear around the existing fractures; (b) Tensile cracks appears along the direction of shear load at one tip of the fracture ﬁrst and then appear at the other tip;

Physical Modelling of Jointed Rock Mass

17

Table 2.7. Materials and fracture parameters. Model material

Dry density (g/cm3) Porosity (%) Uniaxial compressive strength (MPa) Compressive modulus (MPa) Poisson’s ratio Tensile strength (MPa) Tensile modulus (MPa) Fiction angle ( ) Cohesion (MPa)

Fracture

Fracture length (mm) Normal stiffness (MPa/cm) Shear stiffness (MPa/cm) Friction angle ( ) Cohesion (MPa)

0.68 68.9 2.05 1.3 103 0.2 0.27 2.6 103 35 0.53 20, 40, 60, 80 100 20 10 0.1

Figure 2.14. Model test arrangement.

(c) A pair of transversal compressive cracks appear along two approximately orthogonal directions between two adjacent tensile cracks; (d) Rotation of lozenge block takes place, in association with shear dilation and small secondary cracks appearing around large cracks; (e) Visible opening of joints takes place; and macro visible shear failure occurs where the block breaks along the lozenge diagonal.

18

Chapter 2

Figure 2.15. Stages of shear failure of the model.

For samples of diﬀerent joint persistence and diﬀerent vertical stresses, the shear failure has three categories [112–119]: (a) For specimens with low joint persistence and medium vertical stress, considerable dilatancy of the specimen occurs. The failure mode is compressive torsional shear failure. (b) For specimens with medium joint persistence and high vertical stress, the deformation and failure of the specimen undergoes the following stages: (i) micro cracks are generated at the fracture tips, (ii) cracks propagate and the secondary compressive torsional cracks occurs, (iii) lozenge blocks are formed, (iv) shear crack appears along the diagonal of the lozenge blocks and connects with tips of the fractures, and (v) the overall failure of the specimen occurs along the existing fractures and the new shear cracks, and the dilatancy takes place within the whole specimen. The failure mode is torsional tensile shear failure. (c) For specimens with high joint persistence and high vertical stress, the deformation and failure of the specimen develop in the following stages: (i) small ‘‘rock bridge’’ is broken due to the vertical stress, (ii) with the increase of the shear stress, overall failure takes place along the existing fractures and the cracks crossing the rock bridges. Although a slight dilatancy can be observed in the shearing process, the specimen as a whole fails basically due to the compression. This is pure shear failure. Results of over 20 specimens are summarised in Figure 2.16. It shows four shear strength envelopes for four groups of joint persistence. The apparent shear strength

Physical Modelling of Jointed Rock Mass

19

Figure 2.16. Change of shear strength with joint persistence.

falls with increasing persistence. When the persistence increases from 0 to 65%, internal friction angle decreases from 35 to about 15 , and the cohesion decreases from over 0.5 MPa to about 0.3 MPa.

2.2.2

Experiment Phase II

In order to have better control of joint persistence in the specimens, a new material mix is used in the second phase of experiments. The properties of the new material mix of the model are close to those of rock. Joint persistence is properly controlled. The material used is the mixture of sand, barite powder, colophony and alcohol. Fractures are created with polythene ﬁlms of diﬀerent lengths. Two types of fractures with diﬀerent shear strength (friction coeﬃcient) are created by two diﬀerent methods: one by a single ﬁlm and another one by two ﬁlms with sandwiched grease. The designed joint persistence is the same as those in the phase I, i.e., 20%, 40%, 60% and 80%. Mechanical tests show that this material has similar properties as some sedimentary rocks and similar dilatancy character before failure. The tensile–compressive strength ratio is about 0.11. The mechanical parameters are summarised in Table 2.8.

20

Chapter 2

Table 2.8. Properties of model material and fracture. Model material

Dry density Uniaxial compressive strength (MPa) Compressive modulus (MPa) Poisson’s ratio Tensile strength (MPa) Friction angle ( ) Cohesion (MPa)

2.21 0.15 1.45 103 1.12 0.122 38.5 0.25

Single fracture

Fracture length (mm) Normal stiffness (MPa/cm) Shear stiffness (MPa/cm) Friction angle ( ) Cohesion (MPa)

20, 40, 60, 80 43 17 30 0.065

Double fracture

Fracture length (mm) Normal stiffness (MPa/cm) Shear stiffness (MPa/cm) Friction angle ( ) Cohesion (MPa)

20, 40, 60, 80 33 11 24.3 0.041

The test equipment, installation and testing methods are the same as that adopted in Phase I. In the tests, failure processes are observed and noted. The crack initiation and propagation are similar to that in Phase I. From test data of over 60 specimens, a series of peak shear strength with corresponding normal stress are obtained by means of the Least Square Method, and are presented in Figures 2.17 and 2.18. The results indicate a systematic decrease of shear strength of the specimen with increasing joint persistence. For specimens with double fracture the decreasing rate is greater. The strength envelopes show relatively good linearity. With increasing joint persistence, the rock mass cohesion and rock mass internal friction coeﬃcient decrease accordingly. The weighted mean method is commonly adopted for predicting the strength of the fractured rock mass. The shear resistance of the shear plane is obtained by summing the shear resistances contributed by the fracture and by the bridges. By assuming the normal stress on the shear plane before failure is uniformly distributed, the shear resistance of the shear plane can be calculated as, t ¼ ½ncj þ ð1 nÞcr þ sn ½nj þ ð1 nÞ r

ð2:1Þ

where n is the joint persistence, cj is the joint cohesion, cr is the cohesion of rock bridge, fj is the internal friction coeﬃcient of joint plane, fr is the internal friction coeﬃcient of rock bridge.

Physical Modelling of Jointed Rock Mass

21

Figure 2.17. Shear strength at diﬀerent normal stress and joint persistence for single fracture model.

Figure 2.18. Shear strength at diﬀerent normal stress and joint persistence for double fracture model.

22

Chapter 2

Figure 2.19. Change of normalised (a) cohesion and (b) friction angle with persistence.

However, the actual distribution of the normal stress on the shear plane before failure is not uniform. The failure mechanism of the fractured rock mass under the shear loading is complex, characterised by multi-stage development of fracturing [117–129]. It is often a combined tensile-shear failure [130–133]. Therefore the shear strength expressed in equation (2.1) diﬀers from the actual strength. The diﬀerences in cohesion and friction angle are analysed by normalising the actual cohesion and friction to that expressed in equation (2.1), and are shown in Figure 2.19. It can be seen that the diﬀerences are little when the persistence is less than 40%. But with the increase of persistence the diﬀerences becomes greater. When the persistence is between 40–80%, the error is about 20%.

2.3.

MODEL TESTS ON STABILITY OF SURROUNDING ROCK OF LARGE-SCALE CAVITY

Numerous model tests on large-scale underground projects have been conducted (e.g., [25,125,134–136]). However, model tests studying the stability of the surrounding jointed rock mass are limited. In this section, a large-scale model test on jointed rock mass stability is performed by modelling a rock cavern project [137]. Site investigation shows that there are three major joint sets: NE and NW sets with steep dip angles, and EW with a gentle dip angle. The surrounding rock mass of the cavern is cut into prisms and polyhedrons by these three joint sets. For simpliﬁcation of the problem, only the NE and EW sets are modelled, as shown in Figure 2.20. The joint sets have cut the surrounding rock mass into cuboids of 8 5 l0 m in size. In the model, these joint distributions and block arrangements are simulated (Figure 2.21).

Physical Modelling of Jointed Rock Mass

23

Figure 2.20. Polyhedron blocks produced by three joint sets cutting through the surrounding rock mass.

Figure 2.21. Potential failure patterns of the opening and sequence of excavation.

24 2.3.1

Chapter 2 Similarity conditions of modelling

A physical simulation model should meet the similarity to the prototype in the following aspects: (i) (ii) (iii) (iv) (v)

Size and conﬁguration of the engineering project, Geometry of the geological structures, Physico-mechanical properties of the surrounding rock mass, The initial stress state, and The construction sequences.

A scale of 1:200 is chosen between the model and the prototype. The rectangular model has a height of 1.7 m and a width of 1.6 m. The model material is made of gypsum, sands of various granular sizes and water. It is modelled by plane stress state condition. An initial horizontal in situ stress of 28.84 MPa and a vertical in situ stress of 13.35 MPa are applied, before the simulation of the excavation. In order to simulate the actual excavation sequence, a ﬁve-stage excavation sequence is adopted from the top to the bottom for the excavation, as shown in Figure 2.21.

2.3.2

Deformation of the surrounding rock mass during excavation

The model is applied with a vertical and a horizontal compressive stress ﬁrst at boundaries to simulate the in situ stresses. The stresses are maintained at constant, while excavations in ﬁve stages are modelled. The deformation and failure are monitored during the excavation.

2.3.2.1 Deformation of the surrounding rock mass during excavation. In the ﬁrst to the third stage of excavation, both vertical and horizontal displacements are not remarkable. From the beginning of the fourth stage through the ﬁfth stage, large inward vertical displacements take place at the crown, with a maximum displacement of about 0.1 cm. The displacements at base are small. Large horizontal displacements are noted on two side walls, with maximum displacement of 0.3 cm, as shown in Figure 2.22. After the ﬁfth stage when the excavation has been completed, the displacement zones extends to 15 m above the crown and 53 m away from the sidewalls, as shown in Figure 2.22. The results show that the horizontal displacement zones are considerably large, about 8 times that of the excavation width.

Physical Modelling of Jointed Rock Mass

25

Figure 2.22. Displacement curve and aﬀected range in rock mass surrounding the excavation.

2.3.2.2 Failure of the surrounding rock mass during excavation. From the ﬁrst to the third stage excavation, there is basically no failure taking place in the rock mass, only small tetrahedronal blocks are loosened at the crown. From the fourth stage to the end of the ﬁfth stage, the tetrahedronal blocks are sliding at the lower walls. In the upper walls, the blocks are falling along the steep dip joint into the opening. The rock mass at crown displays tensile rupture along joint planes and the rock mass at base starts to loose (Figure 2.21). 2.3.2.3 Stability of the surrounding rock mass. Based on the deformation and failure characteristics of the surrounding rock mass, general conclusions on the stability of the rock mass can be drawn. During the excavation from the ﬁrst to the third stage, the surrounding rock mass is basically stable. From beginning of the fourth stage till the completion of the ﬁfth, noticeable instability takes place. The aﬀected displacement zone in the surrounding rock mass is large. The aﬀected displacement zone extends far into the sidewalls (50 m) and through the crown (15 m). The instability is reﬂected in various patterns: tensile rupture in the crown, structural loosening in the bottom; and block sliding and falling on the sidewalls. The instability occurs in the following sequences: ﬁrst in the lower sidewall, followed by the upper sidewall and ﬁnally the rock mass at the top and bottom. It should be noted that the model exaggerates the joint persistence, as 100% persistence is assumed. Simpliﬁcation and approximation are applied to treat the three-dimensional problem by the two-dimensional model. The modelling is based on the law of scaling and the similarity principle, results obtained and observation made on the mechanism are qualitative, but have signiﬁcant applications to research and engineering.

This Page Intentionally Left Blank

Chapter 3

Numerical Modelling of Jointed Rock Mass It is diﬃcult to quantitatively predict the mechanical properties of jointed rock masses due to their complexity, even by expensive, large-scale in situ tests. The parameters obtained from laboratory or ﬁeld tests are usually for joint planes or rock blocks to a very limited scale. In most cases, the scale is limited to a few meters. The overall equivalent properties of rock masses of large size are almost not available through direct measurement. Prediction of jointed rock mass behaviours by numerical modelling is useful to study the eﬀects of loading, the behaviour of rock material, rock joints and rock masses. Therefore, it has wide and promising applications. However, the validity of numerical modelling should be supported by physical simulation and correlation. For jointed rock masses, various types of numerical models have been developed [12,14,15,138–162]. For rocks containing a small number of joints, joint element [25,26] can be adopted to represent the discontinuous planes. Modelling of densely jointed rock masses can be mainly realised by two methods. One is the approach of continuum mechanics, or the equivalence approach, such as material parameter equivalence, energy equivalence, deformation equivalence, composite equivalence, fracture mechanics and damage mechanics [77,83,114,163–167]. The other approach is to take the rock blocks as particles of a discontinuum, to study the mechanical properties of the assembly of these particles, including stress, strain and stability in light of discontinuum theory, and to derive mechanical law for the blocks. Examples of this approach are the rigid block method, discrete element method [15,150,152– 154,168,169], discontinuity deformation analysis method [20,92,156,159]. However, all the methods have their own limitations, for example, incompatibility of joint elements with adjacent continuum elements, the scale eﬀect in equivalence, deﬁnition of damage tensor, equation of damage evolution and discretion error. This chapter attempts to develop a new numerical approach for jointed rock masses and a modelling technique for rapid and accurate prediction of their mechanical behaviour. An equivalent continuum model for jointed rock mass based joint element assembly concept, coupled with damage–fracture mechanics and analytical approach is introduced in the following sections.

27

28 3.1.

3.1.1

Chapter 3 EQUIVALENT CONTINUUM MODEL FOR JOINTED ROCK MASSES

Basic principles

The basic principle of an ‘‘equivalence continuum method’’ is: (a) to have an overall consideration of the inﬂuence of joints on rock mass properties based on equivalence principles; (b) to make the jointed rock mass homogenous and continuous so as to derive a set of constitutive relations and then to obtain mechanical properties of the jointed rock mass by means of numerical analysis. These equivalence models are usually elastic. Elasto-plastic constitutive relations are seldom adopted. As shown in Figure 3.1, in order to model the surrounding rock mass consisting of two joint sets, typical elements should be identiﬁed. The typical elements should be large enough to include the two joint sets and their interaction characteristics (Figure 3.2). It is desired that the size of a typical element should be suﬃciently small compared with the engineering dimension. Therefore numerical analysis of this typical element can lead to understanding of the strength and deformation properties of the typical rock mass containing joints. It further leads to the development of constitutive relation of the jointed rock mass and the analysis and modelling of rock mass stability. The modelling procedure discussed here involves several steps: (a) the typical jointed rock mass is discretised into intact rock elements and rock joint elements; (b) equivalent constitutive relations and strength-deformation properties of the

Figure 3.1. Jointed rock mass surrounding an opening.

Numerical Modelling of Jointed Rock Mass

29

Figure 3.2. Typical element representing the jointed rock masses.

Table 3.1. Principle of the equivalent continuum model. Original rock model

Equivalent continuum model

Geometry mode Deformability

Discontinuous Overall deformation comprises rock deformation and joint deformation

Strength characteristics

Joint strength is dominant when failure takes place along the joint plane, and rock strength is used when failure takes place in the intact rock

Equivalent continuous body Deformation equivalence: deformation equal to that of original jointed model under the same loading Strength equivalence: failure of model takes place as the original jointed model.

jointed rock elements are established; (c) continuum modelling of the typical jointed rock mass element is performed and the overall mechanical properties of the element are obtained. This approach has the ‘generality’ of the equivalence method and the ‘particularity’ of the joint element method. The modelling method has signiﬁcant applications in modelling jointed rock masses. The principle of the equivalent continuum model is illustrated in Table 3.1. The equivalent constitutive relation is established based on deformation and strength. Both approaches are discussed in the following sections.

3.1.2 Deformation equivalence The principle of deformation equivalence approach assumes that the equivalent continuum element and the jointed rock mass element deform exactly the same under the same loading. Based on this principle, the relation of material constants between

30

Chapter 3

the equivalent continuum element and jointed rock mass element is derived. The equivalent continuum element can be treated as an anisotropic medium. Assuming the joint strike is along z axis, then in the plane-stress condition, the anisotropic material has a stress–strain relation of 0

sx

1

0

10

ex

1

c11

c12

c13

Bs C B @ y A ¼ @ c21

c22

CB C c23 [email protected] ey A

c32

c33

txy

c31

ð3:1Þ

gxy

where sx, sy and txy are stresses acting on x, y and xy plane; ex, ey and gxy are strains in x, y and xy direction; cij are elastic constants, called elastic stiﬀnesses. Because cij ¼ cji (i, j ¼ 1, 2, 3), there are six independent elastic constants. Since these six parameters can be determined, elastic ﬁnite element method analysis will not be diﬃcult. However, in situ measurement of these constants is not easy. As a jointed rock mass can be regarded as a composition of isotropic intact rock material and rock joints, the deformation parameters of the equivalent continuum medium can be obtained from the properties of the rock material and the joints, such as the elastic constants of the rock material and rock joints, or joint stiﬀnesses and the joint geometrical parameters (spacing, persistence, orientation, aperture and roughness) [4–7,170–197]. Methods of obtaining the equivalence are discussed in the following sections.

3.1.2.1 Deformation equivalence with no joint dilation. Assume that an elastic medium is generally anisotropic. The elastic constitutive relation can be written in the form of: 0

ex

1

0

1

0

sx

1

s11

s12

s13

Be C B @ y A ¼ @ s21

s22

C B C s23 A @ sy A

s32

s33

gxy

s31

ð3:2Þ

txy

where sij are the elastic constants, called elastic moduli. For a rock mass containing a single joint shown in Figure 3.3, the stress on the joint plane can be obtained from the equilibrium equation: sn ¼ sx sin2 a þ sy cos2 a txy sin 2a t ¼ sy sin a cos a sx sin a cos a þ txy cos 2a

) ð3:3Þ

Numerical Modelling of Jointed Rock Mass

31

Figure 3.3. Mechanical analysis of a jointed rock element.

From the principle of superposition, the deformation components of the rock mass are, 9 sn t dm sin a cos a > = x ¼ s11 sx d þ s12 sy d þ s13 txy d þ ks kn ð3:4Þ sn t > ; ¼ s s l þ s s l þ s t l þ cos a þ sin a dm 21 x 22 y 23 xy y ks kn After substituting equation (3.3) into equation (3.4) and rearranging, dm x

dm y

9 1 1 2 2 > ¼ s11 d þ sin a sin a þ cos a sin a sx > > > kn ks > > > = 1 1 2 2 þ s12 d þ cos a sin a cos a sin a sy > kn ks > > > > > 1 1 ; s13 d sin 2a sin a cos 2a sin a txy > kn ks

ð3:5aÞ

9 1 1 2 2 > ¼ s21 l þ sin a cos a sin a cos a sx > > > kn ks > > > = 1 1 2 2 þ s22 l þ cos a cos a þ sin a cos a sy > kn ks > > > > > 1 1 ; s23 l sin 2a cos a cos 2a cos a txy > kn ks

ð3:5bÞ

32

Chapter 3

The elastic constitutive relation of the equivalent continuum medium is: 0

eex

1

0

se11

B ee C B e @ y A ¼ @ s21 gexy se31

se12

se13

1

0

sx

1

se22

C B C se23 A @ sy A

se32

se33

ð3:6Þ

txy

Under the same loading, the deformation of the equivalent medium is: dex ¼ se11 sx d þ se12 sy d þ se13 txy d

)

dey ¼ se21 sx l þ se22 sy l þ se23 txy l

ð3:7Þ

e m e Following the deformation equivalence principle, i.e. dm x ¼ dx , dy ¼ dy then

se11 se12 se13 se21 se22 se23

9 1 1 2 2 > > sin a þ cos a sin a ¼ s11 þ > > kn d ks d > > > > > > 1 1 > 2 2 > cos a cos a sin a ¼ s12 þ > > kn d ks d > > > > > > 1 1 > cos 2a cos a þ sin 2a sin a > ¼ s13 > = ks d kn d > 1 1 > > sin2 a sin2 a cos a ¼ s21 þ > > > kn l ks l > > > > > 1 1 > 2 2 > > cos a þ sin a cos a ¼ s22 þ > > kn l ks l > > > > > > 1 1 ; cos 2a sin a sin 2a cos a > ¼ s23 þ ks l kn l

ð3:8Þ

From Figure 3.3, l ¼ d tan a and hence se12 ¼ se21 . From the symmetry of the constitutive relation, it is easy to see that se31 ¼ se13 and se32 ¼ se23 . However, s33 is very diﬃcult to derive. To simplify the analysis, it can be assumed that se33 ¼ s33 . Therefore, all six parameters needed for building the constitutive relations are obtained. For rock mass element with two joints, seij of the rock mass element containing one joint can be obtained from equation (3.8). By replacing sij with seij in equation (3.8) and repeating the analysis for the second joint, seij and the constitutive relation of the equivalent medium with two joints can be obtained. As for the case of multiple joints, the method is similar by repeating the above procedure.

Numerical Modelling of Jointed Rock Mass

33

3.1.2.2 Deformation equivalence with joint dilation. Let the joint dilation (normal displacement caused by shearing) be dd, the dilation angle be i, then dd ¼ t/(ks tan i), by considering dilation, equation (3.4) becomes 9 sn t t sin a cos a tan i sin a > > = ks ks kn sn t t > ; dm cos a þ sin a tan i cos a > y ¼ s21 lsx þ s22 lsy þ s23 ltxy þ kn ks ks dm x ¼ s11 dsx þ s12 dsy þ s13 dtxy þ

ð3:9Þ

Let the two terms on the right-hand side of each of the equations (3.9) be: 9 t t t cos a tan i sin a ¼ 0 cos a > > = ks ks ks t t t > ; sin a tan i cos a ¼ 00 sin a > ks ks ks

ð3:10Þ

where k0s ¼ ks =ð1 þ tan i tan aÞ

)

k00s ¼ ks =ð1 tan i cotanaÞ By replacing ks with k0s in the ﬁrst three equations in equation (3.8), and replacing ks with k00s in the last three equations of equation (3.8), then se11 , se12 , se13 , se21 , se22 and se23 can be obtained. Similarly, as an approximation, se33 ¼ s33 , the deformation equivalence formula with joint dilation is then established.

3.1.3

Formula of strength equivalence

The strength of jointed rock mass is governed by the strengths of rock material and of rock joint. The jointed rock mass may undergo two types of failure: the failure of the rock material and the failure of the rock joint.

3.1.3.1 Strength equivalence in the case of a single joint. Assuming that the strength of the intact rock, rock joint and the equivalent rock mass element follow the Mohr–Coulomb criterion and the parameters are (cr, fr), (cj, fj), (ce, fe), respectively. The strength conditions of the rock material element and equivalent

34

Chapter 3

continuum rock mass element are: s1 s3 s1 þ s3 sin jr ¼ cr cos jr 2 2

ð3:11Þ

s1 s3 s1 þ s3 sin je ¼ ce cos je 2 2

ð3:12Þ

and it is evident that ce ¼ cr

) ð3:13Þ

je ¼ jr For jointed rock mass element, let b be the angle between the joint planes and the plane of the major principal stress, as shown in Figure 3.4. When b < bmin or b > bmax, the strength of the jointed rock mass is dominated by the strength of the rock material. The strength of the jointed rock mass element, in this case, is the same as equations (3.13). When bmin b bmax, the failure of jointed rock mass element takes place along the joint plane, the strength is governed by: s1 s3 s1 þ s3 sinð2b jj Þ sin jj ¼ cj cos jj 2 2

Figure 3.4. Jointed rock element containing single joint.

ð3:14Þ

Numerical Modelling of Jointed Rock Mass

35

when bmin b bmax, by loading the jointed rock mass element, the strength envelope (c and f ) can be obtained. If this strength envelope is regarded as the envelope of the equivalent medium, then ce and fe can be obtained. Rewriting equation (3.11) in the following form: sin jj cos jj s1 s3 s1 þ s3 ¼ cj , 2 2 sinð2b jj Þ sinð2b jj Þ and compared to equation (3.12), leading to: 9 sin jj > > je ¼ sin > sinð2b jj Þ = cos jj > > > ce ¼ cj ; sinð2b jj Þ cos je 1

ð3:15Þ

Unlike the intact rock material, when the failure of jointed rock mass is governed by the joint plane, the strength of jointed rock mass is not constant but varies with joint inclination (b).

3.1.3.2 Strength equivalence in the case of two joint planes. Figure 3.5 shows a jointed rock mass element containing two joints. The failure of the rock mass element is dependent on the geometric combination of the two joints, the stress distribution and the shear strength of the joints [187–207]. Failure usually takes place

Figure 3.5. Jointed rock element containing two joints.

36

Chapter 3

along one of the joint planes. The equivalent strength is therefore dependent on the strength of that joint plane where failure occurs. Let the strength criterion of the joint plane be written in another form: s1 ¼

2cj þ 2 tan jj s3 þ s3 ð1 tan jj cotanbÞ sin 2b

ð3:16Þ

Let b1 and b2 be the inclination of the joint plane 1 and the joint plane 2, and both b1 and b2 meet the condition of bmin b bmax. In addition, let 1 , ð1 tan jj cotanb1 Þ sin 2b1 1 : A2 ¼ ð1 tan jj cotanb2 Þ sin 2b2 A1 ¼

When A1 > A2, joint 1 will be the dominant plane governing the strength of the rock mass. Similarly when A2 > A1, joint 2 will be the dominant plane governing the strength. It is easy to obtain the equivalent strength following the method similar to equation (3.15). For a rock mass element containing three joints, the parameters of equivalent rock mass strength can be determined in a similar way. Hoek and Brown [194] pointed out that the strength and deformation properties of a rock mass element containing four joint sets or more can be considered to be isotropic. 3.1.3.3 Analysis of tensile failure. When a jointed rock mass element is subjected to tension, its tensile strength, to a large extent, depends on the tensile strength of joints. Since the rock joint is normally considered having zero tensile strength, the tensile strength of the jointed rock mass is usually taken as zero [186,194–196]. Therefore, non-tension analysis should be adopted in this circumstance. 3.1.4 Treatment of elements with non-persistent joint In a jointed rock mass, the joints are often not persistent throughout the joint planes [175,178,197]. Therefore, relevant modiﬁcation should be made on the fundamental formula of deformation and strength equivalence. For simplicity, let us deﬁne the joint projection length ratios in the x- and y-directions, respectively as: Rx ¼

Ljx Lx

and

Ry ¼

Ljy , Ly

ð3:17Þ

where Ljx and Ljy are the joint projection lengths in the x- and y-directions. Lx and Ly are the element sizes in the x- and y-directions. It is obvious that deformation of

Numerical Modelling of Jointed Rock Mass

37

elements is positively correlative to Rx and Ry. Thus, the following modiﬁcation should be made for equation (3.8): when 0 a 45 , the terms containing Kn and Ks are multiplied by Ry and Rx; when 45 < a 90 , the terms with Kn and Ks are multiplied by Rx and Ry. The modiﬁed formula is veriﬁed in terms of joint length, joint inclination and scale eﬀect. First, deformation d1 (or strain e1) in the direction of s1 can be calculated using ﬁnite element method (e.g., Goodman’s joint element). With d1 as the equivalent deformation, elastic constant s11 ¼ e1/s1 can be obtained, and D11 is deﬁned as sr11 =s11 . Alternatively, s11 and D11 can be derived from the modiﬁed equation (3.8) according to the joint distribution. The values obtained from FEM modelling and from equation (3.8) can be compared. The relative error of the value of D11 can be estimated by treating the FEM results as the exact solutions.

3.1.4.1 Variation of joint length. Assuming that the jointed rock element is square in shape at the XOY plane with a size of 5 5 cm, joint inclination is at 45 , Young’s modulus of material is 68 MPa and Poisson’s ratio of material is 0.25. The joint has a normal stiﬀness of 75.0 MPa/cm and a shear stiﬀness of 5.0 MPa/cm. Six cases, as illustrated in Figure 3.6, are modelled and discussed. The modelling results are summarised in Table 3.2.

3.1.4.2 Variation of joint inclination. As shown in Figure 3.7, three models are studied to examine the eﬀects of joint orientation. In the models, a joint with a length

Figure 3.6. Veriﬁcation of models with diﬀerent joint persistence.

38

Chapter 3

Table 3.2. Influence of joint length distribution on elastic constant D11 ðD11 ¼ sr11 =s11 Þ: Case

Rx

Ry

sr11 =s11 (Analytical)

sr11 =s11 (FEM)

Relative error (%)

a b c d e f

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

1.00 0.83 0.71 0.62 0.55 0.49

1.00 0.94 0.83 0.72 0.62 0.49

0 12 14 14 11 0

Figure 3.7. Veriﬁcation models with diﬀerent joint inclination.

Table 3.3. Influence of joint angle on elastic constant D11 ðD11 ¼ sr11 =s11 Þ: Case

Rx

Ry

sr11 =s11 (Analytical)

sr11 =s11 (FEM)

Relative error (%)

a b c

0.2 0.2 0.2

0.0 0.2 0.3

0.96 0.83 0.81

0.90 0.88 0.93

7 6 13

of 1.0 cm at the centre of the element dips at 0, 45 and 71.56 degrees, respectively. Results obtained from the three models are given in Table 3.3.

3.1.4.3 Scale effect of element. Scale eﬀects are studied using the jointed element models shown in Figure 3.6(d), with the joint dip angle at 45 , Rx and Ry at 0.6. Six diﬀerent model sizes are considered, and the results are presented in Table 3.4. The results show that when the jointed rock element size is greater than 10 cm, the error decreases to 5% or less. In discretisation of the jointed rock mass in engineering practice, the size of the ﬁnite elements is generally much greater than 10 cm. Therefore, the treatment of the deformation equivalence using modiﬁed equation (3.8) is reasonably accurate and easy to be realised in modelling.

39

Numerical Modelling of Jointed Rock Mass Table 3.4. Scale effect on elastic constant D11 ðD11 ¼ sr11 =s11 Þ(a ¼ 45 , Rx ¼ Ry ¼ 0.6). Joint length (cm) 5 10 20 30 40 50

3.1.5

sr11 =s11 (Analytical)

sr11 =s11 (FEM)

Relative error (%)

0.62 0.77 0.87 0.90 0.93 0.94

0.72 0.81 0.88 0.91 0.93 0.94

14 5 1 1 0 0

Verification of the numerical model by physical modelling

The overall mechanical properties of jointed rock masses can be reﬂected when the proposed jointed rock elements are incorporated as the basic elements into the FEM analysis. Veriﬁcations have been performed through correlation with physical modelling [8,9]. The size of the physical model is 50 50 7 cm, containing two sets of orthogonal joints with the same spacing and length of 5 cm. Figure 3.8 shows three types of joint arrangements and the FEM meshes. Due to the complexity of the physical model test, comparisons are only made on strength and deformation properties under the uniaxial stress state for all the three types of joint patterns (Figure 3.8a, b and c) and the properties under the biaxial stress state for one joint pattern (Figure 3.8a). The comparisons are shown in Figures 3.9 and 3.10. In Figure 3.9, e1 and e2 are the strains in the direction of s1 and s2 under uniaxial loading, i.e., s2 ¼ 0. In Figure 3.10, e1 and e2 are the strains under biaxial loading, i.e., when s1 ¼ s2. Figures 3.9 and 3.10 indicate that the stress–strain relations obtained from the physical and the numerical modelling under the uniaxial stress state match well. In the biaxial stress state, the major principal stress–strain relations, peak strengths and strains match well between the physical and the numerical modelling results. But the major principal stress–lateral strain relations dismatch. The reason may be that the boundary friction in physical modelling leads to smaller measured deformations than the expected values.

3.1.6

Examples of engineering applications

3.1.6.1 Prediction of strength of jointed rock mass. The above method is used to model the strength of the rock mass at the Ertan hydropower cavern project in China. The rock masses at the Ertan project site have three major joint sets, two of which dip steeply and the other dips gently. The two orthogonal main joint sets (Figure 3.11) are considered in the modelling.

40

Chapter 3

Figure 3.8. Models of diﬀerent joint arrangement and FEM mesh.

Figure 3.9. Comparison of s–e relations of the physical and the numerical models under uniaxial loading.

The measured length of the joints is between 4 and 7 m, and is taken as 5 m in average in the modelling, with a joint spacing of 1 m, based on the geological investigation. The joint persistence is 50% and 30%, respectively, for the two joint sets. The model dimension is 17.8 17.8 m, which satisﬁes the requirement that each boundary of the rock mass model contains about 10 joints, in order to represent

Numerical Modelling of Jointed Rock Mass

41

Figure 3.10. Comparison of s–e relations of the physical and the numerical models under biaxial loading.

the rock mass adequately. The distribution of joints and the FEM meshes are shown in Figure 3.11. There are 625 elements, 242 of which are jointed rock elements. The number of nodes is 676. The joint arrangement belongs to the type shown in Figure 3.8a. The mechanical parameters of rock and joints are obtained from the laboratory tests as given in Table 3.5. In the study, only strength equivalence is considered. FEM analysis involves three types of elements: intact rock element, and two types of jointed rock elements, as shown in Figure 3.11. The intact rock element has a size of 71.2 71.2 cm. The equivalence of strength property is automatically treated by the program. Loading simulated by the numerical modelling gives the yield values of s1 with respect to diﬀerent lateral pressures s2. Strength parameters of the rock masses are

42

Chapter 3

Figure 3.11. Distribution of joints and meshing of jointed rock.

Table 3.5. Parameters of rock material and rock joints. Material parameters Intact rock Joint plane

E (MPa)

n

c (MPa)

f ( )

35,000 –

0.30 –

14.58 0.5

65.20 36.89

obtained through back analysis, using the Drucker–Prager criterion [208,209]. Results are summarised in Figure 3.12, showing the strength envelopes of the rock masses.

3.1.6.2 Stability analysis of jointed rock masses. Numerical modelling is performed to study the stability of the jointed rock mass surrounding a large opening 400 m below the ground surface. The surrounding rock mass is granite. The separation between the powerhouse and the transformer chamber, the cavern dimension and the excavation sequence are shown in Figure 3.13. During the excavation, deformation measurement with extensometers was taken on several sections of the caverns. The distribution of two major joint sets and three faults is shown in Figure 3.14.

Numerical Modelling of Jointed Rock Mass

43

Figure 3.12. Rock mass strength envelope obtained from the modelling.

Figure 3.13. Layout of the powerhouse, the transformer chamber and excavation sequence.

Deformation properties of the jointed rock mass and the constitutive relations are obtained by the deformation equivalence method. In modelling the jointed rock element is generally assumed anisotropic and the joints have zero aperture and are evenly spaced across the model. The persistence of the joints is taken as 50% based on site investigation. Constitutive relation of the rock mass with two joint sets is derived with the method stated in Section 3.1.2. Mechanical parameters are

44

Chapter 3

listed in Table 3.6. The in situ stresses are sx ¼ 10.19 MPa, sy ¼ 8.51 MPa, and txy ¼ 1.12 MPa. The size of model was 525 320 m, containing 654 elements and 630 nodes. The initial stresses were obtained from back analysis of the displacement data of the fourth excavation stage of the powerhouse and the data of the second excavation stage of the transformer chamber. Using the above mechanical parameters and stress conditions, non-linear FEM modelling has been performed on the stability of the caverns. The Drucker–Prager criterion is adopted. The extent of failure zone around the caverns and displacements are obtained as shown in Figure 3.14. Table 3.7 gives the comparison of the displacements obtained by modelling of the jointed rock mass and by elastic analysis of the rock without considering joints.

Figure 3.14. Over-stressed areas of the surrounding rock mass.

Table 3.6. Mechanical parameters of the rock material and rock joints. Materials Intact rock Rock joint

E (MPA)

n

c (MPa)

f ( )

s1(MPa)

Kn (MPa/cm)

Ks (MPA/cm)

37,000 –

0.20 –

22.27 0.5

48.1 35

5.0 0.0

– 600,000

– 7500

Table 3.7. Predicted displacement convergence values (mm). Measurement Modelling of jointed rock mass Elastic analysis of non-jointed rock mass

1

2

3

4

5

6

7

8

9

11.4 9.7

18.5 15.5

19.5 16.3

6.0 5.0

6.9 5.8

7.3 6.1

10.8 9.0

7.4 6.3

6.9 5.8

45

Numerical Modelling of Jointed Rock Mass 3.2.

EQUIVALENT ANALYSIS FOR ROCK MASSES CONTAINING THICK JOINTS

In this section an equivalent continuous model is proposed for rock mass consisting of thick joint sets. The method is based on strength analysis with the same principle discussed in Section 3.1.

3.2.1

Equivalent deformation principle and method

When a rock mass is cut by two joint sets in direction a1 and a2 respectively, and the rock material is homogeneous, then there are two parameters describing the rock material (Er and sr) and four parameters describing each joint set (Ej, sj, b and a). E is the elastic modulus, s is the strength, b is the ratio of joint aperture to rock mass width and a is the orientation of joint set. Only the basic formulae for plane condition are given here, which are also illustrated in Figure 3.34. In the process, the eﬀect of one joint set is taken into account ﬁrst to produce an ‘equivalent medium’. The other joint set is subsequently taken into consideration to derive the equivalent continuous medium of the rock mass containing two joint sets. As shown in Figure 3.34, considering the ﬁrst joint set, the joint stress–strain relation is as follows: fsgj ¼ ½ sxj

syj

txyj T

fegj ¼ ½ exj

eyj

gxyj T

ð3:18Þ

The stress–strain relation of the rock material is fsgR ¼ ½ sxR

syR

txyR T

fegR ¼ ½ exR

eyR

gxyR T

ð3:19Þ

If the stress–strain state of the above combined joints and material is replaced by the stress–strain state of the jointed rock mass as shown in Figure 3.15, namely fsg1 ¼ ½ sx1

sy1

txy1 T

feg1 ¼ ½ ex1

ey1

gxy1 T

ð3:20Þ

then they are related by the following equations: 3.2.1.1 Equilibrium condition. 9 sx1 ¼ ð1 b1 ÞsxR þ b1 sxj > = sy1 ¼ syR ¼ syj > ; sxy1 ¼ txyR ¼ txyj

ð3:21Þ

46

Chapter 3

Figure 3.15. Representation of rock joints and rock material.

3.2.1.2 Displacement compatibility condition. ex1 ¼ exR ¼ exj ey1 ¼ ð1 b1 ÞeyR þ b1 eyj gxy1 ¼ ð1 b1 ÞgxyR þ b1 gxyR

9 > = > ;

ð3:22Þ

3.2.1.3 Physical equations. Suppose that both the rock material and joint material are homogeneous, isotropic and elastic, then for joint 8 > < sxj ¼ aj1 exj þ bj1 eyj syj ¼ bj1 exj þ cj1 eyj ð3:23Þ > : txyj ¼ dj1 gxyj for rock material: 8 s ¼ aR exR þ bR eyR > < xR syR ¼ bR exR þ cR eyR > : txyR ¼ dR gxyR

ð3:24Þ

47

Numerical Modelling of Jointed Rock Mass

where aR, bR, cR, dR are deﬁned by Er and sr, the elastic constants of rock material. aj1, bj1, cj1, dj1 are deﬁned by Ej1 and mj1, the elastic constants of joint material. In plane strain condition, 8 a ¼ c ¼ Eð1 mÞ=ð1 þ mÞð1 2mÞ > > < b ¼ Em=ð1 þ mÞð1 2mÞ > > : d ¼ E=2ð1 þ mÞ

ð3:25Þ

Based on further derivation, the relation between {s}1 and {e}1 in X1OY1 coordinate system can be obtained fsg1 ¼ ½D0j ½H1 1 feg1 ¼ ½D1 feg1

ð3:26Þ

where 2

1 6 bR bj1 6 b1 6 cj1 ½H1 ¼ 6 6 4 0

0 1 b1 þ b1

3

0 cR cj1

0

0 1 b1 þ b 1

7 7 7 7 7 dR 5

ð3:27Þ

dj1

2

bj1 bj1 6 ð1 b1 ÞaR þ b1 aj1 þ b1 cj1 ðbR bj1 Þ ð1 b1 ÞbR þ b1 cj1 cR 6 ½D0j ¼ 6 6 cR bR 4 0 0

3 0 7 7 7 ð3:28Þ 0 7 5 dR

In the X2OY2 coordinate system, the elastic matrix of the jointed rock mass is ½D1 ¼ ½T1 ½D1 ½TT1

ð3:29Þ

where 2

cos2 ða1 a2 Þ

sin2 ða1 a2 Þ

6 2 ½T1 ¼ 6 cos2 ða1 a2 Þ 4 sin ða1 a2 Þ 1 1 2 sin 2ða1 a2 Þ 2 sin 2ða1 a2 Þ

sin2 ða1 a2 Þ

3

7 sin2 ða1 a2 Þ 7 5 cos 2ða1 a2 Þ

ð3:30Þ

It is followed by the second joint set subsequently taken into consideration. Similarly, the elastic matrices of the bi-directional jointed rock mass in XOY

48

Chapter 3

coordinate system are given as: T ½D2 ¼ ½T2 ½Dij ½H1 2 ½T2

ð3:31Þ

where 2

cos2 a2

6 2 ½T2 ¼ 6 4 sin a2 1 2 sin 2a2

sin2 a2 cos2 a2 12 sin 2a2

2

1 6 d 1 bj2 6 b 21 6 2 c ½H2 ¼ 6 j2 6 0 4 d31 b2 dj2

sin2 a2

7 sin2 a2 7 5 cos 2a2

0 1 b2 þ b2 b2

0 d0 b2 23 cj2

0 d22 cj2

0 d32

dj2

3

1 b2 þ b2

ð3:32Þ

3 7 7 7 7 7 0 5 d33

ð3:33Þ

dj2

2

bj2 0 bj2 0 0 0 6 ð1 b2 Þd11 þ b2 aj2 þ b2 cj2 ðd21 bj2 Þ ð1 b2 Þd12 þ b2 cj2 d22 6 ½Dij ¼ 6 0 0 6 d21 d22 4 0 d31

ð1

0 d32

3 bj2 0 þ b2 d23 7 cj2 7 7 0 7 d23 5

ð3:34Þ

0 b2 Þd13

0 d33

and dij0 (i, j ¼ 1, 2, . . . , 6) are elements of [D]1 as expressed in equation (3.29). Generalised in the spatial condition, the general matrix for anisotropic elasticity is 32 d11 d12 d16 sx 6 sy 76 d21 d22 d26 76 6 .. 6 s 76 ... . 6 z 76 .. 76 .. 6 6 txy 76 . . 76 . 6 .. 4 tyz 54 . . . d61 d62 d66 tzx 2

32

3 ex 7 6 ey 7 76 7 76 e 7 76 z 7 76 7 76 gxy 7 76 7 54 gyz 5 gzx

ð3:35Þ

49

Numerical Modelling of Jointed Rock Mass

Suppose that [D]i1 is the elastic matrix of rock mass in the ith local coordinate system OXiYiZi of the rock mass cut by the (i1)-th group of joint, the elements of [D]i1 are the coeﬃcients (k, l ¼ 1, 2, . . . , 6) in the above expression. Similarly, suppose that the rock mass is cut by the ith joint set and the joints are parallel to plane OXiYi, then 2

1 0

0 1

0 0

0 0 0 0 i1 i1 d34 d35 bi bi cij cij 1 0 d i1 d i1 bi 54 1 bi þ bi 55 dij dij

6 6 6 d i1 b d i1 bij d i1 ij 6 bi 32 1 bi þ bi 33 6 bi 31 6 cij cij cij 6 0 0 0 ½Hi ¼ 6 6 i1 6 d i1 d i1 6 b d51 bi 52 bi 53 i 6 dij dij dij 6 6 i1 i1 4 d d d i1 d i1 bi 61 bi 62 bi 63 bi 64 dij dij dij dij 2

½DTi1j

bij i1 i1 ð1 bi Þd11 þ bi dij þ bi ðd31 bij Þ 6 cij 6 6 6 ð1 b Þd i1 þ b d þ b bij ðd i1 b Þ 6 ij i 12 i ij i cij 32 6 6 6 bij i1 i1 6 þ bi d33 ð1 bi Þd13 6 cij 6 ¼6 b 6 ij i1 i1 6 ð1 bi Þd14 þ bi d34 6 c ij 6 6 bij i1 6 i1 ð1 bi Þd15 þ bi d35 6 cij 6 6 4 b ij i1 i1 ð1 bi Þd16 þ bi d36 cij

bi

i1 d65 dij

0 0 i1 d36 bi cij 0 d i1 bi 56 dij 1 bi þ bi

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 i1 5 d66 dij ð3:36Þ

bij i1 ðd bij Þ cij 31 bij i1 i1 ð1 bi Þd22 þ bi dij þ bi ðd32 bij Þ cij bij i1 i1 ð1 bi Þd23 þ bi d33 cij b ij i1 i1 ð1 bi Þd24 þ bi d34 cij bij i1 i1 ð1 bi Þd25 þ bi d35 cij b ij i1 i1 ð1 bi Þd26 þ bi d36 cij i1 ð1 bi Þd21 þ bi dij þ bi

i1 d31

i1 ð1 bi Þd41

i1 d51

i1 d61

i1 d32

ð1

i1 bi Þd42

i1 d52

i1 d62

i1 d33

i1 ð1 bi Þd43

i1 d53

i1 d34

ð1

i1 bi Þd44

i1 d54

i1 d35

i1 ð1 bi Þd45

i1 d55

i1 d36

i1 ð1 bi Þd46

i1 d56

3

7 7 7 7 7 7 7 i1 7 d63 7 7 7 7 i1 7 d64 7 7 7 i1 7 d65 7 7 5 i1 d66

ð3:37Þ

50

Chapter 3

Having been cut by the ith joint set, the rock mass has the elastic matrix ½Di ¼ ½Di1j ½H1 i

ð3:38Þ

The direction cosines between system OXiþ1Yiþ1Ziþ1 and OXiYiZi are as follows: Original coordinates

New coordinates

Yi

Xi

Zi

direction cosines li1 li2 li3

Xiþ1 Yiþ1 Ziþ1

mi1 mi2 mi3

ni1 ni2 ni3

In the (i þ 1)-th local system OXiþ1Yiþ1Ziþ1, the elastic matrix of the jointed rock mass is ½Di ¼ ½Ti ½Di ½TTi 2

2 li1

6 6 l2 6 i2 6 6 2 6 li3 ½Ti ¼ 6 6 6 li1 li2 6 6 6 li2 li3 4 li3 li1

ð3:39Þ

m2i1

n2i1

2li1 mi1

2mi1 ni1

m2i2

n2i2

2li2 mi2

2mi2 ni2

m2i3

n2i3

2li3 mi3

2mi3 ni3

mi1 mi2

ni1 ni2

li1 mi2 þ li2 mi1

mi1 ni2 þ mi2 ni1

mi2 mi3

ni2 ni3

li2 mi3 þ li3 mi2

mi2 ni3 þ mi3 ni2

mi3 mi1

ni3 ni1

li3 mi1 þ li1 mi3

mi3 ni1 þ mi1 ni3

2ni1 li1

3

7 7 7 7 7 2ni3 li3 7 7 7 ni1 li2 þ ni2 li1 7 7 7 ni2 li3 þ ni3 li2 7 5 ni3 li1 þ ni1 ni3 2ni2 li2

ð3:40Þ i ¼ 1, 2, . . . , 6.

3.2.2

Basic principle of strength equivalence

Suppose that the strengths of the rock material, the joint and the equivalent continuum body all follow the Mohr–Coloumb criterion, the strengths are cr, fr , cj, fi , ce, fe , respectively. The strength of jointed rock mass element consists of the rock

Numerical Modelling of Jointed Rock Mass

51

material strength and the joint strength. Adopting the equivalent strength method described in Section 3.1, the strength condition is s1 s2 s1 þ s2 sin je ¼ Ce cos je 2 2

ð3:41Þ

With diﬀerent angles of the joint plane to the major principal stress plane b, ce and fe are diﬀerent: when b < bmin or b > bmax, Ce ¼ CR

je ¼ jR

ð3:42Þ

when bmin b bmax 9 cos jj 1 > = Ce ¼ Cj sinð2b jj Þ cos je > ; je ¼ arcsinðsin jj = sinð2b jj ÞÞ

ð3:43Þ

If the rock mass contains more than two joint sets, the related results can be obtained by superposition. But it should be noted that the smaller value of c, f should be adopted if two values exist. For diﬀerent stress states, if all the rock material and rock joint parameters are known, the strength of the rock mass can be derived by the above procedure. The equivalent rock mass strength parameters ce and fe can be determined on the basis of Drucker–Prager criterion as shown before. With two stress states (a and b) known, the equivalent rock mass strength parameters (ce and fe ) can be determined through the Drucker–Prager criterion. pﬃﬃﬃﬃﬃ aI1a þ J2a ¼ K qﬃﬃﬃﬃﬃ ð3:44Þ aI1b þ J2b ¼ K From equation (3.44), we have

a¼

qﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ J2b J2a I1a I1b

Because pﬃﬃﬃ 3 sin j a ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ , 3 3 þ sin2 j

ð3:45Þ

52

Chapter 3

the expression below is obtained,

3a je ¼ j ¼ arcsin pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 3a2

ð3:46Þ

From pﬃﬃﬃ 3C cos j K ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 þ sin2 j we have qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 þ sin2 j K pﬃﬃﬃ Ce ¼ C ¼ 3 cos j

ð3:47Þ

The values of ce and fe for the rock mass can be obtained from equations (3.46) and (3.47).

