Interaction Between Soil Foundation and Subway Shield Tunnel 9819968690, 9789819968695

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Zhen-Dong Cui

Interaction Between Soil Foundation and Subway Shield Tunnel

Interaction Between Soil Foundation and Subway Shield Tunnel

Zhen-Dong Cui

Interaction Between Soil Foundation and Subway Shield Tunnel

Zhen-Dong Cui State Key Laboratory of Intelligent Construction and Healthy Operation and Maintenance of Deep Underground Engineering School of Mechanics and Civil Engineering China University of Mining and Technology Xuzhou, Jiangsu, China

ISBN 978-981-99-6869-5 ISBN 978-981-99-6870-1 (eBook) https://doi.org/10.1007/978-981-99-6870-1 This work was funded by the National Natural Science Foundation of China (Grant No. 52378381). © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface

During the operation of the subway, the vibration caused by the subway cyclic loading exists for a long time, which will reduce the strength of the soft soil foundation, generate different degrees of settlement and deformation, affect the deformation of the tunnel axis, and cause ground subsidence. Supported by the National Natural Science Foundation of China (Grant No. 52378381), this monograph studies the interaction between the soil foundation and the subway shield tunnels under the vibration loading, including the dynamic response of the track inside the shield tunnel, the dynamic properties of soil around the subway shield tunnel, the mechanical properties of subway tunnel, and the long-term settlement of the subway tunnel. Chapter 1 is “Introduction”. It introduces the dynamic response of subway track, the vibration induced by subway load with track irregularity, the dynamic response of around soil and subway shield tunnel, the mechanical properties of subway shield tunnel, and the long-term settlement of subway shield tunnel. Chapter 2 is “Dynamic Response of Subway Track”. In this chapter, the infinite Euler–Bernoulli double-beam system was presented, resting on a viscoelastic foundation and subjected to a harmonic moving point load, including two spring-damping systems. The governing equations were solved by means of Fourier transform and residue theory. Chapter 3 is “Vibrations Induced by Subway Load with Track Irregularity”. In this chapter, the vibrations induced by subway random load with excitation of track vertical profile irregularity were studied and the simplified train-track coupled model was established to simulate subway random load with excitation of track vertical profile irregularity. In addition, the 3D finite element analysis model was established to examine the vibration response of soils and structures. Chapter 4 is “Dynamic Response of Around Soil and Subway Shield Tunnel”. In this chapter, the dynamic characteristics of soft clay were studied by the dynamic triaxial apparatus considering the load frequency and the confining pressure. The dynamic finite difference models were conducted to analyze the interaction between the around soil and the shield tunnel.

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Chapter 5 is “Mechanical Properties of Subway Shield Tunnel”. In this chapter, the functionally graded lining of the subway tunnel is modeled as a functionally graded hollow cylinder under non-axisymmetric loads, and its mechanical properties are studied. Chapter 6 is “Long-Term Settlement of Subway Shield Tunnel”. In this chapter, theoretical calculations of the long-term settlement of the shield tunnel were compared with the numerical simulation. Based on in-site monitoring, the settlement trend was predicted by GM (1, 1) and ARMA (n, m) model. The characteristics of the long-term settlement and the settlement trough were studied, and the prediction model for the settlement trough was built. Chapter 7 is “Conclusions and Prospects”. This part comprehensively summarizes the research conclusions. Several controversial issues are discussed and then the further research work and prospects are simply described. This monograph has been prepared with the combined effort of all researchers in the group under Prof. Zhen-Dong Cui’s leading, in which Peng-Peng He, Cheng-Lin Zhang, Shi-Xi Ren, Jun Tan, Tong-Tong Zhang, and Shan-Shan Hua, and some other students all have involved in this comprehensive research work in this monograph. Xuzhou, China August 2023

Prof. Zhen-Dong Cui

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Dynamic Response of Subway Track . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Vibration Induced by Subway Load with Track Irregularity . . . . . . . 1.4 Dynamic Response of Around Soil and Subway Shield Tunnel . . . . 1.5 Mechanical Properties of Subway Shield Tunnel . . . . . . . . . . . . . . . . 1.6 Long-Term Settlement of Subway Shield Tunnel . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 6 7 8 10

2 Dynamic Response of Subway Track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Theoretical Analysis of Track Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Lower Than the First Cut-Off Frequency . . . . . . . . . . . . . . . . 2.2.2 Higher Than the First Cut-Off Frequency and Lower Than the Second Cut-Off Frequency . . . . . . . . . . . . . . . . . . . . 2.2.3 Higher Than the Second Cut-Off Frequency . . . . . . . . . . . . . 2.3 Case Study for Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Dispersion Curve and Cut-Off Frequency . . . . . . . . . . . . . . . . 2.3.2 Critical Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Analysis of Parameters for Theoretical Model . . . . . . . . . . . . . . . . . . 2.4.1 Variations of the Load Frequency . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Variations of the Load Velocity . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Dynamic Response Along the Beams . . . . . . . . . . . . . . . . . . . 2.4.4 Variations of the Bending Stiffness . . . . . . . . . . . . . . . . . . . . . 2.4.5 Variations of the Elastic Coefficients . . . . . . . . . . . . . . . . . . . . 2.4.6 Variations of the Damping Coefficients . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 16 19 21 22 23 24 25 26 26 32 32 36 38 38 41 45

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3 Vibrations Induced by Subway Load with Track Irregularity . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Simulating Subway Random Load with Excitation of Track Vertical Profile Irregularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Numerical Simulation of Dynamic Response . . . . . . . . . . . . . . . . . . . 3.4.1 Time-Domain Characteristic of Vertical and Longitudinal Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Frequency-Domain Characteristic of Vibration Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Analysis of Ground Vibration Attenuation . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 48 51 54 54 57 61 68 68

4 Dynamic Response of Around Soil and Subway Shield Tunnel . . . . . . 69 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Dynamic Triaxial Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.1 Test Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.2 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.3 Test Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2.4 Hysteresis Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2.5 Shear Modulus and Damping Ratio . . . . . . . . . . . . . . . . . . . . . 77 4.3 Numerical Simulation of Dynamic Response of Around Soil and Shield Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.1 Finite Difference Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.2 Dynamic Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.3 Dynamic Response of Soft Soils Around Lining . . . . . . . . . . 92 4.3.4 Dynamic Response of Lining . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.3.5 Influence of Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3.6 Influence of Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5 Mechanical Properties of Subway Shield Tunnel . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Theoretical Model of Functionally Graded Subway Shield Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Building Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Solutions of Internal Force Coefficients . . . . . . . . . . . . . . . . . 5.3 Single Factor Test on the Functionally Graded Subway Tunnel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Orthogonal Test on the Functionally Graded Subway Tunnel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Deformation Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Section Moment Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 107 107 108 109 109 112 120 120 122