3.2.3

Engineering applications

The above equivalent analysis of jointed rock masses is applied to model a large underground cavern complex of a hydropower project, as shown in Figure 3.16. The main work includes the selection of separations between caverns, the excavation sequences and the support schemes. Two-dimensional non-linear analysis and three-dimensional anisotropic analysis are conducted. In the FEM modelling, continuous rock mass is simulated with the commonly used isoparametric elements. Dominant faults are simulated with the isoparametric joint elements. The fractured rock mass containing joint sets is simulated with equivalent medium. In two-dimensional modelling, primarily 4-node quadrilateral elements and 3-node triangle elements are used. The number of elements is 661, and the number of nodes is 603. Six schemes for cavern separation optimisation, three schemes for excavation sequence optimisation and two schemes for shotcrete– rockbolt support optimisation are modelled. In three-dimensional modelling, 8-node brick-shaped elements and 6-node pentahedron elements are commonly used. The number of elements is 1094, and the node is 1368. Two schemes of cavern separation optimisation and two schemes of excavation and support optimisation are modelled. A large number of modelling cases have been conducted. Some important conclusions are outlined here. The maximum displacements in plane elastic analysis

Numerical Modelling of Jointed Rock Mass

53

Figure 3.16. Systematic view of cavern complex of a hydropower station.

and in non-linear analysis occur at diﬀerent locations. The former appears at the middle of the left wall in the powerhouse (Figure 3.16). In non-linear analysis the yield plastic zones between the main powerhouse and the transformer room is joined together, and the maximum displacement appears at the middle of the right wall of the powerhouse with a magnitude of about 6.7 cm. The maximum compressive stresses also appear at the vault and the ﬂoor. The maximal compressive stress at the left vault is nearly 47.0 MPa. With the increase of spacing between the caverns, the areas of yield zones signiﬁcantly decrease, and eventually the yield zones become isolated. The cavern separation has signiﬁcant inﬂuence on the area of yield zone. Non-linear analysis is conducted for three schemes of excavation sequence: Scheme 1: I ! II ! III ! IV ! V

ðfive excavation stepsÞ

! III ! Scheme 2: I ! II ! !V ! IV !

ðfour excavation stepsÞ

! II ! ! III ! Scheme 3: I ! ! ! IV ! !V !

ðthree excavation stepsÞ

Modelling results indicate that the maximum displacement is decreased by 32% in Scheme 1, by 35% in Scheme 2 and by 12% in Scheme 3. As for yield zone, Scheme 1 has the least yield zone compared with Schemes 2 and 3. Through comprehensive consideration, Scheme 1 is recommended for the actual construction.

54

Chapter 3

In the non-linear modelling, the eﬀects of two diﬀerent bolt schemes are compared. New concept and method are developed for modelling the bolt eﬀects. From the model tests, the results show that the main eﬀect of the bolts is to increase the strength and the stiﬀness of the surrounding rock mass. Its eﬀect to produce support reaction is much less signiﬁcant. The results show that the peak strength and modulus of the bolted rock mass increase approximately by 10–20%. However, the residual strength of the bolted rock mass increases considerably as compared with unbolted rock mass. Under the uniaxial compression, the residual strength increases by 60–80%, and is inﬂuenced by the bolt spacing and the stiﬀness ratio between the bolt and the rock mass. The study on bolt is discussed in detail in Chapter 8.

3.3.

STRENGTH CHARACTERISTICS OF FRACTURED ROCK MASS UNDER COMPRESSIVE SHEAR STRESS

The natural rock masses consist of faults, joints, fractures and other discontinuities. Engineering projects often generate compressive or shearing stress on the rock mass [186,194–196,210–218]. It is important to understand the interaction of the rock discontinuities and the strength and failure mechanisms of the rock masses under compressive and shear stresses. Joints and fractures in a rock mass generally distribute intermittently and the strength of the joint and the bridge area in the rock mass diﬀer greatly in diﬀerent rock masses [205,206]. Researchers usually pay close attention to the overall shear strength of a jointed rock mass and to the secondary crack initiation. Many researchers suggested that the compressive strength of collinear fractures and rock bridges can be determined by means of the weighted mean method [219–221]. Others believed that the diﬀerent deformation characteristics of the joint and the bridge in a rock mass should be taken into consideration to assess the rock mass strength. Because of diﬀerent stress distributions in the rock bridge and on the joint surfaces, the friction on the joint surfaces must be multiplied by a mobilising factor [199–200]. Tests conducted on gypsum show that the strength properties of a rock mass with multi-rowed intermittent fractures aligning along the same direction [205]. Brown [206] studied the strength of jointed rock mass by means of fracture mechanics. Horii and Nemat-Nasser [219] gave analytical solution and stress intensity factor of the secondary crack trajectory, and discussed the formation of a failure plane. Reyes and Einstein [220], through tests and damage analyses, studied the stress distribution and failure mode of a rock bridge between two fractures in en-echelon crack arrays. Analytical models for fracturing mechanisms of the rock bridge between adjacent fractures in a bilateral compressive stress ﬁeld, based upon the phenomena observed

Numerical Modelling of Jointed Rock Mass

55

in the experimental studies are proposed and discussed in this section. The models can be used to predict the overall shear strengths of the rock mass. The analytical solutions have been veriﬁed through experiments.

3.3.1 Strength of rock mass containing collinear cracks 3.3.1.1 Fracture propagating at tight cracks. Stress concentration phenomena will take place at crack tips under loading. Although the average stress of the crack is generally less than the threshold stress of crack propagating, the stresses at the tips are by far higher than the threshold stress. As a result, secondary cracks will occur near the tips. The initiating point of secondary cracks can be determined from the fracture toughness of the rock material. For collinear cracks (Figure 3.17) loaded by compound stresses of compression and shear, stress intensity factor at the crack tips can be derived as follows: kI ¼ 0

ð3:48Þ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pa kII ¼ ðt s tan jj cj Þ 2b tan 4b

ð3:49Þ

Where compressive stress is deﬁned as positive and tensile stress as negative; cj, fj are the cohesion and the friction angle of the crack, respectively. The condition for the crack to propagate is KII KIIc where KIIc is type II fracture toughness of the crack. The crack propagation does not indicate that the stress reaches the rock strength. The strength is reﬂected by the stress that makes the cracks join together to form a failure plane. The crack initiation and the propagating path become very important

Figure 3.17. Collinear cracks under compressive shear stress.

56

Chapter 3

for estimating the strength. As shown in Figure 3.18, during the test, rupture initiates at crack tips, it tends to propagate gradually along the direction of maximum principal stress, s1. When a crack propagates to a certain extent and meets the adjacent crack, rock failure occurs. Analysis on the above-observed phenomenon has shown that stress intensity factors at the tip (point A) of a crack under A A compressive and shear stresses are kA I ¼ 0 and kII 6¼ 0. When jkII j kIIc , the crack rupture initiates at point A with an initial rupture angle of approximately 70 (the angle between initial rupture line and line of the crack). However, with the increment of external loading, the direction of the secondary crack gradually turns to the direction of s1. The secondary crack path with direction of turning is schematically described in Figure 3.19. The cracking starts along A B. When the crack propagates to point B, the shear stress between B C exceeds the maximum shear strength of the rock material, resulting in shearing failure on plane BC to form crack B C. Therefore, the fracture ABC becomes the ﬁnal fracture plane.

3.3.1.2 Determination of shearing strength of crack body. At present, the commonly used method to analyse the strength of a jointed rock mass takes into consideration the joint persistence Z, the shear strength parameters of the crack, cf and ff, the shear parameter of rock bridge material, cr and fr [221]. f ¼ Z fj þ ð1 ZÞfr

)

c ¼ Zcj þ ð1 ZÞcr

Figure 3.18. Fracture through bridged rock materials.

ð3:50Þ

Numerical Modelling of Jointed Rock Mass

57

Figure 3.19. Stress analysis on the fracture plane.

t ¼ Zðsfj þ cj Þ þ ð1 ZÞðsfr þ cr Þ

ð3:51Þ

Taking the stress diﬀerence between rock bridge and crack plane into account, the following equation is also used. t ¼ Z½ð1 bÞsfj þ cj þ ð1 ZÞ½bsfr þ cr

ð3:52Þ

where b is a factor concerning the normal stress distribution on failure planes and is to be determined through experiments or numerical modelling. As shown in Figure 3.10, DA and CE are the existing crack surfaces and AB is the secondary crack. Shear strength can be expressed as: sn ¼ s1 cos2 a þ s2 sin2 a

ð3:53Þ

tj ¼ sn Hðsn Þ fj þ cj

ð3:54Þ

then

where ( HðxÞ ¼

1

x>0

0

x0

ð3:55Þ

58

Chapter 3

The equilibrium equations of the crack element (Figure 3.19) are Fx ¼ 0

Fy ¼ 0

ð3:56Þ

Consequently, 2bs2 sin a þ 2aðtj cos a sn sin aÞ þ 2cðtBC cos y sBC n sin yÞ ¼ 0

ð3:57Þ

2bs1 cos a þ 2aðtj sin a þ sn cos aÞ þ 2cðtBC sin y sBC n cos yÞ ¼ 0

ð3:58Þ

where c ¼ (b a)(cos a)/(cos y) Rewriting (3.57) and (3.58), then tBC sBC n tan y ¼

aðsn tan a tj Þ bs2 tan a ba

ð3:59Þ

bs1 aðtj tan a þ sn Þ ba

ð3:60Þ

tBC tan y þ sBC n ¼

Solving simultaneously, equations (3.59) and (3.60) gives sBC n ¼

B0 A0 tan y 1 þ tan2 y

ð3:61Þ

tBC ¼

A0 þ B0 tan y 1 þ tan2 y

ð3:62Þ

aðsn tan a tj Þ bs2 tan a ba

ð3:63Þ

bs1 aðtj tan a þ sn Þ ba

ð3:64Þ

where A0 ¼

B0 ¼ Let

BC F ¼ jtBC j sBC n Hðsn Þ fr cr

Hence, when F 0, shear failure occurs at plane BC and a failure plane is formed. At this moment, @[email protected] ¼ 0, thus @tBC @sBC n HðsBC ¼0 n Þfr @y @y

ð3:65Þ

Numerical Modelling of Jointed Rock Mass

59

based on @tBC B0 ð1 tan2 yÞ 2A0 tan y ¼ @y ð1 þ tan2 yÞ2 cos2 y

ð3:66Þ

@sBC A0 ðtan2 y 1Þ 2B0 tan y ¼ @y ð1 þ tan2 yÞ2 cos2 y

ð3:67Þ

Substituting equations (3.66) and (3.67) into (3.65) gives 1 A0 B0 fr HðsBC r Þ y ¼ arcctg 2 B0 þ A0 fr HðsBC n Þ

ð3:68Þ

3.3.1.3 Verification through model tests. The modelling material made in the physical model is a mixture of gypsum and ceyssatile, which has the following mechanical properties: uniaxial tensile strength ¼ 0.27 MPa, compressive elastic modulus ¼ 1.3 103 MPa, Poisson’s ratio ¼ 0.20, internal friction angle ¼ 35 , cohesion ¼ 0.53 MPa. The cracks are closed and have the following properties: length ¼ 20, 40, 60, 80 mm, internal friction angle ¼ 10 and cohesion ¼ 0.1 MPa. Layout of the direct shear test is shown in Figure 2.14. The rock mass model has crack persistence ranging from 0 to 62%. In the test, the normal stress on shear plane is kept constant and the lateral loads are gradually increased till the model fails. The test results are analysed by least square method, giving shear strength curves as shown in Figure 3.20. It can be seen from the Figure 3.20 that the experimental results agree well with the analytical solution. The comparison is summarised in Table 3.8. The above comparison indicates that the strength criterion proposed reasonably represents the properties of the rock mass containing collinear cracks.

3.3.2

Strength of rock mass containing multiple cracks

Joints, fractures and cracks in a rock mass may be distributed parallel but not necessarily co-linear, as shown in Figure 3.21. The failure of such rock mass is indicated by dotted lines. Because the crack is in compressive shear stress state, the stress intensity factor at the tip is KI ¼ 0 and KII 6¼ 0. Hence under the compressive shear state the condition of the crack initiation should be KII KIIc.

60

Chapter 3

Figure 3.20. Comparison of strength envelopes between measured and calculated results.

As shown in Figure 3.23, crack initiates at fracture tips, and the secondary cracks BC and DE are formed at the tips B and D. The secondary cracks expand in the direction of maximum compressive stress, and at a certain stage, the cracks join each other to form CD. The rock mass containing multiple cracks then fails. With understanding the failure mechanism after multi-fractured rock mass, the following assumptions are used to develop a strength criterion: (i) (ii)

The secondary crack propagates in the direction of the maximum compressive stress, and along a straight line, The failure of multi-fractured rock mass is due to shear.

As shown in Figure 3.23, BC and ED are the secondary crack surfaces. Stresses can be expressed as: sn ¼ s1 cos2 a þ s2 sin2 a tj ¼ sn Hðs0 Þ fj þ cj For the element in Figure 3.23, the force equilibrium condition is Fx ¼ 0 Fy ¼ 0

Numerical Modelling of Jointed Rock Mass

61

Table 3.8. Comparison between measured and calculated results. Persistence (%)

0 0 0 9 9 13 13 13 20 42 43 43 44 45 46 46 50 60 60 62 62 62

Normal stress (0.1 MPa)

3 4 5 0.8 4 3 4 6 5 5 0.8 3.5 3 3.5 4 6 5 0.8 2 3 4 6

Shear strength (0.1 MPa) Measured

Calculated

7.43 8.56 8.99 5.76 7.86 6.43 6.80 8.06 7.49 5.91 4.39 5.43 5.36 5.03 5.63 6.10 5.35 3.52 3.65 4.34 4.14 4.95

7.4 8.1 8.8 5.44 7.53 6.64 7.27 8.54 7.42 5.9 3.84 5.12 4.82 4.99 5.16 6.08 5.34 3.03 3.50 3.76 4.14 4.89

Figure 3.21. Rock mass with a group of parallel cracks.

Theoretical error (%)

0.4 5.4 2.1 5.56 4.2 3.3 6.9 6.00 1 0.17 12.5 5.7 10 0.8 8.3 0.3 0.18 14 4.1 13.36 0 1.2

62

Chapter 3

Figure 3.22. Enechelon cracking trajectory.

Figure 3.23. Force analysis.

Namely, 2cs2 sin a þ 2aðtj cos a sn sin aÞ þ 2mðtCD cos y sCD n sin yÞ ¼ 0 2cs2 cos a þ 2aðtj sin a þ sn cos aÞ þ 2mðtCD sin y sCD n cos yÞ ¼ 0 where m ¼ ðc aÞ

cos a cos y

) ð3:69Þ

Numerical Modelling of Jointed Rock Mass

63

Arrangement of equation (3.69) leads to tCD sCD n tan y ¼

aðsn tan a tj Þ cs2 tan a ca

ð3:70Þ

cs1 aðtj tan a þ sn Þ ca

ð3:71Þ

tCD tan y þ sCD n ¼

Solution of equations (3.70) and (3.71) gives sCD n ¼

B0 A0 tan y 1 þ tan2 y

ð3:72Þ

tCD ¼

A0 þ B0 tan y 1 þ tan2 y

ð3:73Þ

in which A0 ¼

aðsn tan a tj Þ cs2 tan a ca

B0 ¼

cs1 aðtj tan a þ sn Þ ca

It follows from Figure 3.23 by geometrical analysis that y ¼ arctan ctana 4d tan a

4L 1 3c 2a sin a

ð3:74Þ

According to the above analysis, the failure criterion for multi-fractured rock mass under the action of compressive shear stress is F 0

ð3:75Þ

where CD F ¼ tCD sCD n Hðsn Þ fr cr

tCD ¼

A0 þ B0 tan y 1 þ tan2 y

sCD n ¼

B0 A0 tan y 1 þ tan2 y

ð3:76Þ

64

Chapter 3 A0 ¼

aðsn tan a tj Þ cs2 tan a ca

ð3:77Þ

csj aðtj tan a þ sn Þ ca

ð3:78Þ

B0 ¼

4L 1 y ¼ arctan c tan a þ 4d tan a 3c 2a sin a

ð3:79Þ

For uniaxial loading; s1 6¼ 0 L¼

s2 ¼ 0 6:703T 2 2 pK1c

ð3:80Þ

For biaxial loading; s1 6¼ 0 s2 6¼ 0 2qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 32 2 1 4 K1C þ 11:614s2 K1C 5 L¼ p 2:234s2

ð3:81Þ

T ¼ 2aFn cos a

ð3:82Þ

Fn ¼ jðs1 s2 Þ sin a cos aj fj Hðsn Þsn cj ( 1 x>0 HðxÞ ¼ 0 x0

ð3:83Þ

Figure 3.24. Diagram of the model test.

ð3:84Þ

Numerical Modelling of Jointed Rock Mass

Figure 3.25. Comparison of predicted values with the tested values when s2 ¼ 0.

Figure 3.26. Comparison of predicted values with the tested values when s2 ¼ 0.2 MPa.

65

66

Chapter 3

In addition, when L 2d/cos a, connection takes place among the propagated cracks, and at the same time, the rock mass strength reaches the peak. The present criterion for multi-fractured rock mass is veriﬁed by physical model tests [8]. Figure 3.24 shows the schematic layout of the physical model tests. The dimension of the existing fracture are: a ¼ 4.8 cm, b ¼ 8.0 cm, d ¼ 1.0 cm. The model material is gypsum and its main physical parameters are: uniaxial compressive strength ¼ 1.84 MPa, uniaxial tensile strength ¼ 0.38 MPa, modulus of elasticity ¼ 0.238 104 MPa, Poisson’s ratio ¼ 0.20, internal friction angle ¼ 45 , cohesion ¼ 0.38 MPa and the fracture toughness of type I K1C ¼ 74.06 N/cm3/2. The main physical parameters of the fracture are: the friction coeﬃcient ¼ 0.365 and the cohesion ¼ 0. The criterion is also implemented with numerical modelling code. A computer program using the new strength criterion is written in FORTRAN and was used to simulate the physical model tests. Comparisons of the results obtained from the numerical modelling and the physical modeling are shown in Figures 3.25 and 3.26. From the comparisons, it can be seen that the numerical modelling based on the new strength criterion generally agree well with the physical test results.

Chapter 4

Sensitivity Analysis of Rock Mass Parameters At present, a variety of numerical methods are available for stability analysis of rock masses. Among them, the most widely used are the ﬁnite element method (FEM) [138–140,222–232], the boundary element method (BEM) [14,233–241], the discrete element method (DEM) [15,148–154,242–246] and coupled methods [16,247–253]. The rationality and reliability of the results from those methods depend, to a great extent, upon the appropriate selection of computational model and mechanical and mathematical parameters. Once the computational model is determined, the key to success hinges on the rational selection of the computing parameters. There are many factors and parameters that aﬀect the rock mass stability. One has to identify the order of importance of all the parameters [1–3]. In terms of computation, the limited resource may be the primary restriction. This section presents some speciﬁc assessments on the eﬀects of various parameters by means of the sensitivity analysis in systematic approaches.

4.1.

SENSITIVITY ANALYSIS OF COMMONLY USED PARAMETERS

4.1.1 Method of sensitivity analysis Sensitivity analysis method is the method used to analyse the stability of a system [254]. Given a system whose character, P, is governed mainly by n factors of a ¼ {a1, a2 , . . . , an}, P ¼ f {a1, a2 , . . . , an}. Under a reference state of a ¼ fa1 , a2 , . . . , an g, the character is described by P*. The sensitivity analysis is to, let the above individual factors vary within respective possible range and then analyse both tendency and extent to which the character of the system, P, departs from the base state due to the variation of the factors. The ﬁrst step of sensitivity analysis is to establish the system model, i.e., the functional relation between the system character and the factors, P ¼ f {a1, a2 , . . . , an}. This relation should be, if possible, described by analytical expression. In the case of a complex system, it can be expressed by numerical method or through presentation of graphic chart. It is a key step of the eﬀective analysis on parameter sensitivity to establish a model that reﬂects the system with the reality as fully as possible. The basic parameter set should be given after the system model has been established. The basic parameter set should be established to reﬂect the subjects considered. For example, when the sensitivity of the stability of an underground project to the variation of rock mass parameters is to be studied, the suggested rock 67

68

Chapter 4

mass parameters of the in situ rock mass can be used as the basic parameter set. Once the basic parameter set is determined, the sensitivity analysis can be performed on each parameter. When analysing the eﬀect of ak on the characteristic of the system, P, the parameter ak in the set is varied within a possible range while the remaining parameters are kept constant. In this case, the system’s character displays the following relation: P ¼ f ða1 , . . . , ak1 , ak , akþ1 , . . . , an Þ ¼ jk ðak Þ

ð4:1Þ

The character curve of P–ak is plotted from equation (4.1), which roughly describes the sensitivity of P to the disturbance from ak. For example, the rapid change of the curve around ak1 shows a high sensitivity of P to ak, i.e., a slight change of ak will cause a great change of P. On the contrary, the curve is gently around ak2, then the system character of P is less sensitive to the parameter of ak, i.e., P varies slightly with a large change of ak. In other words, when the parameter is near ak1, it is a parameter of high sensitivity; when it is near ak2, then is of low sensitivity (Figure 4.1). The above analysis only gives the sensitivity behaviour of the system character P to a single factor. The character of a real system is generally governed by many factors of diﬀerent physical quantities and units. Therefore, it is diﬃcult to compare the sensitivity of the various factors by the above method. To solve this problem, dimensionless treatment and analysis can be applied. In dimensionless analysis, the sensitivity function and the sensitivity factor are deﬁned in dimensionless terms. The ratio of the relative error (dp), of the system character P (dp ¼ |P|/P) to the relative error of parameter ak (dak ¼ |ak|/ak) is deﬁned as the sensitivity function, Sk(ak), of the parameter ak. P ak jPj . jak j k ¼ 1, 2, . . . , n ð4:2Þ Sk ðaK Þ ¼ P ak ak P When |ak|/ak is small, the function of Sk(ak) can be expressed approximately as djkðak Þ ak Sk ðak Þ ¼ dak P

k ¼ 1, 2, . . . , n

ð4:3Þ

From Equation (4.3), the sensitivity function curve of ak can be obtained, which is shown in Figure 4.2. Given ak ¼ ak , the sensitivity factor Sk of the parameter ak is obtained as djkðak Þ a Sk ¼ Sk ðak Þ ¼ ak ¼ ak k k ¼ 1, 2, . . . , n ð4:4Þ dak P

Sensitivity Analysis of Rock Mass Parameters

69

Figure 4.1. Characteristic curve of system P–ak.

Figure 4.2. Curve of sensitivity function Sk–ak.

where Sk and k ¼ 1, 2, . . . , n are a group of non-negative dimensionless real numbers. The higher Sk is, the more sensitive P is to ak. Based on comparison between diﬀerent Sk values, one can give synthetic assessment on the sensitivity of various factors. When jk (ak) is a sectional function, its derivative may be discontinuous at section boundary point of ak0, which makes Equation (4.3) fail to give the sensitivity of S 0k at ak0. In this case, one of the following methods can be used: (i)

Choose the higher value of the left and the right limits Sk(ak) at ak ¼ ak0 as the sensitivity Sk0 , i.e., Sk0 ¼ maxfSk ðak Þ, Sk ðak þÞg

(ii) (iii)

Smooth the function of jk (ak) using cubic spline function ﬁtting method or other techniques to eliminate the discontinuous point of the derivative; and Set a common relative error e for every parameter, i.e., let jak j=ak ¼ e, k ¼ 1, 2, . . . , n and then calculate S 0k using equation (4.2).

70

Chapter 4

4.1.2 Sensitivity analysis of stability of underground works Examples of using the sensitivity analysis method described in the previous section are given in this section. It analyses various factors that aﬀect the stability of a rock cavern complex of a hydropower project. The project comprises main powerhouse, transformer house and tailrace surge chamber, and the sectional layout is shown in Figure 4.3. The powerhouse measures 64.4 m in height and 27 m in width. The deformations of the side wall and the crown are of interest to the stability analysis. The project is located in the region of high in situ stress ﬁeld where horizontal stresses are greater than the vertical stresses. Due to the high overlying depth above the chamber complex, the in situ stress ﬁeld is assumed to be uniform and the directions of its two principal stresses are oriented horizontally and vertically at sx ¼ 13.3 MPa and sy ¼ 9.5 MPa respectively.

4.1.2.1 Computational model. A two-dimensional non-linear FEM programme has been used to model and analyse the stability of the underground structures. The following simpliﬁcations and assumptions have been made: (i)

(ii)

The surrounding rock mass is homogeneous and continuous with the eﬀect of faults neglected, but the joint eﬀect is considered using the equivalent elastic module, Ee, from the in situ measurements. The initial in situ stress is uniformly distributed within the computational domain and the two principal stresses act in horizontal and vertical directions.

Figure 4.3. Sectional diagrammatic sketch of Laxiwa hydroelectric power station.

71

Sensitivity Analysis of Rock Mass Parameters

The excavation is simulated by unloading process and the eﬀect of the excavation of the tailrace surge chamber on the weakening of the rock is treated with the stiﬀness-reduction method. The system character, namely, the stability of the underground structure complex is reﬂected by the maximum horizontal deformation of the left sidewall of the main powerhouse (or by other indexes, e.g., damage areas). The parameters used for the sensitivity analysis include elastic module (E), Poisson’s ratio (n), cohesion (c), internal friction angle (f), horizontal and vertical principal stresses (sx and sy). For the above exercise, the basic parameter set is given in Table 4.1.

4.1.2.2 Analysing the results. All parameters have been analysed one by one using the method stated earlier. The procedure to analyse the horizontal stress sx and the elastic module of E is described below. From experiences, the possible varying range of those two parameters are determined. E is in the range of 1.0–4.5 104 MPa, and sx is in the range of 4.75–19.95 MPa. The values of E and sx are adjusted step by step to calculate the maximum horizontal displacement, u, of the left sidewall of the main powerhouse. Curves representing u–E and u–sx are plotted from the computing results, as shown in Figures 4.4 and 4.5. The function relations of u–E and u–sx have been obtained from the curves and can be expressed as: u ¼ jE ðEÞ ¼ 8:70=j

ð4:5Þ

u ¼ jsx ðsx Þ ¼ 0:225sx 0:2425

ð4:6Þ

From the equation (4.3) we have two sensitivity functions of SE ðEÞ and Ssx ðsx Þ: SE ðEÞ ¼

8:70

1 Eu

Ssx ðsx Þ ¼ 0:225

ð4:7Þ

sx 0:225sx ¼ u 0:225sx 0:2425

ð4:8Þ

Table 4.1. Basic parameter set. E (104 MPa) 3.2

n

c (MPa)

f ( )

sx (MPa)

sy (MPa)

0.21

25

48

13.3

9.5

72

Chapter 4

Figure 4.4. u–E curve.

Figure 4.5. u–sx curve

Sensitivity Analysis of Rock Mass Parameters

73

The corresponding sensitivity curves of SE –E and Ssx–sx are shown in Figures 4.6 and 4.7 respectively. As shown in Figures 4.6 and 4.7, the sensitivity function of SE ðEÞ 1 means that the sensitivity factor, SE , is constantly equal to 1, and is not inﬂuenced by the basic value of the elastic module E. Ssx ðsx Þ is a decreasing function, and the sensitivity is high when sx is low and decreases with increasing sx. The limit of Ssx is 1. By substituting sx ¼ 1.33 MPa into Equation (4.8), the sensitivity factor, Ss x Ss x , of the parameter sx is 1.088. Similar analysis can be performed on other parameters. Table 4.2 summarises the sensitivity factor of other parameters.

Figure 4.6. SE –E curve.

Figure 4.7. Ssx–sx curve.

Table 4.2. Sensitivity factor of various parameters. SE

Sm

Sc

Sj

Ss x

Ss y

1.0

0.077

0.039

0.020

1.088

0.077

74

Chapter 4

Table 4.2 indicates that when the horizontal displacement of the sidewall of the powerhouse is used to judge the stability of the cavern, then the most sensitive factor that aﬀects the stability is the horizontal stress. It has sensitivity as high as 1.088. In other words, if there exists an error of 15% in sx, the relative error (du) of u is 1.088 15% ¼ 16.32%. The internal friction angle f is the least sensitive factor (Sj ¼ 0.02%). An error of 15% in f only results in a 0.3% (0.2 15%) error of u. As seen from the analysis, the horizontal stress and the elastic module have high sensitivity factors and should be treated with great care. It should be noted that the above conclusion is drawn from a given set of basic parameters that are directly involved in a speciﬁc engineering project. The conclusion diﬀers from problem to problem and project to project. As shown in Figure 4.7, the sensitivity factor, Ss x , of sx varies, depending upon the selection of the basic value of sx . In addition, the sensitivity of some parameters is aﬀected by the interaction of other parameters. For example when a structure is in a perfect elastic state, cohesion and friction have no eﬀect on the deformation of the structure. But when the structure is in an elasto-plastic state, and cohesion and friction aﬀect the deformability, such eﬀect becomes more remarkable with increasing plastic area. The sensitivity of c and f is therefore dependent not only on their basic value but also on the values of sx and sy. Further analysis of such interaction between parameters requires more rock mechanics knowledge of the problem. One can use the Rock Engineering System [1–3] or the Grey System [254] to study the interaction between the parameters. For such cases, when sensitivity analysis is performed on a parameter that has active interaction with others, it is desirable to make the latter varying within their possible ranges.

4.1.3 Application to optimisation of test schemes In order to analyse the stability of the underground rock structures, the mechanical and engineering parameters of the rock mass must be known. Accurate rock mass properties can only be obtained from large in situ tests. Such tests are seldom carried out as they are very expensive and time consuming. Sensitivity analysis of parameters can be applied for the optimisation of testing schemes. Test scheme optimisation is to obtain the rock properties that meet the engineering requirement with the least amount of work for large in situ tests. Studies of sensitivities of all the parameters would identify the parameters of high sensitivities that should be measured during the in situ tests. In the example in Section 4.1.2, the horizontal deformation of the high wall, the horizontal stress and the rock mass module are the key parameters. They must be determined through ﬁeld tests, while the other parameters are less critical and can be obtained through less expensive tests or other methods.

Sensitivity Analysis of Rock Mass Parameters

75

Sensitivity analysis helps to avoid mistakes due to subjective conjecture. Diﬀerent basic parameter sets have diﬀerent orders of sensitivity. For this reason, the sensitivity analysis of parameters should be aimed at speciﬁc engineering problems so as to distinguish key parameters from the rocks. The key parameters may vary from project to project. The amount of ﬁeld tests for rock parameters can be rationalised according to their sensitivity factors. In principle, parameters of higher sensitivity should be subjected to more tests. Selections of appropriate test methods shall be made in accordance with the requirements of the engineering project and based on the sensitivity factors. For example, when there are two options available in measuring the horizontal stress sx: one is cheap but has a relative error of dsx1 ¼ 15%, while another is expensive but has a relative error of dsx2 ¼ 10%. As the sensitivity of Ss x is 1.088, the relative errors of the two methods results in measurement errors of du are 16.32% and 10.88%, respectively. If a relative error du < 15% is required, the second method should be adopted although the ﬁrst one is cheap. If a larger error of du < 20% is allowed, the ﬁrst method can be used. Although the sensitivity analysis provides, in view of accuracy, scientiﬁc basis for the optimisation of testing schemes, the overall optimisation should consider other factors, such as testing cost, duration and availability.

4.2.

ANALYSIS OF THE EFFECT OF JOINT PARAMETERS ON ROCK MASS DEFORMABILITY

In the previous section, the sensitivity analysis is performed by assuming rock masses as isotropic and homogeneous media. However, natural rock masses contain sets of joints, which have inﬂuence on the strength and the deformation characteristics of the rock mass. Chapter 3 has suggested several mechanical models for diﬀerent kinds of jointed rock masses. This section examines the sensitivity of the joint mechanical and geometric parameter on the rock mass stability.

4.2.1

Application of equivalent model for jointed rock mass

The sensitivity analysis of the eﬀect of joints on underground rock structures involves a considerable amount of work. The current study therefore adopts the same analytical model and method described in Section 3.3. The rock mass is assumed to contain two joint sets with diﬀerent orientations (Figure 4.8). The following notations are used in the analysis. For the rock material: Er - elastic module, nr - Poisson’s ratio; and for joint sets (subscripts 1 and 2 for set I

76

Chapter 4

Figure 4.8. Typical rock mass containing two joint sets.

and II respectively): Ej1 and Ej 2 - elastic moduli, nj1, nj 2 - Poisson’s ratio, aj1 and aj 2 - dip angle, Zj1 and Zj 2 - persistence, bj1 and bj2 - speciﬁc width (total joint width percentage in the rock mass). The model is treated under the plane strain condition, and the strike of the joint sets coincides with the z axis, and the joint persistence Cj1 and Cj2 are both equal to 1. Equivalent properties of the rock mass are obtained according to the properties of individual joint sets and the rock material, as discussed in Section 3.3. The stress states of the joint and the rock material are respectively (refer to Section 3.3). j1 j1 r fsgj1 ¼ fs j1 x , s y , t xy g

j1 j1 r fegj1 ¼ fe j1 x , e y , g xy g

ð4:9Þ fsgr ¼ fs rx , s ry , t rxy gr

fegr ¼ fe rx , e ry , g rxy gr

4.2.2 Basic parameter for sensitivity analysis The sensitivity analysis of the joint parameter is conducted for the same case as those in the previous section (Section 4.1). Sensitivity of various parameters to the maximum horizontal displacement of the cavern sidewall and to the maximum settlement of the roof is studied. The project was described in Section 4.1.2 and shown in Figure 4.3. There exists two prevailing joint sets in the region and the elastic constants of the rock material are: elastic module Er ¼ 5.8 104 MPa and Poisson’s ratio nr ¼ 0.21. The parameters of the two joint sets are obtained from the site investigation and are summarised in Table 4.3.

77

Sensitivity Analysis of Rock Mass Parameters Table 4.3. Basic parameter values of joint sets used in sensitivity analysis. Joint set I Joint set II

Ej1 ¼ 0.5 104 MPa Ej1 ¼ 1.4 104 MPa

j1 ¼ 0.23 j1 ¼ 0.23

aj1 ¼ 70 aj2 ¼ 110

Zj1 ¼ 0.6 Zj2 ¼ 0.6

bj1 ¼ 3.210 2 MPa bj2 ¼ 3.2 10 2 MPa

The values of parameter in Table 4.3 are also suggested for the sensitivity analysis. Accordingly, given that each parameter varies within a certain range, the side wall displacement (ux) and the roof settlement (uy) are calculated. The change and trend of ux and uy with the variation of the parameters are examined. A FEM program is used for the analysis.

4.2.3 Computational results 4.2.3.1 Effects of joint elastic moduli on displacement. Figure 4.9 shows the relation curves between the joint elastic modulus (Ej) and the surrounding rock mass displacement (m), for each joint set. ux is the deformation of the high wall and uy is the deformation of the crown. Relative convergence (u/B) and the relative ratio of the joint elastic modulus to the rock mass modulus (Ej/Er) are also presented in Figure 4.9. B is the cavern width in calculating ux, and B is the cavern height in calculating uy. From Figure 4.9, it can be seen that the u–Ej curve can be ﬁtted with the function of Y ¼ A/X þ B. The regression analysis determines the u–Ej relationship for

Figure 4.9. Deformation-joint elastic moduli relations for both joint sets.

78

Chapter 4

the two joints as: For joint set I, ux ¼ 527:6=Ej1 þ 1:5505

ð4:10aÞ

uy ¼ 326:4=Ej1 þ 0:4772

ð4:10bÞ

ux ¼ 675:8=Ej2 þ 1:6098

ð4:10cÞ

uy ¼ 286:2=Ej2 þ 0:5251

ð4:10dÞ

For joint set II,

where ux and uy (cm) stand for the maximum displacement of the sidewall and the maximum vertical deformation of the cavern, respectively. Ej1 and Ej2 (MPa) are the modulus of joint sets I and II, respectively. The following conclusions can be drawn from Figure 4.9 and equation (4.10): (i)

(ii)

(iii)

If the actual elastic moduli of the rock joints are lower than the suggested elastic moduli used in the computation, then the actual deformation of the surrounding rock mass is greater than that computed. For the main powerhouse cavern, the total deformation of the sidewall is greater than that of the roof, for the same change in the joint modulus. However, the relative deformation of the sidewall is less than that of the roof. For example, when the elastic modulus of joint set I changes from Ej1 ¼ 5000 to 2500 MPa (given that the maximum relative error of Ej is 50%), then the absolute values of ux and uy are 0.106 and 0.065 cm, while the corresponding relative changes are 6.37 and 12.04%, respectively. The results show that the displacement of roof is more sensitive than the displacement of sidewall to the change of joint elastic modulus. The same changes of the elastic moduli of the joint sets with diﬀerent initial moduli have diﬀerent eﬀects. Given the same relative errors of 50% of both joint sets whose initial moduli are 5000 and 14,000 MPa, the errors caused are 12.04 and 3.76% at the roof, respectively. It shows that lower elastic moduli has higher sensitivity to the rock mass deformation.

4.2.3.2 Effect of joint Poisson’s ratio on rock mass deformation. Figure 4.10 shows the relation curves between the joint Poisson’s ratio (nj) and the rock mass displacement (u). The Poisson’s ratios of the joint sets (nj1 and nj2) vary between 0.184 and 0.276, and the deformations (ux and uy) remain almost the same. This shows that the rock mass deformation is by far less sensitive to the Poisson’s ratio.

Sensitivity Analysis of Rock Mass Parameters

79

Figure 4.10. Eﬀects of joint Poisson’s ratio on rock mass deformation.

Figure 4.11. Relationship between displacement and dip angle.

Therefore, in analysing the stability of the rock mass, selection of the Poisson’s ratio will not result in signiﬁcant errors.

4.2.3.3 Effect of joint dip angle on rock mass displacement. The eﬀect of joint dip angle to the rock mass deformation is shown in Figure 4.11. From the curves, the following conclusions can be obtained.

80 (i)

(ii)

(iii) (iv)

Chapter 4 The change of rock mass deformation is slightly aﬀected by the joint dip angle varying between 0 and 90 . It suggests that the joint dip angle is a non-sensitive factor aﬀecting the rock mass deformation. The sensitivity of joint dip angle is related to the joint elastic modulus. For the joint set II with a high elastic modulus, the change in the dip angle results in little change of the rock mass deformation. On the other hand, for the joint set I with a low modulus, the change of the dip angle causes increase of the rock mass deformation as shown in Figure 4.11(c). It implies that the sensitivity of the joint dip angle increases with the decreasing joint elastic modulus. The joint dip angle is a sensitive factor when the joint elastic modulus is very low. The vertical settlement of roof, uy, is more sensitive to the change of joint dip angle, as compared with the horizontal deformation of the sidewall. It is noted from Figure 4.11(c) that when a joint set of low modulus and dip angle is between 40 and 60 , the rock mass deformation becomes the greatest, which is most unfavourable to the rock mass stability.

4.2.3.4 Effect of joint persistence on rock mass deformations. Figure 4.12 presents the relation curve of the joint persistence (Zj) and the rock mass displacement (u). The displacement (u) shows the following character as Zj varies: (i)

(ii)

(iii)

When Zj 0.6, ux and uy increase more or less linearly with increasing Zj; when Zj > 0.6, the horizontal displacement, ux, increases acceleratively with increasing Zj, while the vertical displacement, uy, increases at a much lower rate. This indicates that the joints with a persistence greater than 0.6 have higher sensitivity to the sidewall deformation. The sensitivity of joint persistence to the roof displacement, however, is rather low. The sensitivity of the persistence depends on the joint elastic modulus. For example, when Zj ¼ 0.6 and the maximum deviation is 0.2, for joint set I ¼ 5000 MPa, the maximum error of the sidewall displacement is 4.5%; with Ej1 while for joint set II with Ej2 ¼ 14,000 MPa, the error of the sidewall displacement is 1.68%. It can be seen that when the joint elastic modulus is low, the sensitivity of the joint persistence to the rock mass deformation is high. The sensitivity of joint persistence to the sidewall displacement is higher than that to the roof displacement, as shown in Figure 4.12.

4.2.3.5 Effect of joint aperture on rock mass deformations. The relation between the joint aperture and the rock mass deformation is as shown in Figure 4.13.

Sensitivity Analysis of Rock Mass Parameters

81

Figure 4.12. Eﬀects of joint persistence on rock mass displacement.

Figure 4.13. Eﬀect of joint aperture on rock mass deformation.

It leads to the following conclusions: (i) (ii) (iii)

The rock mass deformation increases linearly with the increasing joint aperture. The roof settlement is more sensitive to the change of the joint aperture than the sidewall deformation, as shown in Figure 4.13. The sensitivity of the joint aperture to the surrounding rock mass deformation is also inﬂuenced by the joint elastic modulus. The comparison between Figures 4.13a and 4.13b shows clearly that joint set I, with lower elastic modulus has higher sensitivity factor compared with joint set II.

82

Chapter 4

4.2.3.6 Comparison of sensitivity of different parameters. The previous section discussed the sensitivity of various parameters to the rock mass deformation. In this section, attempt is made to compare the sensitivities of various parameters within the same domain. Since the Poisson’s ratio and the joint dip angle are non-sensitive parameters, only elastic modulus, persistence and aperture of the joint sets are used in the comparison study. The sensitivity of a parameter can be measured by the sensitivity factor Sp [254], deﬁned as the ratio of the relative deviation (dp ¼ jp=p j), by which the system’s character p departs from a certain state of P*, to the relative deviation of the parameter, hence, Sp ðaÞ ¼ p=p =a=a

ð4:11Þ

A higher value of Sp suggests greater eﬀect of the parameter a on the system’s character, P, i.e., the greater sensitivity of a. The relative deviation of a may depend on the problem concerned. In the comparison study below, a relative error of 50% is selected for all parameters for the same underground cavern complex analysis in Figure 4.3. The sensitivity factors of various parameters to the sidewall displacement, the roof settlement and their relative deviations are analysed and summarised in Table 4.4. It can be seen from Table 4.4 for that particular underground project, the sensitivity order of the joint parameters to the cavern sidewall deformation: Zj1 (persistence of joint set I), Ej1 (elastic modulus I), bj1 (joint aperture I), Ej2 (elastic modulus II), Zj2 (persistence II), and bj2 (joint aperture II). Zj1 and Ej1 are the parameters of high sensitivity and the others are of low sensitivity. Similarly, the sensitivity order of the joint parameter to the roof settlement is: Ej1, bj1, Ej2, bj2 and Zj2. Ej1 and bj1 are the parameters of high sensitivity and the rest are of low sensitivity. The synthetic analysis on ux and uy shows that joint set I with a low elastic modulus is the main factor aﬀecting the rock mass deformability. A 50% of combined error of the parameters of this joint set will result in an error of over 10% of the rock mass deformation.

Table 4.4. Sensitivity factors of various parameters to roof and sidewall displacement. Sensitivity factor Sux Suy dSux % dSuy %

Ej1

Zj1

bj1

Ej2

Zj2

bj2

0.127 0.241 6.37 12.03

0.203 0.077 10.15 3.86

0.082 0.118 4.11 5.89

0.058 0.075 2.91 3.75

0.0578 0.013 2.88 0.66

0.036 0.047 1.82 2.34

Sensitivity Analysis of Rock Mass Parameters

83

It should be noted that the above conclusions have been drawn on the basis that each basic parameter varies independently. The interactions between the parameters have not been considered. A more comprehensive study of sensitivity can be conducted by coupling the present method with a parameter interaction study, e.g. the Rock Engineering System approach [1–3]. Nevertheless, the present method is usually suﬃcient to quantify the sensitivity of various parameters.

4.3.

SENSITIVITY ANALYSIS OF ROCK MASS PARAMETERS ON DAMAGE ZONES

This section discusses the sensitivity analysis of the rock material and joint parameters on the magnitude of the damaged zone. The rock material parameters and the rock joint parameters are regarded as dependent variables in the analysis. The stability criterion is based on the magnitude of the damaged zone. The engineering case used is the same cavern complex project as the previous section (Figure 4.3).

4.3.1

Failure criterion for the equivalent jointed rock mass

The principle for analysing the deformation equivalence of the jointed rock mass is similar to that discussed in the previous section. However, each ﬁnite element includes at least one joint. The ﬁnite element may have several possible failure modes or damage patterns including: plastic ﬂow (or shear) of the rock material, tensile or shear failure of joints. The rock mass is considered having been damaged, if any of these phenomena takes place. Drucker–Prager criterion is used to judge if the plastic ﬂow (or shearing) of the intact rock occurs, qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 3 sin I1 þ 3 3 þ sin2 J2 3 3c cos 0

ð4:12Þ

I1 and J2 are the ﬁrst invariant of stress tensor and the second invariant of stress deviator respectively, and c and f are the cohesion and the frictional angle of the rock block. The tensile or shear failure of the joint planes in the rock mass follows the criterion of s 0 jtj stg0j þ Cj0

ð4:13Þ

84

Chapter 4

s is the normal stress on the joint plane (negative sign stands for tension); jtj is the absolute value of the shear stress on the joint plane; c0j and 0j are the cohesion and friction angle of the joint plane of failure. c0j and 0j are calculated through weighted mean method taking account of the cohesion and friction angle of the existing joint and the rock block as well as the joint persistence.

4.3.2 Sensitivity analysis of an underground cavern complex The sensitivity analysis is to study the eﬀect caused by the error in the rock mass parameters on the damaged zones of the underground cavern complex. The underground cavern complex comprises the main power cavern, transforming cavern and tailtrace surge cavern, as shown in Figure 4.3. Details of the project are outlined in the Section 4.1.2. The single-parameter approach is adopted for the sensitivity analysis [255–261]. The method allows each parameter to vary at a time within a possible range, and then derives the corresponding variation of the damage zone in the surrounding rock mass. In order to compare the sensitivities of all the parameters, the dimensionless sensitivity factor is deﬁned as: Ak max A A Ak min SðkÞ max A A

ð4:14Þ

where S(k) is the sensitivity of the parameter k, A* is the area of the damaged zone corresponding to the basic parameter set, Ak max and Ak min are the maximum and minimum areas of the damaged zone within the error domain of the parameter k. The deﬁnition of the sensitivity factor here is not the same as the one deﬁned in the previous section. The sensitivity analysis is aimed at the following parameters: rock material elastic modulus Er, Poisson’s ratio nr, cohesion Cr and internal friction angle fr; rock joint Ej, uj, cj, fj and dip angle aj, persistence Zj and speciﬁc width bj. The basic values and error ranges of the parameters are summarised in Table 4.5.

4.3.3 Result and analysis 4.3.3.1 Effect of parameters on damaged zones. The relationship between parameter’s error and damaged zone area has been obtained through computation. Examples are given in Figure 4.14 showing the relationships between the main parameters (Cr, fr and Ej1) to the damaged zone area. The values of the parameters and the damaged zone area are normalised according to respective basic values.

85

Sensitivity Analysis of Rock Mass Parameters Table 4.5. Basic value and error range of parameters. Parameter Basic value Error range

Parameter

Er (103 MPa)

nr

Cr (MPa) fr ( ) Ej1 (103 MPa)

58.0 46.4 69.6

0.21 0.168 0.252

2.5 1.5 2.5

48.0 28.8 67.2

5.0 1.0 9.0

0.23 0.184 0.276

0.1 0 0.2

Zj1

bj1 (%)

Ej2 (10 MPa)

nj2

cj2 (MPa)

fj2 ( )

aj2 ( )

14.0 2.8 252

0.23 0.184 0.276

0.1 0 0.2

24 19 29

110 90 130

nj1

cj1 (MPa) fj1 ( )

aj1 ( )

24 70 19 29 50 90

Zj2

bj2 (%)

3

Basic value Error range

0.6 0.4 0.8

3.2 1.6 4.8

Figure 4.14. Eﬀects of rock mass parameters on damaged zone area.

0.6 3.2 0.4 1.6 4.8 0.8

86

Chapter 4

It can be seen from Figure 4.14 that the damaged zone area increases linearly with decreasing cr, while it increases abruptly with decreasing fr. Therefore, special attention should be paid to fr with respect to rock stability. The decrease of the joint elastic modulus also results in the increase of the damaged zone area. But its sensitivity is much lower than that of fr and cr.

4.3.3.2 Comparison between sensitivities of various parameters. Based upon the error ranges of parameters listed in Table 4.5, the sensitivity factors of all the parameters have been obtained as deﬁned by equation (4.14). The sensitivity factors are summarised in Table 4.6. A parameter with a sensitivity factor greater than 0.2 (S 0.2) is deﬁned as a highly sensitive parameter. S 0.2 means that 20% of the apparent error in damage zone area is resulted from the parameter error. A parameter with 0.04 < S < 0.2 is a moderately sensitive parameter. A parameter of S < 0.04 is considered non-sensitive parameter. From the results, each parameter can be arranged from high to low sensitivity as follows: Highly sensitive parameters: fr, cr (high to low) Moderately sensitive parameters: Ej1, Zj1, Ej2, and aj1 (high to low) Non-sensitive parameters: bj1, nr, Zj2, bj2, Er, aj2, nj1, cj1, fj1, nj2, cj2, and fj2. The results show that the strength of the rock material is the most critical factor aﬀecting the damaged zones in the surrounding rock mass. Particularly, the internal friction angle, fr, is the most sensitive parameter. On the other hand, the deformation parameter of the rock material has little eﬀect on the damaged zone. The deformation and geometry properties of the joints have certain eﬀects on the size of damaged zone in the surrounding rock mass. The comparison between various parameters of the same joint set shows that the elastic modulus is the most sensitive parameter, the persistence and dip angle have almost the same sensitivity factor. The Poisson’s ratio is non-sensitive. As in the previous sections, the eﬀect of joint parameter with a lower elastic modulus on the damaged zone area is larger than that of the joint set with a higher elastic modulus.

Table 4.6. Sensitivity of various parameters. Rock parameter

S(Er) ¼ 0.014

S(nr) ¼ 0.035

S(cr) ¼ 0.264

S(fr) ¼ 0.837

Joint set I S(Ej1) ¼ 0.101 S(nj1) ¼ 0 S(cj1) ¼ 0 S(fj1) ¼ 0 S(aj1) ¼ 0.047 S(Zj1) ¼ 0.053 S(bj1) ¼ 0.036 Joint set II S(Ej2) ¼ 0.051 S(nj2) ¼ 0 S(cj2) ¼ 0 S(fj2) ¼ 0 S(aj2) ¼ 0.034 S(Zj2) ¼ 0.035 S(bj2) ¼ 0.010

Sensitivity Analysis of Rock Mass Parameters

87

The computation results also show that the joint strength parameters of cj and fj have no eﬀects on the damaged zone, which can be explained from the following two aspects: (i)

(ii)

4.3.4 (i)

(ii)

(iii)

Under the given in situ stresses of sx ¼ 13.3 MPa and sy ¼ 9.5 MPa, the damage of the surrounding rock mass mainly behaves yielding ﬂow and only slight damage takes place along the joint plane. With the joint persistence at 0.6, rock mass strength is governed by rock material properties rather than those of the joint. Therefore, the eﬀect of the change in joint strength parameters on the damaged zone is not being reﬂected.

Summary The strength parameters of rock materials are the main factors aﬀecting the size of the damaged zone. When there is a 20% relative error in fr and cr, they will produce about 83.7% and 26.4% increments in damaged zone areas, respectively. Therefore, careful assessment of the two parameters is important to control damage zone. Strength reinforcement such as systematic rock bolts can be applied to reduce the damaged zone. The elastic modulus of the joint set I also aﬀects the damaged zone area in the surrounding rock mass considerably. The existence of weaker joint set has greater eﬀect on rock mass stability, and vice versa. The eﬀect of the geometric parameters of joint sets on the damaged zone area is relatively low and often can be neglected.

This Page Intentionally Left Blank

Chapter 5

Stability Analysis of Rheologic Rock Mass Rocks and rock masses often exhibit rheologic behaviour, especially weak and soft rocks or highly jointed rock masses. Rheologic behaviour is the time-dependent characteristics of the material deformation and strength [262–275]. For example, the deformation of a rock mass may increase under a constant loading with time elapsing, i.e., creep eﬀect, and the strength may decrease. In the case of underground works, the phenomenon is often found that due to the rheologic behaviour, the loading on support elements gradually increases, leading to the ﬁnal failure of the system [276–286]. In the vicinity of an excavated opening, weak rocks or jointed rock masses can creep and deform visco-elastically [287–301]. Often, stresses in the surrounding rock mass exceed the rock strength, causing the rock mass to become visco-plastic and increasing the visco-plastic composition in the total deformation [302–306]. The understanding of rheologic characteristics is important to the design of underground excavation and support. The reinforcement and support design for the rheologic rock mass must take into consideration the visco-elastic and visco-plastic deformation.

5.1.

RHEOLOGICAL MECHANICAL MODELS FOR ROCKS AND ROCK MASSES

There are two basic approaches to study the rheological phenomena of a rock mass. The ﬁrst approach is from the macro point of view to study synthetical and mechanical behaviour of the rock mass by taking a large volume to represent the whole rock mass containing adequate quantities of discontinuities [17–19,307]. The second is to study the intact rock material and discontinuities individually and then to assemble them together [15,20,21]. Here, emphasis is laid on the ﬁrst approach. There are three basic ideal bodies for common rheological mechanical models: Hooke’s elastic solid, Newton’s viscous liquid and St. Venant’s plastic mass [264,291]. In the rock mechanics, these basic bodies are often used to model a variety of rock masses and to simulate diﬀerent rheological characteristics of the rock masses. Table 5.1 summarises the common rheological models of rock mechanics. It should be noted that all the models are linear. The stress–strain curves of the models describe the relationship between deviator tensors. The equations are mainly applicable to the uniaxial loading state. The equations that describe multi-axial loading state can be derived through the superposition theorem.

89

90

Table 5.1. Typical mechanical model, formula and rock type. Formula

Rheological type

Rock type

Hooke (H)

s ¼ Ee

Elastic

Hard or relatively hard rock

Newton (N) St. Venant (V)

s ¼ 2Z e_ s¼y

Viscous Plastic

Soft rock Soft rock or rock under high

Kelvin (K ¼ H/N)

s ¼ 2Ge þ 2Z e_

Visco-elastic

Most medium to soft rock

Model

Model illustration

confining pressure (elastic post effect)

Maxwell (M ¼ H N) Prandtl (P ¼ H V)

sþ

(PT ¼ H/(H N))

Zm s GM

¼ 2GHe þ Bingham (B ¼ H (N/V))

Burgers (Bu ¼ M K)

Schofield–Scott–Blair (SSB ¼ K (M/V))

Visco-elastic ( plastic)

or rock at depth

Elasto-plastic

Elasto-visco-plastic

Soft rock and soil under high confining pressure

sþ

Visco-elastic ( plastic)

GM Zp þ GK Zp þ GM ZK s _ GM GK Zp ZK 2Zp ZK s€ ¼ y þ 2Zpe_ þ e€ GM GK GK

Elasto-visco-plastic

sþ

Medium strength rock and most sedimentation rock

GM þ GH ZM e_ GM

ZMðGM þ GK Þ þ ZKGM s_ GM GK Z Z Z þ M K s€ ¼ 2ZM e_ þ K e€ GM GK GK

Rock under certain confining pressure

Visco-elasto-viscous

8 < s ¼ZEe when s < y s þ ps_ ¼ y þ 2Zp_e G : when s y

Halite under long-term load

Soft rock such as clayed shale, mudstone

Note: s – deviator component of stress tensor, e – deviator component of strain tensor, G – shearing module and Z – viscosity factor.

Halite under long-term load

Chapter 5

Poynting-Thomson

Z _ ¼ 2Z_e sþ s G s ¼ Ee when s < y s ¼ Eðe epl Þ when s y

Stability Analysis of Rheologic Rock Mass

91

From the testing results on the rheological characteristics of most rock materials, it can be seen that the constitutive relations of rocks and rock masses are all non-linear. This is reﬂected in two aspects: ﬁrstly, the rheological mechanical parameters, such as G, Z, E and n are not constant but the functions of stress level and/or time; and secondly, strain, stress and their rates are also related non-linearly [19,194,308,309]. The universal constitutive relation of rocks and rock masses can be expressed as aðs,tÞðsÞn1 þ bðs,tÞðs_ Þn2 þ cðs,tÞðs€ Þn3 ¼ y þ aðs,tÞðeÞn4 þ bðs,tÞð_eÞn5 þ ðs,tÞð€eÞn6

ð5:1Þ

where n1, n2, . . . , nn are all positive, y is the threshold value of stress at which the rock material enters the plastic state. The above non-linear constitutive relation can be used provided that all parameters concerned are available by suﬃcient testing of rock behaviour, i.e., the functional factors such as a(s,t) and power values nn of stress and strain can be obtained. However, it is diﬃcult to determine so many parameters or variables as the rheologic tests are time consuming. In practice, simpliﬁed linear models are often adopted for engineering applications [308,309]. Equation (5.2), for example, is a common constitutive rheological model adopted in rock mechanics study to simulate a variety of rocks: as þ bs_ þ cs€ ¼ y þ ae þ b_e þ s€

ð5:2Þ

where a, b, c, y, a, b, are specially deﬁned as constants. It can be seen that the mathematical equations of various rheological mechanical models in Table 5.1 are in fact special cases of equation (5.2) or diﬀerent combinations of equation (5.2).

5.2.

VISCO-ELASTIC SURROUNDING ROCK MASS AND SUPPORTING PROBLEM

Most rock masses involved in underground engineering exhibit characteristics of visco-elasticity to diﬀerent extents except some very hard and massive rocks [194,308]. It is therefore of paramount importance to study the behaviour of the visco-elastic rock masses. Nowadays, numerical analyses including ﬁnite element method (FEM) and boundary element method (BEM) have been widely used to study the stress state of the surrounding rock masses. Nevertheless, it is still of great signiﬁcance to use the analytical method as it provides theoretical solutions in limited forms. The ﬁnal expression of this method contributes to direct understanding of the problems under consideration.

92 5.2.1

Chapter 5 General solution for circular visco-elastic media

In this section, the excavated opening, the support lining and the surrounding rock mass are treated as a generalised two-dimensional problem. They are assumed to be homogeneous, isotropic and of ﬁnite deformation. A triple-element model of visco-elasticity (Figure 5.1) is adopted. It is equivalent to the Poynting–Thomson model (see Table 5.1). Assuming that the deformations of each element have their own independent physical equations and the Poisson’s ratio is constant, then the rheological physical equations for the plane problems can be expressed as [8,52]: 9 > > > > > 3n 3n > e þ 2Z e_ Z þ e_ > > > > 1 2n 1 2n = 3n 3n > e þ 2Z e_ X þ e_ > sX þ s_ X ¼ 2G eX þ > 1 2n 1 2n > > > > > > > 3n 3n ; e þ 2Z e_ sY þ s_ Y ¼ 2G 1 2n 1 2n

þ _ ¼ G þ Z_ sZ þ s_ Z ¼ 2G eZ þ

ð5:3Þ

where s_ X , s_ Y , s_ Z and _ are the normal stress rate and shear stress rate along the directions of X, Y, Z; e_ X , e_ Z and _ are the normal strain rate and the shear strain

Figure 5.1. Rheological model of triple-element.

Stability Analysis of Rheologic Rock Mass

93

Figure 5.2. Coordinate system of circular medium.

rate along X, Z; is the relaxation time, G is the elastic shear module (inﬁnitesimal loading rate) and Z is the viscosity of shearing deformation. Writing equation (5.3) in the form of integration equation and substituting it into equilibrium equation and considering geometric equation, one can obtain the Volter integration equation set of the second type with the body force neglected. The equation is integral. The components of the stress in a circular medium derived from the general solution can be expressed in polar coordinates [8,52], as shown in Figure 5.2: r 1 0 0 2 Ur ðtÞ ¼ ð1 2nÞC0 ðtÞ þ b0 ðtÞS þ a01 ðtÞð3 4nÞS ln 2G S 1 3 b1 ðtÞS ð1 4nÞC1 ðtÞq þ d1 ðtÞ Q1 ðjÞ q 1 h X n nþ2 þ Az,n ðnÞan ðtÞS nbn ðtÞS n¼2 i n n2 Qn ðjÞ þ Aw,n ðnÞCn ðtÞq þ ndn ðtÞq ð5:4Þ

W y ðtÞ ¼

r 1 a01 ðtÞS ð3 4nÞS ln þ 1 þ b1 ðtÞS3 þ ð5 4nÞC1 ðtÞq þ d1 ðtÞq1 2G S 1 h dQðjÞ r X þ Bz,n ðnÞan ðtÞSn þ bn ðtÞS nþ2 dj 2G n¼2 i dQ ðjÞ n ð5:5Þ þ Bw,n ðnÞCn qn þ dn ðtÞqn2 dj

94

Chapter 5

2

sr ðtÞ ¼ KT C00 ðtÞ b00 ðtÞS þ ½ð3 2nÞSa01 ðtÞ þ 2b1 ðtÞS3 2C1 ðtÞqQðjÞ Zt 0 0 ½C0 ðt Þ b00 ðt0 ÞS 2 þ ½ð3 2nÞSa01 ðt0 Þ þ 2b1 ðt0 ÞS 3 2C1 ðt0 Þq þ Kð1 KT Þ t0

1 X 0 QðjÞ eKðt tÞ dt0 KT nðn 1Þðn þ 2Þan ðtÞS n nðn þ 1Þbn ðtÞS nþ2 n¼2 n

nðn þ 1Þðn 2Þcn ðtÞq nðn 1ÞdnðtÞq

n2

1 X

Qn ðjÞ Kð1 KT Þ

Z

n¼2

t

nðn 1Þ

t0

ðn þ 2Þan ðt0 ÞS n nðn þ 1Þbn ðt0 ÞS nþ2 nðn þ 1Þðn 2Þcn ðt0 Þqn

0 nðn 1Þdn ðt0 Þqn2 Qn ðjÞeKðt tÞ dt0

ð5:6Þ

sy ðtÞ ¼ KT C00 ðtÞ b00 ðtÞS2 ð1 2nÞa01 ðtÞS þ 2b1 ðtÞS 3 þ 6C1 ðtÞq QðjÞ Zt 0 0

C0 ðt Þ þ b00 ðt0 ÞS2 ð1 2nÞa01 ðt0 ÞS þ 2b1 ðt0 ÞS 3 þ 6C1 ðt0 Þq þ Kð1 KT Þ t0

1 X 0 QðjÞ eKðt tÞ dt0 þ KT nðn 1Þðn þ 2Þan ðtÞS n nðn þ 1Þbn ðtÞSnþ2 n¼2 1 X

nðn þ 1Þðn þ 2ÞCn ðtÞqn nðn 1Þdn ðtÞqn2 Qn ðjÞ þ Kð1 KT Þ n¼2

Z

t

nðn 1Þ

t0

ðn 2Þan ðt0 ÞS n nðn þ 1Þbn ðt0 ÞSnþ2 nðn þ 1Þðn þ 2ÞCn ðt0 Þqn

0 nðn 1Þdn ðt0 Þqn2 Qn ðjÞeKðt tÞ dt0

ð5:7Þ

sy ðtÞ ¼ KTn 2C0 ðtÞ þ ½2a1 ðtÞS 8C1 ðtÞqQðjÞ Zt 0 2C0 ðt0 Þ þ ½2a1 ðt0 ÞS 8C1 ðt0 ÞqQ1 ðjÞeKðt tÞ dt0 KTn þ Kð1 KT Þn t0

1 X

4nðn 1Þan ðtÞS n þ 4nðn þ 1ÞCn ðtÞqn Qn ðjÞ Kð1 KT Þn

n¼2

1 Z X n¼2

t

0

½4nðn 1Þan ðt0 ÞS n þ 4nðn þ 1ÞCn ðt0 Þqn Qn ðjÞeKðt tÞ dt0 t0

dQðjÞ ðtÞ ¼ KT ½ð12nÞa1 ðtÞS 2b1 ðtÞS 3 þ2C1 ðtÞq dðjÞ Zt dQ1 ðjÞ Kðt0 tÞ 0 e þKð1KT Þ ½ð12nÞaðt0 ÞS 2b1 ðt0 ÞS 3 þ2C1 ðt0 Þq dt dj t0

ð5:8Þ

95

Stability Analysis of Rheologic Rock Mass þKT

1 X

nðn1Þan ðtÞS n ðnþ1Þbn ðtÞSnþ2 þnðnþ1ÞCn ðtÞqn þðn1Þdn ðtÞqn2

n¼2

1 Z t X dQn ðjÞ þKð1KT Þ nðn1Þan ðt0 ÞSn ðnþ1Þbn ðt0 ÞSnþ2 þnðnþ1ÞCn ðt0 Þqn dj t n¼2 0

þnðn1Þdn ðt0 Þqn2

dQn ðjÞ Kðt0 tÞ 0 e dt dj

A,n ðnÞ n½n þ 2ð1 2nÞ Bz,n ðnÞ ¼ n þ 4ð1 nÞ an ðtÞ ¼ ½a0n ðtÞ, a00n ðtÞ Cn ðtÞ ¼

½Cn0 ðtÞ, C 00n ðtÞ

Qn ðjÞ ¼ ðcos nj, sin njÞ

ð5:9Þ 9 Aw,n ðnÞ ¼ n½n 2ð1 2nÞ > > > > > > Bw,n ðnÞ ¼ n þ 4ð1 nÞ > > = 0 00 bn ðtÞ ¼ ½bn ðtÞ, bn ðtÞ > > > > dn ðtÞ ¼ ½dn0 ðtÞ, d 00n ðtÞ > > > > ; Q1 ðjÞ ¼ ðcos j, sin jÞ

ð5:10Þ

where R0 r q¼ n ¼ 2, 3, 4 R1 r 1 (reciprocal of the relaxation time) K¼ Z T ¼ (time of post-elastic behaviourÞ G S¼

The functions of cos j [with coeﬃcients of a0n ðtÞ, b0n ðtÞ, C 0n ðtÞ, d 0n ðtÞ] and sin j [with b00n ðtÞ, C 00n ðtÞ, d 00n ðtÞ] are determined from the boundary conditions. Applying the previous equation sets describing the components of displacement and stress, one can solve all problems (which can be expressed by the Fourier series) with visco-elastic medium boundary conditions. a00n ðtÞ,

5.2.2

Interaction of visco-elastic surrounding rock mass and elastic lining

Long tunnels at depth can often be treated as plane problems for stability analysis. When the overlying depth (H) is greater than 30 times of tunnel radius, the gravitational ﬁeld can be replaced by the stress state in an inﬁnite plane approximately, as shown in Figure 5.3. The initial vertical stress is Pz ¼ 0 H and horizontal stress is Px ¼ l1 Pz ¼ l1 0 H. H is the overlying depth, is the unit weight and l1 is the lateral pressure coeﬃcient.

96

Chapter 5

Figure 5.3. Plane problem of tunnel stability analysis.

Before excavation, the undisturbed rock mass is generally in an elastic state. According to the theory of elasticity, its initial stress state is 9 1 > sð2Þ ¼ p ½ð1 þ l Þ ð1 l Þ cos 2j z 1 1 > r,0 2 > > > > ð2Þ 1 = sy,0 ¼ 2 pz ½ð1 þ l1 Þ þ ð1 l1 Þ cos 2j > sð2Þ y,0 ¼ l1 pz 0ð2Þ ¼ 12 pz ð1 l1 Þ sin 2j

> > > > > > > ;

ð5:11Þ

and initial displacement components are 9 r > ð1 l1 Þr0 Hð1 cos 2jÞ > > > 4G2 > = r ð2Þ W y,0 ¼ ð1 l1 Þr0 H sin 2j > 4G2 > > > > ð2Þ ; V ¼0 ð2Þ ¼ Ur,0

ð5:12Þ

y,0

If an opening is excavated, the initial stress state around the opening periphery ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ can be described as: (r ¼ a), S ð2Þ 0 ¼ S 0 ðs r,0 , s y,0 , s y,0 , 0 , U r,0 , W y,0 , V y,0 Þ. Such a disturbed stress state does not meet the boundary conditions, and a compensating

Stability Analysis of Rheologic Rock Mass

97

stress state, S(2), is produced in the same area. The new stress state, Scð2Þ , is the summation of the above two stress states, i.e.,

ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ s ,s ,s , ,U ,W ,V s ,s ,s , ,U ,W ,V S ð2Þ ¼ S c r,c y,c y,c c r,c y,c 0 r,0 y,0 y,0 0 r,0 y,0 y,c y,0

ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ þ S ð2Þ sð2Þ ,s ,s , ,U ,W ,V r y r y y y ð5:13Þ Compared with the stress on the lining after excavation, the stress state produced by the self weight is so small that it can be neglected. The initial stress state in the lining, S ð1Þ 0 , can be considered as zero, so

ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ S ð1Þ sð1Þ sð1Þ ,U ð1Þ c r,c ,sy,c ,sy,c , c ,U r,c ,W y,c ,V y,c ¼ S r ,sy ,sy , r ,W y ,V y ð5:14Þ When the overlying depth above the opening is suﬃciently great and the initial in situ stress state is nearly of hydrostatic stress, Px ¼ Py ¼ Pz ¼ 0H, then from equations (5.11) and (5.12) we have 9 ð2Þ ð2Þ > sð2Þ r,0 ¼ st,0 ¼ sy,0 ¼ P > > = ð2Þ ð5:15Þ 0 ¼ 0 > > > ð2Þ ð2Þ ; U ð2Þ r,0 ¼ W t,0 ¼ V y,0 ¼ 0 The stress state after excavation can be derived accordingly from equations (5.4), (5.5), (5.6), (5.7), (5.8) and (5.9). In consideration of one-dimensional axial symmetry of the stress state, all terms relating to the polar angle j are equal to zero, then Z th h i i 0 2 0 0 0 0 2 k2 ðt0 tÞ 0 sð2Þ þ K ðtÞ ¼ K T C ðtÞ b ðtÞS ð1 K T Þ C ðt Þ b ðt ÞS dt 2 2 0,2 2 2 2 r 0,2 2 0,2 0, 2 2 e h

0

i

0 0 2 sð2Þ y ðtÞ ¼ K2 T2 C 0,2 ðtÞ þ b0,2 ðtÞS 2 þ K2 ð1 K2 T2 Þ 0 sð2Þ y ðtÞ ¼ K2 T2 C 0,2 ðtÞ þ K2 ð1 K2 T2 Þ

Z

t 0

Z th 0

i 0 C 00,2 ðt0 Þþb00, 2 ðt0 ÞS 22 ek2 ðt tÞ dt0

0

C 00,2 ðt0 Þek2 ðt tÞ dt 0

ð2Þ ðtÞ ¼ 0 U ð2Þ r ðtÞ ¼

i 1 h r2 ð1 2nÞC 00, 2 ðtÞ þ b 00,2 ðtÞS 22 2G2

W ð2Þ y ðtÞ ¼ 0 V ð2Þ y ¼0

ð5:16Þ

98

Chapter 5

where K2, T2, G2 are the mechanical parameters of the rheological characteristics of the surrounding rock mass; S2 ¼ a/r, r is the radial distance to any point in the rock mass from the centre of the circular tunnel. Because the second stress state after excavation is the sum of the initial and the compensating stress state (equation (5.13)), then Z th h i i 0 0 0 0 0 0 2 k2 ðt0 tÞ 0 sð2Þ þK ¼K T C ðtÞb ðtÞS ð1K T Þ C ðt Þb ðt ÞS dt P 2 2 2 2 2 2 r,c 0,2 0,2 0,2 0,2 2 e 0 Z th h i i 0 0 2 0 0 0 0 2 k2 ðt0 tÞ 0 ðtÞ¼K T C ðtÞþb ðtÞS ð1K T Þ C ðt Þþb ðt ÞS dt P þK sð2Þ 2 2 2 2 2 0,2 0,2 2 0,2 0,2 2 e y,c 0 Zt 0 0 sð2Þ ðtÞ¼K T C ðtÞþK ð1K T Þ C 00,2 ðt0 Þek2 ðt tÞ dt0 P 2 2 3 3 3 y,c 0,2 0

ð2Þ c ðtÞ¼0 U ð2Þ r,c ðtÞ¼

i 1 h r2 ð12nÞC 00,2 ðtÞþb00,2 ðtÞS 22 2G2

W ð2Þ y,c ðtÞ¼0 ð5:17Þ The stress state in the lining which is assumed to be ideally elastic can be derived from equation (5.16). Since y1 and T1 ¼ Z1 =G1 have the same order of magnitude for the common solid materials, and y1 ! 0, T 1 ! 0 for an elastic body, then K1 T1 ¼ ðT1 =y1 Þ ! 1, i.e., K1 T1 ¼ 1

) ð5:18Þ

K1 ð1 K1 T1 Þ ¼ 0 In this case, equation (5.16) expressing the stress state in the lining is 0 0 2 sð1Þ r ðtÞ ¼ C 0,1 ðtÞ b0,1 ðtÞS 1 0 0 2 sð1Þ y ðtÞ ¼ C 0,1 ðtÞ þ b0,1 ðtÞS 1 0 sð1Þ y ðtÞ ¼ 2n1 C 0,1 ðtÞ

ð1Þ ¼ 0

9 > > > > > > > > > > > > > > =

> > > h i> > 1 0 0 2 > > ðtÞ ¼ ð1 2nÞC ðtÞb ðtÞS U ð1Þ 1 > 0,1 0,1 1 > r > 2G1 > > > > ; ð1Þ ð1Þ W y ðtÞ ¼ V y ðtÞ ¼ 0

ð5:19Þ

Stability Analysis of Rheologic Rock Mass

99

where n1 and G1 are the mechanical parameters of the lining; S 1 ¼ a0 =r1 , r1 is the radial distance to a given point in the lining from the centre of the circular tunnel and a0 is the inner radius of the lining. There are four coeﬃcients in equations (5.17) and (5.19) to be determined; 0 C 0, 2 ðtÞ, b00, 2 ðtÞ, C 00,1 ðtÞ and b00,1 ðtÞ, which are the functions of time. They can be obtained according to the following boundary conditions: 9 ð2Þ ð2Þ sð2Þ r,c ¼ sr,0 þ sr ¼ P > > = ð1Þ ðiiÞ r1 ¼ a sr ¼ 0 > > ; ðiiiÞ r1 ¼ r2 ¼ a ðiÞ

r2 ! 1

ð5:20Þ

ð2Þ ð2Þ sð1Þ r ¼ sr þ sr,0 ð2Þ ð2Þ U ð1Þ r ¼ U r U r,ðt¼0Þ

where the last term is the boundary condition of displacement, implicating that once the lining is completed, the elastic displacement at the periphery of the opening is relieved. The term of U ð2Þ rðt¼0Þ can be easily obtained using the method of elasticity theory. The above four functional coeﬃcients can be obtained by simultaneously solving the simultaneous equation set (5.17), (5.19) and (5.20) that consists of algebraic and integration equations. And ﬁnally, the stress states in the surrounding rock mass and the lining can be derived as follows: for the surrounding rock mass, 9 a2 > 0 b2 t > ¼ P 1 2 ½1 a ð1 e Þ > > > r2 > > > = 2 a ¼ P 1 þ 2 ½1 a0 ð1 eb2 t Þ > > > r2 > > > > > ; ð2Þ ¼ P c ¼ 0

sð2Þ r,c sð2Þ y,c sð2Þ y,c where

a 1 a ¼ 1 a þ 2G2 T2 K2 0

¼ 2G1

a2 a20 1 2 a a0 þ ð1 2n1 Þa2

ð5:21Þ

100

Chapter 5

and for the lining, 9 a20 > 0 b2 t > sð1Þ ¼ a Pð1 e Þ 1 > r,c 1 > r21 > > > = 2 a0 ð1Þ 0 b2 t sy,c ¼ a1 Pð1 e Þ 1þ 2 > > r1 > > > > > ; ð1Þ 0 b2 t sy,c ¼ 2n1 a1 Pð1 e Þ

ð5:22Þ

where

a01

ðT2 K2 1Þ ð1 2n1 Þ þ X02 ¼ T2 K2 ½ð1 2n1 Þ þ X02 þ 1 X02 G

G¼

G1 G2

X0 ¼

a0 a

K2 ð1 2n1 Þ þ X02 þ 1 X02 G

b2 ¼ K2 T2 ð1 2n1 Þ þ X02 þ 1 X02 G It clearly shows that the ﬁnal stress states in the rock mass and the lining depend upon the following parameters: (i) the initial stress state, P (ii) thickness of the lining, X0 ¼ a a0 (iii) elastic shearing moduli of the lining and the rock mass, G ¼ G1 =G2 (iv) relation of post-elastic behaviour time to relaxation time in the rock mass, T2 K2 ¼ T2 =y2 (v) the Poisson’s ratio of the lining, n1. The value of b2 in the equation for stress states only aﬀects the rate of transition from the initial state to the ﬁnal state but does not aﬀect the magnitude of the ﬁnal state. The parameters that are related to b2 include n, X0, G, T2K2 and T2.

5.2.3

Interaction of rock mass and lining of different visco-elastic media

When the surrounding rock mass and the lining are composed of visco-elastic media, but diﬀerent rheological parameters, the rheological mechanical stress state of the rock mass can still be expressed by equation (5.17). The stress state in lining can be

101

Stability Analysis of Rheologic Rock Mass

expressed by the following rheological mechanical equations, Z th 9 h i i ð1Þ 0 0 2 0 0 0 0 2 k1 ðt0 tÞ 0 > sr ðtÞ ¼ K1 T1 C 0,1 ðtÞ b0,1 ðtÞS 1 þ K1 ð1 K1 T1 Þ C 0,1 ðt Þ b0,1 ðt ÞS 1 e dt > > > > 0 > > Z > h i i th > ð1Þ 0 0 2 0 0 0 0 2 k1 ðt0 tÞ 0 > sy ðtÞ ¼ K1 T1 C 0,1 ðtÞ þ b0,1 ðtÞS 1 þ K1 ð1 K1 T1 Þ C 0,1 ðt Þ þ b0,1 ðt ÞS 1 e dt > > > > 0 = Zt ð1Þ 0 0 0 k1 ðt0 tÞ 0 > sy ðtÞ ¼ 21 K1 T1 C 0,1 ðtÞ þ K1 ð1 K1 T1 Þ C 0,1 ðt Þ e dt > > > 0 > h i > > r 1 > ð1Þ 0 0 2 > U r ðtÞ ¼ ð1 2n1 ÞC 0,1 ðtÞ þ b0,1 ðtÞS 1 > > > 2G1 > > ; ð1Þ ð1Þ ð1Þ W y ðtÞ ¼ V y ðtÞ ¼ 0 ðtÞ ¼ 0 ð5:23Þ By assuming that the boundary condition is similar to that given in last section, several functional coeﬃcients in the equations of the rock mass and the lining can be determined. The four boundary equations independent to each other are: (i)

when r2 ! 1 ð2Þ sð2Þ r,0 þ sr ¼ P Rt 0 P þ K2 T2 ½C 00, 2 ðtÞ þ K2 ð1 K2 T2 Þ 0 C 00, 2 ðt0 Þek2 ðt tÞ dt0 ¼ P

(ii)

) ð5:24aÞ

when r1 ¼ a0 sð1Þ r ¼0 Rt 0 K1 T1 ½C 00,1 ðtÞ b00,1 ðtÞ þ K1 ð1 K1 T1 Þ 0 ½C 00,1 ðt 0 Þ b 00,1 ðt0 Þ ek1 ðt tÞ dt0 ¼ 0

ð5:24bÞ (iii)

when r1 ¼ r2 ¼ a ð2Þ ð2Þ sð1Þ r ¼ sr þ sr,0 Z th h i i 0 K1 T1 C 00,1 ðtÞ b00,1 ðtÞX02 þ K1 ð1 K1 T1 Þ C 00,1 ðt0 Þ b00,1 ðt0 ÞX02 ek1 ðt tÞ dt0 0 Zt 0 ¼ K2 T2 ½C 00, 2 ðtÞ b00, 2 ðtÞ þ K2 ð1 K2 T2 Þ ½C 00, 2 ðt0 Þ b00, 2 ðt0 Þek2 ðt tÞ dt0 P 0

(iv)

When r1 ¼ r2 ¼ a ð2Þ ð2Þ U ð1Þ r ¼ U r U r,ðt¼0Þ i a h a2 0 aP 0 0 2 ð1 2n1 ÞC 0,1 ðtÞ þ b0,1 ðtÞX0 ¼ b ðtÞ 2G1 2G2 T2 K2 2G2 0, 2

102

Chapter 5

To obtain the solutions to these four simultaneous equations, it is necessary to solve the following second-order homogeneous diﬀerential equations, ab€00,1 ðtÞ þ bb_00,1 ðtÞ þ cb00,1 ðtÞ ¼ d ð5:25Þ where

a ¼ K1 T1 1 X02 G þ K2 T2 ð1 2n1 Þ þ X02

b ¼ K1 1 X02 Gð1 þ K 2 T2 Þ þ K2 ð1 2n1 Þ þ X02 ðK1 T2 þ 1Þ

c ¼ K1 K2 1 X02 G þ ð1 2n1 Þ þ X02

K1 Gð1 T2 K2 Þ P T2 There are three possible solutions to the integration of equation (5.25), depending on the diﬀerent values of the discriminant of ¼ b2 4ac, i.e., > 0, < 0 and ¼ 0. The main results with the derivation omitted are given below: When 6¼ 0, the stress state of the surrounding rock mass 9

> a2 0 0 r1 t 00 r2 t ð2Þ 0 k2 t > sr,c ¼ P 1 2 a2 b2 e b2 e r2 e > > > r2 > = 2

a ð5:26Þ ð2Þ 0 r1 t 00 r2 t 0 0 k2 t sy,c ¼ P 1 þ 2 a2 b2 e b2 e r2 e > > > r2 > > > ; ð2Þ ð2Þ sy,c ¼ P c ¼ 0 d¼

where a02

1 X02 G þ ð1 2n1 Þ þ X02 T2 K2

¼ T2 K2 1 X02 G þ ð1 2n1 Þ þ X02

K2 ð1 þ T2 r1 Þ b02 ¼ C 1 ð1 2n1 Þ þ X02 K2 þ r1

K2 ð1 þ T2 r2 Þ b2 ¼ C 2 ð1 2n1 Þ þ X02 K2 þ r2

C1 C 2 K2 a02 r2 ¼ ðK2 T2 1Þ ð1 2n1 Þ þ X02 þ K2 þ r1 K2 þ r2 K1 ðT2 K2 1Þr2 T2 K2 1

þ C 1 ¼ T2 ðr2 r1 Þc T2 ðr2 r1 Þ K1 T1 1 X 2 G þ K2 T2 ð1 2n1 Þ þ X 2

9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > =

> > > > > > > > > > > 0 0 > > > > > K ðT K 1Þ r T K 1 > 1 2 2 1 2 2 > >

C2 ¼ > 2 2 T2 ðr2 r1 Þc T2 ðr2 r1 Þ K1 T1 1 X0 G þ K2 T2 ð1 2n1 Þ þ X0 > > > > ) > pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > 2 > r1 b b 4ac > > ¼ > ; 2a r2 ð5:27Þ

Stability Analysis of Rheologic Rock Mass

103

and the stress state of the lining n o9 a20 0 r1 t 00 r2 t 0 0 k1 t > > > ¼ 1 2 GP a1 b1 e b1 e r1 e > > r1 > > > > > > n o 2 > a ð1Þ 0 r1 t 00 r2 t = 0 0 k1 t > 0 sy,c ¼ 1 þ 2 GP a1 b1 e b1 e r1 e r1 > > > n o > > 0 r1 t 00 r2 t ð1Þ 0 0 k1 t > > sy,c ¼ 2n1 GP a1 b1 e b1 e r1 e > > > > > > ; 00 0 ¼ 0

ð5:28Þ

9 T2 K2 1 > >

> > > T2 K2 1 X02 G þ ð1 2n1 Þ þ X02 > > > > > > > c1 K1 1 þ T1 r1 > 0 > > b1 ¼ > = K1 þ r1 > > c2 K1 1 þ T1 r2 > > > b001 ¼ > > > K1 þ r2 > > > > > > c c 1 2 0 0 > K 1 a1 > r1 ¼ ðK1 T1 1Þ þ ; K1 þ r1 K1 þ r2

ð5:29Þ

sð1Þ r,c

where

a01 ¼

When ¼ 0, the derivation procedure is the same. For the rock mass, 9

> a2 0 0 r 00 t 00 0 k2 t r0 t > sð2Þ ¼ P 1 a b e b ðtÞe r e > r,c 2 2 2 > > r22 2 > > > = i 2h a ð2Þ 0 r 00 t 00 r0t 0 0 k2 t sy,c ¼ P 1 þ 2 a2 b2 e b2 ðtÞe r2 e > > > r2 > > > > > ; ð2Þ ð2Þ sy,c ¼ P c ¼ 0

ð5:30Þ

104

Chapter 5

where

a02 b02 b002

r 00 c 001 c 002

1 X02 G þ ð1 2n1 Þ þ X02 T2 K2

¼ T2 K2 1 X02 G þ ð1 2n1 Þ þ X02

c00 K2 ð1 þ T2 r Þ þ c002 K2 ðK2 T2 1Þ ¼ ð1 2n1 Þ þ X02 1 ðK2 þ r Þ2 ( 00

00

2 c2 K2 ðT2 r þ 1Þ 0 ¼ ð1 2n1 Þ þ X0 r2 ðK2 T2 1Þ ð1 2n1 Þ þ X02 : 00 K2 þ r 00 00 00 K2 ½c1 ðK2 þ r Þ c 2 0 a2 ðK2 þ r 00 Þ2 b ¼ 2a T2 K2 1

¼ K2 T2 ð1 2n1 Þ þ X02 þ 1 X02 G ðT2 K2 1Þr 00

¼ K2 T2 1 X02 G þ ð1 2n1 Þ þ X02 T2 K2 1

T2 K2 T2 ð1 2n1 Þ þ X02 þ K1 T1 1 X02 G

ð5:31Þ

and for the lining, 9 0 > a20 0 r 00 t 00 r 00 t 0 k1 t > > sð1Þ ¼ 1 GP a b e b e r e > r,c 1 1 1 1 > r21 > > > > 2 = 0 a0 00 ð1Þ 0 r 00 t 00 r t 0 k1 t sy,c ¼ 1 þ 2 GP a1 b1 e b 1 e r1 e r1 > > > 00 00 > ð1Þ > sy,c ¼ 2n1 GP a01 b01 er t b 001 er t r01 ek1 t > > > > ; ð1Þ c ¼ 0

ð5:32Þ

9 T2 K2 1 > > > > T2 K2 ½ 1 X02 G þ ð1 2n1 Þ þ X02 > > > > 00 00 c1 K1 ð1 þ T1 r ÞðK1 þ r Þ þ c2 K1 ðK1 T1 1Þ > > 0 > > b1 ¼ = 2 00 ðK1 þ r Þ > c2 K1 ðT1 r 00 þ 1Þ > > b 001 ¼ > > > K1 þ r 00 > > > 00 > K1 ½c1 ðK1 þ r Þc2 > 0 0 > r1 ¼ ðK1 T1 1Þ a ; 1 2 ðK1 þ r 00 Þ

ð5:33Þ

where a01 ¼

Stability Analysis of Rheologic Rock Mass

105

In some limiting cases, the above equations can be simpliﬁed. When t ! 1, for the rock mass ( > 0 and ¼ 0),

sð2Þ r,c sð2Þ y,c

¼

a02 P

¼

a02 P

sð2Þ y,c ¼ P

a2 1 2 r2 1þ

a2 r2

9 > > > > > > =

ð2Þ c ¼ 0

ð5:34Þ

> > > > > > ;

and, for the lining ( > 0 and ¼ 0),

sð1Þ r,c sð1Þ y,c

0 sð1Þ y,c ¼ 2n1 a1 GP

5.2.4

9 > > > > > > =

a20 ¼ 1 2 r1 2 a ¼ a01 G 1 þ 20 r1 a01 G

ð1Þ c ¼0

> > > > > > ;

ð5:35Þ

Two-dimensional stress state in surrounding visco-elastic rock mass

In most cases, the in situ stress state of rock mass is anisotropic. The magnitude of the vertical stress generally equals to the overburden stress, i.e., directly proportional to the overlying depth. On the other hand, the horizontal stresses vary, which can approximately be derived using the elasticity theory adopting lateral pressure coeﬃcient of l ¼ n/(1n) (n is the Poisson’s ratio of the rock mass). Generally, it is greater than the vertical stress, i.e., l > 1 [310–313]. Therefore it is of great signiﬁcance to study the stress state in surrounding visco-elastic rock masses for l 6¼ 1. From the general equations (5.4)(5.9) for visco-elastic rock mass given in Section 5.2, it can be seen in principle that the equations can be used to solve any boundary problems of multi-layered circular media. Of course, provided that the lining is circular in shape and composed of a visco-elastic medium while the surrounding rock mass is composed of another visco-elastic medium, such twodimensional problem can be solved, even for initial in situ stress with l 6¼ 1. The solving procedure is tedious. For this reason, this section only deals with a simple case of two-dimensional problems, i.e., the variation of the stress state in the rock mass with an unlined opening.

106

Chapter 5

In this case, the terms with subscript 1 in equations (5.4)–(5.9) are zero, i.e., the ﬁrst circular layer does not exist. The basic equations are given as follows: For stress components, 9 > > > > > 0 > > > > > Zt > >

> 2 4 0 2 0 4 0 > KT 8a2 ðtÞS 6b2 ðtÞS 2d2 ðtÞ þKð1KT Þ 8a2 ðt ÞS 6b2 ðt ÞS 2d2 ðt Þ > > > 0 > > > > > > > 0 > kðt tÞ 0 > > e dt cos2j > > > > > > > Zt > >

> 0 > 0 0 2 0 0 0 0 2 kðt tÞ 0 > c0 ðt Þþb0 ðt ÞS e dt s0 ¼KT ½c0 ðtÞþb0 ðtÞS þKð1KT Þ > > > 0 > = > > KT ½6b2 ðtÞS4 24c2 ðtÞq2 2d2 ðtÞþKð1KT Þ > > > > > > > > Zt > > 0 > 0 4 0 2 0 kðt tÞ 0 > > ½6b2 ðt ÞS 24c2 ðt Þq 2d2 ðt Þe dt cos 2j > > > 0 > > > > > > > sy ¼nðsr,c þsy,c Þ > > > > > >

> 2 4 2 0 > ¼2 KT 2a2 ðtÞS 3b2 ðtÞS þ6c2 ðtÞq þd2 ðtÞq > > > > > > Zt > >

0 > 0 2 0 4 0 2 0 0 kðt tÞ 0 > ; þKð1KT Þ 2a2 ðt ÞS 3b2 ðt ÞS þ6c2 ðt Þq þd2 ðt Þq e dt sin 2j sr ¼KT ½c00 ðtÞb00 ðtÞS2 þKð1KT Þ

Z

t

0 c00 ðt0 Þb00 ðt0 ÞS 2 ekðt tÞ dt0

0

ð5:36Þ For displacement components, Ur¼

r ð1 2n0 Þc00 ðtÞ þ b00 ðtÞS 2 þ ½8ð1 n0 Þa2 ðtÞS 2 2b2 ðtÞS 4 2G þ8n0 c2 ðtÞq2 þ d2 ðtÞq cos2j

9 > > > > > > > > > > > > =

r > 4ð1 2n0 Þa2 ðtÞS2 þ 2b2 ðtÞS 4 þ 4ð3 2n0 Þc2 ðtÞq2 þ 2d2 ðtÞq0 sin2j > > > 2G > > > > > > 0

> n n 0 > 2 2 ; y 2c0 ðtÞ 8a2 ðtÞS þ 24c2 ðtÞq cos2j Vy ¼ 2Gn ð5:37Þ

Wy ¼

107

Stability Analysis of Rheologic Rock Mass For strain components, @U r @r 1 ð1 2n0 Þc00 ðtÞ b00 ðtÞS 2 8ð1 n0 Þa2 ðtÞS2 6b2 ðtÞS 4 ¼ 2G

24n0 c2 ðtÞq2 2d2 ðtÞq0 cos 2j

er ¼

1 @Wt Ur þ r @j r 1 ð1 2n0 Þc00 ðtÞ þ b00 ðtÞS 2 þ 8n0 a2 ðtÞS2 6b2 ðtÞS 4 ¼ 2G

24ð1 n0 Þc2 ðtÞq2 2d2 ðtÞq0 cos 2j

ey ¼

9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > =

> > > > > > > > e ¼ er þ ey > > > > >

> 1 0 0 0 2 0 2 > 2ð1 2n Þc0 ðtÞ 8ð1 2n Þa2 ðtÞS 24ð1 2n Þc2 ðtÞq cos 2j > ¼ > > 2G > > > > > @W t W t 1 @U r > > þ 2r ¼ > > r @j @r r > > > > > 1 > 2 4 2 ; 8a2 ðtÞS 12b2 ðtÞS þ 24c2 ðtÞq þ 4d2 ðtÞ sin 2j ¼ 2G

ð5:38Þ

The above equations indicate that the undetermined coeﬃcients of c00 ðtÞ, b00 ðtÞ only exist in the terms with n ¼ 0 (without trigonometric functional terms), whereas those of a2(t), b2(t), c2(t), d2(t) only exist in the terms with n ¼ 2. Therefore, these undetermined coeﬃcients can be divided into two groups with respect to boundary equations, i.e., a group of n ¼ 0 and a group of n ¼ 2, which can be conveniently solved. The equations for solving these coeﬃcients for the following boundary conditions are given below: (a)

When r ¼ a, sr,c ¼ sr þ sr,0 ¼ 0 ¼ þ0 ¼ 0

(b)

sr,c ðn ¼ 0Þ ¼ 0 sr,c ðn ¼ 2Þ ¼ 0

ðn ¼ 2Þ

When r! 1, sr,c ¼ sr þ sr,0 ¼ sr,0 c ¼ þ0

ðn ¼ 2Þ

sr,c ðn ¼ 0Þ ¼ sr,0 sr,c ðn ¼ 2Þ ¼ sr,0

ðn ¼ 0Þ ðn ¼ 2Þ

ð5:39Þ

108

Chapter 5

Hence, six simultaneous equations can be given according to equation (5.36) and the above boundary conditions: 9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > ðsr,c , n ¼ 2Þ > > > Zt > > > 0 0 0 > > KT ½8a2 ðtÞ 6b2 ðtÞ 2d2 ðtÞ Kð1 KT Þ ½8a2 ðt Þ 6b2 ðt Þ 2d2 ðt Þ > > > 0 > > > > 1 0 > kðt tÞ 0 > e dt þ pz ð1 l1 Þ ¼ 0 > > 2 > > > > > ð c ,n ¼ 2Þ > > > Zt > > 1 > 0 0 kðt0 tÞ 0 > dt pz ð1 l1 Þ ¼ 0 > 2KT ½2a2 ðtÞ 3b2 ðtÞ 2Kð1 KT Þ ½2a2 ðt Þ 3b2 ðt Þe = 2

when r ¼ R0 ¼ a, R0 a r a q¼ ¼ !0 S¼ ¼ ¼1 R1 R1 ! 1 r a ðsr,c ,n ¼ 0Þ Zt 0

0 0

0 1 0 KT c0 ðtÞ b0 ðtÞ þ Kð1 KT Þ c0 ðt Þ b00 ðt0 Þ ekðt tÞ dt0 pz ð1 þ l1 Þ ¼ 0 2 0

0

when r ¼ R1 ! 1 R0 a !0 S¼ ¼ r R1 ! 1 ðsr,c ,n ¼ 0Þ KTc00 ðtÞ þ Kð1 KT Þ

q¼ Z

t 0

ðsr,c , n ¼ 2Þ

r R1 ¼ ¼1 R1 R1

1 1 0 c00 ðt0 Þekðt tÞ dt0 pz ð1 þ l1 Þ ¼ pz ðHl1 Þ 2 2

Z

t

2KTd2 ðtÞ þ 2Kð1 KT Þ 0

1 1 0 d2 ðt0 Þekðt tÞ dt0 þ pz ð1 l1 Þ ¼ pz ð1 l1 Þ 2 2

ð c ,n ¼ 2Þ

Z

2KT ½6c2 ðtÞ þ d2 ðtÞ 2Kð1 KT Þ 0

1 ¼ pz ð1 l1 Þ 2

t

1 0 ½6c2 ðt0 Þ þ d2 ðt0 Þekðt tÞ dt0 pz ð1 l1 Þ 2

> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > ; ð5:40Þ

The coeﬃcients in the form of a function can be completely determined from these six simultaneous equations. The explicit formulae of the coeﬃcients can be obtained by converting the above integral equations into diﬀerential equations in accordance with the initial conditions. Then each component of the stress state is divided into

Stability Analysis of Rheologic Rock Mass

109

two parts: static and rheologic. The latter is related to deformation rate. The two expressions for each stress component are shown below, the superscripts of s and r stand for static and rheologic respectively: for radial stress, sr,c ¼ ssr,c þsrr,c

9 > > > > > > > =

2 2 1 a a4 t=T a t=T 1ð1e Þ 2 þ pz ð1l1 Þ 1ð1e Þ 4 2 3 4 cos2j r r r 2 2 > > > > 2 2 4 > > 1 a 1 a a r t=T t=T > ; sr,c ¼ pz ð1þl1 Þe p cos2j ð1l Þe 4 3 z 1 2 2 4 2 r 2 r r ð5:41Þ

1 ssr,c ¼ pz ð1þl1 Þ

for tangential stress, 9 1 a2 1 a4 > ssy,c ¼ pz ð1 þ l1 Þ 1 þ ð1 et=T Þ 2 pz ð1 l1 Þ 1 þ 3ð1 et=T Þ 4 cos 2j > > = 2 2 r r 1 a2 3 a4 sry,c ¼ pz ð1 þ l1 Þet=T 2 pz ð1 l1 Þet=T 4 cos 2j 2 2 r r

> > > ; ð5:42Þ

for shearing stress 9 2 1 a a4 > t=T > ¼ pz ð1 l1 Þ 1 þ ð1 e Þ 2 2 3 4 sin 2j > > = 2 r r 2 > > 1 a a4 > r t=T > 2 2 3 4 sin 2j c ¼ pz ð1 l1 Þe ; 2 r r sc

ð5:43Þ

The displacement components after excavation surrounding the opening are: 9 Pz r KT 1 t=T a2 Pz r > > ð1 þ l1 Þ 1 e ð1 l þ Þ > 1 > > 4G KT r2 4G > > > > > = 2 4 KT 1 t=T a a 0 e 1 4ð1 n Þ 2 4 cos 2j > KT r r > > > > > > 2 4 > Pz r KT 1 t=T a a > 0 ; ð1 l1 Þ 1 e W#¼ 2ð1 2n Þ 2 þ 4 sin 2j > 4G KT r r Ur ¼

ð5:44Þ

110

Chapter 5

Figure 5.4. Variation of stress components in surrounding rock mass with time at roof (j = 90 ) and sidewall (j = 0 ).