Contents

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5.5 Verification of the Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . 127 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6 Long-Term Settlement of Subway Shield Tunnel . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Theoretical Analysis of Long-Term Settlement of Subway Shield Tunnel and Numerical Verification . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Numerical Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Numerical Simulation of Long-Term Settlement of Subway Shield Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Model Establishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 In-Site Monitoring of Long-Term Settlement of Subway Shield Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Arrangement of Monitoring Benchmarks . . . . . . . . . . . . . . . . 6.4.2 Implementation of Measurement . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Analysis of Settlements of the Whole Subway Line 1 . . . . . . 6.4.4 Analysis of Settlements of Running Tunnels and Subway Stations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Differential Settlements of Running Tunnels . . . . . . . . . . . . . 6.5 Prediction of Long-Term Settlement of Subway Shield Tunnel . . . . 6.5.1 Prediction of Long-Term Settlement by Cubic Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Prediction of Long-Term Settlement by GM (1, 1) and ARMA (N, M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Prediction of Settlement Trough of Subway Shield Tunnel . . . . . . . . 6.6.1 Settlement Behavior of Subway Tunnels . . . . . . . . . . . . . . . . . 6.6.2 Settlement Trough of Subway Tunnels . . . . . . . . . . . . . . . . . . 6.6.3 Prediction Model Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 A Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 142 142 144 149 149 150 160 161 162 163 165 170 171 171 174 175 175 180 183 188 190 191

7 Conclusions and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.2 Prospects for Further Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Appendix A: GM (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Appendix B: ARMA (N, M) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Appendix C: Major Published Works of the Book Author . . . . . . . . . . . . . 207

About the Author

Prof. Dr. Zhen-Dong Cui is the Dean of Institute of Geotechnical Engineering, China University of Mining and Technology. In July 2008, he obtained the Ph.D. from School of Civil Engineering, Tongji University, Shanghai, China. Since then, he had been a postdoctoral research fellow at the Hong Kong University of Science and Technology for one year. He joined Shanghai Institute of Geological Survey in 2009. In July 2010, he joined CUMT as an associate professor and was promoted to full professor in 2013. Supported by China Scholarship Council, from August 2015 to August 2016, as a visiting scholar, he researched and studied in the Department of Civil, Environmental and Architectural Engineering, University of Colorado Boulder. He won the Nomination of 100 Excellent Doctoral Dissertations in China in 2011. He won Shanghai Excellent Doctoral Dissertations in 2010 and Excellent Doctorate Thesis of Tongji University in 2009. In 2015, he was awarded the third prize of Shanghai Natural Science. In 2003, he was awarded the second prize of Natural Science of the Ministry of Education. In 2008, he was awarded the second prize of Progress of Science and Technology in Shanghai. In 2012, he was selected as Qinglan Project for Outstanding Young Teachers of Jiangsu Province, and in 2016, he was selected as 333 Talent Project in Jiangsu Province. In 2014 and 2017, he was twice selected as the Young Academic Leader of China University of Mining and Technology. In 2015, he was awarded as Excellent Innovation Team Leader of China University of Mining and Technology.

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About the Author

His research interests focus mainly on the land subsidence, the urban subway tunnel, the deformation of soft foundation, the soil dynamics, the centrifuge model test, and the artificial frozen soil. In the past 5 years, he took charge of the National Natural Science Foundation of China (NSFC), Jiangsu Natural Science Foundation of China, Outstanding Innovation Team Project at China University of Mining and Technology and Special Fund for China Postdoctoral Science Foundation. He published 77 English papers, in which 57 journal papers indexed by SCI have been published in Engineering Geology, Computers and Geotechnics, Soil Dynamics and Earthquake Engineering, Cold Regions Science and Technology, International Journal of Rock Mechanics and Mining Sciences, Bulletin of Engineering Geology and the Environment, Natural Hazards, Environmental Earth Sciences, etc. He has applied for nine national invented patents, among which seven patents have been awarded. He is the specialized committee member of the soft soil engineering of the geotechnical engineering branch of the Civil Engineering Society of China, the member of International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE), and the member of International Association for Engineering Geology and Environment (IAEG).

List of Figures

Fig. 2.1 Fig. 2.2 Fig. 2.3

Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8 Fig. 2.9 Fig. 2.10 Fig. 2.11 Fig. 2.12 Fig. 2.13 Fig. 2.14 Fig. 2.15 Fig. 2.16 Fig. 2.17 Fig. 2.18 Fig. 2.19

Track model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roots distribution with load frequency lower than the first cut-off frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roots distribution with load frequency higher than the first cut-off frequency and lower than the second cut-off frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roots distribution with load frequency higher than the second cut-off frequency . . . . . . . . . . . . . . . . . . . . . . . . . Dispersion curve at v = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of displacements with load frequency at v = 0 and s = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dispersion curves at two velocities . . . . . . . . . . . . . . . . . . . . . . . . Variations of displacements with load velocity at f = 0 and s = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of displacements with load velocity at s = 0 . . . . . . . . Variations of moments with load velocity at s = 0 . . . . . . . . . . . . Variations of shear force with load velocity at s = 0 . . . . . . . . . . Variations of displacements with load frequency at s = 0 . . . . . . Variations of displacements with distance at f = 0 . . . . . . . . . . . . Variations of displacements with distance at f = 20 Hz . . . . . . . . Variations of displacements with distance at different E 1 I 1 and constant E 2 I 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of displacements with distance at different E 2 I 2 and constant E 1 I 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of displacements with distance at different k 1 and constant k 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of displacements with distance at different k 2 and constant k 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of displacements with distance at different c1 and constant c2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16 20

21 23 24 25 26 27 28 30 33 35 37 39 40 41 42 43 44

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Fig. 2.20 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 3.10 Fig. 3.11 Fig. 3.12 Fig. 3.13 Fig. 3.14 Fig. 3.15

Fig. 3.16

Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8

List of Figures

Variations of displacements with distance at different c2 and constant c1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simplified train-track coupled calculation model . . . . . . . . . . . . . Samples of track vertical profile irregularity . . . . . . . . . . . . . . . . . Variations of exciting force with time . . . . . . . . . . . . . . . . . . . . . . Model and observing points in the middle longitudinal sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subway moving loading with time . . . . . . . . . . . . . . . . . . . . . . . . Schematic diagram of characteristic lengths of subway train . . . Vibration acceleration with time at point B . . . . . . . . . . . . . . . . . . Vibration acceleration with time at point L . . . . . . . . . . . . . . . . . . Ground vibration acceleration with time at point A1 . . . . . . . . . . Ground vibration acceleration with time under exciting force (F2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency spectrum of vertical and longitudinal vibration acceleration at point B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency spectrum of vertical and longitudinal vibration acceleration at point L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency spectrum of ground vertical vibration acceleration under axle load (F1) . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency spectrum of ground vertical vibration acceleration under exciting force (F2) . . . . . . . . . . . . . . . . . . . . . . Vertical peak acceleration (VPA) and vibration acceleration level (VVAL) of ground surface with the distance from the centerline . . . . . . . . . . . . . . . . . . . . . . . Longitudinal peak acceleration (LPA) and vibration acceleration level (LVAL) of ground surface with the distance from the centerline . . . . . . . . . . . . . . . . . . . . . . . Time-history curve of dynamic shear strain and dynamic shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hysteretic curve of dynamic shear stress versus dynamic shear strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GDS dynamic triaxial apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . A 10 hysteresis loops at different dynamic stress of Scheme No. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 10 hysteresis loops at different dynamic stress of Scheme No. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A10 hysteresis loops at different dynamic stress of Scheme No. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elliptical fitting of the 5th hysteresis loop at different dynamic stress of Scheme No. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of the shear modulus and damping ratio with the shear strain of Scheme No. 1 . . . . . . . . . . . . . . . . . . . . . .

45 48 51 52 53 55 55 56 58 60 62 63 64 65 66

67

67 70 71 72 74 76 78 80 83

List of Figures

Fig. 4.9

Fig. 4.10

Fig. 4.11 Fig. 4.12 Fig. 4.13 Fig. 4.14 Fig. 4.15 Fig. 4.16 Fig. 4.17 Fig. 4.18 Fig. 4.19 Fig. 4.20 Fig. 4.21 Fig. 4.22 Fig. 4.23 Fig. 4.24 Fig. 4.25 Fig. 4.26 Fig. 4.27 Fig. 4.28 Fig. 4.29 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 5.9

Comparison between the curves of dynamic shear modulus and damping ratio versus dynamic shear strain under different frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between the curves of dynamic shear modulus and damping ratio versus dynamic shear strain under different confining pressures . . . . . . . . . . . . . . . . . . . . . . . . Fitting of dynamic shear modulus and damping ratio versus dynamic shear strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-dimensional finite difference model . . . . . . . . . . . . . . . . . . . Vibrating loads and the spectrum diagrams under different speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of degradation coefficient of shear modulus with shear strain for different soft soil layers . . . . . . . . . . . . . . . . Curves of cumulative vertical displacement at different points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curves of zero vertical displacement at different time around tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical displacements of different horizontal planes . . . . . . . . . . Variations of horizontal displacements of planes with different distances away from Point C . . . . . . . . . . . . . . . . . . Motion of soils around tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of vertical stress with time . . . . . . . . . . . . . . . . . . . . . . Variations of acceleration and velocity with time below Point B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moment and shear force of lining at 2 s . . . . . . . . . . . . . . . . . . . . Variations of maximum moment and maximum shear force with time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of acceleration and velocity with time at point B . . . . Cumulative vertical displacement with time under different speeds for two points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of vertical stress with time under different speeds for two points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of acceleration and velocity at point B . . . . . . . . . . . . . Cumulative vertical displacement with time for two points . . . . . Variations of vertical stress with time at two points . . . . . . . . . . . Microelement of the circular lining . . . . . . . . . . . . . . . . . . . . . . . . Calculation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radial displacement in the single factor test of parameter a . . . . Section moment in the single factor test of parameter a . . . . . . . . Radial displacement in the single factor test of parameter b . . . . Section moment in the single factor test of parameter b . . . . . . . . Radial displacement in the single factor test of parameter λ . . . . Section moment in the single factor test of parameter λ . . . . . . . Intuitive analysis of influence factors of the radial displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

84

85 86 87 90 91 93 94 95 96 97 98 99 100 100 101 102 102 103 104 104 108 110 113 114 116 117 118 119 123

xvi

Fig. 5.10 Fig. 5.11

Fig. 5.12 Fig. 5.13

Fig. 5.14

Fig. 5.15 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7 Fig. 6.8 Fig. 6.9 Fig. 6.10 Fig. 6.11 Fig. 6.12 Fig. 6.13 Fig. 6.14 Fig. 6.15 Fig. 6.16 Fig. 6.17 Fig. 6.18 Fig. 6.19 Fig. 6.20 Fig. 6.21 Fig. 6.22 Fig. 6.23 Fig. 6.24 Fig. 6.25 Fig. 6.26 Fig. 6.27 Fig. 6.28

List of Figures

Intuitive analysis of influence factors of the section moment . . . Comparison of numerical simulation results and theoretical calculation results of radial deformation of homogeneous cylinder at different positions under different thickness diameter ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The change trend of cylinder mechanical properties with the number of segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between the theoretical and numerical results of radial displacement of functionally graded lining under the condition of elastic modulus function I . . . . . . . . . . . . . Comparison of radial displacement calculation results of functionally graded lining with different elastic modulus parameters under the condition of elastic modulus function I . . . Comparison of mechanical properties of functionally graded lining under two elastic modulus functions . . . . . . . . . . . . Generalized Kelvin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long-term settlement of shield tunnel . . . . . . . . . . . . . . . . . . . . . . Vertical displacement of soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The horizontal displacement of soil . . . . . . . . . . . . . . . . . . . . . . . . Deformation of segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consolidation of soil near the segment . . . . . . . . . . . . . . . . . . . . . Theoretical value and simulated value of settlement . . . . . . . . . . Schematic of finite element model . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of soil distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacement after ground stress balance . . . . . . . . . . . . . . . . . . . Schematic of shield tunnel excavation . . . . . . . . . . . . . . . . . . . . . . Vertical displacement of the model after excavation . . . . . . . . . . . Vertical displacement of lining after excavation . . . . . . . . . . . . . . Schematic of the path of the vertical displacement . . . . . . . . . . . . Vertical displacement of different paths . . . . . . . . . . . . . . . . . . . . Vertical displacement of each excavation cycle on path 2 . . . . . . Vertical displacement of each excavation cycle on path 4 . . . . . . Vertical displacement of upper and lower face of tunnel at 40 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Horizontal displacement of the model . . . . . . . . . . . . . . . . . . . . . . Horizontal displacement of different paths . . . . . . . . . . . . . . . . . . Distribution of the vertical stress . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical stress of different paths . . . . . . . . . . . . . . . . . . . . . . . . . . . Distribution of the pore pressure . . . . . . . . . . . . . . . . . . . . . . . . . . Pore pressure of different paths . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure graph of monitoring benchmarks . . . . . . . . . . . . . . . . . . Location of the monitoring benchmarks along the Subway Line/station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bedrock bench mark schematic diagram . . . . . . . . . . . . . . . . . . . . Route of Shanghai Subway Line 1 (phase I) . . . . . . . . . . . . . . . . .