Figure 5.5. Variation of radial displacement of the opening periphery with time at roof and sidewall.

Stability Analysis of Rheologic Rock Mass

111

Figure 5.4 plots the variations of the tangential stress with time at two points of the circular opening (j ¼ 0 and 90 ), given l1 ¼ 0.5. The curves for the static and the dynamic components are plotted separately. Figure 5.5 shows the variation of radial displacement with time at the same points under the same condition as in Figure 5.4, given KT ¼ 1.25.

5.3.

INTERACTION BETWEEN THE VISCO-ELASTIC–PLASTIC SURROUNDING ROCK AND LINING

This section studies the interaction between the visco-elastic rock mass within plastic zones and lining.

5.3.1

Stress state in plastic zones of rock mass

The occurrence and conﬁguration of plastic zones in the surrounding rock mass are often non-symmetrical and they depend on the rock mass characteristics and in situ stress. However, in order to give an explicit equation using analytical method, some assumptions are made to simplify the solutions. As a typical condition, in situ stress ﬁeld is isometric, and the surrounding rock mass is homogeneous, isotropic, viscoelastic and continuous. Such a problem can be treated as a symmetrical one. If the plastic zone is assumed incompressible, then the expression for strain state, when using polar coordinates, becomes very simple: ey þ er ¼ 0 du er ¼ dr

9 =

ð5:45Þ

u ey ¼ ; r

Rearrangement and integration of the above two equations yield: u¼

AðtÞ r

ey ¼

AðtÞ r2

er ¼

AðtÞ r2

ð5:46Þ

Given that the physical equations of the medium within the plastic zone are sy s ¼ 2Mey sr s ¼ 2Mer where s is the average stress.

) ð5:47Þ

112

Chapter 5

Based upon experimental data available and Mogi’s criterion [128], the plastic condition of surrounding rock mass can be expressed as OCT ¼ f ðs1 þ s3 þ as2 Þ Because of symmetry, 1 sm ¼ ðs1 þ s3 þ s2 Þ ¼ P ¼ const 3 then s1 þ s3 þ as2 ¼ const ðs1 s2 Þ2 þ ðs2 s3 Þ2 þ ðs3 s1 Þ2 ¼ Kp2 ¼ const

ð5:48Þ

Also for a symmetrical problem s1 ¼ sy

s3 ¼ sr

s2 ¼ sy

By substituting equation (5.47) into equation (5.48), we have Kp M ¼ pﬃﬃﬃ 2 6e y

ð5:49Þ

By substituting equation (5.49) into equation (5.47), it gives 9 Kp > sy ¼ s pﬃﬃﬃ > > 6= Kp > > sr ¼ s þ pﬃﬃﬃ > ; 6

ð5:50Þ

2Kp sy sr ¼ pﬃﬃﬃ 6

ð5:51Þ

Therefore,

The equilibrium equation of the plastic zone is r

dsr þ sr sy ¼ 0 dr

ð5:52Þ

Stability Analysis of Rheologic Rock Mass

113

The load applied to the lining at the opening periphery (r ¼ a) is sð1Þ r¼a ðtÞ. By substituting equation (5.51) into (5.52) and through rearrangement and integration, the stress components in this area are: 9 2Kp r ð1Þ > > p ﬃﬃ ﬃ þ s sð2Þ ðtÞ ¼ ðtÞ ln > r¼a r = a 6 > 2Kp r > ð1Þ ; p ﬃﬃ ﬃ ðtÞ ¼ ðtÞ > sð2Þ þ sr¼a 1 þ ln y a 6

5.3.2

ð5:53Þ

Interaction between surrounding rock mass and lining

The area beyond the plastic zone in the surrounding rock mass is still in visco-elastic state. The stress ﬁeld can be determined using the method stated in previous sections, i.e., the new stress ﬁeld (S ð3Þ c ) after excavation is the summation of the initial stress ﬁeld (S ð3Þ ) and the compensating stress (S ð3Þ ) ﬁeld: 0 ð3Þ ð3Þ S ð3Þ c ¼ S0 þS

where the initial stress ﬁeld is the same as in equation (5.23). The compensating stress ﬁeld can be simpliﬁed because of axial symmetry. The global components of the stress state are 9 Z th h i i > 0 0 2 0 0 0 0 2 k3 ðt0 tÞ 0 > sð3Þ þK e ðtÞ¼K T C ðtÞb ðtÞS ð1K T Þ C ðt Þb ðt ÞS dt P > 3 3 3 3 3 r,c 0,3 0,3 3 0,3 0,3 3 > > t0 > > > Z th > h i i > 0 > ð3Þ 0 0 2 0 0 0 0 2 k3 ðt tÞ 0 C 0,3 ðt Þþb0,3 ðt ÞS 3 e dt P > sy,c ðtÞ¼K3 T3 C 0,3 ðtÞþb0,3 ðtÞS 3 þK3 ð1K3 T3 Þ > > > > t0 > > Zt > = ð3Þ 0 0 0 k3 ðt0 tÞ 0 sy,c ðtÞ¼K3 T3 C 0,3 þK3 ð1K3 T3 Þ C 0,3 ðt Þe dt P > t0 > > > > > ð3Þ ðtÞ¼0 > c > > > h i > 1 > ð3Þ 0 0 2 > r3 ð12nÞC 0,3 ðtÞþb0,3 ðtÞS 3 U r,c ðtÞ¼ > > > 2G3 > > > ; ð3Þ W y,c ðtÞ¼0 ð5:54Þ where K3, T3, G3 are the mechanical parameters in the visco-elastic area of the surrounding rocks (as illustrated in Figure 5.6); S 3 ¼ R=r3 , R is the radius of the plastic zone and r3 is the radial coordinate of the point under consideration.

114

Chapter 5

Figure 5.6. Division of diﬀerent damage zones around an opening.

Figure 5.7. Synthetic rheological mechanical model for plastic zone, visco-elastic area and elastic lining, where G – bKp; b – constant related to the frictional coeﬃcient of rock block; A – visco-elastic area; B – plastic zone; C – lining.

The lining material is assumed to be a Hooke’s elastic body, its basic equation is the same as equation (5.23). The synthetic mechanical model for the plastic zone, visco-elastic area and elastic lining can be described approximately using the diagram shown in Figure 5.7. Each coeﬃcient in equation (5.54) can be determined using the following boundary and physical conditions. (a)

At an inﬁnite point, i.e., r3 ! 1 sð3Þ r,c ¼ P

(b)

On the inner boundary of the lining, i.e., r1 ¼ a0, sð1Þ r¼a0 ¼ 0

Stability Analysis of Rheologic Rock Mass (c)

Along the interface between the lining and the rock mass, i.e., r1 ¼ r2 ¼ a, and provided that the lining is applied with the moment of t ¼ t1, then ð2Þ ð2Þ U ð1Þ r ¼ U r U rðt¼t1 Þ

(d)

115

ð5:55Þ

Along the interface of visco-elastic and plastic zones, i.e., r2 ¼ r3 ¼ R,

For plastic condition 2 ð3Þ sð3Þ y sr ¼ pﬃﬃﬃ Kp 6 For condition of continuous displacements ð3Þ U ð2Þ r ¼ Ur

Based upon the above boundary conditions, the corresponding equations can be given. By assuming that the time for applying lining is t ¼ t1, the excavating time is t0 ¼ 0 and solving integral and algebraic equations in connection with equations (5.54), (5.23), (5.45) and (5.55), functions of C 00,3 ðtÞ, C 00,1 ðtÞ, b00,1 ðtÞ, b00,3 ðtÞ and A(t) can be determined. They ﬁnally give the stress components: For the lining 9 Kp ðdK3 T3 1Þ t1 =T3 GR2 t=T3 2 >

pﬃﬃﬃ ¼ 2 ðe e Þð1 S 1 Þ > > > > a ð1 2nÞ þ X02 6K 3 T 3 > > = 2 Kp ðdK3 T3 1Þ t1 =T3 GR ð1Þ t=T 2 3

p ﬃﬃ ﬃ sy ðtÞ ¼ 2 ðe e Þð1 þ S Þ 1 2 > a ð1 2nÞ þ X0 6K 3 T 3 > > > 2 > Kp ðdK3 T3 1Þ t1 =T3 2n1 GR > t=T ð1Þ > 3

; p ﬃﬃ ﬃ ðe e Þ sy ðtÞ ¼ 2 2 a ð1 2nÞ þ X0 6K 3 T 3

sð1Þ r ðtÞ

ð5:56Þ

where G ¼ G1/G3 is the stress state in the visco-elastic zone, when r 5 R, 9 2 Zt > KTKp K 1 R > p t=T3 > p ﬃﬃ ﬃ p ﬃﬃ ﬃ > ð1 KT Þe Kð1 KT Þ ¼ 1 þ > ð3Þ 2 KT r 6 6> sy,c ðtÞ > 0 = 2 1 R 0 0 > ð1 KT Þet =T3 2 eKðt tÞ dt0 P 1þ > > > KT r > > > ; ð3Þ s ðtÞ ¼ P sð3Þ r,c ðtÞ

y,c

)

ð5:57Þ

116

Chapter 5

For the plastic zone 9 2 2 > GR 1 X 2K K ðK T 1Þ r p p 3 3 > 0 >

pﬃﬃﬃ sð2Þ ðet1 =T3 et=T3 Þ > r ðtÞ ¼ pﬃﬃﬃ ln 2 = 2 a ð1 2nÞ þ X a 6 6K3 T3 0 > GR2 1 X02 2Kp K ðK T 1Þ t1 =T3 r >

ppﬃﬃ3ﬃ 3 > ðe et=T3 Þ > sð2Þ ; y ðtÞ ¼ pﬃﬃﬃ ð1 þ ln Þ 2 2 a a ð1 2nÞ þ X0 6K3 T3 6 ð5:58Þ ð2Þ Given sð3Þ r,c ðtÞ ¼ sr,c ðtÞ, then the boundary, R, between the visco-elastic and plastic zones can be obtained from equations (5.53) and (5.57),

pﬃﬃﬃ 6 ð3Þ ð2Þ R ¼ a exp s ðtÞ sr,c ðtÞ 2Kp r,c

ð5:59Þ

It can be seen from the above equations that the loading on the lining is a function of (a) the plastic zone size (R) at the time when the lining is applied (t1), (b) the ratio of the shearing moduli of the lining and the rock mass (G), and, (c) other related physical, mechanical and geometrical parameters of the lining and the surrounding rock mass.

5.4.

STRESS STATE IN VISCO-ELASTIC–VISCO-PLASTIC SURROUNDING ROCK MASSES

The preceding section discussed the stress states in the surrounding rock mass and the lining when there exist plastic zones. The plastic zone is time-dependent, in other words, the plastic zone is visco-plastic, whereas the rock masses at further distance still behaves visco-elastically [320–323]. This section discusses the interaction between the surrounding rock mass and the lining under such condition. Figure 5.8 shows the mechanical model of the whole system. The mechanical equations for

Figure 5.8. Mechanical model for the surrounding rock and the lining.

117

Stability Analysis of Rheologic Rock Mass

the visco-elastic rock mass and the lining zone are the same as that given in the preceding section; and the stress in the visco-plastic zone consists of two parts, i.e., 0

sð2Þ ¼ sð2Þ þ sð2Þ

00

ð5:60Þ

0

where the ﬁrst part, sð2Þ is the stress component of the Maxwell medium, and the 00 second part, sð2Þ is the stress component of the St. Venant medium. The physical equation for the Maxwell medium is

e_ y e_ ¼

ð2Þ sð2Þ s y sy

9 _ ð2Þ s_ > s_ ð2Þ > y s y > > > þ = 2G

2Z2 2 ð2Þ _ ð2Þ s_ ð2Þ s s_ sr sð2Þ r r s r e_ r e_ ¼ þ 2Z2 2G2

> > > > > ;

ð5:61Þ

in which s is the average stress, s_ is the average stress rate, e_ is the average strain rate. The general solutions to the diﬀerential equation of (5.61) can be obtained. The assumptions similar to the last section are made and the visco-plastic zone is approximately regarded as an incompressible medium. By applying the solutions of equations (5.45) and (5.46), equation (5.61) becomes: u¼

AðtÞ r

ey ¼

AðtÞ r2

er ¼

AðtÞ r2

ð5:62aÞ

u_ ¼

A_ ðtÞ r

e_ y ¼

A_ ðtÞ r2

e_ r ¼

A_ ðtÞ r2

ð5:62bÞ

hence

From equation (5.61), e_ y e_ r ¼

0 1 ð2Þ0 1 ð2Þ0 ð2Þ0 _ _ sy sð2Þ s þ s r r 2Z2 2G2 y

ð5:63aÞ

Substituting (5.62a) into (5.63a), then 1 @ðsy sr Þ sy sr 2 @AðtÞ þ ¼ 2 2G2 @t r @t 2Z2

ð5:63bÞ

118

Chapter 5

Examining the visco-elastic zone and its boundary with the visco-plastic zone, the condition of continuous displacements in the boundary leads to ð3Þ U ð2Þ r¼R ¼ U r¼R

ð5:64Þ

and the plastic condition (see equation (5.51)) leads to ð3Þ sð3Þ y sr ¼ dKp

ð5:65Þ

A solution can be obtained by using equations (5.16), (5.62), (5.63) and (5.65) for stress and displacement in the visco-elastic zone, i.e.,

AðtÞ ¼

R2 dKp 4G3

1

1 et=T3 1 K3 T3

ð5:66Þ

By substituting equation (5.66) into equation (5.63a) and solving the corresponding diﬀerential equation, we have sy sr ¼

M t=T3 e þ c3 eðG2 =Z2 Þt r2

ð5:67Þ

where

M¼

R2 G2 dKp 1 1 1 K3 T3 G3 T3 ðG2 =Z2 Þ ð1=T3 Þ

ð5:68Þ

In equation (5.67), c3 is an undetermined constant. In consideration of t ¼ 0, the stress state in the surrounding rock mass after excavation starts transiting from elastic state to visco-plastic state, i.e.,

sy srjt ¼ 0 ¼ 2P

a2 r2

ð5:69aÞ

In substitution of this equation into equation (5.67), we have c3 ¼ 1=r2 ð2Pa2 þ MÞ. Again, upon substitution of the so-obtained c3 into equation (5.67), we have sy sr ¼

1 Mðet=T3 et=T2 Þ 2Pa2 et=T2 r2

ð5:69bÞ

Stability Analysis of Rheologic Rock Mass

119

It can be obtained according to the equilibrium condition of the medium, sy sr ¼ r

@sr @r

ð5:69cÞ

Solving equations (5.69b) and (5.69c), by (5.69c) separating variables and integrating, we have the solution in the form of 9 d > > sr ¼ BðtÞ þ c 2 TðtÞ > = r > d > sy ¼ BðtÞ þ c þ 2 TðtÞ > ; r

ð5:70Þ

Subtraction of these two equations gives sy sr ¼

2dTðtÞ r2

By substituting this equation into equation (5.69), we obtain TðtÞ ¼

1 t=T3 M e et=T2 2Pa2 et=T2 2d

ð5:71Þ

According to the condition of continuous displacements at the boundary between the lining and the rock mass, we have ð2Þ ð2Þ U ð1Þ r¼a ðtÞ ¼ U r¼a ðtÞ U r¼a ðt1 Þ

ð5:72Þ

where t1 is the time of applying the lining. Making use of equations (5.62), (5.66) and (5.64) and substituting them into equation (5.72), we have sð1Þ r¼a ðtÞ ¼

t=T3 1 X02 dKp G2 R2 1 2 et1 =T3 e 1 2 K3 T3 2 G3 a ð1 2n1 þ X0 Þ

In the case of unlined opening, i.e., t t1, sð2Þ r¼a ðtÞ BðtÞ ¼

d ¼ bðtÞ þ c 2 TðtÞ ¼ 0 a

1 c t=T3 et=T2 2Pa2 et=T2 M e 2 2a 2d

ð5:73Þ

120

Chapter 5

Substituting the expressions for B(t) and T(t) gives sð2Þ r ðtÞ

)

sð2Þ y ðtÞ

1 c t=T3 2 M e et=T2 2Pa2 et=T2 ¼ 2 2a 2r

ðt t1 Þ

ð5:74Þ

while in the case of lined opening, i.e., t t1, sð2Þ r¼a ðtÞ

d ¼ bðtÞ þ c 2 TðtÞ ¼ sð1Þ r¼a ðtÞ a

Substitution of the results from equations (5.73) and (5.71) into the above equation gives t=T3 1 X02 dKp G2 R2 1 BðtÞ ¼ e 1 et1 =T3 K3 T3 2 G3 a2 ð1 2n1 þ X02 Þ

1 c t=T3 þ M e et=T2 2Pa2 et=T2 2a2 2d Substituting this equation and equation (5.71) into equation (5.70) gives sð2Þ r ðtÞ sð2Þ y ðtÞ

)

t=T3 1 X02 dKp G2 R2 1 ¼ e 1 et1 =T3 K3 T3 2 G3 a2 ð1 2n1 þ X02 Þ

1 1 t=T3 þ M e et=T2 2Pa2 et=T2 for t t1 2 2 2a 2r

ð5:75Þ

According to the plastic condition (5.65) and the boundary condition, sð3Þ r ! P for r ! 1, we have at the boundary of r2 ¼ r3 ¼ R 9 dKp > = 2 dKp > ; sð3Þ y ð0Þ ¼ P 2

sð3Þ r ð0Þ ¼ P þ

Let the stress sðtÞ r remain continuous on the boundary of r ¼ R, then when t t1,

dKp 1 1 t=T3 et=T2 2Pa2 et=T2 ¼ P þ M e 2 2 2a 2R 2

ð5:76Þ

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121

and when t t1,

dKp G2 R2 1 X02 1 1 t=T3 t=T2 2 t=T2 e M e 2Pa e þ 2a2 2R2 2 G3 a2 ð1 2n1 þ X02 Þ t=T3 dKp 1 1 e : ð5:77Þ et1 =T3 ¼ P þ K3 T3 2

Equations (5.76) and (5.77) describe the change of the radius (R) of the viscoplastic zone, with time elapsing in the form of an explicit function. And equations (5.74), (5.75), (5.76) and (5.77) comprise the complete solution to the stress state of the visco-plastic zone. For example, given that d ¼ 0.8, Kp ¼ 1.5 P, a ¼ 2 m, lining thickness ¼ 40 cm, n ¼ 0.25, T2 ¼ T3 ¼ 12 days, K2 ¼ K3 ¼ 1/8 days and G2/G3 ¼ 0.5, then for t ¼ 0, R ¼ 2.31 m and for t ! 1, R ¼ 2.623 m (as shown in Figure 5.9). It is easy to prove that the boundary conditions are met for t ¼ 0 and t ! 1 under limit condition. Given t ! 1 in equation (5.75), we have the stress state in the viscoplastic zone in inﬁnite time after applying lining: 9 sð2Þ r ðtÞ =

1 X02 dKp G2 R2 1 et1 =T3 2 1 ¼ 2 ; ð2Þ K 2 G a T þ X 1 2n 3 3 3 1 0 s ðtÞ

ð5:78Þ

y

Figure 5.9. Stress distribution in various zones in the surrounding rock mass under limit condition.

122

Chapter 5

It can be seen from equation (5.78) that the rock mass in the visco-plastic zone is unable to bear long-term shearing loads because its physical property is close to ﬂuids. Finally, the two main stress components in the whole region tend to balance and the ﬁnal loading on the lining is the same as that given in the above equation, i.e., ð2Þ sð1Þ r¼a ðt ! 1Þ ¼ sr ðt ! 1Þ

It can be seen from above that for this type of visco-elastic and visco-plastic rock mass, the ﬁnal loading on the lining is related not only to the physical and geometrical parameters of the visco-elastic zone in the rock mass and the lining but also to the time when the lining is applied (t1) and the magnitude of the radius of the visco-plastic zone. It is diﬀerent from the previously stated viscoelastic-plastic media. In the present case the ﬁnal loading on the lining is related to the ratio of the shearing deformation moduli G2/G3 of the visco-plastic and visco-elastic zones, but not related to the shearing modulus G1 of the lining itself. Additionally, it also can be seen that to reach a mechanical equilibrium state, the visco-plastic zone extends beyond the lining into the rock mass with the radius reaching R.

5.5.

RHEOLOGICAL ANALYSIS WITH DILATION AND SOFTENING OF THE ROCK MASS

The previous sections assume that the plastic medium is incompressible. However, in most cases there exists a broken-swelling eﬀect after plastic zone produced by excavation. The purpose of this section is to study the stress state accompanied by dilation eﬀect and meanwhile to consider the softening phenomenon and its eﬀect in the post-failure region in the rock mass [324–329]. For this reason, a more complete rheological mechanical model is proposed.

5.5.1

Mechanical model of surrounding rock mass

The assumption of continuum mechanics is still adopted. The media involved is considered to be isotropic, and the initial stress ﬁeld is assumed to be uniform, so axisymmetrical analysis is allowable. Assume that the rock mass in the far region after excavation is in elastic state while the rock mass in the near region is in the visco-plastic state in which certain residual strength still exists. The lining is assumed to be an elastic medium, as shown in Figure 5.10.

Stability Analysis of Rheologic Rock Mass

123

Figure 5.10. Schematic division of zones.

On the boundary between visco-elastic and visco-plastic zones, the stress state is assumed to meet the Mohr–Coulumb criterion, i.e., sy sr ¼ ðsy þ sr Þ sin j þ 2c cos j

ð5:79Þ

The stress components in the visco-elastic zone are the same as in equation (5.54). It is known from equation (5.54) that when r ! 1, Ur(t) ¼ 0; then c0(t) ¼ 0 and the following equation is valid: 1 ðsy þ sr Þ ¼ P 2 By substitution of this equation into Mohr–Coulumb criterion, gives sy sr ¼ 2P sin j þ 2c cos j ¼ Kp

ð5:80Þ

Supposing the minimum strength in the residual strength region is d times (d < 1) Kp, i.e., equal to dKp, then dKp ¼ 2dðP sin j c cos jÞ

ð5:81Þ

By substituting equation (5.80) into equation (5.54), we can obtain the undetermined functional coeﬃcient and then obtain the radial and tangential stresses 9 R22 > = ðP sin j c cos jÞ r2 2 R > sy ¼ P 22 ðP sin j c cos jÞ ; r sr ¼ P þ

ð5:82Þ

124

Chapter 5

and the radial displacement R22 1 t=T3 ðP sin j c cos jÞ 1 e 1 ur ¼ K3 T3 4Gr

ð5:83Þ

5.5.2 Visco-plastic model considering dilation and softening 5.5.2.1 Physical model. The rock mass in the post-failure region displays viscoplastic softening behaviour when the peak strength is reached and the dilation phenomenon takes place simultaneously [325–327]. For this reason, the diﬀerential equation for the Maxwell medium that describes the visco-plastic behaviour is adopted and applied to describe the softening and dilation characteristics, it is expressed as follows: ðsy sr Þ þ

Z 2nG ðs_ y s_ r Þ ¼ 2Zð_ey e_ r Þ þ ðey þ er Þ 2Gðey er Þ þ D G 1 2n

ð5:84Þ

This equation not only reﬂects the softening and relaxation characteristics with visco-plastic property of the rock mass (the third term on the right-hand side of the equation), but also guarantees the stress diﬀerence in the rock mass.

5.5.2.2 Geometric equation. Assuming that the problem under consideration meets the small deformation concept, we have, for an axisymmetrical problem, er ¼

@u @r

ey ¼

u r

If the dilation phenomenon takes place in the visco-plastic zone and the radial strain is l/r times of the tangential strain, i.e., dilation occurs near to the opening periphery, then l er ¼ ey r @u l u u ¼ ¼ l 2 @r r r r Integrate u with respect to r1, then 9 u ¼ AðtÞel=r > > > > = l l=r er ¼ 2 e AðtÞ r > > > 1 l=r > ey ¼ e AðtÞ ; r

ð5:85Þ

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125

and strain rate 9 l l=r _ > = A ðtÞ e r2 > 1 e_ y ¼ el=r A_ ðtÞ ; r e_ r ¼

5.5.2.3 Equilibrium equation. r

5.5.3

ð5:86Þ

For an axisymmetrical problem, we have

9 @sr > > ¼ sy sr = @r Z 1 > ; sr ¼ ðsy sr Þdr þ f ðtÞ > r

ð5:87Þ

Stress components in each zone

5.5.3.1 Visco-plastic zone. By substitution of equations (5.85) and (5.86) into equation (5.84) and further interpretation, we have s_ y s_ r þ

G ðsy sr Þ ¼ A A_ ðtÞ B AðtÞ þ C Z

where 9 1 l > > 1 þ el=r A ¼ 2G > > r r > > > = 2 2G 1 2n l l l=r 1 e B ¼ þ 1þ 1 2n r r > Z r > > > > > G > ; C¼ D Z

ð5:88Þ

Equation (5.88) is the inhomogeneous partial diﬀerential equation of ðsy sr Þ, with the following solution Z G Z AðtÞeðG=ZÞt dt þ A eðG=ZÞt AðtÞ þ C eðG=ZÞt þ D ðrÞ sy sr ¼ eðG=ZÞt A þ B Z G ð5:89Þ

126

Chapter 5

By substituting equation (5.89) into equilibrium equation (5.87) and integrating it with respect to r, we have 4G2 4n 2 3n 1 l=r sr ¼ þ e e ðG=ZÞt l r Zð1 2nÞ Z Z 2 1 l=r Z 1 ðG=ZÞt AðtÞe dt 2G þ e AðtÞ þ C ln r þ DðrÞ dr þ f ðtÞ l r G r

ð5:90Þ

It is easy to determine D ðrÞ in equation (5.90), if analysing the transient response. The equation of D ðrÞ ¼ D is given by Zhu et al. (1988). Considering this result in equation (5.90), we obtain Z t G 0 sy sr ¼ eðG=ZÞt A þ B Aðt0 ÞeðG=ZÞt dt0 þ A eðG=ZÞt AðtÞ þ D eðG=ZÞt 1 Z 0 ð5:91Þ Substituting equation (5.91) into equilibrium equation (5.87) and integrating the result with respect to r gives Zt 4G2 4n 2 3n 1 l=r ðG=ZÞt 2 1 0 þ e þe sr ¼ eðG=ZÞt Aðt0 ÞeðG=ZÞt dt0 AðtÞ 2G þ el=r l r l r Zð1 2nÞ 0 ðG=ZÞt þD e 1 lnr þ f ðtÞ ð5:92Þ

Applying the condition sr,nt ¼ sr,np on the boundary between R2 visco-elastic and visco-plastic zones and making equality between equations (5.82) and (5.92) results in ðG=ZÞt

4G2 4n 2 3n 1 l=R2 þ e l R2 Zð1 2nÞ

f ðtÞ ¼ p þ ð p sin j c cos jÞ þ e Zt 2 1 l=R2 ðG=ZÞt 0 ðG=ZÞt0 0 ðG=ZÞt e Aðt Þe dt 2G þ e AðtÞ Dðe 1Þ ln R2 l R2 0 ð5:93Þ By substituting equation (5.93) into equation (5.92), we can derive the explicit formula of sr. Applying equations (5.91), (5.92) and (5.93) gives Z t G 0 Aðt0 ÞeðG=ZÞt dt0 þ A eðG=ZÞt AðtÞ þ D eðG=ZÞt 1 þ sr sy ¼ eðG=ZÞt A þ B Z 0 ð5:94Þ

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127

5.5.3.2 Residual strength zone. Because the residual strength is related to the conﬁning pressure, the strength is higher at greater depth [328–334]. Owing to the small size of this zone, it can be assumed that the residual strength and r have a linear relationship. By examining equations (5.80) and (5.81), the physical equation to describe this zone can be approximately expressed as sy sr ¼ c0 r þ dKp

ð5:95Þ

Substituting equation (5.95) into equation (5.87) and integrating the result, we have sr ¼ c0 r þ dKp ln r þ jðtÞ

9 =

sy ¼ 2c0 r þ ð1 þ ln rÞdKp þ jðtÞ ;

ð5:96Þ

In the case of unlined opening, r ¼ a, sr ¼ 0, then, from equation (5.96), 1 c0 ¼ ½dKp ln r þ jðtÞ a

ð5:97Þ

The term of j(t) in the above equation can be determined from the condition that the radial stresses sr along the boundary R1 between the visco-plastic zone and the residual strength zone are equal, i.e., the equation (5.92) is equal to equation (5.96). The stress components in the visco-elastic zone are expressed by equations (5.82) and (5.83) respectively.

5.5.4

Stress state without lining

5.5.4.1 Visco-plastic zone, R1 < r < R2. In the previous equations, there is a function A(t) that is undetermined. It can be derived from the condition of continuous displacements on the boundary between the visco-elastic and the viscoplastic zones (R2). By letting unp ¼ une, then we have its expression from equations (5.83) and (5.89),

AðtÞ ¼

R2 l=R2 1 e et=T3 1 ðP sin j C cos jÞ 1 K3 T3 4G

ð5:98Þ

128

Chapter 5

By substituting equation (5.98) into equations (5.92) and (5.94) and integrating the result, we have the expression of the stress state in the visco-plastic zone, GR2 1ð1=K3 T3 Þ l=R2 sr ¼ PþðpsinjccosjÞþe ðpsinjccosjÞ e G=ðZ1=T3 Þ Zð12nÞ Z 4n2 3n1 l=R2 4n2 3n1 l=r eððG=ZÞð1=T3 ÞÞt 1 ðeðG=ZÞt 1Þ e þ þ e G l R2 l r R2 1 2 1 l=R2 2 1 eðG=ZÞt el=R2 ðpsinjccosjÞ 1 þ þ el=r et=T3 1 e K3 T3 l R2 l r 2 DðeðG=ZÞt 1ÞðlnR2 lnrÞ ð5:99Þ ðG=ZÞt

G R2 l=R2 e sy ¼ e A þB ð p sin j c cos jÞ Z 4G Z ðG=ZÞt 1 ð1=K3 T3 Þ ððG=ZÞð1=T3 ÞÞt e e 1 1 ðG=ZÞ ð1=T3 Þ G ð5:100Þ 1 ðG=ZÞt R2 l=R2 t=T3 e e þA e ð p sin j c cos jÞ 1 1 K3 T3 4G þD eðg=ZÞt 1 þ sr ðG=ZÞt

5.5.4.2 Residual strength zone, a < r < R1. As stated previously, the explicit formula j(t) can be derived from equations (5.96), (5.97) and (5.99) by applying the condition of equal sr on the boundary between two adjacent zones (R1):

GR2 el=R2 ðpsin j ccosjÞ jðtÞ ¼ P þ ðp sin j c cos jÞ e Zð1 2nÞ Z ðG=ZÞt 1 ð1=K3 T3 Þ ððG=ZÞð1=T3 ÞÞt e 1 e 1 ðG=ZÞ ð1=T3 Þ G 4n 2 3n 1 l=R2 4n 2 3n 1 l=R1 1 þ þ e e R2 l R2 l R1 2 1 2 1 l=R2 þ eðt=T3 Þ 1 e eðG=ZÞt eðl=R2 Þ ðp sin j ccosjÞ 1 K3 T3 l R2 2 1 l=R1 R1 R1 1 DðeG=Zt 1Þðln R2 ln R1 Þ dKp ðlnR1 lnaÞ þ e l R1 a a ðG=ZÞt

ð5:101Þ

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129

The stress components in this zone are,

9 r r > sr ¼ d Kp ln r ln a þ jðtÞ 1 = a a 2r 2r > ; sy ¼ 1 þ ln r ln a dKp þ jðtÞ 1 a a

ð5:102Þ

5.5.4.3 Determination of boundary R2. Since the rates of displacement components on the boundary between the visco-elastic and visco-plastic zones are equal, we can determine the value of R2 by letting _ ð3Þ u_ ð2Þ r ¼ u r , and derive the corresponding displacement components in equations (5.83) and (5.85) with respect to t and let the results be equal, then R2 ¼ l

ð5:103Þ

Although this equation is extremely simple, its physical meaning is rationally clear. The assumption of er ¼ l=rey in the geometric equation in the previous sections, when r ¼ R2, er ¼ ey, which is the starting point where the volumetric deformation in the visco-plastic zone begins to dilate in volume. Therefore, the result of R2 ¼ l coincides with the assumption made initially. 5.5.4.4 Determination of boundary R1. Similarly, the stress diﬀerences between two sides of the boundary are equal when r ¼ R1, R1 can be determined. For the residual strength zone, we have, from equation (5.102)

r r ð5:104Þ sy sr ¼ dKp 1 ln a jðtÞ a a By comparing the stress diﬀerence of (sysr)vp in the visco-plastic zone shown in equation (5.104) to equations (5.91), (5.93) and (5.98), we have R1 R1 dKp 1 lna jðtÞ a a 4G 1 l=R1 n l l e 1 ¼ eðG=ZÞt þ 1þ Z R1 1 2n R1 R1 Z R2 l=R2 1 ð1=K3 T3 Þ ððG=ZÞð1=T3 ÞÞt e e ðpsinj ccosjÞ 1 eðG=ZÞt 1 ðG=ZÞ ð1=T3 Þ G 4 1 R2 l 1 elðR2 R1 Þ=R1 R2 ðpsinj ccosjÞ 1 et=T3 1 þ 1þ 2 R1 R1 K3 T3 ðG=ZÞt ðG=ZÞt e 1 ð5:105Þ þ De

130

Chapter 5

By substituting the values of R2 and j(t) into equation (5.105), we can obtain the value of R1 using trial and error method. In the limit case, we can derive the following equation for t ¼ 0 R1 R1 1 ln a p þ ð p sin j c cos jÞ R2 el=R2 ð p sin j c cos jÞ dKp 1 2 a a 1 2 1 l=R2 2 1 l=R1 R1 R1 þ þ ln a 1 e e dKp ln R1 K3 T3 l R2 l R1 a a R2 l 1 elðR2 R1 Þ=R1 R2 ð p sin j c cos jÞ ð5:106Þ 1þ R1 K3 T3 2R1 when t ! 1 R1 R1 R2 dKp 1 ln a p þ ð p sin j c cos jÞ þ el=R2 ð p sin j c cos jÞ a a 1 2n 4n 2 3n 1 l=R2 4n 2 3n 1 l=R1 þ þ e e l R2 l R1 R2 l=R2 e ð p sin j c cos jÞ 2 2 1 2 1 l=R1 þ þ el=R2 e þ Dðln R2 ln R1 Þ l R2 l R1 R1 R1 ln a 1 dKp ln R1 a a R2 lðR2 R1 Þ=R1 R2 n l l 1 ¼ þ 1þ e ð p sin j c cos jÞ 1 2n R1 R1 R1 R2 lðR2 R1 Þ=R1 R2 l e 1þ ð p sin j cos jÞ þ D ð5:107Þ R1 2R1

5.5.5

Stress state with lining

Assume that circular lining of elastic media is applied soon after the excavation, i.e., t ¼ 0, the stress state in this case can be expressed as [8,52]: 9 a20 > sr ðtÞ ¼ cðtÞ bðtÞ 2 > > r = > a2 > > sy ðtÞ ¼ cðtÞ þ bðtÞ 20 ; r

ð5:108Þ

131

Stability Analysis of Rheologic Rock Mass

If the inner surface of the lining is free from loading, i.e., sr ¼ 0 when r ¼ a0, then, cðtÞ ¼ bðtÞ So equation (5.108) becomes, 9 a20 > > sr ðtÞ ¼ cðtÞ 1 2 > r = a2 > > sy ðtÞ ¼ cðtÞ 1 þ 20 > ; r

ð5:109Þ

Again, sr and sy at the boundary (r ¼ a) of two adjacent zones are equal, and the values of stress diﬀerences (sy sr) are also equal at r ¼ R1. The following undetermined parameter or function can be determined when solving the simultaneous equations (5.102), (5.109), (5.91), (5.95) and (5.98), 2G 1 n1 l l c0 ¼ eðG=ZÞt el=R1 þ 1þ 1 Z R1 R1 R1 1 2n R2 l=R2 1 ð1=K3 T3 Þ ððG=ZÞð1=T3 ÞÞt Z ðG=ZÞt e ðe ðpsinj ccosjÞ 1Þ ðe 1Þ ðG=ZÞ ð1=T3 Þ G 2 R2 l 1þ þ elðR2 R1 Þ=R1 R2 R1 2R1

t=T3 ðG=ZÞt ðG=ZÞt R1 ðpsinj ccosjÞ ð1 ð1=K3 T3 ÞÞe 1 þ De ðe 1Þ dKp ð5:110Þ 9 > > > =

a2 0 ðc a þ dKp Þ 2a20 2 2 > a 3 a 1 > 0 > jðtÞ ¼ þ dK ac ln a ; p 2 2 2a0 2 2a0 2 cðtÞ ¼

ð5:111Þ

The term of f(t) in the stress component in the visco-plastic zone can be determined using the condition of equal sr at r ¼ R1 of the two adjacent zones, R2 G 4n 2 3n 1 ðpsinj ccosjÞ þ f ðtÞ ¼ c R1 þ dKp lnR1 þ jðtÞ þ elðR2 R1 Þ=R1 R2 Zð1 2nÞ l R1

Z ðG=ZÞt 1 ð1=K3 T3 Þ ððG=ZÞð1=T3 ÞÞt R2 ðG=ZÞt ðe 1 ðe 1Þe elðR2 R1 Þ=R1 R2 ðG=ZÞ ð1=T3 Þ G 2 2 1 1 1 et=T3 1 ðpsinj ccosjÞ þ ð5:112Þ l R1 K3 T3 0

132

Chapter 5

The stress components in the visco-plastic zone, residual strength zone and the lining can be obtained from the function or parameter determined using equations (5.110), (5.111) and (5.112). The locations of R1 and R2 are then determined according to the boundary condition that the stress or displacement components are equal at r ¼ R2 of two adjacent zones. For simpliﬁcation, only the stress component in the lining for r!1 is given by the equation below, 2 3 sr a a 1 R2 n l l lðR2 R1 Þ=R1 R2 1 ðpsinj ccosjÞe ¼ 20 þ 1 þ 1 2n R1 R1 r 2a20 R1 R1 sy 2 R2 l a þ D dKp þ 2 dKp ð5:113Þ elðR2 R1 Þ=R1 R2 ðpsinj ccosjÞ 1þ R1 2 2a0

5.6.

EFFECT OF BOLT REINFORCEMENT IN VISCO-ELASTIC ROCK MASS

Bolt and shotcrete support is to reinforce surrounding rock masses, and to control deformation. The technique eﬀectively controls the rock deformation, block loosening and cave-in. It reinforces the complete rock mass system by adjusting the stress distribution in the rock mass, and by mobilising the self-supporting ability of the surrounding rock mass [66,194–196,330–339]. The mechanism of bolting technique has not yet been thoroughly understood, although the technology has been well developed and advanced in recent years [180,218,340–381]. The purpose of this section is to examine the interaction between the rock mass and the bolts, to study the use of bolt reinforcement and its eﬀect on the stress state of the rock mass.

5.6.1

Stress state in different zones

Similar to the previous cases, the rock mass surrounding an opening in far region is regarded as a visco-elastic medium, whereas the rock mass in the bolted region is considered to be another visco-elastic medium having diﬀerent parameters, as shown in Figure 5.11. The assumptions on the continuum mechanics adopted in the previous sections are still used, and the initial stress state is assumed to be uniform and the surrounding rock mass is assumed to be isotropic and homogeneous. The visco-elastic stress state in the far region is the same as in equation (5.54). Assume that bolts are applied when t ¼ t1, the bolts are (Ra) in length and create a bolted circular ring. The parameters of T, K, G in the bolted region have basically

Stability Analysis of Rheologic Rock Mass

133

Figure 5.11. Schematic representation of surrounding rock mass zones.

the same value as those in the visco-elastic zone, the bolts only restrain the displacement of the bolted rock mass. According to the principle of superposition, the components of stress and displacements in the bolted region should meet the following conditions: sð2Þ r

¼

K3 T3 ½c2 ðtÞ b2 ðtÞS 22 þ K3 ð1 K3 T3 Þ

Z

t

t1

0 ½c2 ðt0 Þ b2 ðt0 ÞS 22 eK3 ðt tÞ dt0 P 1 S 22 ð5:114Þ

2 sð2Þ y ¼ K3 T3 ½c2 ðtÞ þ b2 ðtÞS 2 þ K3 ð1 K3 T3 Þ

U ð2Þ r

Z

t

t1

0 ½c2 ðt0 Þ þ b2 ðt0 ÞS 22 eK3 ðt tÞ dt0 P 1 þ S 22

ð5:115Þ 1 1 1 ¼ r½ð1 2nÞc2 ðtÞ þ b2 ðtÞS 22 þ PrS 22 1 ð5:116Þ et=T3 1 2G3 2G3 K3 T3

where S2 ¼ a/r and S3 ¼ R/r. It should also satisfy the boundary condition below. (i)

When r ! 1, because U ð3Þ r ¼ 0, we have, from equation (5.54) c3 ðtÞ ¼ 0

(ii)

ð5:117Þ

When r ¼ a, the stress condition is ð2Þ ð2Þ sð2Þ r ¼ aU þ s0 ¼ aU r¼a U r¼R þ s0

ð5:118Þ

134

Chapter 5

where a is a constant related with the length, cross-sectional area of rock bolt, and bolting density and mechanical property of the bolts, i.e.,

a¼

E1 B1 ðR aÞB2

ð5:119Þ

In the above equation, E1 is the elastic modulus of the bolts; B1 is the sum of the cross-sectional areas of all bolts in unit length of the opening; B2 is the total area of the opening face in unit length; s0 is the equivalent normal stress converted from the ð2Þ axial stress in the bolt; jU ð2Þ r¼a U r¼R j is the displacement diﬀerence in the bolted circular ring after bolts installed at t t1. This boundary condition reﬂects the action of the bolt, i.e., if relative displacements take place in the bolted region after bolting, they will be restrained by bolts. In addition, the bolt’s bearing force in the surrounding rock mass is proportional to the displacement. Substituting equation (5.116) into equation (5.118), we have sð2Þ r¼a ¼

a ½ð1 2nÞða RÞc2 ðtÞ þ að1 X2 Þb2 ðtÞ 2G3 1 þ Pa 1 ð1 X2 Þðet=T3 et1 =T3 Þ þ s0 K3 T3

ð5:120Þ

where X2 ¼ a/R Substituting r ¼ a into equation (5.114) gives

sð2Þ r¼a ¼ K3 T3 ½c2 ðtÞ b2 ðtÞ þ K3 ð1 K3 T3 Þ

Z

t

0

½c2 ðt0 Þ b2 ðt0 ÞeK3 ðt tÞ dt0 t1

From equations (5.120) and (5.121), we have Z

t

0

½c2 ðt0 Þ b2 ðt0 ÞeK3 ðt tÞ dt0

K3 T3 ½c2 ðtÞ b2 ðtÞ þ K3 ð1 K3 T3 Þ t1

a ¼ ½ð1 2nÞ ða RÞc2 ðtÞ þ að1 X2 Þb2 ðtÞ 2G3 1 þ Pa 1 ð1 X2 Þðet=T3 et1 =T3 Þ þ s0 K3 T3

ð5:121Þ

Stability Analysis of Rheologic Rock Mass

135

Multiplying the two sides of the above equation by eK3 t , deriving the result with respect to t and eliminating the term of eK3 t , we have

a a _ ð1 2nÞða RÞ K3 T3 c_2 ðtÞ þ ð1 X2 Þ þ K3 T3 Þb2 ðtÞ 2G3 2G3 a a ð1 2nÞða RÞ 1 c2 ðtÞ þ K3 að1 X2 Þ þ 1 b2 ðtÞ þ K3 2G3 2G3 aPa 1 1 t=T3 t1 =T3 ð1 X2 Þ K3 e e K3 s0 ¼ 1 K3 2G3 K3 T3 T3

(iii)

ð5:122Þ

When r ¼ R, the displacement condition is ð3Þ U ð2Þ r ¼ Ur

According to equations (5.54), (5.116) and (5.117), we have b3 ðtÞ ¼ ð1 2nÞc2 ðtÞ þ X22 b2 ðtÞ (iv)

ð5:123Þ

When r ¼ R, the stress condition is ð2Þ ð2Þ sð3Þ r¼a ¼ sr¼R X2 sr¼R

The last term in this equation reﬂects the action of the bolts. Substituting equations (5.54), (5.114) and (5.117) into the above equation and interpreting the results, we have K3 T3 ½ðX2 1Þc_2 ðtÞ þ X2 ðX2 1Þb_2 ðtÞ b_3 ðtÞ þ K3 T3 ½ðX2 1Þc2 ðtÞ þ X2 ðX2 1Þb2 ðtÞ b3 ðtÞ ¼ 0 By solving this diﬀerential equation and substituting the result into equation (5.123), we have

2Rðn 1Þ b2 ðtÞ ¼ 1 þ c2 ðtÞ þ C 1 et=T3 a where C1 is an undetermined constant.