128

132 135

136

137 138 144 145 146 147 147 148 148 149 150 151 151 152 153 153 154 155 155 156 157 157 158 159 159 160 161 161 163 164

List of Figures

Fig. 6.29 Fig. 6.30 Fig. 6.31 Fig. 6.32

Fig. 6.33 Fig. 6.34 Fig. 6.35 Fig. 6.36 Fig. 6.37 Fig. 6.38 Fig. 6.39 Fig. 6.40 Fig. 6.41

Accumulated settlement of Subway Line 1 (from May 1995 to December 2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Settlements of running tunnels (between April 1999 and June 2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cumulative settlements of subway stations (between April 1999 and June 2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Settlement course curve of maximum and minimum accumulated settlement monitoring points of stations and interval tunnels of line 1 for uplink (1999.4–2007.6) . . . . . . Variations of differential settlements of running tunnels with time between December 1995 and June 2007 . . . . . . . . . . . . Settlement prediction result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accumulated settlement of subway tunnels in Shanghai . . . . . . . Schematic diagrams of settlement trough . . . . . . . . . . . . . . . . . . . Accumulated settlement difference from 1999 to 2007 . . . . . . . . Variations of accumulated settlement with time at Hengshan Road Station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seasonal settlement of Subway Line 1 at Hengshan Road station from 2000 to 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the measured accumulated settlement with the predicted one of settlement trough . . . . . . . . . . . . . . . . . Prediction precision of the whole settlement trough . . . . . . . . . . .

xvii

164 166 167

169 171 176 178 181 182 182 184 189 189

List of Tables

Table 2.1 Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8 Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 5.5 Table 5.6 Table 5.7 Table 6.1 Table 6.2 Table 6.3 Table A.1

Parameters of a double-beam system . . . . . . . . . . . . . . . . . . . . . . Subway train parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical and mechanical properties of clay . . . . . . . . . . . . . . . . . Material models and parameters of structures . . . . . . . . . . . . . . . Boundary parameters adopt in model . . . . . . . . . . . . . . . . . . . . . . Test scheme of dynamic triaxial tests . . . . . . . . . . . . . . . . . . . . . . Selection of k value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A σv , σh and σm corresponding to the five test schemes . . . . . . . Gmax corresponding to five test schemes . . . . . . . . . . . . . . . . . . . Physical and mechanical properties of various silty clay layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters of shield lining and ballast used in model . . . . . . . . . Mechanical properties of interface used in model . . . . . . . . . . . . Parameters of damping in different materials . . . . . . . . . . . . . . . Factor levels of the single factor test . . . . . . . . . . . . . . . . . . . . . . Level values of every factor in orthogonal test . . . . . . . . . . . . . . Orthogonal test scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radial displacement of functionally graded lining . . . . . . . . . . . Variance analysis of influence factors of radial displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section moment of functionally graded lining . . . . . . . . . . . . . . . Variance analysis of influence factors of section moment . . . . . . Material parameters of the segment and the soil . . . . . . . . . . . . . Parameters of each layer soil and structures . . . . . . . . . . . . . . . . Segmentation curve simulation parameters of Subway Line 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predicted precision indexes of model . . . . . . . . . . . . . . . . . . . . . .

24 49 53 53 54 73 85 85 86 88 88 88 92 112 120 120 121 126 127 131 145 150 172 201

xix

Chapter 1

Introduction

1.1 Background In China, with the continuous growth of population and the acceleration of urbanization, especially in economically developed areas along the eastern coast, the population of cities is rapidly increasing, and the urban diseases increasing population, increasing housing density, relatively narrow streets, and crowded vehicles in various regions are becoming increasingly serious, It is urgent to solve the urban disease by developing underground spaces such as shopping malls, pedestrian streets, parking lots, and railways. The first subway in China was the Beijing subway, which was opened in 1969. Subsequently, subways in Shanghai, Guangzhou, Tianjin, and Shenzhen have also been built, and some medium-sized cities in the eastern and central regions are planning subway construction. The construction of these subways will further alleviate the urban transportation environment and reduce urban pollution. Subway construction has broad prospects in China and also marks the peak period of subway development in China. Although the construction of subways brings great convenience to urban residents, the engineering problems caused by subways are also becoming increasingly prominent. Subway excavation causes cracking of surrounding buildings and ground subsidence; the impact of vibration and noise caused by subway operation, as well as the liquefaction of foundation caused by vibration, among which the environmental problems caused by subway vibration are particularly serious. Subway vibration can have an impact on surface building structures, human physiological health, animal habits, and the normal operation of some precision instruments. The external environmental impact on buildings mainly comes from two aspects. One is traffic loads, such as road vehicles, trains, and subway vibrations; the second is the construction activities around the building, such as piling, excavation of foundation pits, and water pumping. The impact of subway vibration on buildings cannot be ignored, especially for historical buildings with strict protection.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Z.-D. Cui, Interaction Between Soil Foundation and Subway Shield Tunnel, https://doi.org/10.1007/978-981-99-6870-1_1

1

2

1 Introduction

Therefore, it is of great significance to study the interaction between the soil foundation and the subway shield tunnel, which can provide theoretical basis and guidance for the effective control of long-term subway settlement and maintaining the safety of subway operation.

1.2 Dynamic Response of Subway Track A beam is a common model to characterize many moving-load problems, including tracks (Koh et al. 2003), vehicles travelling on a bridge (Lou 2005; Lou and Zeng 2005), high-speed aircrafts (Zhao et al. 2015), fluids flowing in a pipe (Stein and Tobriner 1970; Chen 1972) and nano-electromechanical systems (NEMS) (Darijani and Mohammadabadi 2014). To model infinite beams, Euler–Bernoulli beams and Timoshenko beams are the main two physical models at present. The Euler–Bernoulli beam theory has been extensively used due to its relatively simpler mathematical formulation. The steady-state solution is necessary for a beam model subjected to a moving load. However, there are certain existing inherent difficulties involved in the mathematical formulation to solve relevant problems. Considering the effects of axial load and linear damping, Sheehan and Debnath (1972) gave an analytical steady-state solution of an Euler–Bernoulli beam on an elastic foundation. Similarly, Kenney (1954) studied the analytical steady-state solution of the Euler–Bernoulli beam on a Winkler foundation, including the effect of damping. Recently, a new method was developed to calculate the dynamical response of Euler–Bernoulli beams under large deflections (Ordaz-Hernandez and Fischer 2008). Mallik et al. (2006) derived the closed-form steady-state response of a uniform beam resting on an elastic foundation and subjected to a moving concentrated load with constant speeds by Fourier transform. One-parameter and two-parameter foundation are modeled and compared. In track engineering, critical velocities are typical characteristics of a beam subjected to a moving load. Nechitailo and Lewis (2006) calculated critical velocities by utilizing analytical and finite element models. A closed-form analytical solution for critical velocities was obtained, which was based on the Euler–Bernoulli model of a beam resting on an elastic foundation regardless of damping. In some special cases, analytical solutions can be derived. Sun (2001) constructed the form of the convolution of the Green function and obtained the steady-state response of a beam on a viscoelastic foundation subjected to a harmonic line load. Sun (2003) derived a closed-form deflection response of a beam on an elastic foundation for different combinations of load frequency and velocity using the integral transform method. In order to consider the damping of foundations, viscoelastic foundations had been commonly used instead of elastic ones (Sun 2001, 2002). Whereas homogeneous foundations are ideal, foundations with randomly varying support stiffness are more practical. Andersen and Nielsen (2003) solved the dynamic response of an Euler– Bernoulli beam placed on a Kelvin foundation with random stiffness and subjected to a constant moving load by proposing a first-order perturbation approach. Recently, Koroma et al. (2014) investigated the vibration of a beam that was continuously

1.2 Dynamic Response of Subway Track

3

supported on an elastic foundation with nonhomogeneous stiffness and damping and subjected to a harmonically excited mass. The results showed that a beam resting on a nonhomogeneous foundation may exhibit different resonances for different individual sections with different foundation stiffness. The dynamic response of finite and infinite beams on a piece-wise homogeneous viscoelastic foundation was studies by Dimitrovová (2010). Furthermore, the Euler–Bernoulli beam theory can be applied to underground subway (Forrest and Hunt 2006) and track engineers (Sheng et al. 2004a, b). However, the Euler–Bernoulli beam theory just considers deflections of the beam, indicating that the shear deformation and rotatory inertia are ignored. In some practical cases, a Timoshenko beam was more accurate than an Euler–Bernoulli beam (Massalas and Kalpakidis 1984). Ruge and Birk (2007) derived and compared the dynamic response of both Timoshenko and Euler–Bernoulli beams on a Winkler foundation in the frequency- and time-domain, who aimed to compare the differences between Timoshenko and Euler–Bernoulli beams and to confirm the numerical advantages of the Timoshenko beam theory. Chen et al. (2001) constructed a dynamic stiffness matrix of an infinite Timoshenko beam on a viscoelastic foundation and calculated the corresponding critical velocities and the resonant frequencies. Liu and Li (2003) proposed a high-order artificial boundary condition to solve elastic wave propagation in an infinite Timoshenko beam resting on a viscoelastic foundation. As damping properties is a mechanism of vibration reduction or suppression, damped beam structures have been proposed (Capsoni et al. 2013). Carvalho and Zindeluk (2001) studied the wave solution for an infinite Timoshenko beam under a concentrated applied load and adopted two actuator array models (force and pair of moments). It should be mentioned that most papers have assumed foundations to be linear due to its simple model. Whereas nonlinearity of viscoelastic foundations was also discussed by former researchers. Kargarnovin et al. (2005) obtained a closedform solution of infinite Timoshenko beams on nonlinear viscoelastic foundations subjected to a harmonic moving load using a perturbation method in conjunction with complex Fourier transform. Moreover, Sim¸ ¸ sek (2010) derived the nonlinear equations and investigated the dynamic response of a Timoshenko beam with pinned– pinned ends subjected to a moving harmonic load, pointing that the geometrical non-linearity and the power-law exponent had great influence on the stresses of the beam. Some other practical cases, such as thermoelastic vibrations of a Timoshenko beam (Manoach and Ribeiro 2004), delaminated beams (Kargarnovin et al. 2013) and saturated poroelastic beams (Kiani et al. 2015), were also investigated through Timoshenko beam model. It should be noted that all the references mentioned above just discussed a single beam regardless of their beam models (Euler–Bernoulli beams or Timoshenko beams). In fact, a double-beam system may be more accurate to describe a real track system, especially at high frequencies (Wu and Thompson 1999). The main reason is that the rail has distinct features (e.g., mass per unite length and bearing stiffness) from the slab, resulting in different cut-off frequencies. Due to the smaller mass of the rail per unite length, the rail normally has a considerably higher cut-off frequency than the slab. Therefore, a single beam is not so proper and accurate to represent the

4

1 Introduction

track system as a double-beam with two different beams, which can model the rail and the slab respectively. Vu et al. (2000) studied the dynamic response of a doublebeam system and decoupled the differential equations. However, the flexural rigidity and the mass per unit length of the two beams were assumed to be identical, which was just a special case and not so practical in most applications. Zhang et al. (2008) investigated the dynamic properties of a double-beam subjected to an axial load on the basis of the Euler–Bernoulli beam theory. Wu and Thompson (1999) developed a double Timoshenko beam to study the vertical vibration at high frequencies.

1.3 Vibration Induced by Subway Load with Track Irregularity For the study on the vibration caused by train moving load, different approaches were well developed. Metrikine and Vrouwenvelder (2000) established a two dimensional model by considering the tunnel as Euler beam embedded in a viscoelastic soil layer with a rigid base. The vibrations at the ground surface were investigated for constant, harmonic, and random loads moving along the beam, respectively. Sheng et al. (2003) studied the dynamic response of ground under harmonic loads moving in a circular tunnel. The effect of the tunnel on vibration responses at the ground surface and the difference between lined tunnel and unlined tunnel were presented. Subsequently, they established a rail-layered ground coupled model that included the effects of track irregularity. The moving axle load and track irregularity were used as input excitation to obtain the wheel-rail interaction force as well as the displacement spectra of the rail and the ground surface (Sheng et al. 2004a, b). In addition, a coupled periodic FE-BE model had been developed that exploits the longitudinal invariance or periodicity of the track-tunnel-soil system (Clouteau et al. 2005; Degrande et al. 2006; Gupta et al. 2008). Then Gupta et al. (2009) used this model and the pipe-in-pipe model, considering the dynamic interaction between the train, the track, the tunnel and the soil to study the influence of parameters on the free field response induced by underground railways. Yaseri et al. (2014) presented the application of the scaled boundary finite-element method (SBFEM) to the 3D analysis of ground vibrations induced by underground trains. Zhai et al. (2009) established a three-dimensional train-rail coupled vibration model, and focused on analyzing the rail and train vibration responses when the train ran on irregular tracks. Galvín and Domínguez (2009) used the three-dimensional finite element–boundary element methods to calculate the vibration response of a coupled vehicle-rail-subsoil system. Lombaert and Degrande (2009) established a three-dimensional vehicle-rail coupled model and calculated the vibration response for train speeds below the critical velocity. It was found that the quasi-static axle load determines the track response, whereas the dynamic load determines the free field response; the track response caused by the quasi-static axle load increases slightly as train speed increases. Kouroussis et al. (2012) investigated numerically the influence of vehicle and track parameters on ground vibrations