ð5:124Þ

136

Chapter 5

Substituting equation (5.124) back into equation (5.122) gives

a 2R K3 T3 ðn 1Þ c_2 ðtÞ ð1 X2 Þða þ 4nR 3RÞ þ 2G3 a a 2R ðn 1Þ c2 ðtÞ þ K3 ð1 X2 Þ ða þ 4nR 3RÞ þ 2G3 a aPa 1 1 t=T3 1 ð1 X2 Þ K3 et=T3 K3 e ¼ 2G3 K3 T3 T3 aa 1 K3 s0 þ ð1 X2 Þ K3 C 1 et=T3 2G3 T3

then solving this equation gives a PaðK3 T3 1Þ C 1 et=T3 et=T3 3R a 4nR K3 T3 ða þ 4nR 3RÞ ð1 X2 ÞaPK3 a2 1 2K3 G3 a et1 =T3 þ 1 s0 K3 T3 N2 N2

c2 ðtÞ ¼ C 2 eðN2 =N1 Þt þ

ð5:125Þ

where N1 ¼ aað1 X2 Þða þ 4nR 3RÞ þ 4G3 K3 T3 ðn 1ÞR

) ð5:126Þ

N2 ¼ K3 aað1 X2 Þða þ 4nR 3RÞ þ 4G3 RK3 ðn 1Þ and C1, C2 are undetermined constants. For equation (5.114), when t ¼ t1, r ¼ a and r ¼ R, we have respectively c2 ðt1 Þ b2 ðt1 Þ ¼ 0

and

c2 ðt1 Þ X22 b2 ðt1 Þ ¼ 0

From this relation, we have c2 ðt1 Þ ¼ b2 ðt1 Þ ¼ 0 If we substitute this result back into equations (5.124) and (5.125), then the undetermined constants of C1 and C2 can be obtained: C 1 ¼ 0 PaðK3 T3 1Þ aPK3 a2 ð1 X2 Þ 1 C2 ¼ 1 K3 T3 ða þ 4nR 3RÞ N2 K3 T3 2K G as 3 3 0 ðN2 =N1 Þt1 e eððN2 =N1 Þð1=T3 ÞÞt þ N2

9 > > > > = > > > > ;

ð5:127Þ

Stability Analysis of Rheologic Rock Mass

137

Substituting equation (5.127) back into equations (5.124) and (5.125) gives PaðK3 T3 1Þ aPK3 a2 ð1 X2 Þ 1 et=T3 et=T3 þ 1 c2 ðtÞ ¼ C 2 eðN2 =N1 Þt K3 T3 ða þ 4nR 3RÞ N2 K3 T3 2K3 G3 a s0 ð5:128Þ N2 2Rðn 1Þ PðK3 T3 1Þða þ 2nR 2RÞ t=T3 e b2 ðtÞ ¼ 1 þ C 2 eðN2 =N1 Þt a K3 T3 ða þ nR 3RÞ aPK3 að1 X2 Þða þ 2nR 2RÞ 1 2K3 G3 ða þ 2nR 2RÞ þ 1 s0 et1 =T3 N2 K3 T3 N2 ð5:129Þ Furthermore, by substituting equations (5.128) and (5.129) back into equations (5.114), (5.115) and (5.116) and integrating the results, we have the expressions for the stress and strain components in the bolted region: N1 T3 N2 2S 22 Rðn 1Þ N1 2 C 2 eðN2 =N1 Þt sð2Þ ¼ K 1 S K3 ð1 K3 T3 Þ 3 r 2 a K3 N1 N2 K3 N1 N2 2S 22 Rðn 1Þ N2 PðK3 T3 1Þ 1 S 22 t1 K 3 t c2 exp K3 a a þ 4nR 3R N1 1 aPK 3 að1 X2 Þ ½a S 22 ða þ 2nR 2RÞ exp K3 t1 K 3 t þ T3 N2 1 aPK að1 X 3 2 Þð1 K3 T3 Þ ½a S 22 ða þ 2nR 2RÞ 1 et1 =T3 K3 T3 N2 1 1 2K3 G3 s0 2 exp K3 t þ K3 t1 ½a S 2 ða þ 2nR 2RÞ 1 K3 T3 T3 N2 2 2 ½a S 2 ða þ 2nR 2RÞ½1 ð1 K3 T3 Þ exp½K3 ðt1 tÞ P 1 S 2 ð5:130Þ N1 T3 N2 2S 22 Rðn 1Þ N1 ð2Þ 2 K3 1 þ S 2 þ K3 ð1 K3 T3 Þ sy ¼ C 2 eðN2 =N1 Þt a K3 N1 N2 K3 N1 N2 2S 22 Rðn 1Þ N2 PðK3 T3 1Þ 1 þ S 22 þ C 2 exp K3 t1 K 3 t a a þ 4nR 3R N1 1 aPK3 að1 X2 Þ 2 t1 K 3 t þ ½a þ S 2 ða þ 2nR 2RÞ exp K3 T3 N2 1 aPK3 að1 X2 Þð1 K3 T3 Þ ½a þ S 22 ða þ 2nR 2RÞ 1 et1 =T3 K3 T3 N2 1 1 2K3 G3 s0 ½a þ S 22 ða þ 2nR 2RÞ 1 exp K3 t1 K3 t K3 T3 T3 N2 ½a þ S 22 ða þ 2nR 2RÞf1 ð1 K3 T3 Þ exp½K3 ðt1 tÞg Pð1 þ S 22 Þ ð5:131Þ

138

Chapter 5 2S 22 Rðn 1Þ PðK3 T3 1Þ C 2 eðN2 =N1 Þt 1 2n þ S 22 þ a K3 T3 ða þ 4nR 3RÞ aPK að1 X2 Þ 3 ½að1 2nÞ þ S 22 ða þ 2nR 2RÞet=T3 þ N2 1 2K3 G3 s0 2 t1 =T3 ½að1 2nÞ þ S 2 ða þ 2nR 2RÞ 1 e K3 T3 N2 1 1 ½að1 2nÞ þ S 22 ða þ 2nR 2RÞ þ PrS 22 1 et=T3 1 2G3 K3 T3

U ð2Þ r ¼

r 2G3

ð5:132Þ By substituting equations (5.128), (5.129) back into equation (5.123), we can obtain the explicit formula of b3(t). By further substituting b3(t) and equation (5.117) back into equation (5.54) and integrating the results, we have the expression for the stress and strain components in the visco-elastic zone as following: N1 T3 N2 sð3Þ K3 S 23 r ¼ K3 N1 N2

2X 2 Rðn1Þ 12nþX22 þ 2 a

C 2 eðN2 =N1 Þt

N1 2X 2 Rðn1Þ C2 K3 S 23 ð1K3 T3 Þ 12nþX22 þ 2 a N1 K3 N2 N2 PðK3 T3 1Þ 2 S ½að12nÞþX22 ðaþ2nR2RÞ t1 K3 t þ exp K3 aþ4nR3R 3 N1 1 aPK3 að1X2 Þ 2 exp K3 t1 K3 t S 3 ½að12nÞþX22 ðaþ2nR2RÞ T3 N2 1 aPK3 að1X2 Þ 2 S 3 ½að12nÞþX22 ðaþ2nR2RÞ 1 et1 =T3 þ K3 T3 N2 1 1 2K3 G3 s0 2 S 3 ½að12nÞþX22 ðaþ2nR2RÞ 1 exp K3 t1 K3 t þ K3 T3 T3 N2 f1ð1K3 T3 Þexp½K3 ðt1 tÞgP 1S 22 ð5:133Þ þ

2 2 2X2 Rðn1Þ 12nþX2 þ C 2 eðN2 =N1 Þt a 2 N1 2 2 2X2 Rðn1Þ K3 S 3 ð1K3 T3 Þ 12nþX2 þ C2 a K3 N1 N2 N2 PðK3 T3 1Þ 2 S ½að12nÞþX2 ðaþ2nR2RÞ t1 K3 t exp K3 aþ4nR3R 3 N1 1 aPK3 að1X2 Þ 2 t1 K3 t þ S 3 ½að12nÞþX22 ðaþ2nR2RÞ exp K3 T3 N2

N1 T3 N2 K3 S 23 sð3Þ y ¼ K3 N1 N2

Stability Analysis of Rheologic Rock Mass 139 1 aPK3 að1 X2 Þ 2 et1 =T3 1 S 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ K3 T3 N2 1 1 exp K3 t1 K 3 t 1 K3 T3 T3 2K3 G3 s0 2 S 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ N2 i h 1 ð1 K3 T3 ÞeK3 ðt1 tÞ Pð1 þ S 22 Þ ð5:134Þ 2X 2 Rðn 1Þ PðK3 T3 1Þ 1 2n þ X22 þ 2 C 2 eðN2 =N1 Þt a K3 T3 ða þ 4nR 3RÞ aPK að1 X2 Þ 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞet=T3 þ N2 1 2K3 G3 s0 2 t1 =T3 e ½að1 2nÞ þ X2 ða þ 2nR 2RÞ 1 K3 T3 N2 1 1 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ þ PrS 22 1 et=T3 1 2G3 K3 T3

U ð3Þ r ¼

rS 23 2G3

ð5:135Þ By allowing t ! 1, the expression of stress and strain components can be obtained for the diﬀerent zones. (i)

In the bolted region

sð2Þ r

sð2Þ y

aPK3 að1 X2 Þ 1 2 et1 =T3 ¼ ½a S 2 ða þ 2nR 2RÞ 1 N2 K3 T3 2K3 G3 s0 ½a S 22 ða þ 2nR 2RÞ P 1 S 22 N2

aPK3 að1 X2 Þ 1 2 ¼ ½a þ S 2 ða þ 2nR 2RÞ 1 et1 =T3 N2 K3 T3 2K3 G3 s0 ½a þ S 22 ða þ 2nR 2RÞ Pð1 þ S 22 Þ N2

aPK3 að1X2 Þ U ð2Þ r½að12nÞþS 22 ðaþ2nR2RÞ r ¼ 2G3 N2

ð5:136Þ

1 1 et1 =T3 K3 T3

K3 s0 1 r½að12nÞþS 22 ðaþ2nR2RÞ PrS 22 2G3 N2

ð5:137Þ

140

Chapter 5

(ii)

In the visco-elastic zone

aPK3 að1 X2 Þ 2 1 2 ¼ S 3 ½að1 2nÞ þ X2 ða þ 2nR 2RÞ 1 et1 =T3 N2 K3 T3 2K3 G3 s0 2 þ S 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ P 1 S 22 N2 aPK3 að1 X2 Þ 2 1 2 ¼ S ½að1 2nÞ þ X ða þ 2nR 2RÞ 1 sð3Þ et1 =T3 3 2 y N2 K3 T3 2K3 G3 s0 2 S 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ Pð1 þ S 22 Þ ð5:138Þ N2 aPK3 að1 X2 Þ 2 1 2 U ð3Þ et1 =T3 ¼ rS ½að1 2nÞ þ X ða þ 2nR 2RÞ 1 r 3 2 2G3 N2 K3 T3 K3 s0 2 1 rS 3 ½að1 2nÞ þ X22 ða þ 2nR 2RÞ PrS 22 ð5:139Þ 2G3 N2 sð3Þ r

It can be seen clearly from equation (5.139) that the expression of stress or displacement comprises three terms, each having distinct physical meaning. The ﬁrst term is the eﬀect of the time (t1) of bolt installation on the surrounding rock mass. It indicates that the earlier the installation the better to improve the stress state in the rock mass, i.e., reducing t1 as fully as possible. The second term reﬂects the eﬀect of prestress of the bolts (s0). The higher s0 gives the more eﬀective improvement of the rock mass. The third term reﬂects the eﬀect of the initial in situ stress, P.

5.6.2

Discussion and application

Figure 5.12 shows stress distribution in the surrounding rock mass reinforced with bolts. According to equations (5.126) and (5.119), we have N2 ¼ K3 aað1 X2 Þða þ 4nR 3RÞ þ 4G3 RK3 ðn 1Þ

E1 B1 a K3 a 3 4n 4G3 RK3 ð1 nÞ ¼ R B2 Because B2 B1 , 4G3 RK3 ð1 nÞ >> ðE1 B1 =B2 ÞK3 að3 4n ða=RÞÞ: The above equation can be approximately reduced to N2 ¼ 4G3 RK3 ð1 nÞ

ð5:140Þ

141

Stability Analysis of Rheologic Rock Mass

Figure 5.12. Stress distribution in the surrounding rock mass after bolting.

Substituting equations (5.119) and (5.140) into equations (5.136) and (5.137) gives the approximate solution to the stress and strain in the bolted zones:

E1 B1 P a ha a i 1 þ S 22 2 2n 1 et1 =T3 4B2 G3 ð1 nÞ R R R K3 T3

s0 h a a i þ S 22 2 2n þ P 1 S 22 R 2ð1 nÞ R

sð2Þ r ¼

sð2Þ y

E1 B1 P a h 2 a ai 1 S 2 2n et1 =T3 1 ¼ 4B2 G3 ð1 nÞ R 2 R R K3 T3

s0 h a a i S 22 2 2n þ P 1 S 22 R 2ð1 nÞ R

ð5:141Þ

ð5:142Þ

r E1 B1 P a ha a i 1 ð1 2nÞ S 22 2 2n et1 =T3 1 2G3 4B2 G3 ð1 nÞ R R R K3 T3

s0 h a a i ð1 2nÞ S 22 2 2n þ PS 22 ð5:143Þ R 2ð1 nÞ R

U ð2Þ r ¼

From the plastic condition of equation (5.51) given in Section 5.3, we have 2 sy sr ¼ pﬃﬃﬃ dKp 6

ð5:144Þ

142

Chapter 5

Considering equations (5.141) and (5.142), we have sð2Þ y

sð2Þ r

E1 B1 PS 22 a a 1 2 2n 1 et1 =T3 ¼ R K3 T3 2B2 G3 ð1 nÞ R

s0 a S 22 2 2n 2PS 22 þ R ð1 nÞ

ð5:145Þ

Therefore, in order to prevent the opening wall (S2 ¼ 1) from becoming plastic state after an inﬁnite long time, we may apply the following criterion according to equations (5.144) and (5.145): E1 B1 P a a 1 s0 a 2 2 2n 2 2n et1 =T3 þ 1 2P pﬃﬃﬃ dKp 2B2 G3 ð1 nÞ R R K3 T3 R ð1 nÞ 6 ð5:146Þ To meet the condition in equation (5.146), it is necessary to enlarge the ﬁrst and the second terms. To enlarge the second term, however, it needs to increase both prestress and length of the bolt, i.e., s0 and R. However, there is a limit to increase the prestress of bolts, because a bolt of a certain cross-sectional area can bear only a limited prestress corresponding to the tensile strength. Therefore, eﬀorts should be made to increase the ﬁrst term as far as possible. There are several ways to increase this term, for instance, install bolts as early as possible (decreasing t1), or enlarge the bolt cross-section area (i.e. increase B1) and increase the bolting density (i.e. decrease B2), or change the bolt length. The ﬁrst two measures are more eﬀective. When the prestress (s0) is not taken into consideration or the bolt is not prestressed, equation (5.145) becomes sð2Þ y

sð2Þ r

E1 B1 P a a 1 2 2n 1 et1 =T3 2P ¼ 2B2 G3 ð1 nÞ R R K3 T3

ð5:147Þ

This equation takes the status of sidewalls of the opening into account, i.e., S2 ¼ 1. ð2Þ The relationship between sð2Þ y sr and a=R is expressed in diagram shown in Figure 5.13. Given that E1 ¼ 2.1 105 MPa, B1 ¼ 1 cm2, B2 ¼ 50 cm2, G3 ¼ 4 103 MPa, ¼ 0.25, T3 ¼ 12d, K2 ¼ 1/8d, t1 ¼ 0, then from the ﬁgure, D1 ¼ (0.01 2)P, D2 ¼ (0.006 2)P, D3 ¼ (0.004 2)P. ð2Þ It can be seen from equation (5.147) or from Figure 5.13 that sð2Þ y sr has the minimum value when a=R ¼ 1 n. The above analysis leads to the conclusion that in order to avoid the surrounding rock mass damage due to stress concentration or to make the stress diﬀerence in

Stability Analysis of Rheologic Rock Mass

143

Figure 5.13. Eﬀect of relative bolt length on stress diﬀerence of surrounding rock mass.

opening walls reach the lowest level: (a) the bolts must be installed as early as possible, (b) the prestress in the bolts must be suﬃciently large, (c) the bolts must have suﬃcient length and cross-sectional area, (d) bolting should be installed with suﬃcient density. The optimal bolt length is n=ð1 nÞ times opening radius, i.e., a=R ¼ 1 n, when non-prestressed bolts are used. If bolts are installed late, the ﬁrst term in equation (5.145) becomes basically non-functional. The second term in the equation shows that the higher R, the better, i.e., long bolts should be selected, up to the optimal length. If failure is determined by the magnitude of displacements, the conditions of which the displacement of the opening wall becomes in minimum should be examined. From equation (5.143) and substituting S2 ¼ 1, r ¼ a into equation (5.143), we can obtain U

ð2Þ r¼a

a E1 B1 P a a 1 a t1 =T3 1 1 P e ¼ þ ðs0 Þ 1 2G3 2B2 G3 R R K3 T3 R

If the prestress s0 ¼ 0, then the above equation becomes, U ð2Þ r¼a ¼

a E1 B1 P a a 1 1 1 et1 =T3 P 2G3 2B2 G3 R R K3 T3

Under the conditions given in this section, ½Dn is proven to be symmetrical and to have the expression similar to that of matrix [D]. In the case of ! ¼ 0, ½Dn becomes

144

Chapter 5

Figure 5.14. Eﬀect of relative bolt length on displacement of surrounding rock mass.

an elastic matrix and for ! ¼ constant with no damage development occurring, the above proposed algorithm constitutes the constant stiﬀness algorithm. In Figure 5.14, D4 ¼ ð0:0025 1Þ ðaP=2G3 Þ and D5 ¼ ð0:0018 1Þ ðaP=2G3 Þ. Figure 5.14 indicates that U ð2Þ r reaches the minimum value when a=R ¼ 1=2, i.e., when bolt length equals to the opening radius, opening wall has minimum displacement. Similarly, in order to minimise the displacement of opening walls, bolts should be installed as early as possible and the prestress of bolts should be increased, in addition to enlarge bolt cross-sectional area and to increase bolting density. In the case of non-prestressed bolts, the optimum length of bolt should have the ratio of a=R ¼ 1=2. If bolts are installed late, the same conclusion can be drawn from equation (5.147) that long bolts should be used. In summary, the optimum time for installing bolts and the magnitude of prestress applied in the bolts can be determined quantitatively using the method presented in this section. A clear conclusion can be drawn that the most rational bolt length (R a) for visco-elastic surrounding rock mass should be a=ð1 nÞ, where a is the opening radius, R is the radius of the visco-elastic zone, n is the Possion’s ratio of the rock mass.

5.7.

RHEOLOGICAL DAMAGE ANALYSIS OF THE ROCK MASS STABILITY

Previous sections have studied the problems of the stress state in surrounding rock masses and the interaction of the rock mass and support using various rheological models. The present section combines damage mechanics and rheological mechanics

Stability Analysis of Rheologic Rock Mass

145

to study the stability of rock mass. When excavation is in soft rock or at great depth, the rock mass deforms. A great portion of the deformation is caused by the loosening due to relatively high stresses as compared to the strength of rock around the opening. It produces cracks in the surrounding rock mass, and results in remarkable dilatancy phenomena reﬂected by obvious volume increment in the rock mass [39,206,382–389]. As a consequence, the conventional analysis methods are not adequate to study the rock mass exhibiting the above behaviour. The concept and method of damage mechanics combining with rheological mechanics are applied to analyse the rock mass in this section. 5.7.1

Damage evolution equation

The basic concepts and assumptions of damage mechanics have been discussed in Chapter 3. This section adopts the damage equation by Desai [291,292,382] based upon the isotropic damage concept to express the eﬀective stress and apparent stress, i.e., sij ¼ ð1 !Þsij þ

! dij skk 3

ð5:148Þ

where ! is the damage variable. Considering that the rupture failure planes have residual strength, a speciﬁc maximum damage value !p less than 1 of the rock mass can be assumed. The value of !p depends upon the type of the rock. Most rocks have brittle characteristics. Under loading, microcracks initiate and propagate and a failure plane forms. The failure can be caused by tensile, shear or combined tensile and shear. The rock dilates before the ﬁnal failure as cracks are growing in number and extending gradually. The dilatancy phenomenon is the most important warning sign of rock failure. There are three main types of damage evolutions for brittle material such as rocks: (i) (ii) (iii)

Evolution and development of damages are governed mainly by strains, especially by the tensile strain. Damage evolution and development are governed mainly by stresses, especially by the deviator stress or the major principal stress. Such evolution is governed by energy or relief rate of energy.

Tan [39] studied the mechanism and criterion of rock dilatancy phenomenon and suggested the condition of dilatancy generation, soct k2 1 f3

ð5:149Þ

146

Chapter 5

where f3 ¼ k mp is the strength envelope curve (or surface) for a critical surface, which is roughly corresponding to the long-term envelope (surface); p is the mean spherical stress; k and m are the strength index; k2 is a real number greater than unity that is determined from test, and soct is the second variant of deviator stress tensor. The damage evolution equation adopted in the present section that follows the assumption of isotropic damage expressed by equation (5.149) is, soct ! ¼ K1 ð1 !Þ F 1 f3

ð5:150Þ

and, 8 soct k2 > > < 1 soct f3 ¼ F > f3 > :0

soct >1 f3 soct when 1 f3

when

in which tensile stress is taken as positive. The above damage evolution equation is associated to both increments in number and extension of the cracks and the dilatancy of the material.

5.7.2

Viscoelastic–viscoplastic-damage constitutive equation and FEM method

5.7.2.1 Constitutive equation. The common constitutive equation of non-damaged viscous materials has the following increment expression, fdsg ¼ ½Dðfdeg fdeV gÞ

ð5:151Þ

where [D] is the elastic matrix; fdsg,fdeg and fdeV g are the apparent stress increment, overall strain increment and viscous strain increment in the form of column matrix respectively. Based on the concept of equivalent strain, i.e., the eﬀect of damage on the strain behaviour is reﬂected only by the equivalent stress. This means that the constitutive relation of a damaged material can use equation (5.151) by replacing the stress term with the equivalent stress. Hence, we have the following constitutive equation for viscoelastic–viscoplastic-damage rock masses: fds g ¼ ½Dðfdeg fdeV gÞ

ð5:152Þ

Stability Analysis of Rheologic Rock Mass

147

Figure 5.15. A rheological model (ss is the plastic limit).

A typical rheological model with visco-elastic and visco-plastic characteristic is shown in Figure 5.15. In the case of one-dimensional problem, the constitutive equation of the model can be divided into two stages according to the stress level, sþ

Z2 E1 E2 E1 Z2 e_ s_ ¼ eþ E1 þ E2 E1 þ E2 E1 þ E2

for s ss

ð5:153Þ

and s ss þ

Z3 Z2 þ Z3 Z Z Z Z s_ þ 2 3 s€ ¼ Z2 e_ 2 3 e€ þ E1 E2 E1 E2 E2

for s > ss

ð5:154Þ

in the case of creep, the equation becomes e¼

s s þ 1 eðE2 =Z2 Þt E1 E2

for s ss

ð5:155Þ

and e¼

s ss s s þ 1 eðE2 =Z2 Þt þ t for s > ss E1 E2 Z3

ð5:156Þ

and in the case of creep damage, s s þ 1 eðE2 =Z2 Þt for s ss E1 E2

ð5:157Þ

s ss s s þ 1 eðE2 =Z2 Þt þ t for s > ss E1 E2 Z3

ð5:158Þ

e¼ and e¼

148

Chapter 5

5.7.2.2 FEM method. According to the virtual work principle, we have the following equilibrium equation, Z

½BT fsgn dV ¼ ff gn

ð5:159Þ

V

where [B] is the strain matrix; fsgn , ff gn are the stress increment and the exterior R R loading with the time interval of tn; ff gn ¼ S t ½NT ftgn dS þ v ½NT fpgn dV; [N] is the matrix of shape functions; {t}n is the surface force exerted on the boundary of St within tn; and {p}n is the body force within tn. Integration on equation (5.148) gives 1 dsij ¼ ð1 !Þdsij di !j dskk S ij d! 3

ð5:160Þ

where S ij is the eﬀective deviator stress. Equation (5.160) can be written in the matrix form, fdsg ¼ ½C ! fds g d!fS g

ð5:161Þ

where [C!] is the matrix related to the damage variable !, called damage matrix, for a three-dimensional problem, it is expressed as 1!þ! 3 ½C ! ¼

! 3

! 1!þ 3 symmetry

! 3 ! 3

! 1!þ 3

0

0

0

0

0

0

0

0

0

ð1 !Þ

0 ð1 !Þ

0

ð1 !Þ ð5:162Þ

Writing the diﬀerential expression of equation (5.186) in the increment form gives fsgn ¼ ½C ! n fs gn !n fS gn

ð5:163Þ

Equation (5.163) shows that within the time interval of tn, the apparent stress increment is correlated to the eﬀective stress state, the damage state at the current moment, as well as the increment of damage and the increment of eﬀective stress within tn.

Stability Analysis of Rheologic Rock Mass

149

Within tn, from equation (5.152), we have fs gn ¼ ½Dðfegn feV gn Þ

ð5:164Þ

Substituting equations (5.163) and (5.164) into the equilibrium equation (5.159), we have the equilibrium equation that describes the damage eﬀect, Z

Z

½BT ½C ! n ½D½BdVfugn ¼ ff gn þ V

!n ½BT fS gn dVþ V

Z

½BT ½C ! n ½Df!V gn dV V

ð5:165Þ By letting ½D n ¼ ½C ! n ½D

ð5:166Þ

fS ! gn ¼ !n fS gn

ð5:167Þ

we can rewrite equation (5.165) as ½kfugn ¼ ff gn þ fF gn þ fF 0 gn

ð5:168Þ

where Z

½BT ½Dn ½B dV

½k ¼

ð5:169Þ

V

is the stiﬀness matrix. Z

fF gn ¼ V

½BT fS ! gn dV

ð5:170Þ

is the additional force caused by damage evolution. The following equation fF 0 gn ¼

Z

½BT ½Dn feV gn dV

ð5:171Þ

V

is the additional force caused by the viscous stress increment. Under the conditions given in this section, ½Dn is proven to be symmetrical and to have the expression similar to that of matrix [D]. In the case of ! ¼ 0, ½Dn becomes

150

Chapter 5

an elastic matrix and for ! ¼ constant with no damage development occurring, the above proposed algorithm constitutes the constant stiﬀness algorithm. 5.7.3

Application to stability analysis of an underground opening

Analysis of underground excavation project using damaged rheological model is conducted and compared with non-damage analysis. The underground project model described is a metal mine located in a very complicated geological environment where the old metamorphic rock mass has experienced many times of tectonic movement. In addition to the faults of various sizes, the joints and fractures are densely distributed in the mining area. The surrounding rock mass displays remarkable rheological characteristics. In order to study the surrounding rock mass stability and to rationalise the supporting scheme, a special testing gallery has been excavated, with a variety of monitoring instrumentation. Large-size in situ triaxial rheological tests have been conducted in the gallery to study the rheological behaviour. The gallery has overlying depth of 500 m and a span of 3 m with arch roof. The testing gallery has been supported using initial shotcrete-bolting that follow the advance of working face closely and secondary shotcrete-steel net supporting. The grouted bolts are 1.5–1.8 m in length and the total thickness of the shotcreting layer is about 25 cm. Two sections for multi-point borehole extensometer were installed in the testing segment, each measuring 10 m in depth with six monitoring points. The observed deformation curve shows obvious 3-stage rheological deformability. Owing to the uniformly cracked or fractured structure of the surrounding rock mass, it can be considered approximately as a quasi-continuous and homogeneous rheological medium. The rock mass stability around the opening is studied using the viscoelastic– viscoplastic-damage method discussed in Section 5.7.2, incorporated in a FEM program. The results from the analysis are compared with the in situ measurements to check the eﬀectiveness of the model and the method. To make comparison between the methods, calculations have also been carried out using the rheological model with no damage evolution. 5.7.3.1 Decomposition of the rheological deformation. According to the model (Figure 5.15), the overall deformation comprises three parts. (a) Transient elastic deformation

It can be obtained according to Hooke’s law

fee g ¼

1 ½Afsg E1

ð5:172Þ

Stability Analysis of Rheologic Rock Mass

151

where E1 is the transient elastic module; [A] is the constant matrix related to Poisson’s ratio. In the case of plane strain, 2

1 n2

½A ¼ 4

nð1 þ nÞ 1 n2

symmetry

(b) Visco-elastic deformation. time interval tn,

feve gn ¼

3 0 5 0 2ð1 þ nÞ

ð5:173Þ

It is calculated from the following equation, for a

1 ½Afsgn ð1 eðE2 =Z2 Þtn Þ ð1 eðE2 =Z2 Þtn Þfeve gn E2

ð5:174Þ

where E2 is the delay elastic modulus; Z2 is the visco-elastic coeﬃcient; [A] is the constant matrix related to Poisson’s ratio. (c) Visco-plastic deformation. When the material is proven having entered the yield state according to the criterion of viscoplastic yield, this part of deformation can be calculated by the following methods.

(i) Visco-plastic yield criterion. It adopts Mohr–Coulomb yield criterion, whose expression under a complex stress state is 1 1 F ¼ sin jI1 þ cos y pﬃﬃﬃ sin y sin j ðJ2 Þ1=2 C cos j ¼ 0 3 3

ð5:175Þ

where I1 ¼ sij , is the ﬁrst variant of stress tensor; J2 ¼ 12 sij sij , is the second variant of deviator stress tensor; y is Lode parameter, pﬃﬃﬃ 3 3 J3 ; sin 3y ¼ 2 ðJ2 Þ3=2

1 J3 ¼ sij sjk s, 3

is the third variant of deviator stress tensor. (ii) Visco-plastic stress increment. It can be expressed by:

f_evp g ¼

@Q 1 fðF Þ Z3 @fsg

ð5:176Þ

152

Chapter 5

where Z3 is the visco-plastic ﬂowing coeﬃcient; and

fðFÞ ¼

fðF Þ 0

ðF > 0Þ ðF 0Þ

fðFÞ has the form of fðFÞ ¼ F/F0 where F0 is a reference value. The yield function value of F is dimensionless, and F0 ¼ C cos f. Q is the plastic potential function. Q 6¼ F stands for irrelevant ﬂowing, and Q ¼ F stands for relevant ﬂowing; the relevant ﬂowing law is used here. From Equation (5.20), the visco-plastic strain increment that occurs in the time interval of tn ¼ tnþ1 tn can be calculated using the following equation, fevp gn ¼ tn ½ð1 Þf_evp gn þ f_evp gnþ1

ð5:177Þ

When ¼ 0, we have Euler’s time integration method and the strain increment is determined by the strain rate at the current moment, tn; when ¼ 1, we have the complete implicit integration method and the strain increment is determined by the strain rate at the end of the time interval; when ¼ 1=2, The implicit trapezoid method can be adopted. The above equation is for strain increment of non-damaged material. For damaged material, the equation is similar, except that the eﬀective stress term is used instead of the apparent stress term, based on the eﬀective strain concept.

5.7.3.2 Determination of Model’s parameters. The Model’s parameters used for analysis are obtained through ﬁeld triaxial compressive rheological tests on the rock mass. Each parameter used in the damage evolution equation is obtained by ﬁtting the elastic module of specimens from multi-loading-stage creep tests, i.e., !¼1

E E0

ð5:178Þ

where E0 is the elastic modulus of the tested body with no damage evolution; the failure damage value is calculated using equation (5.178) from the elastic modulus. The values of cohesion (C) and friction angle (j) are determined from the stressvolumetric strain curve at maximum volumetric strain. At this stress level, the swelling deformation initiates. It can be seen from equation (5.150) that the criterion for damage development is to judge if dilatancy of the rock mass occurs. Therefore, the method also reveals that the yield condition of the rock mass is the damage evolution condition. The property parameters are listed in Table 5.2 and the initial in situ stress ﬁeld has the vertical and horizontal components of sx ¼ 20.0 and

153

Stability Analysis of Rheologic Rock Mass Table 5.2. Material’s properties. Material

Parameters E2 E1 (MPa) (MPa)

Rock mass 7500 Shotcrete layer 22000

6500

0.3 0.18

¼ 1/Z3 C Z2 (MPah) [l/(MPah)] (MPa) f( ) 174400

4

0.45 10

1.0

K1

K2 3

33.0 5.87 10

!p

4.82 0.40

Figure 5.16. Layout and section view of displacement monitoring in the test gallery.

sy ¼ 14.0 MPa, respectively. The shotcrete concrete layer is treated as a linear elastic medium.

5.7.3.3 Comparison between calculated and in situ measured results. The layout of monitoring points in the testing gallery in the mining under consideration is shown as in Figure 5.16. The measured results obtained from extensometer observation have been compared with the calculated results.

154

Chapter 5

Figures 5.17–5.20 compare the calculated results and the measured results. From these results, we can see that the displacement of the gallery periphery calculated by taking the damage evolution of the rock mass into account is somewhat higher than the result from conventional viscous calculation. With respect to the attenuation law of the displacement rate at the periphery and to the distribution of displacements in the depth of the surrounding rock mass, the computed results by damage model are better coincident with the measured one. On the distribution of displacements along depths, the diﬀerence between the conventional calculation and the damage computation lies in the plastic zone. In the plastic zone around the gallery, the

Figure 5.17. Displacement–time curve of the gallery roof.

Figure 5.18. Displacement–time curve of the sidewall.

Stability Analysis of Rheologic Rock Mass

155

Figure 5.19. Calculated displacement–time curve of the sidewall with damage evolution of rock mass.

Figure 5.20. Displacement distribution with depth on the sidewalls.

displacement gradient obtained from the damage analysis is greater than that from the conventional analysis. The conventional value from the damage analysis is closer to the ﬁeld measurement. It shows that the visco-elasto-plastic calculation with rock mass damage evolution can describe more realistically the loosening failure characteristics in the rheological process of the surrounding rock mass than the conventional analytical method.

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Chapter 6

Back Analysis and Observational Methods In recent years, the back analysis method has been widely applied in geotechnical engineering especially in the underground works. Since Peck [390] proposed this method in various related analytical methods and numerical methods have been developed [53–58,391–400]. As a relatively new branch of mechanics, rock mechanics has been developed by adopting traditional analytical methods of mechanics based on continuum mechanics. The typical mathematical and mechanical analytical model is somewhat to ﬁnd the stress ﬁeld and deformation ﬁeld in material by applying the known external loads (or internal loads) with the known material constitutive relationship, geometrical shape and mechanical parameters [53–56]. However, back analysis is a reverse procedure, which is to solve the external load or partial material parameters, based on the known deformation and stresses at limited points and the partially known material parameters. In the stability analysis of the geotechnical engineering problems, it is often necessary to know the in situ stress ﬁeld, material mechanical parameters and even the mechanical model, by utilising the monitored physical information such as deformation, strain, stress and pressure during constructions. Such a methodology is called back analysis method. The back analysis method, based on the required input physical information, can be divided into deformation back analysis method, stress back analysis method and coupled back analysis method [53]. The physical information in the coupled back analysis method requires both deformation and stress. The back analysis has been applied to various geotechnical engineering projects, particularly to the rock engineering projects. The reason is due to the fact that the rock masses are very complicated and inhomogeneous. Excavation may disturb the rock mass to diﬀerent extents. Therefore, the construction process is not a close system behaviour, and is aﬀected by the environment and at the same time aﬀects the environment [63–70,401,402]. In underground excavation and stability analysis, we face two problems. Firstly, the information available is generally ‘grey’ and uncertain. It is therefore very diﬃcult to obtain explicitly the exact solution by using the conventional mathematical and mechanical analytical methods. Secondly, since the excavation is a process of forming an underground opening, the exchange of energy would occur between the rock mass and its surrounding environment.

157

158

Chapter 6

In other words, the rock mass would absorb some external energy and at the same time, release some energy to the external environment. This would result in some energy concentration and energy relaxation zones. Generally, the energy concentration and relaxation zones are obtained by using conventional continuum-based mechanical methods and they do not account for the energy dissipation. In fact, actual rock masses do not obey the continuity assumption, because some cracks and fractures are induced due to stress concentration and blast eﬀect during the construction. The inhomogeneity and relative slips at fracture interfaces cause energy dissipation, e.g., thermal energy. It is often diﬃcult to obtain exact results by using the continuum-based analytical method to such a discontinuous medium. Therefore, the analysis of rock excavation must combine both structure analysis and behaviour analysis, since the information monitored on-site is related to the rock behaviour during the excavation. The information can supplement the results obtained by using the conventional analytical method. The deformation monitoring and back analysis is to identify the ‘grey’ problems by using the information of rock mass behaviour. The rock engineering problems usually involve several to several thousand meters. It is very diﬃcult to determine representative or average external loads (in situ stresses in most cases) and material parameters (such as the Young’s modulus, the Poisson’s ratio, cohesion and friction coeﬃcient), by testing rock samples. Large-scale in situ tests are generally very expensive, and often conducted at very few numbers. Hence, it is diﬃcult to ensure the result as a representative one reﬂecting the whole project based on in situ tests. On the other hand, the information used for back analysis is monitored directly from the actual site. As compared with the in situ large-scale mechanical tests, the deformation monitoring and back analysis have following features and advantages: (a) Monitored deformation is the average response of a rock mass in a large scale, from several to several tens of meters, sometimes several hundreds of meters. (b) It corresponds the actual engineering response to the excavation. (c) It gives large quantity of information, as each monitoring location produces diﬀerent information. (d) It can be correlated with laboratory tests to generate the correlation relationship. (e) It can monitor the response of rock outside the blast inﬂuence zone. (f) The deformation monitoring is easy to perform. (g) Monitoring is cost-eﬀective and can be widely used in projects of various sizes. With reliable monitoring data obtained from construction site, it is possible to back analyse the external load and mechanical parameters, by using proper physical material model and the analytical method.

Back Analysis and Observational Methods 6.1.

6.1.1

159

ELASTIC BACK ANALYSIS AND STRESS DISTRIBUTION ANALYSIS

Elastic back analysis

Elastic deformation back analysis method can be classiﬁed into three types: force method, mathematical regression method and graphic method [53–56]. The force back analysis method can be analytical or a numerical method. The analytical method is usually used when the excavation shape is simple and analytical solution can be found. The numerical method, ﬁrstly proposed by Sakurai et al. [53,54], is to assume the deformation at a point in a direction being sum of the individual deformations produced by individual loads at the same point in the same direction. Corresponding equation set can then be created. The number of equations or the number of monitored deformations should be equal to or more than the number of variables. The variables can then be solved by using damping least square method. For homogeneous media, the deformation modulus can also be solved. This is termed the inverse approach of the back analysis method.

6.1.1.1. Basic formulation.

The basic formula can be expressed as Z fP 0 g ¼

½BT s0 dV

ð6:1Þ

v

where {s0} is the in situ stress; {P0} is the nodal load acting on the excavation face; V is the excavated volume; [B] is a matrix relating to the element geometry. The relationship of nodal load {P} and nodal deformation {U} is fPg ¼ ½KfUg

ð6:2Þ

where [K ] is the stiﬀness matrix. Using ER and EL to represent the modulus of rock and lining, respectively, we have ½K ¼ E

R

K

R

EL L

þ R K ¼ E R ½K E

ð6:3Þ

where [KR] and [KL] denote the stiﬀness matrixes when ER ¼ 1 and EL ¼ 1, respectively. The nodal displacement {U} in equation (6.2) can be divided into two parts: displacements at monitoring points and displacements at other points. They are: fU m g ¼

1 ½A s0 ¼ ½Afs 0 g R E

ð6:4Þ

160

Chapter 6

where [A] is the compliance matrix which is a function of the Poisson’s ratio and the co-ordinates of monitoring points, s 0 is the normalised in situ stress expressed as n oT s0 ¼ s0x =E s0y =E t0xy =e

ð6:5Þ

Replacing the absolute displacements in equation (6.4) with relative monitored displacements, then fU m g ¼ ½T½Afs 0 g ¼ ½A fs 0 g or

1 fs 0 g ¼ ½A T ½A ½A T fU m g

ð6:6Þ

6.1.1.2. Deformation monitoring and back analysis. An example is given in the following sections to illustrate deformation monitoring and back analysis technique. (i) Deformation monitoring of surrounding rock mass. A large underground hydropower complex is located in a high in situ stress zone in southwest China. Before the construction, a rectangular test tunnel of 2.5 m wide, 5 m high and 30 m long was excavated for deformation monitoring and stability analysis. The layout of testing tunnel is shown in Figure 6.1. The tunnel is parallel to the exploratory tunnel. The distance between the two tunnels is 15 m. Three monitoring sections are set in the test tunnel. Smooth blasting is used around the monitoring sections. Partial monitoring results obtained from a section are illustrated and discussed in this example. Three multiple-point borehole extensometers (MPBX) including two bar electrical transducer and one wire electrical transducer were installed at the roof of the section. Two MPBXs of steel wire electrical transducer were installed at sidewalls. The MPBXs are of 10–13 m in length, for roof and sidewalls, respectively. The monitoring layout is shown in Figure 6.2. The end points are located in the undisturbed zone and have no deformation. In addition, several monitoring holes of 5 m length are set at the location close to the sections for ultrasonic measurements and convergence monitoring. Figure 6.2 shows the monitored ﬁnal displacements at diﬀerent points of the MPBXs. Figure 6.3 shows the displacement-time curves obtained from the MPBX in the left sidewall. From Figure 6.2, it can be seen that the maximum displacement reaches about 1.3 mm on the sidewalls, but less than 0.3 mm at the roof. This may be

Back Analysis and Observational Methods

161

Figure 6.1. Layout plan of the monitoring in experimental tunnel.

Figure 6.2. Displacement measurements by multiple-point borehole extensometers (Section 1 of Figure 6.1).

caused by the high horizontal in situ stress, and high walls of rectangle shape. In addition, the longitudinal ultrasonic wave velocity measured from the boreholes before and after the construction does not change apparently. It indicates that the surrounding rock has not been damaged or disturbed. The rock is at perfectly elastic state. From Figure 6.3, it can be seen that the curve tends to be ﬂat when the excavation approaches Section II (v/v = 1).

162

Chapter 6

Figure 6.3. Changes of displacement with excavation.

The displacement monitored at Section I is the deformation after the 1m excavation from Section I towards Section II. Three-dimensional ﬁnite element modelling is carried out to simulate the total displacement and displacements of diﬀerent excavation stages. The total displacements at left and right walls are computed to be 2.54 mm and 2.45 mm, respectively. (ii) Back analysis and results. The rock mass located around the monitoring sections of the underground complex is a seinite. The rock mass is fractured and the fractures are generally closed. The rock mass is under high in situ stresses. The rock mass is assumed to be uniform, homogeneous and linearly elastic. The in situ stress in rock is assumed to be uniform. Therefore, the linear elastic theory can be used to perform back analysis. If the Young’s modulus and the Poisson’s ratio are known, it is not diﬃcult to back analyse the in situ stress. By inputting the monitored displacements and adopting the least square method for solving equation (6.6), the in situ stress can be computed. Taking E ¼ 40,000 MPa and n ¼0.15 and treating it as a plane strain problem, the maximum principal in situ stresses are computed as s1 ¼ 23.7 MPa and s2 ¼ 18.7 MPa. a1 is at 31.5 to the horizontal plane (Figure 6.4). Figure 6.4 shows comparison of monitored displacement with the back analysis result at left wall. From the ﬁgure, it can be seen that the back analysis agrees well with the monitoring. It indicates that back analysis using linear elastic theory is applicable in this particular case. It is also found that the in situ stress back analysed based on the monitored displacements is very close to in situ stress measured. In other words, the 2-D rock deformation back analysis incorporating rock mechanics parameters E and n and in situ stresses measured on-site, has veriﬁed that the monitored data are reliable.

Back Analysis and Observational Methods

163

Figure 6.4. Monitored and back-analysed displacements at left wall.

6.1.2 Back analysis of in situ stress distribution In situ stress is generally a necessary input parameter in the analysis of rock stability. However, for a large underground complex such as the hydroelectric caverns, the in situ stress cannot be treated to be uniform, as the rock covers very large area (often more than 300 m). The in situ stress ﬁeld changes with depth and distance. A stress function interpolation method [8,52,137] is adopted. Its principle is to use the least square method to obtain the in situ stress ﬁeld satisfying stress equilibrium and deformation condition, based on-site monitoring data and boundary conditions. The following describes the method for the in situ stress ﬁeld of the same hydroelectric complex as in Section 6.1.1.

6.1.2.1. Computational zone and monitored in situ stress. The computational zone of the underground complex is from the ground surface to a depth of 800 m and about 200 m radius from the boundary of the opening in horizontal direction. Six sets of measured in situ stresses are taken [8,52].

6.1.2.2. Determination of stress function. The equilibrium equation at a point in rock is a set of inhomogeneous linear diﬀerential equations. The solution of the equation set is the sum of its corresponding general solution of the homogenous equation set and speciﬁc solution of the inhomogeneous equation set. The speciﬁc solution is taken as sx ¼ sy¼ txy ¼ tyx¼sz¼ 0, txz ¼ gx, and the general solution of the homogeneous equation set is expressed by the stress functions f1 (x, y, z), f2 (x, y, z) and f3 (x, y, z).