1.3 Vibration Induced by Subway Load with Track Irregularity

5

induced by the railway traffic in the ballast track. They observed that the vehicle speed has a strong influence on the ground vibrations notably when the train speed becomes larger than the Rayleigh wave velocity of the ground. In the homogenous ground the vibratory level notably increases at the train speed larger than the Rayleigh wave velocity, whereas the critical velocity in the layered ground is influenced by the depth of the layers and their dynamic characteristics (Kouroussis et al. 2013). By using the 2.5D finite element method, Yang and Hung (2008) performed a parametric study on soil vibrations caused by underground moving trains. Bian et al. (2015) developed a 2.5-dimensional (2.5D) finite element model combining with thin-layer elements was applied to establish a vehicle-track-foundation coupled dynamic analysis model. A quarter-car model was used to derive the equation for wheel-rail interaction force considering cosine track irregularity. Model testing in laboratory, as an effective alternative to field measurement, provides valuable data to understand railway’s dynamic behaviors under train moving loads. Bian et al. (2014) present comprehensive experimental results on track vibration and soil response of a ballastless high-speed railway from a full-scale model testing with simulated train moving loads at various speeds. Comparisons with the field measurements show that the proposed model testing can accurately reproduce dynamic behaviors of the track structure and underlying soils under train moving loads. Jiang et al. (2014) also established full-scale physical model to investigate the dynamic responses of the slab track-subgrade system under train moving loading. Currently, the dynamic response of a tunnel buried in a two-dimensional poroelastic soil layer subjected to a moving point load was investigated theoretically. The tunnel was simplified as an infinite long Euler–Bernoulli beam, which was placed parallel to the traction-free ground surface (Yuan et al. 2015). Gharehdash and Barzegar (2015) adopted the theory of simulated deterministic method (Zhang and Bai 2000) to calculate the metros dynamic wheel-rail forces, which a conversion expression is presented by using FFT algorithm to transform the measured acceleration of rail, and then dynamic expression system based on a vehicle-track coupled model is established to simulate the dynamic wheel-rail interaction forces P(t). Zhai et al. (2015) et al. conducted a field experiment of ground vibration on high-speed railway. It was shown that the periodic exciting action of high-speed train bogies can be identified in time histories of vertical accelerations of the ground within the range of 50 m from the track centerline. The first dominant sensitive frequency of the ground vibration acceleration results from the wheelbase of the bogie, and the center distance of two neighboring cars plays an important role in the significant frequencies of the ground vibration acceleration. With long term operation of the subway train, the rail surfaces will inevitably generate wear which will cause greater dynamic forces between the wheels and rail resulting in settlement of subway subgrade; meanwhile uneven subgrade settlement will also lead to track irregularity. The dynamic excitation is due to random track unevenness, and the influence of the train speed on the free-field vibrations depends on the power spectral density (PSD) of the track unevenness (Lombaert and Degrande 2009). Therefore, track irregularity should be taken into consideration as an essential factor causing vibration.

6

1 Introduction

1.4 Dynamic Response of Around Soil and Subway Shield Tunnel Numerical analysis as an effective method had been conducted to study the vibration induced by the train, such as boundary element method (BEM) and finite element method (FEM). Luo et al. (1996) simulated the infinite soil under the track with a finite FE mesh with rigid boundary conditions; the soil mesh was made large enough that waves cannot return from the boundaries within the time considered. Hall (2003) built a two-dimensional finite element model perpendicular to the track with stationary loading that was used to study the ground response outside the embankment. Ju (2009) proposed a finite element model to investigate the characteristics of the building vibrations induced by adjacent moving truck crossing random irregularities. Çelebi and Göktepe (2012) dealt with the non-linear 2D finite element modeling for the prediction of shielding performance. Nicolosi et al. (2012) used a FEM approach to build the propagation model as the examined railway line is underground. The geometry and layout of the 3D FEM model were decided following a calibration phase and according to the stratigraphic characteristics of the test site. Results showed a good agreement between the numerical and the experimental data. BEM for soilstructure interaction analysis are based on integral formulations and fundamental solutions of a wave propagation problem. Such formulations implicitly satisfied the radiation boundary condition of infinite domains, and were appealing alternatives to the FEM (Rizos and Wang 2002). Lombaert and Degrande (2001, 2003) used the boundary element method in the frequency-wave number domain together with the Betti-Rayleigh reciprocal theorem to compute the dynamic response induced by a moving vibration source. A good agreement was obtained between the numerical results and the field data. Researchers have also used alternative methods to model the soil and the tunnel. Gardien and Stuit (2003) proposed a modular model, consisting of three sub-models (the Static Deflection Model, the Track Model and the Propagation Model). FEM was used to investigate the dependence of the results on the accuracy of the model inputs. Bian et al. (2007) used a 2.5D finite element method with absorbing boundary conditions to model wave propagation from subway traffic. FEM analysis of tracks and soil, often with BEM, has also frequently been used to predict vibration generated by railways. A three-dimensional FEM–BEM treatment of a railway comprising two rail beams on rigid sleeper footings on a half-space had been conducted by Mohammadi and Karabalis (1995). The coupled periodic FE-BE model was based on a subdomain formulation, where a boundary element method was used for the soil and a finite element method for the tunnel (Degrande et al. 2006; Gupta et al. 2007). Then Gupta et al. (2009) used this model and the pipe-in-pipe model, considering the dynamic interaction between the train, the track, the tunnel and the soil to study the influence of parameters on the free field response induced by underground railways. The finite difference method (FDM) has become a significant and general numerical tool considering the ground heterogeneity, the non-linear soil behavior and the soil-structure interaction. Mhanna et al. (2012) built a car model with four degrees

1.5 Mechanical Properties of Subway Shield Tunnel

7

of freedom for the determination of the load due to the road traffic. The load was then introduced in a 3D FDM for the determination of the traffic-induced ground vibrations. It was proved that train speed had an impact on amplitude and frequency of vibration. Gharehdash and Barzegar (2015) built a complex 3D dynamic finite difference model fully considering the joints to show the dynamic response of the shield tunnel buried in soft soil under the vibrating load. Numerical results demonstrated that an operating metro train induced significant dynamic response in the structure of the lining of the shield tunnel and its soft foundation.

1.5 Mechanical Properties of Subway Shield Tunnel In order to improve the supporting performance and waterproofness of subway lining, two methods are usually applied: either to increase the thickness of the lining or to enhance the concrete strength (Zhang et al. 2017). However, both are extravagant in that concrete strength and thickness are the same in every position, which will lead to a waste of concrete in corresponding positions. The functionally graded materials (FGMs) are introduced to the design of the lining of the subway tunnel, and the concrete with different elastic modulus is arranged at different positions to reduce its cost on the premise of ensuring structural safety. The concept of functionally graded materials (FGMs) was proposed in 1984 by materials scientists in the Sendai area as a means of preparing thermal barrier materials (Koizumi 1997). The composition of the functional graded materials changes continuously from one direction to another, resulting in a continuous change of the material properties (such as elastic modulus) (Gupta and Talha 2015). Extensive applications of FGMs have extended to many fields where the operating conditions are severe, including aerospace, chemical plants, nuclear energy reactors and so on (Jha et al. 2013). Efforts are also made to study the performance of construction materials, such as metal and cement-based materials (Yang et al. 2003). Liu et al. (2018) developed the sustainable structure constructed with functionally graded concretes using fibers and recycled aggregates. Ahmadi et al. (2017) studied the mechanical properties of the graded concrete specimens composed with recycled aggregates and steel wires recycled from waste tires. Dias et al. (2010) established the concept of functionally graded fiber cement, and the use of statistical mixture designs was discussed to choose formulations and present ideas for the production of functionally graded fiber cement components. Shen et al. (2008) employed a functionally graded material system to make fiber more efficient in a fiber reinforced cement composite with four layers, each with a different fiber volume ratio. Differently, the study introduced in this monograph aims to develop a functionally graded cement-based lining of the subway tunnel. To achieve this, the elastic modulus function is adopted, which is a power law function with respect to angle α.