164

Chapter 6

Based on equilibrium equation and stress function, following relations can be derived. 9 @2 j 3 > > > txy ¼ > @[email protected] > > > > = 2 @ j1 tyz ¼ @[email protected] > > > > > @2 j 2 > > > ; tzx ¼ @[email protected]

@2 j3 @2 j2 sx ¼ þ 2 @y2 @z sy ¼

@2 j1 @2 j3 þ 2 @z2 @x

sz ¼

@2 j2 @2 j1 þ 2 @x2 @y

ð6:7Þ

By considering the ﬂuctuation and deviation of the measurement data of in situ stresses, stress functions are assumed to be polynomial of power 4 and the corresponding stress components to be polynomial of power 2. The general expression of stress function j1(x, y, z) is: j1 ðx, y, zÞ ¼ a1 x2 þ a2 y2 þ a3 z2 þ a4 xy þ a5 yz þ a6 zx þ a7 x3 þ a8 y3 þ a9 z3 þ a10 xyz þ a11 xy2 þ a12 x2 y þ a13 yz2 þ a14 y2 z þ a15 zx2 þ a16 z2 x þ a17 x4 þ a18 y4 þ a19 z4 þ a20 xyz2 þ a21 xy2 z þ a22 x2 yz þ a23 xy3 þ a24 x3 y þ a25 yz3 þ a26 y3 z þ a27 zx3 þ a28 z3 x þ a29 x2 z2 þ a30 y2 z2 þ a31 z2 x2

ð6:8Þ

where ai is the extrapolation parameter to be determined. The stress components can be expressed as sx ¼ b0 þ b1 x þ b2 y þ b3 z þ b4 x2 þ b5 y2 þ b6 z2 þ b7 xy þ b8 yz þ b9 zx

ð6:9Þ

with b0 b2 b4 b6 b8

¼ 2ða002 þ a03 Þ ¼ 2ð3a008 þ a016 ¼ 2ða0026 þ a027 Þ ¼ 2ða0028 þ a019 Þ ¼ 6ða0025 þ a023 Þ

b1 b3 b5 b7 b9

¼ 2ða0013 þ a015 Þ ¼ 2ða0014 þ 3a09 Þ ¼ 2ð6a0018 þ a028 Þ ¼ 2ð3a0020 þ a031 Þ ¼ 2ða0030 þ 3a021 Þ

where a0 and a00 are the corresponding coeﬃcients of j2 and j3. The expressions of stress functions j2 and j3 and other stress components are similar to equations (6.8) and (6.9) and are not listed here.

Back Analysis and Observational Methods

165

By substituting equation (6.7) into six deformation equations (other components are ignored here), we have ð1 þ nÞr2 sx þ

@2 ¼0 @x2

ð6:10Þ

where ¼ sx þ sy þ sz ; n is the Poisson’s ratio. The six extrapolation functions can be eliminated. By considering the relationship between stress functions and stress components above, each stress function can eliminate some extrapolation coeﬃcients, and the amount of independent extrapolation coeﬃcients to be determined becomes 55. The corresponding boundary conditions must be determined to obtain the extrapolation functions by using least squares method. They include the stress measurements at the ground surface and at selected locations. The normal and tangent stresses at ground surface with outward normal cosine directions of l, m and n are equal to zero, which can be expressed as 9 X0 ¼ sx l þ txy m þ txz n ¼ 0 = Y0 ¼ tyx l þ sy m þ tyz n ¼ 0 ; Z0 ¼ tzx l þ txy m þ sz n ¼ 0

ð6:11Þ

Substituting the stress components with those extrapolation function coeﬃcients as listed in equation (6.9) into equation (6.11), all the boundary conditions can be expressed by extrapolation function coeﬃcients. Therefore, each stress measurement point below ground has six equations and each measurement at ground surface point has three equations. These equations have a general form as A1 Q1 ðxi Þ þ A2 Q2 ðxi Þ þ þ An Qn ðxi Þ ¼ Bi

ð6:12Þ

where An is the extrapolation coeﬃcient to be determined (n ¼ 55); xi is the co-ordinate (x, y, z) of the point; Qi is the exponent function of co-ordinate ( j ¼ 1, 2, . . ., n); Bi is the stress measurement values at the measurement points or zero values at ground surface points; i ¼ 1, 2, . . . , n, m ¼ 6 I þ 3 K, I is number of stress measurement points, K is the number of ground surface points. Based on the above equations, all the coeﬃcients of stress functions can be solved by using least squares method to obtain the co-ordinates of the measurement points. The in situ stress distribution in the whole zone can thus be obtained. The stress distribution computed satisﬁes force equilibrium conditions, continuous deformation conditions and minimum value in the squares error with actual measurements

166

Chapter 6

Figure 6.5. Comparison of stresses measured and computed.

and the given boundary conditions. Figure 6.5 shows the comparison of stresses between the actual measurements and the computed results using the above method. It can be seen that in general they agree well with each other. The results obtained by this method were compared with the results obtained by multiple-variable regression method. It is found that both methods provide similar results, and the result obtained by this method generally agree better with the actual measurements. This method is simpler and has the advantage in computer running time and the preparation work. This method also eliminates the human error in determining stress ﬁeld. Figure 6.6 shows the stress distribution at a section in the middle of the underground complex.

6.2.

6.2.1

VISCO-ELASTIC BACK ANALYSIS AND ITS ENGINEERING APPLICATIONS

Method of site deformation monitoring and its application results

This section introduces the back analysis method for visco-elastic media using a case study on a shallow tunnel.

Back Analysis and Observational Methods

167

Figure 6.6. Stress distribution at a section in the middle of the underground complex.

Figure 6.7. Tunnel section and multiple-stage excavation procedure.

A large railway tunnel is located in a loess layer. The tunnel is a double-lane heavy vehicle tunnel of 11.5 m high, 12 m wide and with 12 m thick overburden. The tunnel is constructed by the New Austrian Tunnelling Method (NATM) and has an experimental section to conduct monitoring studies. Monitored vertical deformation at the crown due to excavation is presented here to illustrate the application of back analysis. Figure 6.7 shows the tunnel section and multiple-stage excavation procedure.

168

Chapter 6

Figure 6.8. Ground condition and instrumentation layout.

Figure 6.8 shows the installation of instrumentation. It can be seen that the multiple-point borehole extensometers (MPBX) are installed in the boreholes before the excavation to monitor the absolute deformation in the soil during the tunnel construction. Relationship between ground settlement and MPBX monitored displacement are shown in Figure 6.9. The monitoring results and analysis of MPBX 3 with a period of 65 days during the excavation of the upper part of the tunnel are shown in Figure 6.10. The largest deformation of 15.29 mm is at the end point (point 1). The deformation can be divided into three stages: (a) Initial deformation stage (compression deformation stage) in which the deformation occurs during the tunnel excavation from the beginning to the monitoring section, with a period of 24 days. The maximum compression deformation is 1.91 mm and displacement rate is 0.08 mm/day. All the six points at diﬀerent depths have negative deformations. Initially the monitoring data has small ﬂuctuation in the ﬁrst 10 days. Then the negative deformation increases and the curves tend to be close together, indicating that deformations occurs mainly due to the ground settlement. (b) Sharp deformation stage. As the tunnel excavation passes through the monitoring section, the deformations become positive immediately and increase sharply for 10 days before reaching a maximum of 15.19 mm. The average displacement rate is 1.72 mm/day.

Back Analysis and Observational Methods

169

Figure 6.9. Notation of negative displacement in pre-installed extensometers.

Figure 6.10. Monitored displacement in borehole 3 with a period of 65 days during the excavation.

(c) Stable deformation stage. When the tunnel excavation is 14 m away from the monitoring sections, the deformations increase slightly (displacement rate was 0.06 mm/day) for 1 month and then become stable. However, the displacements at two deeper points (points 1 and 2) decrease in the ﬁrst several days, indicating that there may be saturated sand-lens at these locations. While the lower part is excavated completely, the displacement increased again about 1–5 mm and the average displacement rate is 0.14 mm/day.

170 6.2.2

Chapter 6 Visco-elastic back analysis

6.2.2.1. Computation method. As noted in the observation, the deformation in ground above the tunnel develops with the excavation of the tunnel. Therefore, theoretically it is a 3-D problem. To save the computation time, it is often simpliﬁed as a virtual 3-D problem or a plane problem, but uses the change of a virtual support force to simulate the unloading due to excavation. Because the monitoring data were obtained from the excavation of the upper part of the tunnel, analysis is performed only for the upper part of the excavation. Finite element elastic analysis is conducted. Figure 6.11 shows the generated mesh. Because of the symmetry, only half of the domain is taken as the computational model with 144 elements (4-node isotropic element) and 170 nodes. The in situ stress is obtained from gravity of the overburden soil. The top of the tunnel is 12 m below the ground surface. The elastic solution of the stress boundary condition is

uðx, y, zÞ ¼

f ðx, y, zÞ P E

ð6:13Þ

where u(x, y, z) is the displacement at a point in the 3-D space; f (x, y, z) is the displacement induced by unit load at unit elastic modulus which can be solved by the

Figure 6.11. Generated mesh in modelling.

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171

ﬁnite element analysis; P is the excavation load as a function of time. 1 P ¼ Pðx,y,zÞ 0:5 arctan ðt t0 2Þ

ð6:14Þ

6.2.2.2. Visco-elastic analysis results. Using three-unit visco-elastic model shown in Figure 6.12 and assuming the Poisson’s ratio as a constant, the visco-elastic displacement solution can be expressed as follows by transforming equation (6.14) with visco-elastic responding principle: Uðt, x, y, zÞ ¼ f ðx, y, zÞMðtÞ

ð6:15Þ

where, MðtÞ ¼

1 1 1 E0 arctan ðt t0 2Þ 1 þ eE2 = ðtt0 Þ E1 2 E2

f(x, y) can be computed using 2-D FEM. Subsequently back analysis regression can be performed for the monitored data by using equation (6.15). Figure 6.13 shows the

Figure 6.12. The three-unit visco-elastic model.

172

Chapter 6

Figure 6.13. Regression curves of measurements at points 1 and 4.

regression curves of measurement points 1 and 4. The monitored data are the absolute displacements combining the ground surface settlements and relative displacements. It can be seen that the regression data agree well with the monitored data after the excavation face passed the monitoring sections. Nevertheless, the overall visco-elastic parameters of the soil can also be obtained from the regression of the latter part of the curves. The analysis gives E1 ¼ 15.2 MPa and E2 ¼ 45.8 MPa, and are seemingly reliable compared with the empirical values.

6.3.

BACK ANALYSIS AND OPTIMISED METHODS IN TRANSVERSE ISOTROPIC ROCK

The layered rocks such as sedimentary formation are often encountered, so it is very important to investigate the back analysis method for such media.

6.3.1

Basic formulae of transverse isotropic mechanics

In continuum mechanics, if the media satisﬁes the assumptions of uniformity, continuity, small deformation and linear elasticity, the constitutive equations can be expressed as sij ¼ Dijkl "kl

ð6:16Þ

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173

Considering the symmetry of the strain tensor and stress tensor, the elastic parameters can be reduced from 81 to 36. The matrix form is fsg ¼ ½Df"g

ð6:17Þ

Due to the symmetry of the elastic matrix, the elastic independent parameters can be reduced to 21. This can be used in general for anisotropic material that has three perpendicular elastic symmetric planes. If the axes are located in the elastic symmetric planes, the general Hooke’s law can be expressed in the form of matrix as 9 38 8 9 2d "x > d12 d13 sx > > 11 > > > > > 6 d21 d22 d23 > sy > > "y > > 7> 0 > > > > > > > > 7 6 = 6 < < = 7 " d d d sz z 31 32 33 7 6 ¼ ð6:18Þ 7> xy > 6 txy > d66 > > > > 7> 6 > > > > > 4 > txz > > xz > > 5> 0 d55 > ; : > > : ; tyz yz d44 containing only nine independent elastic parameters. The stress–strain relationships in the components of the anisotropic media are 9 sx ¼ d11 "x þ d12 "y þ d13 "z > > > sy ¼ d21 "x þ d22 "y þ d23 "z > > > = sz ¼ d31 "x þ d32 "y þ d33 "z ð6:19Þ txy ¼ d66 xy > > > > txz ¼ d55 xz > > ; tyz ¼ d44 yz Assuming that there is a plane parallel to xoz plane at each point in the elastic body as shown in Figure 6.14, the elastic properties in any direction are equivalent. The body is called orthotropic medium. The general Hooke’s law has a form of: 9 38 8 9 2d "x > d12 d13 sx > > 11 > > > > > 6 > > > 7> "y > d21 d22 d23 0 > > > > sy > > > > > 7 6 < < = 6 = 7 " d d d sz z 31 32 33 7 6 ¼ ð6:20Þ 1 7 6 txy > xy > ðd11 d12 Þ > > > > > > 2 6 7 >t > > > 4 > > 5> xz > 0 d44 > > > : xz > ; > > : ; tyz yz d44 In this case, only ﬁve independent elastic parameters are unknowns. Therefore, equation (6.17) can be modiﬁed to be f"g ¼ ½Afsg where [A] is the compliance matrix.

ð6:21Þ

174

Chapter 6

Figure 6.14. Transverse isotropic elastic body.

As the xz plane is an elastic isotropic plane, thus Ex ¼ Ez ¼ Es

Ey ¼ En

nxz ¼ ns nxy ¼ nyz ¼ nn Gxz ¼ Gs ¼ 0:5 Es =ð1 þ ns Þ Gxy ¼ Gyz ¼ Gsn where [A] becomes 2

1 6 Es 6 6 nn 6 6 En 6 n 6 s 6 6 Es ½A ¼ 6 6 6 6 6 6 6 6 4

nn En 1 En nn En

0

ns Es nn En 1 Es

3

0

1 Gsn

1 Gsn

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 2ð1 þ ns Þ 5 Es

ð6:22Þ

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175

If the main axes of the medium are not the same as the coordinate axes, then [A] needs to be transformed in coordinate and new compliance matrix is ½A0 ¼ ½T½A½TT

ð6:23Þ

where [T ] is the coordinate transforming matrix. For plane strain problems, [A] can be further simpliﬁed as 2

1 n2s nnn ð1 þ ns Þ 6 1 6 nnn ð1 þ ns Þ nð1 nn2n Þ ½A ¼ 4 Es 0 0

3 0 0 7 7 Es 5 Gsn

ð6:24Þ

The stiﬀness matrix is 2

nð1 nn2n Þ 6 E n 6 nnn ð1 þ ns Þ ½D ¼ ½A1 ¼ m4 0

nnn ð1 þ ns Þ 1 nn2n 0

3 0 0 7 7 mGsn 5 En

ð6:25Þ

where n ¼ Es/En, m ¼ (1þns)(1 ns 2nn2n ) Let us discuss a special and useful case. Assuming the layer plane orientation is parallel to the tunnel direction (z axis), the excavation face is xoy plane, x axis is horizontal and is parallel or oblique to the layer plane, the analysis can be conducted using equations (6.24) and (6.25). If there is a joint set parallel to the layer plane and with the same mechanical properties, and shear stiﬀness is Ks, normal stiﬀness is Kn and joint spacing is b, the elastic parameters in equations (6.24) and (6.25) can be determined by using the following equation: 9 Es ¼ E,ns ¼ n > > > > > 1 > > En ¼ > > > 1=Es þ 1=Kn b = 1 > > Gsn ¼ Gn ¼ > 2ðð1 þ ns Þ=Es Þ þ 1=Ks b > > > > > > n > ; nn ¼ En Es

ð6:26Þ

176

Chapter 6

6.3.2 Optimisation analysis method As stated earlier, the inverse approach of back analysis method can be used in a number of cases with simple conditions. In most cases with complicated conditions, only the direct approach of back analysis method can be used. This method is actually a regression method. The most important issue of this direct approach method is the appropriate understanding of the subject to be back analysed. It depends extensively on the experience and knowledge. For example, in underground excavation analysis, the mechanical model (elastic isotropic or anisotropic, viscoelastic or elasto-plastic models), mechanical analysis method or computational program have to be determined. When the mechanical model and analytical method are determined, the next step is to rationalise the regression computation to make it scientiﬁc and fast. This is the optimisation of the direct back analysis to be discussed in this section. The objective of optimisation, based on monitored convergence displacement, is to obtain the best sets of material parameters and stress parameters. Assuming the computational displacement is ui(X ), monitored convergence displacement is u0i (i ¼ 1, 2, . . . , n), the objective function is deﬁned as FðXÞ ¼

n X

ui ðXÞ u0i

2

ð6:27Þ

i1

where X ¼ ½sx , sy , txy , E, n, c, f where sx, sy and txy are three components of in situ stress, E is the elastic modulus, n is the Poisson’s ratio, c is the cohesion and f is the friction angle. The optimisation method is to make the objective function approach gradually to minimum in the direct approach. Generally, giving one set of initial parameters X0 and their allowable ranges, the optimisation program searches automatically one set of parameters X to make the objective function satisfying the given accuracy requirement. There are many optimisation methods with or without restraints available [49–51]. Brief outlines of the common optimisation methods are presented below. (a) Pure shape speeding method: Pure shape acceleration method was developed from the pure shape method. The pure shape method is to compare the function values at (n þ 1) peak points of an initial pure shape in n-D space, replacing the points of maximum function value, with a new pure shape, and approaching gradually to the points of minimum value through iterations. (b) Composite shape acceleration method: This method is used to solve optimisation problems of multiple variables (normally not more than 20) with constraints of

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177

unequal equations. The optimisation procedure is to take 2n peak points from non-linear constraints in n-D space to form a pure shape. It is then to compare function values at each peak point one by one, replace the worst point with a new point to improve the objective function and satisfy constraint conditions, and approach gradually to the optimised point. (c) Mixing penalty function method: This is also termed as sequential unconstrained minimisation technique (SUMT). It is to add constraint function with equal equations and unequal equations in penalty, respectively, to the objective function to form a new objective function with no constraint (penalty function). So that it converts the original minimisation of objective function with constraints into a new minimisation of penalty function with no constraints. (d) New Powell method: This is an advanced optimisation method to solve the minimisation of complex functions. The iterative procedure is to solve the minimisation of the objective function along a series of conjugate directions starting from the initial point (X0). The forming of the conjugate directions uses only the objective function values at the iteration points. Therefore, it is a direct method. This method is useful when the objective function is a non-convex function. In the case of multiple minimum value points within a certain range, this method can ﬁnd the optimal point. For complicated non-linear problems, the objective function is often non-convex function, and the minimum point is not unique. To obtain the optimal solution, the optimal searches in one dimension and multiple dimensions are added to seek minimum points of objective function along conjugate directions. Meanwhile, the step should be decreased in the area with sharp slope of objective function. After obtaining the minimum point, the searches will be performed for its two sides to obtain second and third class minimum points. Comparing them with each other gives the optimal solution. An optimisation generally has three loops to execute diﬀerent functions: optimisation methods, model types (linear, non-linear or isotropic models) and selection of diﬀerent back analysis parameters section. Figure 6.15 illustrates a typical program ﬂow.

6.3.3

Examples of engineering applications

The following example is to illustrate the application of the optimisation method in back analysis, for a large hydroelectric project. The power complex is located underground in sedimentary rocks. The surrounding rocks are mainly siltstone interlayered with clay slate. Finite element displacement back analysis coupled with an optimal program was used to examine the displacement distribution, in situ stresses and rock parameters.

178

Chapter 6

Figure 6.15. A typical program ﬂow.

As shown in Figure 6.16, Zones 1 and 3 are the siltstone and Zones 2 and 4 are the clay slate. The siltstone has the Young’s modulus of 28 GPa and 35 GPa in directions perpendicular to and parallel to the rock layer, and Poisson’s ratio of 0.25. The clay slate has the Young’s modulus of 21 GPa and 27 GPa in directions perpendicular to and parallel to the rock layer, and Poisson’s ratio of 0.27. The monitored convergence curves are shown in Figure 6.17. The tunnel was constructed by using one-stage excavation. The in situ stresses are uniformly distributed in the surrounding rock masses. This example is treated as a plane strain problem. The ﬁnite element mesh is generated into four-node four-side isotropic parameter elements with 217 elements and 252 nodes. The rock masses are assumed to be isotropic. The modelling assumes that the Young’s modulus between the measured modulus in the two directions, and the Poisson’s ratio does not change. The back analysis results, obtained using four optimisation methods, are summarised in Table 6.1. The analysis results of in situ stresses obtained by the Powell method are sx ¼ 9.27 MPa, sy ¼ 4.80 MPa. The measured stresses are sx ¼ 9.39 MPa,

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179

Figure 6.16. Cross section of the tunnel and monitoring holes.

Figure 6.17. Monitored displacement histories.

sy ¼ 4.75 MPa. The results from back analysis with optimisation are very close to the actual measured values.

6.3.4 Discussions The discussions below are based on the experience in using the four optimisation methods to a variety of engineering problems involving visco-elastic and

180

Chapter 6

Table 6.1. Results for different analysis methods. Method

Pure shape Composite shape Mixing penalty Powell method

No of iteration steps

15 50 15 43

Target function (105)

3.2 2.5 2.2 3.6

Back analysis results (MPa) sx

sy

E1

E2

9.04 9.00 9.04 9.27

4.74 4.80 4.74 4.80

21829.6 21700.0 21721.6 21700.0

30294.0 30560.0 30224.5 30800.0

elasto-plastic back analysis. (a) Pure shape acceleration method: It has a fast convergence and a high accuracy, especially for elastic, isotropic problems with more than two parameters. But initial value of back analysis parameters must be properly determined in advance. (b) Composite shape acceleration: Similar to the pure shape acceleration method, it has a fast convergence and a high accuracy, but has constraint functions. Giving constraint functions and an allowable value range of parameters, the program is able to perform back analysis automatically under the constraint conditions. (c) Mixing penalty function method: It has a constraint optimisation function after the Powell method. It does not require to set properly initial values of back analysis parameters in advance and can ﬁnd most optimal solutions satisfying constraint conditions automatically. It converges faster than the Powell method because of the 1-D search method feature. (d) Powell method: It converges slower than that of pure shape and composite shape, but is applicable when the objective functions are complex non-convex functions. It can ﬁnd the most optimal point when minimum points are not unique in a certain range, especially in the cases of non-linear problems with less than three parameters. In summary, for the back analysis of general elastic, anisotropic problems, the pure shape acceleration method and composite shape acceleration method are the best choices since they have a fast convergence and a high accuracy. However, for complex elasto-plastic problems, the Powell method or the mixing penalty function method should be used to conduct optimisation search, and then use either pure shape or composite shape methods to perform global optimisation back analysis to obtain the ﬁnal results. So the unique result can be obtained with fast convergence.

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181

The criteria to select optimisation methods with or without constraint conditions are: (a) For the cases when the parameters cannot be determined in range, but can be estimated as a possible value, the optimal back analysis approach with no constraint conditions can be used. (b) However, when the parameters have constraints (equal or unequal constraints), the optimal back analysis approach with constraints will be a better alternative. 6.4.

BACK ANALYSIS OF JOINTED ROCK MASS AND STABILITY PREDICTION

The sections earlier introduced the back analyses of elastic, visco-elastic, anisotropic and elasto-plastic problems, but does not attempt the back analysis of jointed rock mass. This section discusses the back analysis of jointed rock mass. It is illustrated by an engineering example of a pumped storage hydroelectric complex.

6.4.1

Description of the project and monitoring

6.4.1.1. Description of the project. The large-scale pumped storage hydroelectric facilities include main powerhouse, transformer house, ventilation and transport tunnels located underground 200–400 m in depth. The surrounding rock mass is mainly a medium-grained granite. There are mainly two joint sets at the crown of the powerhouse, with dip direction of 020–030 and 300–320. The joint spacing is usually 1–2 m. The average persistence is about 50%. Two cases are considered in the analysis. One is to treat the rock mass as an isotropic medium. Another is to consider the joint sets and treat the rock mass as an approximately perpendicularly anisotropic medium. During construction, displacements are monitored at three sections of ventilation tunnel and transport tunnel. The positions of measurement Sections I and II in a ventilation tunnel and the layout of MPBX monitoring is shown in Figure 6.18.

6.4.1.2. Data processing and modification. Due to the ﬂuctuation of the monitored data, data were processed. In addition, as most instruments are installed after the face of excavation, the data do not include the displacements occurring before the installation. The lost displacements are determined using 3-D ﬁnite element analysis for all the points and corresponding modiﬁcation coeﬃcients are derived. Therefore, the ﬁnal displacement based on the 2-D back analysis should be d ¼ dm þ dl ¼ dm,

182

Chapter 6

Figure 6.18. Monitoring layout in ventilation tunnel.

where dm is monitored displacements, dl is lost displacement, is modiﬁcation coeﬃcient which is determined from 3-D ﬁnite element analysis. The instruments (multiple-point extensometers or convergence meters) were installed behind the excavation face about 1 m in the two measurement tunnels. Therefore, the displacement measurements reﬂect partial displacement during the tunnel construction. Three-dimensional ﬁnite analysis is to obtain the modiﬁcation coeﬃcient and determine total displacement at each measurement point. The overburden thickness of the tunnels is 405 m, so the in situ stresses are calculated as sx ¼ 6:89 MPa sy ¼ 9:16 MPa sz ¼ 13:1 MPa

txy ¼ 0:49 MPa tyz ¼ 0:65 MPa tzx ¼ 0:37 MPa

The Young’s modulus is taken as 3 104 MPa, and the Poisson’s ratio as 0.20. Three-dimensional ﬁnite element modelling is carried out for each monitoring point to estimate the total displacement. Details of the modelling can be found in [8,9]. Based on the computed total displacement, the monitored displacement will be modiﬁed. It was found that the modiﬁcation coeﬃcient between the monitored and the computed displacements was 1.27 (average). The monitored displacements and modiﬁed displacements are presented in Table 6.2.

6.4.2

Back analysis using pure shape acceleration method

The approach of ordinary back analysis method is to: (a) create constitutive model of the rock mass, (b) assume an initial value of parameter to be back analysed,

183

Back Analysis and Observational Methods Table 6.2. The monitored and modified displacements. Measurement lines

MP-4 MP-3 MP-2 MP-1

Remark

Measurement points (mm) 1

2

3

4

0.29 0.35 0.21 0.25 0.25 0.30 0.45 0.58

0.34 0.50 0.33 0.39 0.39 0.52 0.60 0.77

0.54 0.82 0.38 0.49 0.41 0.62 0.77 0.99

0.62 0.99 0.40 0.55 – – 0.85 1.09

monitored modified monitored modified monitored modified monitored modified

(c) conduct numerical modelling, (d) solve displacements at measurement points, and (e) compare the computed displacement with monitored displacements. The ﬁnal parameter obtained from the back analysis is the parameter giving minimum diﬀerence between the computed displacement and the monitored displacement. Normally the diﬀerence between the computed displacement and the monitored displacements is expressed by the error objective functions as shown in equation (6.27). The pure shape acceleration method is used here to conduct optimisation analysis, because it is an eﬀective method to make the error objective function be minimum.

6.4.2.1. Computational procedure. Displacement analysis procedure consists of optimisation analysis and positive modelling. The optimisation analysis is discussed earlier and its program ﬂow is listed in [403]. The ordinary positive modelling program is modiﬁed from a commercial program, capable to model anisotropic material. The modelling program ﬂow is given in [404]. The whole program ﬂow is presented in Figure 6.19, to explain the concept of back analysis.

6.4.2.2. Computational results. The back analysis considers two mechanic models. One is to assume rock as uniform and isotropic medium with no joint. The material parameters are E, n, c and f. The other is to treat rock as an anisotropic media with one joint set. The parameters are E, Kn and Ks to be obtained by back analysis using corresponding constitutive relationship. (a) Access tunnel. In case of isotropic rock, two sets of input displacements are taken in back analysis. Upon optimisation back analysis, the relevant parameters are

184

Chapter 6

Figure 6.19. The complete cycle of program ﬂow.

obtained as listed below: E ¼ 3:70 104 MPa s1 ¼ 12:87 MPa

v ¼ 0:15 s3 ¼ 5:47 MPa

F ¼ 0:6 103 a ¼ 50

In case of anisotropic rock, in situ stresses are assumed constant, the back analysis is conducted by inputting in situ stress and corresponding displacement set. The empirical formulae Ks ¼ 1/5 1/10Kn (Belytschko et al. 1984), is taken as a constraint condition. The back analysis results are list below: E ¼ 6:419 104 MPa F ¼ 0:59 103

Kn ¼ 7:25 105 MPa=cm Ks ¼ 0:91 105 MPa=cm

(b) Ventilation tunnel In case of isotropic rock, the back analysis uses modiﬁed displacements at measurement holes MP-4, MP-2. Assuming the Poisson’s ratio n ¼ 0.15, optimised back analysis gives: E ¼ 3:77 104 MPa F ¼ 0:1 102

Back Analysis and Observational Methods

185

s1 ¼ 10:02 MPa s3 ¼ 6:78 MPa a ¼ 25 In case of anisotropic rock, taking the above in situ stress as input data, the back analysis results are: E ¼ 5:91 104 MPa F ¼ 0:2 102 Kn ¼ 48:65 104 MPa=cm Ks ¼ 6:08 104 MPa=cm The measured displacements and back analysis results for the access tunnel and the ventilation tunnel are compared as shown in Figures 6.20 and 6.21. It can be seen that the results agree well with each other. Inputting the above back analysis results for each case into the ordinary ﬁnite element modelling, the results are compared with the modiﬁed displacement measurements as listed in Tables 6.3 and 6.4. It can be seen that the results obtained from back analysis by combining the access tunnel and the ventilation tunnel are better. For each MPBX, the correlation of measured and back-analysed results is better at greater depth than that at shallower depth. This may be because the rock at shallower depth is greatly disturbed by blasting vibration.

6.4.3

Stability prediction of powerhouse and transformer chamber

The powerhouse was constructed by using multiple-stage excavation. In the ﬁnite element modelling, however, only the ﬁrst and last stages are involved. In this section, only the latter is discussed. Two rock mechanical models are considered in the modelling: uniform linear elastic rock with no joint and anisotropic rock mass with joints. In adopting with joints, the joint inﬂuences are estimated by equivalent method discussed in Chapter 3. Two joint sets are considered and equivalent deformation modulus and equivalent strength parameters are obtained. Three key faults are involved in each modelling. The damage distribution is shown in Figure 6.22. The mesh of the computational model has 596 elements and 589 nodes. The in situ stresses of ventilation tunnel are obtained by using ﬁnite element modelling as: sx ¼ 9.42 MPa, sy ¼ 7.369 MPa, txy ¼ 1.245 MPa.

186

Chapter 6

Figure 6.20. Comparison of monitored and back-analysed displacement of the access tunnel.

Material parameters of rock are: E ¼ 3.7 104 MPa; n ¼ 0.20; c ¼ 22.27 MPa; f ¼ 48.1o; st ¼ 5.0 MPa. Joint stiﬀness are estimated from back analysis, and joint strength parameters are estimated from a similar project. The joint parameters are: Kn ¼ 6.0 105 MPa/cm; Ks ¼ 7.5 103 MPa/cm; c ¼ 0.5 MPa; f ¼ 35 .

Back Analysis and Observational Methods

187

Figure 6.21. Comparison of monitored and back-analysed displacements for the ventilation tunnel.

Faults material parameters are: E ¼ 9.0 103 MPa; n ¼ 0.25; c ¼ 0.4 MPa; f ¼ 30 ; st ¼ 1.0 MPa. The layouts of convergence measurements in the powerhouse and transformer chamber are shown in Figure 6.23. The displacement prediction results are listed in

188

Chapter 6

Table 6.3. Comparison of measured and computed results in access tunnel. Measurement holes

MJ-1

MJ-2

MJ-3

MJ-4

Remark

Measurement points (mm) 1

2

3

4

0.28 0.38 0.22 0.25 0.13 0.08 0.20 0.34 0.02 0.03 0.21 0.11 0.08 0.35 0.26 0.25

0.56 0.64 0.47 0.46 0.35 0.25 0.39 0.79 0.12 0.08 0.60 0.21 0.11 0.62 0.51 0.47

0.65 0.72 0.64 0.61 0.63 0.54 0.56 1.29 0.16 0.18 0.97 0.28 0.74 0.74 0.71 0.68

0.73 0.76 0.77 0.78 0.76 0.93 0.75 1.82 0.45 0.29 1.34 0.31 0.84 0.75 0.91 0.84

monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2)

Table 6.4. Comparison of measured and computed results in ventilation tunnel. Measurement holes

MP-4

MP-3

MP-2

MP-predrilled hole

Remark

Measurement points (mm) 1

2

3

4

0.35 0.45 0.38 0.36 0.25 0.07 0.15 0.13 0.30 0.10 0.23 0.20 0.58 0.12 0.10 0.09

0.50 0.75 0.63 0.60 0.39 0.28 0.60 0.50 0.52 0.39 0.57 0.53 0.77 0.34 0.30 0.29

0.82 0.95 0.95 0.92 0.49 0.64 1.08 0.87 0.62 0.50 0.63 0.59 0.99 0.77 0.69 0.67

0.99 1.02 1.13 1.10 0.55 0.79 1.07 1.02 – – – – 1.09 0.98 1.06 1.04

monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2) monitored anisotropic group (1) group (2)

Back Analysis and Observational Methods

189

Figure 6.22. Distribution of damage zones.

Figure 6.23. Layout of convergence measurements.

Table 6.5. From the table, it can be seen that the convergence in the case of considering the joint inﬂuence is about 10% greater than that in the case of no joint. The displacements on tunnel walls and rock failure zone by equivalent model are shown in Figure 6.22. It can be seen that in addition to fault zone failure, shear failure occurs at top and bottom of the tunnel, and tensile failure on the sidewalls.

190

Chapter 6

Table 6.5. Predicted displacement of convergence (mm). Measurement lines Equivalent approach Linear elastic

6.5.

1

2

3

4

5

6

7

8

9

11.4 9.7

18.5 15.1

19.5 16.3

6.0 5.0

6.9 5.8

7.3 6.1

10.8 9.0

7.4 6.3

6.9 5.8

THREE-DIMENSIONAL BACK ANALYSIS OF ANISOTROPIC ROCK

The example using 3-D back analysis of anisotropic rock mass is shown in this section, in a large-scale hydroelectric power cavern complex. The hydroelectrical power is 70 m high and 30 m wide and constructed underground at the downstream of the right side of a river. The overburden depth is 30–107 m. Two sides of the river are symmetric with average slope of 45–50 . The project is very complicated with the following problems. (a) Poor engineering geology conditions: The rocks encountered are mainly killas and chorismite with intrusion of other types of rock which are heavily jointed. Breaking of rock in the tunnel occurs easily due to the excavations. The rocks are divided into three types. There are several faults intersecting the powerhouse that may induce instability to the tunnel walls. (b) High horizontal in situ stress: The site investigation shows that the horizontal in situ stress is more than twice of the vertical in situ stress. (c) Faults: There are a number of faults intersecting the complex and in the vicinity. Experimental and numerical analyses show that the faults intersecting the powerhouse cavern may be slipped during excavation. In summary, this project is constructed under high in situ stresses and complicated ground conditions with non-uniformity, anisotropy and discontinuities. Therefore, displacement monitoring and back analysis are performed to understand the rock mass behaviour and in situ stresses [8,52].

6.5.1

Displacement monitoring in trial tunnel and results

6.5.1.1. Set-up of displacement monitoring. In order to understand the behaviour of underground rock masses during excavation, a pilot tunnel of 1/6 of the cavern size was excavated at the cavern position. The pilot tunnel is 5 m wide, 9.5 m high and 54 m long. The pilot tunnel crosses all the three rock types, and three sections are set up for displacement monitoring. Seven boreholes for MPBX installation are drilled in Section I and II separately, and four are drilled in Section III. Each section has a

Back Analysis and Observational Methods

191

Figure 6.24. Setting up of MPBX in Section II.

pre-embedded MPBX to monitor whole deformation process. The pre-embedded MPBX is drilled from the surge chamber to the wall of the pilot tunnel as shown in Figure 6.24. Each monitoring section has 5–7 measurement points on the wall to form more than six convergence monitoring lines to correlate with results obtained by extensometers. In addition, some tests are conducted in the pilot tunnel including measurements of shotcrete and rock bolt stresses, rock bolt pulling test and seismic measurement of rock disturbance zone. 6.5.1.2. Monitoring results. The displacement monitoring results are given here with a period of 250 days until all the displacements become stable. It shows: (a) Displacement curve tends to be stable in 60–80 days. A typical displacement set is shown in Figure 6.24. A convergence measurement set is shown in Figure 6.25. (b) The smallest deformation occurs in Section II due to the good rock quality, while the largest displacement is observed in Section III due to the weak rock. Deformation in Section I is remarkable due to the faults. However, in overall, all deformations are quite uniform and not extensive. (c) The disturbance zone of surrounding rock is about 1–1.5 times of the tunnel size.

6.5.2

Back analysis

Three-dimensional back analysis is conducted by considering anisotropic rock, complex ground condition and that the in situ stress axis does not coincide with the

192

Chapter 6

Figure 6.25. A typical displacement set.

Figure 6.26. A convergence measurement set.

tunnel axis. Only the monitored displacements at middle measurement points are used in the back analysis, because it is found that the monitored data at two ends of MPBXs have relatively large error and low reliability. The procedure of back analysis modelling is similar to that outlined in Figure 6.19 of Section 6.4.2. The comparison of modelled results and monitored data for Section II are summarised in Table 6.6. The back-analysed material parameters of the anisotropic rock mass and in situ stresses are presented in Tables 6.7 and 6.8. The results indicate that the backanalysed results are lower than the results obtained from laboratory tests. The results also reﬂect the scale eﬀects. In general, large-size rock masses have lower strength and modulus. The back-analysed in situ stresses agree well with those measured, as shown in Table 6.7. From the above results, it can be seen that satisfactory results are obtained from 3-D non-linear regression based on monitoring displacements. Figure 6.27 shows modelled displacement curves compared with monitored

193

Back Analysis and Observational Methods Table 6.6. Comparison of calculated and monitored results in Section II. Measurement lines Relative displacement

Calculated Monitored Calculated Monitored Calculated Monitored Calculated Monitored Calculated Monitored

u1 u0 u2 u0 u3 u0 u4 u0 u5 u0

II-1

II-2

II-3

II-4

II-5

II-6

II-7

0.18 0.19 0.44 0.46 0.68 0.64 1.13 1.13 2.46 2.78

0.14 0.18 0.36 0.40 0.76 0.67 1.21 1.23 2.66 3.14

0.10 0.56 0.58 0.86 0.90 1.22 1.53 1.48 2.98 3.27

0.15 0.15 0.76 0.70 1.09 1.03 1.67 1.57 2.73 2.50

0.15 0.16 0.68 0.37 0.49 0.67 2.03 2.06 2.90 2.99

0.24 0.17 0.67 1.59 1.64 1.78 2.36 2.25 3.57 3.21

0.23 – 0.74 – 1.58 – 2.34 – 3.36 –

Table 6.7. Mechanical parameters of rock mass obtained by back analysis. Classification of rock mass A B1 B2 C f27 f25

E||( 103 MPa)

E?( 103 MPa)

u||

u?

17.0 13.5 11.0 8.5 1.0 2.0

11.5 9.0 7.4 4.3 – –

0.27 0.28 0.28 0.29 0.3 0.3

0.27 0.28 0.28 0.29 – –

Table 6.8. Comparison of calculated and measured in situ stresses. In situ stress

sx

sy

sz

txy

tyz

ca

Measured Calculated

5.9 5.4

7.2 6.7

7.7 8.0

0.3 0.9

0.2 0.4

0.6 0.5

Figure 6.27. Modelled displacement curves compared with monitored displacement curves in two typical MPBXs.

194

Chapter 6

displacement curves obtained from multiple-point extensometers for two typical MPBXs. Again, good matches are observed. 6.6.

THREE-DIMENSIONAL BACK ANALYSIS OF JOINTED ROCK MASS AND STABILITY ANALYSIS

A 2-D back analysis of jointed rock mass for a pumped storage hydroelectric power station was conducted in Section 6.4. The analysis was based on the monitoring information of two branch tunnels, before the excavation of the powerhouse cavern. This section is to illustrate, for the same project, the 3-D back analysis based on monitoring of the powerhouse cavern. Brief concept of the method will be introduced and some main results will be discussed in this section. The theory has been discussed in the previous sections. 6.6.1

Mechanic model

Similar to the example in Section 6.4, the equivalent model of jointed rock is also used in this 3-D back analysis case. For the joint persistence, the 3-D back analysis uses area equivalence, while 2-D uses line equivalence. The material parameters of rock and joints are obtained from approximate weighted average based on the joint arrangement. The joint persistence is taken as 80% from site investigation results. The material parameters of rock element used in the 3-D modelling are E, n, cr, fr and st, where st is tensile strength of rock. The parameters used for joints are Kn, Ks, cj, fj and aj, where aj is the joint dip angle. Equivalently transferring jointed rock mass into quasi-continuous rock mass, new stiﬀness matrix [D] can be obtained. It is a 6 6 full matrix for 3-D (3 3 matrix for 2-D). The new matrix will replace elastic matrix in the ﬁnite element modelling for the rock mass.

6.6.2

Summary of site monitoring data

Five monitoring sections are set up in the powerhouse cavern, while two monitoring sections are set in the transformer chamber cavern. Altogether, a total number of 36 monitoring MPBXs and 144 monitoring points are installed in those sections. Upon the completion of powerhouse and transformer chamber constructions, it is found that only data at 90 monitoring points in 28 MPBXs give eﬀective data. After data sorting, the data from 15 points in 3 MPBXs have clean trends and are considered reliable, and these data are used in the 2-D and 3-D back analysis.

195

Back Analysis and Observational Methods

Figure 6.28 shows the multiple-stage excavation of the powerhouse and the transformer chamber and the layout of instruments. The instrument details and monitoring data in powerhouse are shown in Tables 6.9 and 6.10. Data in Table 6.9 are used for 2-D analysis and data in Table 6.10 are used for 3-D analysis. From the monitoring data it can be seen that the displacements are small with a maximum value of 6 mm close to the right side of powerhouse cavern. In Table 6.10, the n horizontal lines refer to (n 1) stage excavation. The vertical lines across the horizontal lines represent excavation face position and

Figure 6.28. Excavation stage of the caverns and layout of instrumentation.

Table 6.9. Measured displacement by MPBX (mm). 1

2

3

4

Original status

M1-3 M1-5

– –

2.73 –

1.12 2.44

3.11 4.12

M1-7 M1-9

0.14 0.17

1.01 –

1.04 0.90

3.51 1.72

M2-2 M2-6

1.31 3.57

1.47 –

– 5.25

2.28 6.05

After excavation of Stage IV 1.5 m away from excavation face in Stage I

M5-1 M5-2

– –

0.11 0.72

0.30 2.44

0.60 5.05

After excavation of Stage II

After excavation of Stage II 5 m away from excavation face in Stage I After excavation of Stage I After excavation of Stage V

196

Chapter 6

Table 6.10. Instrumentation set-up and measured displacement. Arrangement

Point

Depth (m)

Measured value (mm)

M13 5

8

2.44

M14

5

23

4.12

M21 6

1

3.57

M23 6

8

5.25

M24 6

23

6.05

CM11 2

1

0.09

CM12 2

3

0.63

CM14 2

19

0.85

4

CM14 3

19

0.71

5

CM11 4

1

0.42

CM13 4

8

1.14

CM14

4

19

2.41

6

CM22 2

3

35

7

CM23 3

8

0.34

8

CM21 4

1

0.91

CM23

8

1.72

M11 7

1

0.14

M13 7

3

1.01

M13 7

8

1.04

M14

7

23

3.51

M12 3

3

2.73

M13 3

8

1.12

M14 3

23

3.11

M52 1

3

0.11

M53 1

8

0.30

M54 1

23

0.60

1

2

3

9

10

11

4

Location

1 5m

1 2

1.5m

2

1 1m

1

2 2

1 1

5 5

(continued)

197

Back Analysis and Observational Methods Table 6.10. Continued. Arrangement

12

13

14

15

Point

Depth (m)

Measured value (mm)

Location

M52 2

3

0.72

M53 M54

2

8

2.44

2

23

5.05

CM21 5

1

0.09

CM23 5

6

0.34

CM24 5

12

1.05

2

M21 2

1

1.31

2

M23 M24

2

3

1.47

2

12

2.28

M11 5

1

0.17

M13 M14

5

6

0.90

5

12

1.72

2

2 1 1

that non-crossing the horizontal lines mean that the excavation stage has been completed. 6.6.3 Finite element back analysis of underground powerhouse complex Two-dimensional modelling uses 630 nodes and 635 elements, while 3-D modelling uses 4304 nodes and 3994 elements. Rock material parameters adopted in the 2-D back analysis are given in Section 6.4. To reduce the mesh preparation work, all the computations use the same mesh arrangement as shown in Figure 6.29. However, mesh relating to the excavation will be changed to simulate the multiple-stage excavation process. Upon the back analysis, the ﬁnal rock material parameters are obtained as following: E ¼ 3:7 104 MPa n ¼ 0:24 c ¼ 1:29 MPa f ¼ 41 st ¼ 3:5 MPa

198

Chapter 6

Figure 6.29. 3-D FEM model mesh.

Joint parameters are: Kn ¼ 6:0 105 MPa Kt ¼ 7:5 104 MPa c ¼ 0:5 MPa f ¼ 35 In situ stresses are: sx ¼ 8:95 MPa sy ¼ 12:84 MPa sz ¼ 6:89 MPa txy ¼ 0:58 MPa tyz ¼ 0:38 MPa tzx ¼ 0:41 MPa Comparing above results with results obtained in Section 6.4, there is no much diﬀerence, except the rock material strength parameters.

Back Analysis and Observational Methods

199

The results from the 2-D back analysis and from the monitoring in the powerhouse and the transformer chamber are compared in Figures 6.30 and 6.31. The 3-D back analysis results and the monitoring data in the powerhouse are compared in Figure 6.32.

Figure 6.30. 2-D back analysis and monitoring in the powerhouse.

Figure 6.31. 2-D back analysis and monitoring in the transformer chamber.

200

Chapter 6

Figure 6.32. 3-D back analysis and monitoring data in the powerhouse.

Figure 6.33. Displacement distribution and failure zone in the rock mass around the cavern.

6.6.4

Stability of powerhouse and transformer chamber

Non-linear analysis of powerhouse and transformer chamber using above back-analysed parameters provides the displacement distribution in the surrounding rock mass. The displacement distribution and failure zone in the rock mass around the cavern at a particular section are shown in Figure 6.33. The displacements at roof are smaller than those at the walls and the bottom of the powerhouse. The upward displacement at the lower corner of powerhouse is the largest. Shear failure occurs locally. The above results are obtained in

Back Analysis and Observational Methods

201

the case of no supports. The rock mass can be stabilised by reinforcement and support. From the above analysis results, it can be concluded that the mechanics models, especially the equivalent continuous model for the jointed rock mass proposed, are applicable and validated. In addition, the rock mechanic parameters from the back analysis using those analytic models have given good results. Therefore, the analytical methods are further veriﬁed.

6.7.

APPLICATIONS OF STATISTICS MODEL IN DEFORMATION PREDICTION

In underground constructions, monitoring of rock mass deformation becomes more and more important and is widely applied. Especially when the observational construction method is adopted in tunnelling. The deformation monitoring provides the basic information for back analysis. The information obtained from deformation monitoring can be used in at least three aspects. Firstly, it is used to predict the safety during construction. The deformation monitoring can reﬂect the total deformation and deformation rate. Secondly, ordinary conditions including in situ stress and material parameters can be back analysed. Lastly, the monitored deformation can be used to predict the rock behaviour in subsequent construction [53–56,398,405,406]. This section is to discuss the last application. There are several methods to predict rock parameters and behaviour in subsequent construction, for example, the mathematical analytical method. In usual analytical method, the rock model with material parameters and in situ stresses must be determined in advance and provided to the model as input data. In back analysis method, the in situ stresses and material parameters are obtained from back analysis with monitored data and computational work. However, the statistic method is aimed at saving computational work. The method relies on the observed rock behaviour and monitored data, develops the inter-relation between the properties and behaviour, and predicts the behaviour of the rock [407–413]. This method can accordingly avoid inﬂuences of many mechanical uncertainties and provides results in short time and with high accuracy. There are many methods to be used for data regression in statistics approach, such as Laglongi, power sliding, spline, regression model, time series analysis and grey system theory [408,410,412,414–417]. In this section, a method combining non-linear regression method and modiﬁed grey system model is introduced through an example to predict displacement in a project.

202 6.7.1

Chapter 6 Non-linear regression model

As a statistical model, the regression model requires to make judgement on the data distribution and to assume a model in advance based on the user’s experience and skills. Then the parameters of the given model are recognised by least square estimation of error square sum function. Finally, statistic checking is conducted and prediction is made. Presently, linear regression model is well developed. For general non-linear regression model, the least square estimation of parameters is often carried out by using numerical modelling [24,30,33]. It is transformed to general linear model by mathematical transformation under certain conditions, and then obtain least square estimation of parameters. From the features of rock deformation curves, the regression model is assumed as hyperbolic function

u¼

t A þ Bt

ð6:28Þ

where {ti} denotes time sequence, {ui}, (i ¼ 1, 2, . . . , n) is the measured displacements. Let y ¼ t/u, then, Y ¼ A þ Bt

ð6:29Þ

So it is transformed to be a linear regression model. Setting n 1X t¼ ti n i¼1

y¼

Ltt ¼

n 1X yi n i¼1

n X

ðti tÞ2

9 > > > > > > > > > > > > > > > > > > > > > > > > =

> > > > > > > n > X > 2 > > Lyy ¼ ðyi yÞ > > > > i¼1 > > > > > n > X > > Lyt ¼ ðti tÞðyi yÞ > ; i¼1

i¼1

ð6:30Þ

Back Analysis and Observational Methods

203

It is known that B ¼ Lyt =Ltt A ¼ y Bt The linear relationship of variable y and t can be expressed by using relation coeﬃcient, Lyt r ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ltt Lyy Obviously |r| 1. When |r| ¼ 1, it becomes fully linearly related. When |r| ¼ 0, it becomes fully linearly unrelated. The nearer |r| to 1, the more linearly it is related. Maximum displacement is given as: u1 ¼ lim

t!1 A

t 1 ¼ þ Bt B

ð6:31Þ

It is the predicted ﬁnal displacement. The initial displacement rate is given as: u0t!0 ¼ 1=A From the above two formulae, the physical meaning of A and B can be recognised.

6.7.2

Grey system theory model

Because all the grey models GM have the same basic conditions and principles, the basic grey model GM(1,1) is used here for analysis [414,415]. The grey theory assumes ð0Þ ð0Þ discretised source data series as uð0Þ ¼ fuð0Þ 1 , u2 , . . . , un g, by conducting one accumulað1Þ ð1Þ tion forming treatment (AGO), generates a forming series uð1Þ ¼ fuð1Þ 1 , u2 , . . . , un g, and then creates a one-stage deviation equation, GM(1,1) is created as duð1Þ þ auð1Þ ¼ b dt

ð6:32Þ

b at b ð0Þ ¼ u1 e þ a a

ð6:33Þ

The solution of GM(1,1) is: u^ ð1Þ tþ1

204

Chapter 6

Equation (6.33) is the grey prediction formulae. Deviating uð1Þ tþ1 , we can get the prediction formulae of source data as:

ð0Þ at u^ ð0Þ tþ1 ¼ au1 þ b e

ð6:34Þ

GM(1,1) requires equal time step. However, monitoring data do not satisfy this requirement. Local internal insertion or smooth treatment can be used to form equal time step. The GM(1,1) model of equal time step is subsequently improved, to make it applicable for solving problems of non-equal time step. The improved model is applied to predict single pile capacity and showed that it was eﬀective. The key feature of non-equal time step model is to replace grey deviation by the diﬀerence form: dxð1Þ xð1Þ ðk þ 1Þ xð1Þ ðkÞ xð0Þ ðk þ 1Þ ¼ 0 ¼ ð1Þ t ðk þ 1Þ t0 ðkÞ t ðk þ 1Þ dt

ð6:35Þ

The problem now is how to take the background value of dx(1)/dt. It was noted that this value does not change signiﬁcantly when x changes from x(1)(k) to x(1)(kþ1) if t(0)(kþ1)t(0)(k) is suﬃciently small. Therefore, we have zð1Þ ðk þ 1Þ ¼ 0:5xð1Þ ðk þ 1Þ þ 0:5xð1Þ ðkÞ as the background value of dx(1)/dt between t(0)(k) and t(0)(k þ 1). The background value of dx(1)/dt can further be improved based on the features of rock deformation curves that change sharply in the initial stage. Usually u–t curves is up convex, the background value of dx(1)/dt can be taken as:

zð1Þ ðk þ 1Þ ¼ xð1Þ ðkÞ þ xð1Þ ðk þ 1Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ xð1Þ ðkÞxð1Þ ðk þ 1Þ

ð6:36Þ

From equations (6.31), (6.34) and (6.36), we have, xð0Þ ðk þ 1Þ þ atð1Þ ðk þ 1Þzð1Þ ðk þ 1Þ ¼ btð1Þ ðk þ 1Þ

ð6:37Þ

205

Back Analysis and Observational Methods Based on least squares method, we have, 1 a ¼ BT B BT Yn b

ð6:38Þ

where 2 6 6 6 6 6 6 ½B ¼ 6 6 6 6 6 6 4

tð1Þ ð2Þ

0 tð1Þ ð3Þ : : :

0 2

3 7 7 7 7 7 7 7 7 7 7 7 7 5

tð1Þ ðnÞ h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃi xð1Þ ð1Þ þ xð1Þ ð2Þ xð1Þ ð1Þxð1Þ ð2Þ h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃi xð1Þ ð2Þ þ xð1Þ ð3Þ xð1Þ ð2Þxð1Þ ð3Þ

6 6 6 6 6 6 6 : 6 6 6 : 6 4 h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃi xð1Þ ðn 1Þ þ xð1Þ ðnÞ xð1Þ ðn 1Þxð1Þ ðnÞ

1

3

7 7 7 17 7 7 7 7 7 7 7 5 1

3 xð0Þ ð2Þ 6 xð0Þ ð3Þ 7 7 6 6 : 7 7 6 Yn ¼ 6 7 6 : 7 4 : 5 xð0Þ ðnÞ 2

The measured displacements in underground excavation consist of time series {ti} and corresponding displacement series {ui}. The displacement prediction formulae is obtained from the monitored data as b b u€ kþ1 ¼ u1 eaðt1 tkþ1 Þ þ a a

ð6:39Þ

206

Chapter 6

where 2 6 6 6 ½B ¼ 6 6 6 4

3 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðu1 þ u2 u1 u2 Þ 1 7 6 ðu2 þ u3 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ u2 u3 Þ 17 7 7 6 7 7 6 : 7 76 7 7 6 : 7 7 6 5 5 4 : p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðun1 þ un un1 un Þ 1 tð1Þ ðnÞ

tð1Þ ð2Þ

0

tð1Þ ð3Þ : : : 0

3

2

3 u2 u1 6 u3 u2 7 7 6 7 6 : 7 6 Yn ¼ 6 7 : 7 6 5 4 : un un1 2

The stability of the model is often examined by backward error ratio c and small error ratio p [111]. Error is given as: "ðiÞ ¼ uðiÞ u^ ðiÞ i ¼ 1, 2, . . . , n Mean error is given as: " ¼

n 1X "ðiÞ n i¼1

Variance is given as: s21 ¼

n 1X ½"ðiÞ " n i¼1

Mean of source data is calculated by: u ¼

n 1X uðiÞ n i¼1

Variance of source data is calculated by: s22 ¼

n 1X ½uðiÞ u n i¼1

207

Back Analysis and Observational Methods Table 6.11. Classification of accuracy grade. Accuracy c p

Excellent

Good

Fair

Bad

0.95

0.80

0.70

0.65 0.70

Posterior error ratio is estimated by: c¼

s1 s2

Probability of small error is estimated by: p ¼ p "ðiÞ " < 0:6745s2 A small value of c represents a good model. A small value of c generally leads to a large s2 and a small s1. Small s1 represents small error deviation, and small c means small diﬀerence between the computational data and the actual data. A large value of p represents a good grey model, and means more points with small diﬀerence between the error and the mean of error (< 0.6745s2). The accuracy can be divided into several grades based on c and p value as shown in Table 6.11. If the c and p values are within the allowable range, the model can be used to make prediction. Otherwise, the model needs to be modiﬁed until the accuracy is in a satisfactory range.

6.7.3

Engineering application

The grey model is applied to a trial tunnel of a hydroelectric power station. The layout of extensometers is shown in Figure 6.34. Some regression displacement curves and monitoring curves are shown in Figure 6.35. The monitoring displacements and prediction results in later days are listed in Table 6.12. From Figure 6.35 and Table 6.12, it can be seen that hyperbolic model and grey model all agree well with monitoring data. The ﬁnal monitoring displacements are between the two predictions. Therefore, the use of both models is suggested.

6.7.4

Discussion

The grey model can be created only when the source series is a smooth discrete function. It requires the source series determined or has a determinable trend.

208

Chapter 6

Figure 6.34. Layout of extensometers.

Figure 6.35. Regression displacement curves and monitoring curves.

However, the feature of the grey model is to treat source series and to generate forming series, and to describe x(1) by using the grey amplitude and indirectly describe x(0). Therefore, in practice, the grey model does not require x(0) to be a smooth discrete function, but allows x(0) to have certain ﬂexibility. This extends

209

Back Analysis and Observational Methods Table 6.12. Comparison of measured and predicted displacements. Time (day)

I-3-1 M

140 149 167 173 Infinity

5.28 5.31 5.33 5.30

I-3-2

HM GM 5.29 5.30 5.32 5.32 5.49

5.21 5.21 5.21 5.21 5.21

M 5.46 5.47 5.49 5.46

II-1-1 94 103 118 128 Infinity

2.29 2.32 2.31 2.32

2.26 2.29 2.33 2.36 2.68

I-3-3

HM GM 5.38 5.39 5.42 5.43 5.66

5.26 5.27 5.28 5.28 5.29

M 5.16 5.18 5.21 5.14

II-1-2 2.14 2.22 2.32 2.37 2.77

2.57 2.59 2.59 2.62

2.58 2.60 2.62 2.64 2.83

I-3-4

HM GM 5.07 5.10 5.13 5.14 5.47

4.92 4.93 4.95 4.96 4.98

M 5.10 5.11 5.13 5.03

II-1-3 2.49 2.50 2.51 2.52 2.52

1.44 1.46 1.43 1.41

1.46 1.47 1.48 1.50 1.62

I-3-5

HM GM 4.81 4.84 4.87 4.88 5.15

4.94 4.98 5.05 5.09 5.21

M 4.17 4.16 4.18 4.15

II-1-4 1.50 1.51 1.52 1.53 1.54

2.15 2.14 2.13 2.10

2.41 2.15 2.16 2.17 2.26

HM GM 4.04 4.05 4.08 4.09 4.31

4.15 4.19 4.28 4.31 4.60

II-1-5 2.19 2.19 2.20 2.20 2.21

1.66 1.65 1.63 1.61

1.56 1.57 1.59 1.60 1.71

1.57 1.59 1.62 1.63 1.68

Note: (1) M – Measured; HM – Hyperbolic Model; GM – Grey Model; (2) I-3-1 refers to measurement point 1 in Section I MPBX 3, similarly for other points.

the application range of the grey model. The behaviour of rock displacement in underground projects makes the grey model more suitable and applicable. However, all models are created based on the existing monitoring information. If geological conditions of the excavation are changed, the models need to be re-created or otherwise the expected prediction accuracy will become low. Generally the grey model is applicable in predicting ﬁnal displacements in underground engineering. However, hyperbolic model is applicable for describing monitoring displacements that follows a simple function. It is not suitable for other situations.

This Page Intentionally Left Blank

Chapter 7

Construction Mechanics and Optimisation of Excavation Schemes Construction of rock engineering projects, especially large-scale underground excavations in rock masses, usually requires a long period to complete, i.e., a few months to a few years. To the extreme, a mining project may take tens or hundreds of years. The construction of these rock engineering projects will disturb the initial stable state of the rock masses. After that, various rock mass parameters interact in a dynamic interactive process until the rock mass reaches a new stable state [1–3]. The construction is therefore a dynamic interactive process in time and in space. The success in constructing and managing a rock project not only depends on the eventual state of the project, but also on the interim process and the construction methods adopted [63–65,69,70]. Construction of large-scale rock projects is implemented by continual excavation of new working faces. Each newly excavated face interacts dynamically with the existing excavated space in time and in space [63,64,66,69]. Rock mass is a geological medium, which is generally discontinuous, inhomogeneous and anisotropic. The rock mass has uncertain and variable parameters, which are further changed by the engineering activities. It is very important to consider the eﬀects of engineering activities, especially under complex geological conditions such as high rock stress, weakness zones, faults, joints, and ground water [69,180,348,350,406]. For any large-scale rock engineering project, the eﬀects of engineering activities must be taken into account for the design and construction. From the viewpoint of mechanics, the dynamic interactive process of rock engineering constructions is non-inverse and non-linear. Its eventual state (or solution) is not unique but changeable with the interim process [1–3]. In other words, the eventual state is strongly dependent on the stress paths or stress histories. This leads to the necessity of the optimisation of construction process. It is generally impossible to complete the construction of any large-scale projects just at one stage of full face excavation, especially for large underground caverns. In practice, they are constructed step by step following a certain sequence of excavation [67,68]. The excavation scheme is decided according to the layout of access tunnels, types of tunnelling machines and characteristics of rock masses. In the series of sequential construction, each step of construction corresponds to a certain type of temporary cavern geometry, i.e., diﬀerent sequences of construction correspond to the diﬀerent temporary loading conditions. During and after the 211

212

Chapter 7

construction, rock mass parameters are aﬀected by the continually variable cavern geometry and loading conditions. Typical rock mass parameters are rock stress, rock damage and circumferential displacement around the cavern. The disturbance to the surrounding rock mass and the redistribution of rock stress are primarily due to excavation activities. Therefore, excavation sequences and associated methods have widely crucial implication on the deformation and stability of underground excavation [418–437]. Many engineering measures have been proposed to stabilise the large-scale underground excavations. Of them, the most eﬀective measure is to adopt a proper excavation sequence and to install eﬀective rock reinforcement [418–426]. This forms the principle of the construction mechanics. The initiation and development of the construction mechanics are presented and discussed in various literatures [8,9,52,63,68,404,431]. It was proposed that the excavation steps should be reduced to a minimum number, because the rock mass is of very poor strength against cyclic loads and movements [68,404]. Some researchers studied the eﬀects of excavation sequences on the rock stress distribution at diﬀerent points of surrounding rock mass [68]. Some suggested that the excavated geometry should match the initial distribution and orientation of ﬁeld stresses [194,212,432]. Zhu et al. [52] conducted comparative studies on multiple schemes of diﬀerent excavation sequences, and obtained an optimum scheme for improving the surrounding rock mass. Yu and Yu [433] presented a new concept of rock memory and emphasised the importance of analyses on the functions of surrounding rock masses. The basic principles and applications of construction mechanics are brieﬂy discussed in the following sections.

7.1.

7.1.1

BASIC PRINCIPLES OF INTERACTIVE CONSTRUCTION MECHANICS

Basic principles

It has been commonly recognised that the schemes and sequences of rock excavations and supports have signiﬁcant eﬀects on the stabilisation and cost of underground rock engineering projects. The approach of the New Austrian Tunnelling Method (NATM) actually reﬂects the signiﬁcant eﬀects imposed by excavation activities. NATM emphasises the eﬀects of excavation methods and attends to the economical aspects of construction [438]. It does not, however, examine the mechanisms of construction from a wider viewpoint of philosophy. It is necessary to examine a new concept in rock mechanics and engineering – interactive construction mechanics of rock engineering. The concept of interactive construction mechanics applies the rock mechanics theory to rock engineering practice, by examining the interactive mechanics in the

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213

excavation and support process [8,9,434]. The principles of the interactive construction mechanics are outlined in detail in the following paragraphs. (a) Rock engineering construction under complex rock mass conditions is a complete open system. This open system is aﬀected by the uncertainty in natural geological factors. As a result, analyses of rock mass stability and estimation of construction cost become a systematic work. In order to take the full consideration of the eﬀects of various factors, it is required to study not only the natural factors (e.g., geological conditions, initial stress and mechanical properties of rock mass), but also human factors (excavation sequence, excavation method and geometry). The main idea of this principle is to view a rock engineering project as an open and interactive system rather than a close and static system. The interactive viewpoint emphasises both the natural factors and the human activities. For example, optimisation methods may be adopted in advance at the design stage to minimise the potential problems. At construction stage, pre-measures may be taken to reduce the possibility of anticipated problems. Optimisation analyses should be conducted in advance. (b) Rock mass stability and economic aspects during and after construction are not only related to the eventual states, but also to the excavation sequences and methods. This is because the boundaries of excavated rocks change in time and in space. From the mechanics point of view, construction is a non-linear process that is related to both the eventual states and stress paths as well as stress histories. The sequences of large cavern construction under complex geological conditions can signiﬁcantly inﬂuence the safety and economy of the caverns during excavation and operation. The cavern excavation is actually a process of loading and unloading the rock masses at diﬀerent time and positions. As the surrounding rock mass is a non-linear mechanical medium, the diﬀerent excavation sequences imply diﬀerent stress paths and histories imposed on the rock mass. Hence, it produces very diﬀerent damage to the rock mass. The diﬀerence in damage produced by the various sequences of excavation may be very signiﬁcant. (c) The approach should be adopted before the construction to determine the optimum excavation schemes for ﬁnal decision-making. In the optimisation analyses, rock supports should be considered as an important factor. Optimisation analyses on the excavation sequences should be taken after the cavern design but before the actual construction, in view of the mechanics of excavation process. (d) The design and control of construction process have signiﬁcant eﬀects on the rock engineering projects. It is important to understand the interaction and the response of surrounding rock masses, to adopt proper excavation methods and

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rock supporting measures, and to minimise the potential instability. It emphasises the control on the construction process. Careful considerations must be taken to stabilise the rock masses by choosing very suitable construction technologies and methods, such as charge weight of blasting, instant rock bolting and shotcreting, and groundwater drainage. That is, the rock mass damages resulting from the excavating and supporting activities should be minimised. (e) The optimum construction scheme should be continually modiﬁed during the construction based on observation and monitoring of the surrounding rock mass. The information on rock mass conditions shall be updated during the construction, so that the existing scheme can be continually assessed and improved. The actual rock mass conditions may be diﬀerent from expected conditions during excavation due to the high uncertainty in natural geological environment. The traditional site investigation techniques can only access to a very small quantity of rock mass. Hence, in situ monitoring can be used to verify and modify the predictions of the rock mass. Furthermore, the new information obtained in monitoring can be used for back analyses, or statistical analyses. Accordingly, modiﬁcations can be made to the originally proposed mechanical model of rock masses, geological and mechanical parameters, and eventually to the rock structure design and rock supporting measures. The modiﬁcations should be continually conducted during the excavation. (f) Site investigation, design, construction and research shall be fully combined within an integrated system. It is unwise to strictly follow a speciﬁc scheme and to ignore the changes in rock mass conditions. Instead, modiﬁcation of the existing scheme should be continually carried out according to the current conditions. This principle emphasises the policy guarantee for the implementation of the above principles. It is because the construction of large-scale complex rock projects requires interactive management and close cooperation between geologists, design engineers, contractors, researchers, clients and site engineers. One of the diﬀerences between rock engineering projects and other civil engineering projects is that rock engineering projects require continual updating of the site data and modiﬁcations to the initially proposed design schemes. The excavation methods and schemes should have the ﬂexibility for timely updating and modiﬁcations, which need to be reﬂected in the design codes and the construction speciﬁcations.

7.1.2

Engineering applications

Applications of the construction mechanics to an engineering project are illustrated in this section to demonstrate the necessity and importance of optimisation analyses on excavation sequences.

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7.1.2.1 Description of the project. The hydroelectric power project is a large underground cavern complex, consisting of a main powerhouse cavern, a transforming cavern and a tailrace cavern. The powerhouse cavern is 31.2 m wide and 72.6 m high, the transformer chamber is 17.4 m wide and 36.0 m high, and the surge chamber is 21.7 m wide and 76.9 m high. Bolts and shotcrete are needed for the rock reinforcements. The rock type is syenite. The in situ horizontal ﬁeld stress at the main cavern is about 25 MPa. The cavern complex is 200 m deep below ground [52]. The initial in situ stress distribution is analysed by the method described in Chapter 6. The plane strain model is used to analyse the changes in stress ﬁelds disturbed by diﬀerent excavation sequences. The ﬁnite element mesh is composed of 815 quadrilateral elements and a few triangular elements. The central part of the mesh around the caverns is shown in Figure 7.1. An improved two-dimensional non-linear ﬁnite element program, is used in the computation. The mechanical properties of the rock masses are summarised in Table 7.1. Figure 7.2 shows the constitutive model of the rock mass. The strength criterion shown in Figure 7.3 is mainly based on the

Figure 7.1. FEM meshes of the surrounding rock masses.

Table 7.1. Mechanical properties of the rock mass. Mechanical properties Young’s modulus E (MPa) Poisson’s ratio m Internal frictional angle j ( ) Cohesion C (MPa) Tensile strength (MPa)

Elastic properties

Residual properties

3.5 104 0.2 60 5.0 1.7

54 1.25 0

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Figure 7.2. Constitutive model of rock masses.

Figure 7.3. Strength criterion.

Prager–Drucker criterion, which is further modiﬁed by increasing the cohesion to a reasonable value. The distributions of bolts and pre-stressed cables are shown in Figure 7.4 [52]. The contributions of rock bolts to rock mass stiﬀness are treated by equivalent area and stiﬀness in the computational model, while the contributions to rock mass strength are treated by increasing the cohesion (internal frictional angle remains unchanged). These treatment methods adopted have been discussed in Section 3.3 of Chapter 3.

7.1.2.2 Computational implementation and results for different excavation sequences. A total number of 11 computational schemes are compared to examine the inﬂuences of diﬀerent excavation sequences on the rock mass stability. This includes the schemes of one-, ﬁve- or six-stage excavation. The detailed explanations

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Figure 7.4. Installation scheme of rock bolts and cables.

to the diﬀerent schemes are shown in Table 7.2 and Figure 7.5. The computational results on some of the schemes are compared and discussed in the following sections: (i) Non-linear one-stage excavation As an extreme case, the scheme of non-linear one-stage excavation is ﬁrstly considered. The stress in the surrounding rock mass increases, and becomes greater than the strength. The over-stressed area continually expands into the surrounding ﬁelds. The damage zone is larger than that obtained from elastic analysis. A large plastic zone connecting the three caverns eventually forms, as shown in Figure 7.6. (ii) Non-linear five-stage excavation (5–I and 5–II) The excavation sequence is given in Table 7.2. The computational results indicate that the damage area is signiﬁcantly reduced in comparison with the one-stage excavation scheme. The diﬀerence in displacements at cavern corners also decreases. However, the damage area around each cavern is still connected, forming a continuous area. After the rock bolts are taken into account in the computation, the results indicate that the damage areas around the left two caverns are disconnected, but those around the right two caverns are still inter-connected, as shown in Figure 7.7. This implies that the results are not perfect. (iii) Non-linear six-stage excavation (6–I, 6–II and 6–III) This is the suggested excavation scheme by the design engineers. It is divided into three sequences of excavation: 6–I, 6–II and 6–III (Table 7.2). The eventual stress state for Scheme 6–I is shown in Figure 7.8. Although the total damage area is smaller than that of onestage scheme, the damage areas between the three caverns are still connected. In this six-stage scheme, the damage area is in fact slightly larger than that of the ﬁve-stage scheme. This implies that more excavation stages do not always lead to less damage

218

Table 7.2. Schedule of different excavation schemes and steps for three caverns. No 1

No 2

No 3

No 4

No 5

No 6

Note Elastic one-step excavation

Elasto-plastic one-step excavation

(5–III) (6–I)

A,B,C,D,E, F,G,H,I,J, K,L,M,N,P A,I A

B,F,J I

G B,F,J

H,K,L,M C,D,E

C,D,E,N,P G,K,L,M,N,P

5 6 7 8

(6–II) (5–III) (5–II) (5–II)

A,I A,I A,I A,I

B,F,J B,J B,F,J B,F,J

G C,K G G

H,K,L,M D,F,L H,K,L,M H,K,L,M

N,P E,G,M,N N,P C,D,E,N,P

9

(5–II)

A,I

B,F,J

G

H,K,L,M

C,D,E,N,P

10

(6–III)

A,I

B,J

C,K

D,F,L

E,G,M,N

H,P

11

(6–III)

A,I

B,J

C,K

D,F,L

E,G,M,N

H,P

(1–I)

2

(1–II) (1–III)

3 4

Note: assume that tunnels are completed at the first step of excavation.

H C,D,E H,P C,D,E

The originally proposed scheme

No tunnels Having rock bolts (only short bolts render support to rock mass) Having condensed rock bolts (both short and long bolts render support, but they are not pre-stressed) Having condensed rock bolts and pre-stressed rock cables Having condensed rock bolts and pre-stressed rock cables; subject to additional load induced by crane beams

9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > ;

Chapter 7

A,B,C,D,E, F,G,H,I,J, K,L,M,N,P

1

Elasto-plastic

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Figure 7.5. Diﬀerent schemes for multiple-stage excavation.

Figure 7.6. Damage zones and displacements of one-stage excavation without support (non-linear analysis).

Figure 7.7. Damage zones and displacements after ﬁve-stage excavation with anchoring of rock bolts (non-linear analysis).

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Figure 7.8. Damage zones and displacement of last stage of six-stage excavation without anchoring (non-linear analysis).

Figure 7.9. Damage zones of Scheme 6-III, (a) at excavation stage 2, (b) at excavation stage 4, (c) at excavation stage 6, (d) at excavation stage 6 with reduced bolt spacing.

of the surrounding rock mass. Scheme 6–III is suggested based on the analysis and experience. The two caverns at the left and right sides are ﬁrst excavated, and the cavern at the centre is excavated later. Figure 7.9 illustrates the damage area and displacement distribution at Stages 2, 4 and 6 in the excavation process. It can be seen that the eﬀects are very signiﬁcant in isolating the damage areas between the three caverns, reducing the damage zones around each cavern (within 20 m thickness) and reducing the displacements at the cavern perimeter (half the displacement in the previous case). Figure 7.9d indicates that the rock mass stability

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is further improved after reducing the spacing of rock bolts and cables. The damage areas are therefore signiﬁcantly reduced such that many of them become isolated individual small areas.

7.1.2.3 Discussions. (a) The results from numerical analyses show that diﬀerent excavation sequences produce very diﬀerent eﬀects on the rock mass stability. The one-stage excavation may generate large damage areas and displacements at the cavern perimeter. This implies that the stress history plays an important role, and the optimisation of excavation sequences is necessary. (b) In the situation of multiple cavern excavations in rock mass with high horizontal stress, in order to minimise the damage zone, excavations should be conducted at diﬀerent times. In concurrent excavations, the excavations should be kept at far distance to reduce the interaction between the excavations. (c) The excavated volume at each stage of the excavation should be limited to a reasonable amount in order to reduce the area of over-stressing. In addition, rock support measures such as rock bolt and shotcrete should be applied instantly. All of these have positive eﬀects on the rock mass stability. (d) At the connections between caverns and tunnels, three-dimensional numerical analyses should be conducted, because the two-dimensional analyses may not give quantitative results with the simpliﬁed model. (e) Since the above method of excavation optimisation cannot consider the energy loss in rock mass, the damage areas are generally over-estimated. So they should be normally used as the reference values for comparisons between diﬀerent excavation schemes. Back analysis from monitoring can overcome the shortcoming of the above method.

7.2.

APPLICATIONS OF INTERACTIVE PROGRAMMING IN OPTIMISATION OF CAVERN CONSTRUCTION

The concepts and principles of the interactive construction mechanics of rock engineering, together with the applications to an engineering project have been presented in the previous section. However, the method used has its limitations. For example, the schemes for optimisation analyses are selected based on engineering experiences. It is possible that the optimal scheme obtained with this method is a local optimal scheme within the limited number of proposals rather than a global optimal scheme. In this section, the interactive programming principle and associated methods are discussed and applied to construction optimisation.

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7.2.1 Principles of interactive programming The method of interactive programming, originally proposed by Bellman [439] is a speciﬁc technique solving the optimisation problems. Generally, optimisation is the process of ﬁnding the best solution to the problems from a group of possible solutions according to the given requirements. Correspondingly, the interactive programming is a multistage sequential decision-making process of ﬁnding the best solution. An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the ﬁrst decision. This is known as Bellman’s Principle of Optimality [439]. The main advantages of the interactive programming method are: (a) An optimisation problem of multiple (n) dimensions can be changed into n problems of one dimension, which may be solved in a sequential manner. This solution-searching procedure cannot be implemented with the traditional optimisation methods. (b) The global maximum or minimum value can be directly determined. By contrast, the traditional optimisation methods possibly give the local maximum or minimum value, which leads to the diﬃculty of judging the globality or locality of the solutions obtained. In fact, the interactive programming method is not an ‘algorithm’ but an approach to break and reconstruct the problems so that a suitable optimisation method can be applied. The following technical terms and statements are commonly used in the interactive programming method in describing the optimisation problems. A physical system can always be taken as a hierarchical system in a certain sequence. The interactive programming method divides the optimisation problem into a series of stages, which may represent, for example, time or space. At each stage, the state of system can be depicted by a relatively small group of variables. These variables are termed as state variables or state vectors (or simply state). Single or multiple decisions need to be made in any state of system at each stage. These decisions may depend on either the stage or the state, or both, while the history during which the system comes to the current state or stage is not important. In other words, the decision-making is only based on the current stage or state. After a decision is made, a beneﬁt is obtained correspondingly. At the same time the system changes to the next stage. The beneﬁt is governed by a known single-value function of a certain given state. Similarly, the state of the system after alteration is governed by a single-value function of the current state altered. The eventual objective of hierarchical development of the system stages is to ﬁnd the maximum or minimum values of the function of state variables. The key elements related to the interactive programming are stage, state, decision, alteration and beneﬁt.

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(a) Stage: The concept of stage is required in representing the sequence of decisionmaking. The function of state is to give the number of the sequences involved in the system developing process. Many problems seemingly without stage nature can still be analysed in a stage manner. For example, during the process of excavation, several excavation phases may be combined into one excavation stage. The optimisation problems of multiple variables can be treated as a process of multiple decision-making according to a certain sequence. During this process, each step in the sequence can be represented by a stage, which depends on the current state. (b) State: State space can be represented as a non-nil combination 1 . An element l 2 . This element is called as state, which is the description of a variable (or a group of variables). The state space is composed of all the state variables. For example, during the process of excavation, the state variables describe the current step in excavation sequences within the excavation system. The corresponding state space is composed of all the possible excavation steps. (c) Decision: The system state must include all the information required for determining all possible decisions. Therefore, for each state l 2 , there is a non-nil combination Xl. It is termed as decision combination of l. One of the elements, X3 ðlÞ 2 Xl is a decision or a decision variable. It represents an allowable choice when the system is at the state l. Decision combination Xl is composed of all the possible choices when the system is in the state l. For example, as shown in Figures 7.10 and 7.11, the decision under the state P1T1T2T3 is T4 or P2.

Figure 7.10. Excavation stages of the powerhouse and transformer chamber.

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Figure 7.11. Possible combinations of excavation sequences.

(d) Beneﬁt: Because a problem of dynamic programming is an optimisation problem, there may be an objective function to assess the given decisions. The total beneﬁt is a combination (sum or product) of beneﬁts at all the stages. It is the accumulation of state alteration (from one stage to another). The beneﬁt function is changeable from one stage to another. There is a requirement for proper deﬁnition of the beneﬁt at each stage. For example, for the optimisation of the cavern construction, the beneﬁt function can be deﬁned as the area of rock damage around the caverns.

7.2.2

Applications to the optimisation of cavern construction

A hydropower station is chosen to demonstrate the applications of the interactive programming method. The station mainly consists of a powerhouse cavern (20 m span and 45 m height) and a transformer cavern (18 m span and 28 m height). The design excavation scheme includes six stages (numbered as P1, P2, . . . , P6) for the power house cavern and in four stages (numbered as T1, T2, T3, T4) for the transformer cavern, respectively, as shown in Figure 7.10. To simplify the optimisation problem, some assumptions are made as follows: (a) Caverns are excavated in the sequence from top to bottom within each stage; (b) Each excavation stage is considered as one excavation phase; and

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(c) Stage P1 corresponds to the ﬁrst phase of excavation for the powerhouse cavern, and so on. A total of 168 possible construction schemes can be obtained. However, it is impossible to conduct ﬁnite element computations for all the 168 excavation schemes. In the past study, a limited number of schemes were selected for the ﬁnite element computations, and the computing results are compared to obtain a local optimal scheme. In the present study, a global optimal scheme is determined from n number of possible schemes, and, n ¼ X sNc where Xs is the total number of excavation sequences, and Nc is the number of caverns. For the case in this study, n ¼ 10 2 ¼ 20. That is, only 20 ﬁnite element (FEM) computations are required to obtain the global optimal scheme. Figure 7.11 schematically illustrates some of the possible combinations of excavation sequences. In the FEM computation, rock mass is treated as an isotropic elastic-plastic medium. The mechanical properties are shown in Table 7.3. The in situ ﬁeld stress is calculated based on the back analysis conducted in Section 6.5 of Chapter 6, sx ¼ 9:43 MPa, sy ¼ 7:43 MPa, xy ¼ 1:24 MPa To investigate the eﬀects of in situ stress on selecting the construction sequences, four conditions of stress ﬁeld are assumed by changing the angle a between s1 and horizontal direction. These are: (a) a ¼ 25.2 , (b) a ¼ 0 , (c) a ¼ 90 , and (d) a ¼ 25.2 , as shown in Figure 7.12. The beneﬁt function is deﬁned by the area of rock damaged around the cavern. Figure 7.13 illustrates the optimisation process for Case (a), where the value in the bracket is the beneﬁt. Figure 7.14 shows the distribution of damage zones around the cavern corresponding to the four cases in Figure 7.12. Table 7.4 shows the computational results for the four conditions of ﬁeld stresses of Figure 7.12.

Table 7.3. Mechanical properties of the surrounding rock mass. Parameter Value

E (MPa)

n

c (MPa)

f ( )

Rt (Mpa)

35000

0.22

1.30

50.20

5.0

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Figure 7.12. Diﬀerent distributions of principal stresses.

Figure 7.13. Optimisation process for excavation case (a) in Figure 7.12.

Figure 7.14. Damage zones in surrounding rock masses corresponding to cases in Figure 7.12.

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Table 7.4. Computational results for different stress field in Figure 7.12. Stress field Optimal scheme Damage area (m2) Maximum horizontal relative displacement

(a)

(b)

(c)

(d)

P1. . .P5T1 . . .T4P6 836 1.24

The same as (a) 1119 1.46

P1P2T1T2P3 T3T4P5P6 1370 0.87

The same as (a) 1282 1.44

Table 7.5. The eventual damage area and displacements for different construction schemes. Schemes Damage area (m2) Maximum horizontal relative displacement

1

2

3

4

5

1730 1.90

859 1.21

928 1.36

938 1.23

836 1.24

To compare the results for diﬀerent construction schemes under the same stress ﬁeld (a), the following schemes are ﬁrst selected: Scheme 1: PT (one-step full-face excavation) Scheme 2: P1-P2-P3-P4-P5-P6-T1-T2-T3-T4 Scheme 3: P1-T1-P2-T2-P3-T3-P4-T4-P5-P6 Scheme 4: P1-P2-T1-P3-P4-T2-P5-P6-T3-T4 (actual construction) Scheme 5: P1-P2-P3-P4-P5-T1-T2-T3-T4-P6 (optimal construction) The damage area and the displacement at the cavern edges for the above ﬁve schemes are compared in Table 7.5.

7.2.2.1 Discussions. The interactive programming method is more eﬃcient than the traditional enumerating method in the optimisation of large volume underground excavation. Although it cannot be totally ensured that the obtained optimal scheme is the real global one due to constraints from the assumptions and the specially deﬁned beneﬁt function, the result can at least be taken as the relatively global optimal scheme under the assumed conditions. Improvement is still required in the future. The following conclusions can be drawn from this study: (a) After the excavation steps are determined, the optimal excavation scheme is signiﬁcantly inﬂuenced by the in situ stress magnitude and direction. (b) The optimal scheme searching method used in the present study largely shortens the computation time in comparison with the traditional enumerating method.

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The required computing time is proportional to the product of the total excavation steps and cavern number. (c) The beneﬁt function needs to be properly pre-determined. For example, it can be the convergent displacement, damage area, or both. On the other hand, because the beneﬁt function adopted in the present study is the damage area at each step of excavation rather than the accumulated damage area after the last excavation step, it may not be ensured to ﬁnd the global optimal scheme. Global optimal scheme can be obtained by combining the beneﬁt functions. (d) In the present study, only stress ﬁeld is taken into consideration. In the cases of presence of major geological structures, further investigation is required. (e) The results in this study are obtained through two-dimensional numerical computation. They provide qualitative basis for three-dimensional FEM problems of cavern construction. In fact, it is very diﬃcult to fully solve the optimisation problems by three-dimensional numerical modelling. The possible approach to the three-dimensional optimisation problems is to compare the computational results for a few representative construction schemes. (f) It can be seen from Table 7.5 that the cavern layout and excavation steps have been arranged in such a way that an optimal construction scheme can be achieved. However, the actual construction may not follow the optimal scheme. The damage area in the actual construction is 12% larger than that in the optimal construction scheme, while the maximum horizontal displacement at sidewalls of the cavern is almost the same. It should be noted that the method in this study provides the optimal scheme for construction sequences in view of rock mass stability of the caverns. Before the method is used, the excavation steps have been pre-determined. Cavern construction optimisation shall be conducted by considering both the rock stability and the economics, based on the site conditions. 7.3.

ARTIFICIAL INTELLIGENCE TECHNIQUES IN CONSTRUCTION OPTIMISATION

In the previous section, the interactive programming theory is applied and engineering case studies are conducted to illustrate the method of cavern construction optimisation. However, there are still two problems to be solved: (a) determining all the possible construction sequences; and (b) creating the input data ﬁles required by the numerical computations according to the selected construction sequences. Traditionally, this work is manually implemented with very low eﬃciency. The artiﬁcial intelligence technique has the capabilities of reasoning, understanding, planning, decision-making and learning [1–3,440–443]. Its applications to cavern

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construction optimisation can help to solve the above problems. In this section, the artiﬁcial intelligence technique is utilised in such aspects as automatic determination of construction sequences, automatic generation of input data ﬁles and automatic decision-making for construction optimisation.

7.3.1 Artificial intelligence language prolog Prolog is a very important language tool in artiﬁcial intelligence programming, and is mainly used to deal with logic problems. Prolog language enables computers to have reasoning capability in problem solving. Since it was produced in 1970s, it has quickly become a popular language of artiﬁcial intelligence. Prolog is similar to natural languages, and is therefore easy to learn and utilise by the users. The computer program written with this language is simple and understandable. The reasoning capability of prolog has lead to wide applications in the ﬁelds of database, logic algorithm, expert system, natural language interpretation, and automatic programming. One of the typical prolog languages is Turbo Prolog. Turbo Prolog has kept the main advantages of the traditional explanation type prolog, with enhanced runningspeed improved functions. In comparison with other computer languages such as Pascal, Turbo Prolog is a more advanced language. The program written with Turbo Prolog is 10 times shorter than that with Pascal for the same problem. This is because Turbo Prolog has an internal pattern-matching mechanism, and a simple but eﬀective method to deal with recurrent process. These two advantages are utilised in this study for construction sequence determination.

7.3.2 Problem solving algorithm in cavern construction optimisation 7.3.2.1 Automatic determination of cavern excavation sequences. Determining cavern excavation sequence is to ﬁnd out all the possible excavation sequences. Suppose that a cavern complex comprises Cavern A and Cavern B. Caverns are divided into diﬀerent excavated faces a1, a2 , . . . , a6 and b1, b2 , . . . , b4, as shown in Figure 7.15. The basic rules for excavation are (Zhu et al. 1992): (a) Excavation face a1 is the ﬁrst step; (b) One face is excavated as one step; (c) Each cavern is excavated from top to bottom. A data table is the basic structure in Turbo Prolog language. It is equivalent to the data group in Pascal language. This table consists of a set of sequential elements. The order of data in the table is an important feature of table. In this context, the

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Figure 7.15. Stepwise excavation of cavern complex.

excavated faces in each excavation step are represented by the elements of the table, and the excavation sequence is represented by the element order in the table. For example, a table L ¼ [a1, b1, a2, b2, a3, b3, a4, b4, a5, a6] represents that: (a) the excavated face at ﬁrst excavation step is a1, (b) the face at the last step is a6, (c) the element sequence in the table is the excavation sequence, and (d) there are a total of 10 excavation steps. For the initially generated table [a1, a2, a3, a4, a5, a6, b1, b2, b3, b4], the element order in the table is quickly determined according to the above three rules. The results are written into the output ﬁle ‘order.out’, which includes the combination of all the possible excavation sequences. To implement the above process of sortation in the table, eleven predicates are deﬁned. It is commonly known that a sentence in computer program is of two-fold characteristics: description and operation. When a computer program is constructed, people always pay attenuation to its operation. However, when the program needs to be veriﬁed and explained, people always concern about its description. When a logic algorithm is constructed, one of the methods for combining the two characteristics is to construct the program by considering its operation at ﬁrst instance, and then translate it into descriptive sentences. A computer program is generally constructed from top to bottom according to the sequential steps of problem solving. In other words, a general problem is ﬁrstly broken into several sub-problems, and then the sub-problems are individually solved. This is a natural process to construct computer programs. Describing the logic in the table order is to ﬁnd the sequential position exchange of the elements in the table. The logic module is ‘sort (Xs, Ys)’, where Ys is the table obtained when all the elements in Xs satisfy the ‘ordered’ condition. sort (Xs, Ys): – permutation (Xs, Ys), ordered (Ys) The statement ‘sort (Xs, Ys)’ has been broken into ‘permutation (Xs, Ys)’ and ‘ordered (Ys)’, which are further explained as below.

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Checking whether a table is arranged in the right order according to given requirements can be implemented with the following sentences: ordered ([-]). ordered (Ys): — is_after (a2, a1, Ys), / *a2 and Ys are after a1 */ is_after (a3, a2, Ys), is_after (a6, a5, Ys), is_after (b1, a1, Ys), is_after (b2, b1, Ys), is_after (b4, b3, Ys), ‘Fact’ indicates that only the tables consisting of one element have been arranged in order, while ‘rule’ indicates that the order must satisfy the three assumptions. Permutation represents a manner of element position exchange in the table. That is, an element is randomly selected, then it is taken as the ﬁrst element in the table, and then the other elements are iteratively exchanged. Its basic fact indicates that a nil table is the only exchange. Permutation ([a1 , . . . , an], L) feeds back the result that L has a total number of n! solutions. Permutation (Xs, [ZjZs ]): —select (Z, Xs, Ys), permutation (Ys, Zs) Permutation ([ ], [ ]) Determining the relative positions of two elements involves three predicates: member, position and is_after. Member is to determine whether an element is involved in the table: member (X, [XjXs ]), member (X, [YjYs ]): —member (X, Ys). The above program means: if X is the ﬁrst part of the table, or X is the element of the last part of the table, then X is an element in the table. Position is to calculate the position order of an element in the table: position (A, position (A, member (A, position (A, N ¼ No þ 1

[A|_ ], 1), [ _|T], N): — T), T, No),

It is speciﬁed that A is placed at the ﬁrst position if A is the ﬁrst element in the table. The meaning of ‘is_after (A2, A1, L)’ is: if the position number of A2 is bigger than that of A1, A2 is after A1 in the table.

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The result of selecting X from [XjXs ] is Xs, or, if the result of selecting X from Ys is Zs, then the result of selecting X from [YjYs ] is [YjZs ], i.e., select (X, [XjXs ], Xs) select (X, [YjYs ], [YjZs ]): — select (X, Ys, Zs) ‘Writelist (L)’ and ‘write10 (list, N)’ mean that a table is exported and each line of the table contains N symbols. ‘Run’ means that the results of order arrangement are saved in the ﬁle ‘order.out’. ‘Make-cut (L): L ¼ L0’ (cut/fail method) means that the program is repeatedly run until L is equal to L0. L0 can be displayed on the screen in advance. In general, the above method can be described as: the initial table of excavation sequence is processed through Predicate ‘sort’, and then a table of order arrangement is produced according to the ‘ordered’ rule. The program Autoorder.PRO for generating the excavation sequence is described as below: ———————— Autoorder.PRO code ¼ 10000 trail ¼ 4000 errolevel ¼ 0 domains A ¼ symbol N ¼ integer ﬁle ¼ myﬁle list ¼ symbol* predicates run ordered (List) member (A, List) make_cut (List) write10 (List) sort (List, List) writelist (List) is_after (A, A, List) position (A, List, N) select (A, List, List) permutation (List, List)

Construction Mechanics and Optimisation of Excavation Schemes clauses position (A, [A|_ ], 1). position (A, [ _|T], N): — member (A, T), position (A, T, NO), N ¼ NO þ 1. is_after (A2, A1, L): — position (A1, L, K1), position (A2, L, K2), K2 > K1. member (Name, [Name|_ ]). member (Name, [ _|Tail]): — member (Name, Tail). select (P, [P|Xs], Xs). select (P, [Y|Ys], [Y|Zs]): — select (P, Ys, Zs). permutation (Xs, [Z|Zs]): — select (Z, Xs, Ys), permutation (Ys, Zs). permutation ([ ], [ ]). writelist (NL): — write10(NL, 0), nl. write10 (TL, 10): — !, write10 (TL, 0). Write10 ([H|T], N): — !, write (H, ‘‘ ’’), N1¼Nþ1, write10 (T, N1). Write10 ([ ], _ ). ordered ([ _ ]). ordered (Ys): — is_after (a2, a1, Ys), is_after (a3, a2, Ys), is_after (a4, a3, Ys), is_after (a5, a4, Ys), is_after (a6, a5, Ys), is_after (b1, a1, Ys), is_after (b2, a1, Ys), is_after (b3, b2, Ys), is_after (b4, b3, Ys),

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sort (Xs, Ys): — permutation (Xs, Ys), ordered (Ys). run: — openwrite (myﬁle, ‘‘order.out’’), sort ([a1, a2, a3, a4, a5, a6, b1, b2, b3, b4], L), writedevice (myﬁle), writelist (L) make_cut (L), !, closeﬁle (myﬁle). Make_cut (L): — L ¼ [a1, b1, b2, b3, b4, a2, a3, a4, a5, a6]. goal run. ———————— It can be seen that the program is simple and easy to understand. Figure 7.16 shows the computational results on all the possible construction sequences (a total of 126 schemes). It only requires about one minute to run the program in a 486 computer. By comparison, the manual method is time-consuming and some schemes may be missed. Figure 7.17 schematically illustrates partial manual results by manual sorting.