8

1 Introduction

There have been many analytical results about functionally graded hollow cylinder subjected to mechanical stresses. Shi et al. (2007) defined the elastic modulus as a linear function varying with radius, and Poisson’s ratio is set as a constant, then, the exact solutions of the hollow cylinder with continuously graded properties was obtained. Dai et al. (2006) assumed the elastic modulus as a simple power law function varying through the wall thickness, and the exact solution for displacement and stress is determined when Poisson’s ratio is assumed constant. Similarly, Batra (2008) and Batra and Iaccarino (2008) analyzed the deformation of functionally graded cylinders composed of incompressible isotropic linear elastic materials, with the variation of the shear modulus in the radial direction given by a power law relation and constant Poisson’s ratio. In the other case, the exact elasticity solution of a radially nonhomogeneous hollow cylinder was derived, whose elastic modulus varied in an exponential and power law function (Theotokoglou and Stampouloglou 2008). Beyond those above, the radial elastic modulus was assumed as an arbitrary function (Chen and Lin 2010). In these works, material parameters such as elastic modulus or shear modulus were assumed in advance to be a function of the radius, and Poisson’s ratio was a constant. The stresses and displacement distributions are calculated under the given elastic modulus E(r) or the shear modulus G(r). Differing from the mentioned works above, Zhang et al. (2017) pre-assumed the desired stress distribution with the elastic modulus E(r) being undermined, then the E(r) is confirmed by back- calculations according to the stress distribution and loadings. However, all the above-mentioned works are based on axisymmetric loads, and the situation under non-axisymmetric loads has not been researched yet, like linings of the subway tunnel.

1.6 Long-Term Settlement of Subway Shield Tunnel The long-term settlement of shield tunnel has significant effect on the safety of the subway operation. In order to make the settlement calculation more accurate, the concept of ground loss was proposed by Peck (1969). The application of ground loss to some typical problems, such as soft ground tunneling (Aoyagi 1995) and pile driving (Hwang et al. 2001) or extraction (Zhao et al. 2012), shows that the calculated movements agree well with the experimental observations (Sagaseta 1987). Gonzalez and Sagaseta (2001) presented the analytical solutions to the deformation of soil. The stress functions in general series form were proposed to analyze the ground movements induced by tunneling (Bobet 2001; Chou and Bobet 2002; Park 2004). Model tests were conducted to study the settlement because of the theory limitation. Centrifuge model tests have been widely used in the study on tunnel settlement (Mair and Taylor 1993; Chambon and Corté 1994; Nomoto et al. 1999; Meguid et al. 2008; Yang et al. 2013). More feasible ways are put forward by various tests, which have a good prospect in application and make us understand the tunnel settlement deeply. Chen et al. (2013) studied the face stability of shield tunnel on test, which can effectively control the ground settlement and ensure the safety in tunnel construction.

1.6 Long-Term Settlement of Subway Shield Tunnel

9

Numerical modeling allows one to conduct more realistic analyses that take into account the tunnel-lining interaction, construction sequence and 3D face effects. In order to clarify the mechanism of tunnel settlement, the complex three-dimensional model was established to simulate the deformation characteristics of tunnel (Hudoba 1997; Mroueh and Shahrour 2008; Zhao et al. 2012; Mollon et al. 2013). The EPB tunneling process was simulated in a finite element analysis using two-dimensional models (Richard et al. 1985). Lee and Rowe (1990, 1991a, b) took the twodimensional plane strain finite element method to analyze the influence of parameters on tunneling. It’s an effective way to analyze the deformation of shield tunnel with the help of large finite element software. Rowe (1983) developed an elastic–plastic finite element calculation program to predict the surface settlement of different soil and workmanship caused by tunnel excavation. The finite element method is an effective way to simulate the tunnel deformation, which provides a better design and construction for tunnels (Nam and Bobet 2007; Liao et al. 2009; Verma and Singh 2010). The settlements were often described by the empirical formulae based on field observations, for instance, a normal (Gaussian) distribution curve (Peck 1969; Attewell et al. 1982). Many analytical solutions were presented by some researchers (Verruijt et al. 1996; Bobet 2001; Park 2005). Numerical simulations were also an effective way to analyze deformation of subway shield tunnel (Lee and Rowe 1991a, b; Phienwej and Hong 2006). Model tests were also widely used to analyze the tunnel settlement (Nomoto et al. 1999; Kamata and Mashimo 2003; Jeon et al. 2004; Wang et al. 2012). But the study of field measurement data of shield tunnel is relatively rare because of financial problems, and it is difficult to get enough information that may be applied to actual shield tunneling design (Nomoto et al. 1999). It is of important to know the field measurement data of shield tunnel to prevent the tunnel settlement. Bowers and Hiller (1996) found that significant additional movements occurred after the construction. Clough et al. (1983) measured the surface and subsurface settlements of an EPB shield tunnel (915 m in length and 3.7 m in diameter) constructed in soils. The soil around tunnel moved outward and the lateral displacement of soil was relatively large. Analysis of geodetic monitoring records of certain recently constructed tunnels indicated that, occasionally, large deformation from specific “source” sections was transferred along the tunnel axis to “host”, neighboring sections, causing additional time-delayed deformation of previously stabilized sections and necking of the tunnel (Kontogianni and Stiros 2005). In order to study the response of the soft clay and the ground movement during and after the advance of the tunneling machine, the surface and subsurface ground displacements, pore water pressure changes, and earth pressure development around the concrete lining were monitored during construction (Lee et al. 1999). In the normally consolidated clay, the ratio of the long-term settlement to the total settlement was quite large (Shirlaw 1993, 1994, 1995).