7.3.2.2 Automatic generation of data files for finite element computation. The ﬁnite element program used in this study is written by the authors. The structure of data ﬁles required for determining the excavation sequences is shown in Figure 7.18. Each data ﬁle above is divided into two ﬁles: const.dat and excav.dat. The ﬁrst one is the same for each excavation scheme, and the second one is changeable with diﬀerent excavation schemes. The information on multi-stage excavation is saved into data ﬁles named according to the excavated faces. For example, Data ﬁle a1 contains the information on excavating face a1. Data ﬁles a1, a2, a3, a4, a5, a6, b1, b2, b3, b4 are then processed to form a complete ﬁle excav.dat, which represents excavation scheme [a1, a2 , . . . , a6, b1, b2 , . . . , b4]. Two predicates are used in the processing: ‘ﬁle_str (Filename, Text)’ and ‘form_fort.ﬁle (L)’. The ‘ﬁle_str (Filename, Text)’ reads symbols from a data ﬁle and sends them into a variable or a ﬁle. The ‘form_fort ﬁle (L)’ deﬁnes a data ﬁle as the symbol at the top of the table (the ﬁrst excavation step); and then sends the content of data ﬁle to a variable termed as ‘text’. The ‘text’ is ﬁnally written into the data ﬁle ‘excav.dat’. This process continues until

Construction Mechanics and Optimisation of Excavation Schemes a1 a2 a3 a4 a5 a6 b1 b2 b3 b4 a1 a2 b1 b2 b3 a3 a4 a5 b4 a6 a1 a2 a3 a4 a5 b1 b2 b3 a6 b4 a1 a2 b1 b2 b3 a3 a4 b4 a5 a6 a1 a2 a3 a4 b1 a5 b2 a6 b3 b4 a1 b1 a2 a3 a4 a5 a6 b2 b3 b4 a1 a2 a3 a4 b1 b2 a5 a6 b3 b4 a1 b1 a2 a3 a4 a5 b2 b3 b4 a6 a1 a2 a3 a4 b1 b2 b3 a5 a6 b4 a1 b1 a2 a3 a4 b2 a5 b3 b4 a6 a1 a2 a3 b1 a4 a5 a6 b2 b3 b4 a1 b1 a2 a3 a4 b2 b3 b4 a5 a6 a1 a2 a3 b1 a4 a5 b2 b3 b4 a6 a1 b1 a2 a3 b2 a4 a5 b3 b4 a6 a1 a2 a3 b1 a4 b2 a5 b3 b4 a6 a1 b1 a2 a3 b2 b3 b4 a4 a5 a6 a1 a2 a3 b1 a4 b2 b3 b4 a5 a6 a1 b1 a2 a3 b2 b3 a4 b4 a5 a6 a1 a2 a3 b1 b2 a4 a5 b3 b4 a6 a1 b1 a2 b2 a3 a4 a5 b3 a6 b4 a1 a2 a3 b1 b2 a4 b3 b4 a5 a6 a1 b1 a2 b2 a3 a4 b3 a5 b4 a6 a1 a2 a3 b1 b2 b3 a4 b4 a5 a6 a1 b1 a2 b2 a3 b3 a4 a5 b4 a6 a1 a2 b1 a3 a4 a5 b2 a6 b3 b4 a1 b1 a2 b2 b3 a3 a4 a5 a6 b4 a1 a2 b1 a3 a4 b2 a5 a6 b3 b4 a1 b1 a2 b2 b3 a3 b4 a4 a5 a6 a1 a2 b1 a3 a4 b2 b3 a5 a6 b4 a1 b1 b2 a2 a3 a4 a5 b3 a6 b4 a1 a2 b1 a3 b2 a4 a5 a6 b3 b4 a1 b1 b2 a2 a3 a4 b3 a5 b4 a6 a1 a2 b1 a3 b2 a4 b3 a5 a6 b4 a1 b1 b2 a2 a3 b3 a4 a5 b4 a6 a1 a2 b1 a3 b2 b3 a4 a5 a6 b4 a1 b1 b2 a2 b3 a3 a4 a5 a6 b4 a1 a2 b1 a3 b2 b3 b4 a4 a5 a6 a1 b1 b2 a2 b3 a3 b4 a4 a5 a6 a1 a2 b1 b2 a3 a4 a5 b3 b4 a6 a1 b1 b2 b3 a2 a3 a4 a5 b4 a6 a1 a2 b1 b2 a3 a4 b3 b4 a5 a6 a1 b1 b2 b3 a2 b4 a3 a4 a5 a6

a1 a2 b1 b2 a3 b3 a4 b4 a5 a6 a1 a2 a3 a4 a5 b1 b2 a6 b3 b4 a1 a2 b1 b2 b3 a3 a4 a5 b4 a6 a1 a2 a3 a4 b1 a5 a6 b2 b3 b4 a1 a2 b1 b2 b3 b4 a3 a4 a5 a6 a1 a2 a3 a4 b1 a5 b2 b3 b4 a6 a1 b1 a2 a3 a4 a5 b2 b3 a6 b4 a1 a2 a3 a4 b1 b2 a5 b3 b4 a6 a1 b1 a2 a3 a4 b2 a5 b3 a6 b4 a1 a2 a3 a4 b1 b2 b3 b4 a5 a6 a1 b1 a2 a3 a4 b2 b3 a5 b4 a6 a1 a2 a3 b1 a4 a5 b2 b3 a6 b4 a1 b1 a2 a3 b2 a4 a5 b3 a6 b4 a1 a2 a3 b1 a4 b2 a5 b3 a6 b4 a1 b1 a2 a3 b2 a4 b3 a5 b4 a6 a1 a2 a3 b1 a4 b2 b3 a5 b4 a6 a1 b1 a2 a3 b2 b3 a4 a5 b4 a6 a1 a2 a3 b1 b2 a4 a5 b3 a6 b4 a1 b1 a2 b2 a3 a4 a5 a6 b3 b4 a1 a2 a3 b1 b2 a4 b3 a5 b4 a6 a1 b1 a2 b2 a3 a4 b3 a5 a6 b4 a1 a2 a3 b1 b2 b3 a4 a5 b4 a6 a1 b1 a2 b2 a3 b3 a4 a5 a6 b4 a1 a2 b1 a3 a4 a5 a6 b2 b3 b4 a1 b1 a2 b2 a3 b3 b4 a4 a5 a6 a1 a2 b1 a3 a4 a5 b2 b3 b4 a6 a1 b1 a2 b2 b3 a3 a4 b4 a5 a6 a1 a2 b1 a3 a4 b2 a5 b3 b4 a6 a1 b1 b2 a2 a3 a4 a5 a6 b3 b4 a1 a2 b1 a3 a4 b2 b3 b4 a5 a6 a1 b1 b2 a2 a3 a4 b3 a5 a6 b4 a1 a2 b1 a3 b2 a4 a5 b3 b4 a6 a1 b1 b2 a2 a3 b3 a4 a5 a6 b4 a1 a2 b1 a3 b2 a4 b3 b4 a5 a6 a1 b1 b2 a2 a3 b3 b4 a4 a5 a6 a1 a2 b1 a3 b2 b3 a4 b4 a5 a6 a1 b1 b2 a2 b3 a3 a4 b4 a5 a6 a1 a2 b1 b2 a3 a4 a5 b3 a6 b4 a1 b1 b2 b3 a2 a3 a4 a5 a6 b4 a1 a2 b1 b2 a3 a4 b3 a5 b4 a6 a1 b1 b2 b3 a2 a3 b4 a4 a5 a6 a1 a2 b1 b2 a3 b3 a4 a5 b4 a6

235

a1 a2 a3 a4 a5 b1 a6 b2 b3 b4 a1 a2 b1 b2 b3 a3 a4 a5 a6 b4 a1 a2 a3 a4 a5 b1 b2 b3 b4 a6 a1 a2 b1 b2 b3 a3 b4 a4 a5 a6 a1 a2 a3 a4 b1 a5 b2 b3 a6 b4 a1 b1 a2 a3 a4 a5 b2 a6 b3 b4 a1 a2 a3 a4 b1 b2 a5 b3 a6 b4 a1 b1 a2 a3 a4 b2 a5 a6 b3 b4 a1 a2 a3 a4 b1 b2 b3 a5 b4 a6 a1 b1 a2 a3 a4 b2 b3 a5 a6 b4 a1 a2 a3 b1 a4 a5 b2 a6 b3 b4 a1 b1 a2 a3 b2 a4 a5 a6 b3 b4 a1 a2 a3 b1 a4 b2 a5 a6 b3 b4 a1 b1 a2 a3 b2 a4 b3 a5 a6 b4 a1 a2 a3 b1 a4 b2 b3 a5 a6 b4 a1 b1 a2 a3 b2 b3 a4 a5 a6 b4 a1 a2 a3 b1 b2 a4 a5 a6 b3 b4 a1 b1 a2 a3 b2 b3 b4 a4 a5 a6 a1 a2 a3 b1 b2 a4 b3 a5 a6 b4 a1 b1 a2 b2 a3 a4 a5 b3 b4 a6 a1 a2 a3 b1 b2 b3 a4 a5 a6 b4 a1 b1 a2 b2 a3 a4 b3 b4 a5 a6 a1 a2 a3 b1 b2 b3 b4 a4 a5 a6 a1 b1 a2 b2 a3 b3 a4 b4 a5 a6 a1 a2 b1 a3 a4 a5 b2 b3 a6 b4 a1 b1 a2 b2 b3 a3 a4 a5 b4 a6 a1 a2 b1 a3 a4 b2 a5 b3 a6 b4 a1 b1 a2 b2 b3 b4 a3 a4 a5 a6 a1 a2 b1 a3 a4 b2 b3 a5 b4 a6 a1 b1 b2 a2 a3 a4 a5 b3 b4 a6 a1 a2 b1 a3 b2 a4 a5 b3 a6 b4 a1 b1 b2 a2 a3 a4 b3 b4 a5 a6 a1 a2 b1 a3 b2 a4 b3 a5 b4 a6 a1 b1 b2 a2 a3 b3 a4 b4 a5 a6 a1 a2 b1 a3 b2 b3 a4 a5 b4 a6 a1 b1 b2 a2 b3 a3 a4 a5 b4 a6 a1 a2 b1 b2 a3 a4 a5 a6 b3 b4 a1 b1 b2 a2 b3 b4 a3 a4 a5 a6 a1 a2 b1 b2 a3 a4 b3 a5 a6 b4 a1 b1 b2 b3 a2 a3 a4 b4 a5 a6 a1 a2 b1 b2 a3 b3 a4 a5 a6 b4 a1 b1 b2 b3 b4 a2 a3 a4 a5 a6

Figure 7.16. Sorting results of cavern construction (126 schemes).

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Figure 7.17. Hand-sorted construction sequences.

Figure 7.18. Structure of data ﬁles.

the bottom of the table (the last excavation step) is reached. The program for automatic generation of data ﬁles is shown below: ———————— Autoform.Pro domains ﬁle¼myﬁle L¼symbol* A¼symbol predicates form_fort_ﬁle (L)

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form_fort_ﬁle (H|T): — ﬁlename¼H, ﬁle_str (Filename, Text), write (Text). run (L): — openwrite (myﬁle, ‘exc.dat’), writedevice (myﬁle), form_fort_ﬁle (L), closeﬁle (myﬁle). ————————

7.3.2.3 Implementation of excavation scheme optimisation. The automatic optimisation of cavern excavation schemes is implemented based on the results of an order arrangement on excavation sequence and data ﬁles generated in the last section. The whole process of excavation optimisation is schematically shown in Figure 7.19. The process can be simply described as: (a) make known the ﬁrst step a1 through reading ﬁle ‘order.out’; (b) make known the second step a2 or b1; (c) form two tables L1 ¼ [a1, a2] and L2 ¼ [a1, b1]; (d) call Auto_form.PRO and the ﬁnite element program; and (e) search for the optimal scheme according to the beneﬁt function. By repeating this process, an optimal scheme is obtained, that is, Lopt ¼ [a1, a2, a3, a4, a5, T1, T2, T3, T4, a6].

Figure 7.19. Optimisation process of cavern complex construction.

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7.3.3 Discussions Earlier in this chapter, a new method for the optimisation of cavern construction schemes is outlined. The interactive programming theory is used for cavern construction optimisation, based on the concepts of interactive construction mechanics associated with rock engineering. The traditional optimisation approach, which is based on the simple comparisons among multiple schemes from ﬁnite element computations, is improved in this method based on multi-stage decisionmaking. To further improve the eﬃciency of optimisation process, the artiﬁcial intelligence Prolog language is utilised. Accordingly, the optimisation is automatically implemented. The computer program is simple and easy to understand. Decision-making in cavern construction optimisation is to select the optimal scheme from all the possible schemes according to the given requirements. After all the possible schemes are evaluated, an optimal scheme is then determined as the ﬁnal option [8,9]. If there is only one index for evaluating the schemes in a system, the system is called single index decision-making system. Optimisation in this system is relatively easy, and can be implemented by comparing the indices of all the possible schemes. However, cavern construction is a complex system with multiple indices for evaluation. In other words, it is a multiple indices decision-making system. In such a system, the optimal scheme cannot be obtained directly from the evaluation results on a single index. The comprehensive evaluation on the multiple indices is required. In the study illustrated in this chapter, damage area around the cavern is taken as the beneﬁt function (evaluation index). This means that the optimisation of cavern construction is still based on a single index. Further improvement on the interactive optimisation method can adopt multiple indices for construction optimisation.

7.4.

ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE OPTIMISATION METHODS

An engineering example is given to demonstrate the eﬀectiveness of applying the construction mechanics and artiﬁcial intelligence methods in cavern construction optimisation. The project is an underground hydroelectrical power station excavated in rocks.

7.4.1 Description of the project The power station is excavated in a Triassic sandstone. The geological structure is uniclines. The dip direction of rock strata is NEE-SE, and dip angle is about 10 . No large fault is found except for bedding fractures. Groundwater level is deep below ground surface, and the permeability of the rock masses is low. The cavern complex

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Table 7.6. Mechanical parameters of rock layers. Layer

T12

T131

T132

T14

T151

T152

T153

Young’s modulus (GPa) Poisson’s ratio Cohesion (MPa) Friction angle ( )

11.5 0.22 1.0 35

12.8 0.21 1.17 35

12.0 0.22 1.0 35

13.5 0.2 1.5 35

9.5 0.24 0.83 35

12.8 0.21 1.25 35

10.5 0.22 1.0 35

is approximately 70–100 m below the ground level. According to the in situ measurement, the horizontal stress is about 3 MPa; and the vertical stress is equal to the overburden stress. Underground cavern complex is housed in fractured mudstones (T131 , T132 and T14 ). Among them, T131 comprises thick layers (about 30 m) of calc-silicon ﬁne-grained quartz sandstone and calc sandstone with small amount of mudstone or siltstone. T132 comprises thick layers (about 30 m) of calc or argillaceous-calc siltstone and ﬁne-grained sandstone. T14 comprises very thick layers (about 60 m) of silicon or calc sandstone containing little amount of mudstone or siltstone. The mechanical properties of each layer of rock formations are presented in Table 7.6.

7.4.2

Layout of cavern group and arrangement of step excavation

The layout of the cavern complex is shown in Figure 7.20, together with the distribution of rock layers. Excavation was designed to be completed in 10 stages. This excavation scheme is used in the study of cavern construction optimisation. In addition, several assumptions are made: (a) C0 is the ﬁrst excavated; (b) Excavation sequence is from top to bottom; (c) Each stage is excavated in one run.

7.4.3

Optimisation of excavation sequence

Two stages of study have been performed. At the ﬁrst stage, the surrounding rock mass is assumed isotropic and the same mechanical properties are used. At the second stage, the rock mass is divided into seven layers of diﬀerent mechanical properties (locally isotropic medium) according to the geological conditions (Figure 7.20 and Table 7.6). The results obtained from the two diﬀerent studies are diﬀerent and outlined in the following sections.

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Figure 7.20. Cavern layout and excavation stage.

7.4.3.1 Rock mass assumed as isotropic medium. The studies are conducted using the method outlined in Section 7.3.2. The analysis is based on two dimensional ﬁnite element modelling. The average depth of cavern complex to ground surface is 85 m. The material parameters for modelling are: ¼ 2610 kg/m3, E ¼ 11000 MPa, n ¼ 0.20, c ¼ 0.3 MPa, f ¼ 33 . Two cases of in situ stress ﬁelds are modelled, one based on the elastic theory and the other based on the in situ stress measurement. The in situ stresses are estimated as: Case I: elastic theory sv ¼ s1 ¼ 2.61 85/10 ¼ 2.22 MPa (vertical) n s1 ¼ 0.56 MPa (horizontal) sh ¼ s3 ¼ 1n Case II: In situ stress measurement sv ¼ s1 ¼ 2.22 MPa, sh ¼ s3 ¼ 3.0 MPa.

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Figure 7.21. Mesh of the computation model.

To simulate the excavation process, the whole cavern area is divided into a mesh of triangle and quadrilateral shapes. The total number of elements is 1326, and the total number of nodes is 1284. The central part of the model is shown in Figure 7.21. The artiﬁcial intelligence language Turbo Prolog 2.0 Version is used (see Section 7.32). The results present 1260 possible excavation sequence schemes. It is impossible and unnecessary to carry out ﬁnite element computation for so many schemes. However, by using the construction mechanics theory, optimum schemes can be quickly sorted out. The searching processes for the optimum scheme with two diﬀerent cases of in situ stress ﬁelds are shown in Figure 7.22. The optimum excavation sequences using optimisation method, are diﬀerent from the original scheme, C0 ! C4 þ W3 ! b1 þ W1 þ C1 ! C2 þ b2 þ W2 ! C3 , where C4 þ W3 means that C4 and W3 are simultaneously excavated (Figure 7.20). Assessment of damage areas around the caverns are conducted, and comparisons between the diﬀerent schemes are shown in Figures 7.23 and 7.24, and Table 7.7. Usually, in a large underground cavern complex, there are many possible excavation schemes. However, by using the construction mechanics together with interactive programming theory and artiﬁcial intelligence method, the optimum scheme can be obtained. Compared with the originally proposed scheme, the damage area from the original scheme is 3–10 times larger than that of the optimum scheme. Although this is obtained with certain assumptions, it however indicates the importance of optimisation of cavern construction scheme.

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Figure 7.22. Optimisation search processes for two cases of in situ stress ﬁelds, (a) Case I, and (b) Case II.

Figure 7.23. Damage distribution of the stress ﬁeld case I, (a) original excavation scheme, and (b) optimised excavation scheme.

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Figure 7.24. Damage distribution of the stress ﬁeld case II, (a) original excavation scheme, and (b) optimised excavation scheme.

Table 7.7. Comparisons between the optimised and original excavation schemes. Scheme

Stress case I Stress case II

Optimised Original Optimised Original

Total damage area (m2)

Vertical convergence of powerhouse (mm)

Horizontal convergence of powerhouse (mm)

513 5139 690 2540

9.64 11.75 4.41 6.67

1.68 0.69 20.88 12.51

7.4.3.2 Rock mass assumed as layered isotropic medium. According to the geological conditions, the rock mass is divided into seven layers. Each layer is treated as an isotropic medium. The density of rock mass in all the layers is 2610 kg/m3. The mechanical properties adopted for each rock layer are the same as those shown in Table 7.6.

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Figure 7.25. Optimisation searching process of construction schemes.

The average depth of cavern complex to ground surface is 85 m. The average rock mass density is 2610 kg/m3. The initial stress ﬁeld is determined from the in situ stress measurement. The horizontal stress obtained from measurement is 3.0 MPa. Therefore, sv ¼ s1 ¼ 2.61 85/10 ¼ 2.22 MPa (vertical) sh ¼ s3 ¼ 3.0 MPa. The ﬁnite element computation is based on plane strain elastic-plastic model. The total number of ﬁnite elements is 1506, and the node number is 1211. The program and the method are the same as those in the previous modelling. 1260 possible schemes are obtained, and subsequently reduced to 9 schemes. If the beneﬁt function is deﬁned by the damage area of surrounding rock mass, the smaller damage area is then taken as the optimum scheme. The optimum scheme searching process is shown in Figure 7.25. The excavation sequence in the original scheme is C0 ! C4 þ W3 ! b2 þ W2 þ C1 ! C2 þ b2 þ W2 ! C3 ; where C4 þ W4 represents that C4 and W3 are simultaneously excavated. The comparisons between the optimised scheme and original scheme are shown in Figure 7.26 and Table 7.8. It can be seen that the damage area of the optimised excavation scheme is reduced by 90% in comparison with the original scheme. A substantial improvement can be achieved by the optimisation. It should be noted that the optimised scheme obtained in this study is diﬀerent from that for rock mass assumed to be isotropic in the previous study. This implies that the optimisation results depend on the rock mass conditions.

7.4.3.3 Summaries. (a) Applications of the interactive construction mechanics theory together with the artiﬁcial intelligence method can optimise the construction schemes of cavern complex scientiﬁcally and automatically, as a result high eﬃciency is achieved.

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Figure 7.26. Damage distribution of (a) original excavation scheme, and (b) optimal excavation scheme.

Table 7.8. Comparisons between the optimised and the original excavation schemes. Scheme

Optimised Original

Total damage area (m2)

Vertical convergence of powerhouse (m)

Horizontal convergence of powerhouse (mm)

329 4533

0.16 1.13

9.6 1.9

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(b) Advantages of the optimisation for construction are signiﬁcant for large-scale rock engineering projects. The damage area of the optimised scheme can sometimes be reduced to 1/10 of that of the unoptimised scheme. (c) The application of the optimisation method of combining interactive construction mechanics theory and artiﬁcial intelligence technique to the heavily fractured rock mass and to the three-dimensional problems needed to be studied.

Chapter 8

Reinforcement Mechanism of Rock Bolts Rock bolts have been used in rock engineering from end of the 19th century [445]. In 1890, reinforced steel was used in reinforcing the rock mass in mines in north Wales. After the 1940s, this support technique was adopted worldwide. The use of shotcrete together with rock bolts further promotes the application of bolt as a main supporting method. Based on the rock bolt technique, a new construction method combining the use of rock bolts and shotcrete gradually formed in Europe in the 1950s and 1960s. This method sets a milestone in the underground construction technology. In recent years, many researchers and engineers have studied the eﬀects of bolts in rock reinforcement. This chapter summarises some of the research ﬁndings.

8.1.

EFFECT OF BOLTS ON SUPPORTING THE ROCK MASS

8.1.1 Effects of rock bolts Two main types of rock bolts are widely employed in the underground excavation in rocks. One is the end-anchored bolts varying from wedge to resin anchoring. The other one is the full-length anchored type varying from fully grouted to split-set bolts. The eﬀects of rock bolts on the surrounding rock are mainly two-fold: providing the reaction force and the reinforcement to the surrounding rock mass. The former prevents deformation occurring in the surrounding rock mass and provides the tensile force to resist the movement of rock blocks. The latter reinforces the rock mass to be an integrated medium and to easily form arches around the excavation [446–477].

8.1.1.1 Reinforcement. The eﬀorts of other support methods such as timber frame, segment lining, and cast concrete lining are mainly external supports. As the rock bolts are inserted into the rock masses, the eﬀects of rock bolts on the rock mass are similar to that of steel bars in reinforced concrete. They increase the integrity and overall strength of the rock masses. The bolts prevent the rock masses from sliding and failing and increase the load-bearing capacity eﬀectively. The rock mass is reinforced by the bolts and becomes an eﬀective self-support medium [446–473].

247

248

Chapter 8

8.1.1.2 Post effect of pre-stressing. The deformation modulus of the rock bolt (usually steel) is generally greater than that of the surrounding rock masses. The diﬀerence between the moduli restricts the surrounding rock mass to deform (usually tensile deformation towards the centre of the opening) after excavation. The restriction occurs at the ends of anchored rock bolts and along the full length of grouted rock bolts. This restriction is mobilised once deformation starts to occur and it can be regarded as a complementary pre-stress on the excavation surface after the excavation. Rock bolts signiﬁcantly increase the integrity of the rock mass due to this pre-stressing eﬀect [446–448,474].

8.1.1.3 Prompt prevention. Rock bolts and shotcrete are often applied immediately after excavation. In some cases, rock bolts can also be implemented prior to the excavation, to restrain the relaxation of the surrounding rock and to improve the rock mass strength early. The early application of rock bolts provides prompt prevention of failure.

8.1.1.4 Good match to deformation. Rock bolts possess better match ability to the deformation of rock mass than other support methods. It is often referred as ‘ﬂexible support’. This support method is also very suitable for tunnels in soft rock.

8.1.1.5 Flexibility in construction. The size and distribution of rock bolts can be adjusted according to the geological conditions of the surrounding rock. Bolting and shotcreting can be completed in several steps [475,476]. Additional bolts can be installed at a later stage. The application of bolts has ﬂexibility during the construction.

8.1.2 Reinforcement mechanism of rock bolts The reinforcement mechanisms of rock bolts are very complex. In this section, they are discussed by examining the shear strength and other properties of the bolted rock masses. A series of studies on the reinforcement mechanism of rock bolts have been carried out. Test results indicated that application of rock bolts increases the peak compressive strength of the rock mass by 50100%, compared with that of the rock mass without bolts. For a single opening, the shear strength of the surrounding rock mass can be expressed by the Mohr–Coulomb strength condition, as shown in Figure 8.1.

Reinforcement Mechanism of Rock Bolts

249

The reaction force contributed by rock bolts to the surrounding rock mass (s2) can be estimated from the allowable tensile stresses in the rock bolts. For example, for a tunnel at depth of 500 m with hydro-static in situ stress condition, and assuming the density of rock mass of 2500 kg/m3, the in situ stress ﬁeld can be obtained as: six ¼ siy ¼ siz ¼ 12:5 MPa

ð8:1Þ

The uniaxial compressive strength of the rock material (sc) is assumed as 15.0 MPa, the uniaxial compressive strength of the rock mass strength (smass) is assumed as 5.0 MPa (1/3sc) and the internal friction angle (fmass) is 35 . The cohesion of rock mass can be derived as: cmass ¼ smass ð1 sin fmass Þ=2 cos fmass ¼1:31 Mpa

ð8:2Þ

If the compressive strength of the properly bolted rock mass (c0 mass) is 50% greater than that of unbolted rock mass, then c0 mass ¼ 1.5 cmass ¼ 1.97 MPa. Assuming that the length of bolt is 60 cm and the allowed tensile load is 100 kN, the reaction stress on the rock surface provided by the bolt can be calculated as s2 ¼ 0.28 MPa. For the end-anchored bolts, the reinforcement is applied through a reaction force at the rock wall. The reinforcement eﬀect of a bolt is therefore on the rock wall and the bolt anchor while the other parts of the bolt are not in contact with the rock mass. For full-length grouted or frictional bolts, the reinforcement bolts increase the shear-bearing capacity of the rock masses, in addition to the reaction force at the rock wall generated by the bolts. Assuming that the stresses s1 and s2 on the opening wall reach the rock mass shear strength, the stress condition of the rock wall can be expressed by the Mohr circle s2As1 with centre O. As shown in Figure 8.1, this circle is tangential to the strength envelope BCA at A. The reaction force of rock bolts to the rock wall can be deﬁned as s2 for the end-anchored bolt. In this case the stress circle change to the small Mohr circle s0 2A0 s1 with centre O0 , and the strength envelope is B0 C0 A0 . The new minor principal stress is s20 ¼ s2 þ s2. From Figure 8.1, the shift of the strength envelope, OA ¼ ðc0mass cmass Þ cos f

ð8:3Þ

For the full-length grouted bolts, the shear strength of surrounding rock is enhanced by the bolts, and the shear envelope shifts up to the position of B00 C00 A00 (assuming no change in friction angle). The increased cohesion equals to c00 c0 . The strength

250

Chapter 8

Figure 8.1. Strength analysis of rock bolt eﬀects.

envelope of full-length grouted bolt is increased by an oﬀset O0 A0 ¼ (c00 c0 ) cosf from the end-anchored bolt. The analysis above is under the assumption that the strength oﬀset is based on the shortest relative distance from centre of the Mohr circle to the envelope. The analysis clearly shows that the full-length grouted bolts achieve high strength storage than the end-anchored bolts. The diﬀerence of the eﬀects of two types of rock bolts can be signiﬁcant, due to the eﬀects of grouting.

8.2.

PHYSICAL MODELLING OF ROCK BOLTS

Rock bolts enhance and reinforce the rock masses. However, the mechanisms of reinforcement, and the deformation and damage of bolts in the surrounding rock mass subjected to change of stress ﬁeld have not been fully understood [449,463,467,477]. This section presents studies on the rock bolt mechanism using physical modelling. When the surrounding rock mass is soft and/or subjected to high in situ stresses, displacement of the tunnel wall can often be as high as 50–100 cm. It is of great interest to have a scientiﬁc approach in selecting the rock bolt to overcome such large deformation problems. At present, the common way of applying bolts is to install bolts perpendicular to the rock surface. The systematic studies are conducted on the installation angle, density and other parameters, and results are of interest to engineers.

8.2.1 Similarity of model materials Rock masses close to the surface are treated as separable blocky materials. The model materials should have similar characteristics as the real rock masses, such as stress–strain relation, properties of dilation and softening. After testing diﬀerent model materials, the selected model material (made of sand and white glue) has elastic modulus of 230 MPa. The bolt is modelled by the bamboo material with

Reinforcement Mechanism of Rock Bolts

251

Figure 8.2. Results of compression tests.

Figure 8.3. Results of tensile tests.

elastic modulus of 100–170 MPa. A series of tests are conducted including uniaxial, biaxial and triaxial compression, tensile test, and cyclic loading test for the model specimens with and without bolts, as shown in Figures 8.2–8.4. From the results, it can be seen that the ratio of modulus to strength of the model material is 178 and that of the rock mass is 200. The ratio of modelling material compressive strengths to rock mass strength is 1/15. From the bulk deformation curves, it also can be seen that the modelling material has the similar dilation property as the modelled rock mass.

8.2.2 Comparison of different bolting methods The eﬀects of diﬀerent bolt materials, bolting methods, bolt densities and installation angles are systematically compared in the tests. The strength of the material

252

Chapter 8

Figure 8.4. Results of cyclic compression tests.

Figure 8.5. Diﬀerent bolt arrangement used in the tests.

modelling the rock mass is 1.0 MPa. Bamboo (E ¼ 10 103 MPa) and plexiglass (E ¼ 3 103 MPa) are used as modelling materials of the bolts. Both full-length grouting and end-anchored bolts are modelled. Diﬀerent densities are used at 8, 10, 12, and 36 bolts per 200 cm2, equivalent to the actual bolt spacing of 1.2 to 0.4 m. The bolting methods in the tests include end-anchored and fully grouted parallel bolts perpendicular to the rock wall, fully grouted perpendicular, vertically and horizontally intersected bolts. The details of various arrangements are summarised in Figure 8.5 and Table 8.1. Figures 8.6–8.8 indicate the various load-deformation curves. From the ﬁgures, it can be seen that high density of bolts can improve the strength of surrounding rock mass. The horizontally obliquely intersected full-length grouted bolts give high strength of the rock mass and small number of bolts. Figure 8.8a illustrates that for the same bolt spacing and length, the rock mass can have the highest

253

Reinforcement Mechanism of Rock Bolts Table 8.1. Various types of bolts and their parameters. Type

No bolt

Normal parallel

Normal parallel

Normal parallel

Vertically oblique

Vertically oblique

Symbols NB NP-E10 NP-F10 NP-F36 VO-F8 VO-F12 – 10 10 36 8 12 Bolt density (per 200 cm2) 90 90 73 68 Inclination to wall – 90 Anchor and grouting – Two ends Full-length Full-length Full-length Full-length Bolt material – Bamboo Bamboo Bamboo Bamboo Bamboo Peak strength (MPa) 1.0 1.5 1.5 3.0 1.54 2.2 Softening behaviour Yes Yes No No Yes No

Horizontally oblique HO-F12 12 75 Full-length Bamboo 2.06 No

Figure 8.6. Axial load – axial deformation of diﬀerent bolt arrangement.

strength and the lowest dilation when the bolts are installed at 22.5 and 67.5 , to the rock wall surface. For the same bolting angle and the same total lengths of bolts, the test results shown in Figure 8.8 suggest that mixed length bolts (total 9 bolts of 5.41 and 10.82 cm long and evenly spaced) provide the best reinforcement results, while short bolts with high density (total 12 bolts of 5.41 cm long and evenly spaced) give the

254

Chapter 8

Figure 8.7. Axial load – volumetric strain for diﬀerent bolt arrangement.

Figure 8.8. Axial load – volumetric deformation for diﬀerent (a) bolting angle, and (b) bolt length.

Reinforcement Mechanism of Rock Bolts

255

worst reinforcement results, as compared with uniformly long bolts (total 6 bolts of 10.82 cm long and evenly spaced).

8.2.3 Analysis of test results From the test results shown in Figures 8.6, 8.7 and 8.8 and Table 8.1, following observations are obtained: (a) The peak uniaxial compressive strength of the rock mass with bolts increases signiﬁcantly (up to 20% from the model tests) compared with that without bolts while the residual strength and the tensile strength increase up to 100%. In biaxial case, the peak compressive strength can increase by 50–100% (Figure 8.6) when the density of bolts is high, rock mass strength can increase by 3 times with little volumetric dilation. (b) Under the plane strain condition, the strength of the models is aﬀected not only by the bolt densities but also by bolt installation angles, types, and shear strength and lateral stiﬀness of bolt. (c) The obliquely intersected bolt distribution enhances the rock mass strength and limits the dilation signiﬁcantly. The best arrangement appears to the bolts at 65 angle to the wall. But the bolts are required to possess a high lateral stiﬀness in this case. (d) Compared to the end-anchored bolts, the improvement of full-length grouted bolts on the peak strength of the surrounding rock mass is not obvious. However, the bulk displacement curves are diﬀerent for the two diﬀerent bolt types. The dilation of the fully grouted bolts starts at a later stage. The post-peak softening phenomenon is not obvious for the fully grouted bolts while it is signiﬁcant for the end-anchored bolts. Therefore, end-anchored bolts have low post-peak strength.

8.3.

NUMERICAL MODELLING OF BOLT

In this section, non-linear ﬁnite element numerical modelling is applied to study eﬀects of rock bolts on the stability of tunnels in soft rocks. Parameters obtained from physical modelling are used as input to the numerical modelling.

8.3.1 Basic parameters of the numerical model A simple circular opening model is used in the parametric study. Two in situ stress conditions are modelled: (a) far ﬁeld stresses s1 ¼ s2 ¼ 20 MPa, and (b) far ﬁeld

256

Chapter 8

Figure 8.9. Element division near the opening.

stresses s1 ¼ 2s2 ¼ 20 MPa. The surrounding rock mass is regarded as an elastoplastic material. The problem is treated as a plain stress one. Based on the symmetry, one quarter of problem is analysed by a ﬁnite element program. A total of 136 elements and 162 nodes are used in the model, as shown in Figure 8.9. Three diﬀerent cases are studied: opening without bolts, opening with bolts that are normal to the wall surface, and opening with obliquely intersected bolts. The radius of the opening is 2 m and the thickness of bolted region equals to the opening radius. The mechanical parameters of surrounding rock mass are adopted from the physical model tests described in the previous section, and is summarised in Table 8.2.

8.3.2 Models with far field stresses s1 ¼ s2 ¼ 20 MPa Stress distributions of models I, II and III when s1 ¼ s2 are presented in Figure 8.10, and the largest displacements of the wall are summarised in Table 8.3. The obliquely intersected bolting gives the smallest displacement, and it is less than half of that of the opening without bolts. Figure 8.11 and Table 8.3 illustrate the range of damage zones in the surrounding rock mass occurring near the wall. The results show that the surrounding rock mass with bolts is generally less damaged than that without bolts. The surrounding rock mass with obliquely intersected bolt system is the least damaged.

257

Reinforcement Mechanism of Rock Bolts Table 8.2. Mechanical parameters of rock mass. Parameter

Model I: No bolts

Model II: Normal bolts

Model III: Obliquely intersected bolts

sc (MPa) st (MPa) E (MPa) (MPa) f (MPa) fr (MPa) sr (MPa) c (MPa) cr (MPa)

26 1/15sc ¼ 1.73 0.4 104 0.25 35 35 200.8 3.58 2.86

1.5sc ¼ 39 1.2st ¼ 2.03 1.15E¼0.58104 0.25 35 35 39 5.36 5.36

2.0sc ¼ 52 1.6st ¼ 2.77 1.5E ¼ 0.75 104 0.25 35 35 52 7.15 7.15

Figure 8.10. Stress distribution of models I, II and III, at roof, when s1 ¼ s2 .

Table 8.3. Largest displacement and damage of models I, II and III when far field s1 ¼ s2. Model

Maximum displacement (cm) Number of damage elements

No bolts

Normal bolts

Oblique intersection bolts

1.80 24

1.11 16

0.84 8

258

Chapter 8

Figure 8.11. Damage around the opening of models I, II and III when s1 ¼ s2 .

8.3.3

Models with far field stresses s1 ¼ 2s2 ¼ 20 MPa

The modelling results for the models I, II and III are presented in Figures 8.12 and 8.13, showing the stress distribution in the roof and sidewall respectively. Figure 8.14 shows the damaged zones near the wall surface for the three models. Again, it can be concluded that obliquely intersected bolt system gives the best results. Table 8.4 shows the maximum displacement on the wall surfaces and the numbers of damaged elements of the three diﬀerent models. The obliquely intersected bolted rock mass sustains less than one-third of the damaged area than that of the unbolted rock mass. The bolt quantity of the obliquely intersected is 20% more than that of normal bolt system. However, the eﬀects of the obliquely intersected bolts on reducing the damaged zones and the displacements of the surrounding rock are signiﬁcant compared with that of normal bolt system. The ﬁnite element modelling uses elastoplastic approach, and post-failure dilation of the rock mass is not considered. By considering the post-failure dilation, the eﬀects of the obliquely intersected bolt system are expected to be more signiﬁcant.

8.4.

SCALED ENGINEERING MODEL TEST

In order to study the reinforcement eﬀects of rock bolts on the surrounding rock mass in an underground excavation, a series of biaxial compression tests on engineering physical model were conducted. The eﬀects of various bolt parameters are compared. The modelling materials and mechanical parameters are selected according to modelling scale to model an excavation in a rock mass. The actual excavation is in a sedimentary rock. The surrounding rock mass is faulted and fractured. The in situ horizontal stresses are 19.5 MPa and 12.8 MPa, respectively. The estimated uniaxial compression strength of the rock mass is about 30 MPa. The elastic modulus and Poisson’s ratio are approximately 1.0 104 MPa and 0.25 respectively. Both stress and geometry scale ratios are 40. The model is 50 50 cm, and the bolts are modelled by bamboo of 2 mm diameter and 40 mm long, to match the stiﬀness ratio. The opening shape is an arched roof with

Reinforcement Mechanism of Rock Bolts

259

Figure 8.12. Stress distribution of models I, II and III, at roof, when s1 ¼ 2s2 .

Figure 8.13. Stress distribution of models I, II and III, at sidewall, when s1 ¼ 2s2 .

vertical walls, as shown in Figure 8.15. The tests are performed under the plain strain condition. Three diﬀerent bolt distributions are simulated: (a) obliquely intersected with bolt at 67.5 to the opening surface, and in the same vertical plane; (b) bolts perpendicular to the opening surface, and (c) no bolts, as shown in

260

Chapter 8

Figure 8.14. Damage around opening of models I, II and III when s1 ¼ 2s2 .

Table 8.4. Largest displacement and damage of models I, II and III when s1 ¼ 2s2 . Model Maximum displacement (cm) Number of damage elements

No bolts

Normal bolts

Oblique intersection bolts

1.45 20

1.23 11

1.03 8

Figure 8.15. Scaled engineering model with dimensions.

Figure 8.16. Bolts are fully grouted in the model tests. The test follows the following procedures: (a) Stresses are applied to the horizontal and vertical directions simultaneously, with vertical stress kept twice that of the horizontal stress. The increment of stress is 0.1 MPa horizontally at each step.

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261

Figure 8.16. Arrangement of (a) no bolt, (b) normal bolts and (c) obliquely intersected bolts.

Figure 8.17. Convergent displacements of the tunnel monitored during the loading.

(b) Loading is stopped when the horizontal and vertical stresses reach 1.0 MPa and 2.0 MPa, respectively. (c) The vertical stress is increased gradually to 4.0 MPa while the horizontal stress is kept at 1.0 MPa. The convergent displacements of the tunnel are monitored during the loading. The monitored displacements are presented in Figure 8.17. It is observed that: (a) The convergent displacement of the rock mass reinforced by the obliquely intersected bolts is generally smaller than that of the rock mass reinforced by normal bolts and the displacement of the roof is reduced by 1640%. The plastic

262

Chapter 8

damage zone occurring in the sidewalls are also smaller for the normal bolting system. (b) Bolting in general, as compared to no bolting, improves the surrounding material properties. The convergent displacement of the opening decreases signiﬁcantly and the plastic damage is greatly reduced by bolting. Normal bolt reduces the displacement of the roof by 1443%. (c) Scaled model tests were also performed on circular opening (Figure 8.15) and similar results were obtained.

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Index A aperture 30, 43, 45, 80–82 artiﬁcial intelligence technique 246

damage mechanics 27, 144 damaged rheological model 150 damaged zone 83–87, 258 damage tensor 27 deformation back analysis 157 deformation equivalence 27, 29, 30, 32, 33, 38, 43, 83 deformation modulus 8–10 deformation monitoring 158, 160, 166, 201 deformation stiﬀness 14 dilation 17, 33, 122, 124, 250, 251, 253, 255, 258 discontinuity deformation analysis 27 discontinuum theory 27 discrete element method 27, 67 displacement prediction 187, 205 displacement series 205 displacement zone 24, 25 Drucker–Prager criterion 42, 44, 51, 83, 216

4, 228, 229,

B back analysis 2, 4, 42, 44, 157–163, 166, 167, 171–173, 176, 178–187, 190–194, 197, 199–201, 221, 225 2-D back analysis 181, 194, 197, 199 3-D back analysis 190, 194, 199, 200 beneﬁt function 224, 225, 227, 228, 237, 238, 244 biaxial stress 39 block sliding 25 bolt 4, 52, 54, 87, 132–134, 137, 139–144, 150, 191, 214–217, 219–221, 247–262 bolting method 251, 252 boundary element method 67, 91 bridged joint 14, 16 C character curve 68 collinear crack 55, 59 compressive strength 8–10, 14, 17, 19, 20, 54, 66, 248, 249, 251, 255 compressive torsional shear failure 18 construction mechanics 4, 211–214, 221, 238, 241, 244, 246 construction process 3, 157, 211, 213, 214 convergence displacement 176 convergent displacement 261 coupled back analysis 2, 157 crack initiation 16, 20, 54, 55, 59 crack propagation 55 creep 89, 147, 152 crown 24, 25, 70, 77, 167, 181

E elastic back analysis 4, 159 elastic constant 30, 47, 76 elastic shearing modulus 100 elasto-plastic back analysis 180 end eﬀect 14 end-anchored bolt 247, 249, 255 equivalence approach 27 equivalence continuum 28 equivalent continuum model 27–29 excavation scheme 211, 213, 217, 221, 224, 225, 227, 234, 237, 239, 241–245 excavation scheme optimisation 237 excavation sequence 3, 4, 24, 42, 43, 53, 212–217, 221, 223–225, 229, 230, 232, 234, 237, 239, 241, 244 extensometer 42, 150, 153, 160, 161, 168, 169, 182, 191, 194, 207, 208

D damage analysis 144, 150, 155 damage evolution equation 145, 146, 152

287

288 F faults 1, 5, 42, 52, 54, 70, 150, 185, 187, 190, 191, 211 FEM 37–39, 44, 52, 67, 70, 77, 91, 146, 148, 150, 171, 198, 215, 225, 228 ﬁnite element method 30, 37, 67 fracture mechanics 27, 54, 99 fracture toughness 55, 56 fracture 1, 5, 16, 18, 19, 54, 59, 150, 158, 162, 238 frictional bolt 249 full-length grouted bolt 249, 250, 252, 255 G global optimal scheme 221, 225, 227, 228 grey system 74 grey system theory 201, 203 H Hooke’s elastic body 114 horizontal shear load 16 I in situ stress 24, 44, 70, 87, 97, 105, 111, 140, 152, 157, 159–161, 163, 170, 176, 184, 185, 190, 191, 193, 201, 215, 225, 227, 240, 241, 244, 249, 255 in situ test 5, 27, 74, 158 intact rock elements 28 interactive programming 221, 222, 224, 227, 228, 238, 241 internal friction angle 19, 59, 66, 71, 74, 84, 86, 249 J joint dip angle 38, 79, 80, 82, 194 joint distribution 6–8, 14–16, 22, 37 joint elastic modulus 77, 78, 80, 81, 86 joint element 27–29, 37, 52 joint spacing 6, 40, 175, 181 joints 1, 5, 6, 8, 9, 14, 17, 27, 28, 30, 32, 35, 36, 39, 40–46, 49, 54, 59, 75, 78, 80, 83, 86, 150, 185, 194, 211 L lateral deformation 8, 9 lateral tensile crack 9, 11

Index local optimal scheme 221, 225 local shearing failure 9 longitudinal deformation 8 M maximum displacement 52 maximum volumetric strain 152 Maxwell medium 117, 124 modelling material 5, 6, 59, 251, 252, 258 Mohr–Coulomb criterion 33, 50, 123 multiple crack 59, 60 multiple-point borehole extensometer 160, 161, 168 multiple-stage excavation 167, 185, 195, 197, 219 N New Austrian Tunnelling Method 167, 212 non-equal time step 204 non-linear regression 192, 201, 202 normal stress 15, 16, 20–22, 57, 59, 61, 84, 92, 134 O optimisation analysis 176, 183 optimisation process 225, 226, 237, 238 orientation 5, 30, 37, 45, 75, 175, 212 P peak strength 9, 10, 12, 14, 39, 54, 124, 253, 255 persistence 5, 7, 10, 11, 13, 16, 18–22, 25, 30, 37, 40, 43, 56, 59, 61, 76, 80, 82, 84, 86, 87, 181, 194 physical model test 5, 39, 66, 256 plane strain 3, 13–15, 47, 76, 151, 162, 175, 178, 215, 244, 255 plane strain modelling 13 plane stress 3, 5, 14, 16, 24, 30 plastic zone 53, 111, 113–116, 122, 154, 217 post-failure 122, 124, 258 Principle of Optimality 222

289

Index R reinforcement mechanism 4, 247, 248 relative convergence 77 relative error 37–39, 68, 69, 74, 75, 78, 82, 87 residual strength zone 127–129, 132 rheologic behaviour 3, 89 rock bridge 14, 16, 18, 20, 54, 56, 57 rock engineering system 74, 83 rock mass cohesion 14, 20 rock mass dimension 6 rock mass friction angle 14 rock mass strength 9, 11–13, 36, 43, 51, 54, 66, 87, 216, 248, 249, 251, 255 rock reinforcement 3, 212, 215, 247 roughness 30 S scale eﬀect 27, 37–39, 192 secondary tensile crack 9 sensitivity analysis 2, 3, 67, 68, 70, 74–77, 83, 84 sensitivity factor 68, 73, 74, 81, 82, 84, 86 sensitivity function 68, 69, 71, 73 sensitivity order 82 shear dilation 17 shear failure 9, 18, 22, 58, 83, 189, 200 shear failure plane 9 shear sliding 9 shear strength 14, 18, 19, 20, 22, 35, 54–59, 61, 248, 249, 255 shotcrete 52, 132, 150, 153, 191, 215, 221, 247, 248 sidewall 24, 25, 142, 155, 160, 189, 228, 262 similarity 5, 6, 24, 25, 250 similarity ratio 6 size eﬀect 6, 14 softening 122, 124, 250, 253, 255 spacing 4, 6, 30, 39, 40, 53, 175, 181, 220, 221, 252 St. Venant medium 117

stability analysis 3–5, 42, 67, 70, 89, 96, 150, 157, 160, 194 statistic method 201 stiﬀness-reduction method 71 strength equivalence 3, 29, 33, 35, 36, 41, 50 stress back analysis 2, 157 stress concentration 55, 142, 158 stress intensity factor 56 stress-volumetric strain curve 152 structural loosening 25 superposition theorem 89 swelling 152 system character 67, 68, 71 T tensile crack 9, 16, 17 tensile rupture 25 tensile–compressive strength ratio 19 threshold stress 55 time series 201, 205 time series analysis 201 time-dependent characteristics 89 torsional tensile shear failure 18 transversal compressive crack 17 transverse deformation 8 transverse isotropic 4, 172, 174 U unloading process

71

V visco-elastic back analysis 4, 166, 170 visco-elastic deformation 89, 151 visco-elastic zone 115, 116, 118, 122, 123, 126, 127, 129, 133, 138, 140, 144 visco-plastic deformation 89, 151 visco-plastic zone 117, 118, 121–129, 131, 132 Y yield zone

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