10

1 Introduction

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Koizumi M (1997) FGM activities in Japan. Compos B 28B:1–4 Kontogianni VA, Stiros SC (2005) Induced deformation during tunnel excavation: evidence from geodetic monitoring. Eng Geol 79(1):115–126 Koroma SG, Hussein MFM, Owen JS (2014) Vibration of a beam on continuous elastic foundation with nonhomogeneous stiffness and damping under a harmonically excited mass. J Sound Vib 333(9):2571–2587 Kouroussis G, Verlinden O, Conti C (2012) Influence of some vehicle and track parameters on the environmental vibrations induced by railway traffic. Veh Syst Dyn Int J Veh Mech Mobil 50(4):619–639 Kouroussis G, Conti C, Verlinden O (2013) Investigating the influence of soil properties on railway traffic vibration using a numerical model. Veh Syst Dyn 51(3):421–442 Lee KM, Rowe RK (1990) Finite element modelling of the three-dimensional ground deformations due to tunnelling in soft cohesive soils: Part 2—results. Comput Geotech 10(2):111–138 Lee KM, Rowe RK (1991a) An analysis of three-dimensional ground movements: the Thunder Bay tunnel. Can Geotech J 28:25–41 Lee KM, Rowe RK (1991b) An analysis of three-dimensional ground movements: the Thunder Bay tunnel. Can Geotech J 28(1):25–41 Lee KM, Ji HW, Shen CK, Liu JH, Bai TH (1999) Ground response to the construction of Shanghai Metro Tunnel-Line2. Soils Found 39(3):113–134 Liao SM, Liu JH, Wang RL (2009) Shield tunneling and environment protection in Shanghai soft ground. Tunn Undergr Space Technol 24(4):454–465 Liu T, Li Q (2003) Transient elastic wave propagation in an infinite Timoshenko beam on viscoelastic foundation. Int J Solids Struct 40(13):3211–3228 Liu XZ, Yan MP, Galobardes I, Sikora K (2018) Assessing the potential of functionally graded concrete using fibre reinforced and recycled aggregate concrete. Constr Build Mater 171:793– 801 Lombaert G, Degrande G (2001) Experimental validation of a numerical prediction model for free field traffic induced vibrations by in situ experiments. Soil Dyn Earthq Eng 21(6):485–497 Lombaert G, Degrande G (2003) The experimental validation of a numerical model for the prediction of the vibrations in the free field produced by road traffic. J Sound Vib 262(2):309–331 Lombaert G, Degrande G (2009) Ground-borne vibration due to static and dynamic axle loads of InterCity and high-speed trains. J Sound Vib 319:1036–1066 Lou P (2005) Vertical dynamic responses of a simply supported bridge subjected to a moving train with two-wheelset vehicles using modal analysis method. Int J Numer Methods Eng 64(9):1207– 1235 Lou P, Zeng Q (2005) Formulation of equations of motion of finite element form for vehicletrack-bridge interaction system with two types of vehicle model. Int J Numer Methods Eng 62(3):435–474 Luo Y, Yin H, Hua C (1996) The dynamic response of railway ballast to the action of trains moving at different speeds. Proc IMechE Part F J Rail Rapid Transit 210:95–101 Mair RJ, Taylor RN, Bracegirdle A (1993) Subsurface settlement profiles above tunnels in clays. Geotechnique 43(2):315–320 Mallik AK, Chandra S, Singh AB (2006) Steady-state response of an elastically supported infinite beam to a moving load. J Sound Vib 291(3):1148–1169 Manoach E, Ribeiro P (2004) Coupled, thermoelastic, large amplitude vibrations of Timoshenko beams. Int J Mech Sci 46(11):1589–1606 Massalas CV, Kalpakidis VK (1984) Coupled thermoelastic vibrations of a Timoshenko beam. Int J Mech Sci 22(4):459–465 Meguid MA, Saada O, Nunes MA (2008) Physical modeling of tunnels in soft ground: a review. Tunn Undergr Space Technol 23(2):185–198 Metrikine AV, Vrouwenvelder A (2000) Surface ground vibration due to a moving train in a tunnel: two-dimensional model. J Sound Vib 234(1):43–66

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Mhanna M, Sadek M, Shahrour I (2012) Numerical modeling of traffic-induced ground vibration. Comput Geotech 39(39):116–123 Mohammadi M, Karabalis DL (1995) Dynamic 3-D soil-railway track interaction by BEM-FEM. Earthq Eng Struct Dyn 24(9):1177–1193 Mollon G, Dias D, Soubra AH (2013) Probabilistic analyses of tunneling-induced ground movements. Acta Geotech 8:181–199 Mroueh H, Shahrour I (2008) A simplified 3D model for tunnel construction using tunnel boring machines. Tunn Undergr Space Technol 23(1):38–45 Nam SW, Bobet A (2007) Radial deformations induced by groundwater flow on deep circular tunnels. Rock Mech Rock Eng 40(1):23–39 Nechitailo NV, Lewis KB (2006) Critical velocity for rails in hypervelocity launchers. Int J Impact Eng 33(1):485–495 Nicolosi V, D’Apuzzo M, Bogazzi E (2012) A unified approach for the prediction of vibration induced by underground metro. Proc Soc Behav Sci 53:62–71 Nomoto T, Imamura S, Hagiwara T, Kusakabe O, Fujii N (1999) Shield tunnel construction in centrifuge. J Geotech Geoenviron Eng 125(4):289–300 Ordaz-Hernandez K, Fischer X (2008) Fast reduced model of non-linear dynamic Euler-Bernoulli beam behaviour. Int J Mech Sci 50(8):1237–1246 Park KH (2004) Elastic solution for tunneling-induced ground movements in clays. Int J Geomech 4(4):310–318 Park KH (2005) Analytical solution for tunnelling-induced ground movement in clays. Tunn Undergr Space Technol 20(3):249–261 Peck RB (1969) Deep excavations and tunneling in soft ground. In: Proceedings of the seventh international conference on soil mechanics and foundation engineering. State of the Art Volume, Mexico City, pp 225–290 Phienwej N, Hong CP (2006) Evaluation of ground movements in EPB-shield tunnelling for Bangkok MRT by 3D-numerical analysis. Tunn Undergr Space Technol 21(3):273 Richard J, Finno G, Clough W (1985) Evaluation of soil response to EPB shield tunneling. J Geotech Eng 111(2):155–173 Rizos DC, Wang Z (2002) Coupled BEM–FEM solutions for direct time domain soil–structure interaction analysis. Eng Anal Bound Elem 26(10):877–888 Rowe RK, Lo KY, Kack GJ (1983) A method of estimating surface settlement above tunnels constructed in soft ground. Can Geotech J 20(1):11–22 Ruge P, Birk C (2007) A comparison of infinite Timoshenko and Euler–Bernoulli beam models on Winkler foundation in the frequency-and time-domain. J Sound Vib 304(3):932–947 Sagaseta C (1987) Analysis of undrained soil deformation due to ground loss. Geotechnique 37(3):301–320 Sheehan JP, Debnath L (1972) On the dynamic response of an infinite Bernoulli-Euler beam. Pure Appl Geophys 97(1):100–110 Shen B, Hubler M, Paulino GH, Strublex LJ (2008) Manufacturing and mechanical testing of a new functionally graded fiber reinforced cement composite. AIP Conf Proc 973:519–524 Sheng X, Jones C, Thompson D (2003) Ground vibration generated by a harmonic load moving in a circular tunnel in a layered ground. J Low Freq Noise Vib Active Control 22(2):83–96 Sheng X, Jones C, Thompson D (2004a) A theoretical study on the influence of the track on train-induced ground vibration. J Sound Vib 272(3–5):937–965 Sheng X, Jones CJC, Thompson DJ (2004b) A theoretical study on the influence of the track on train-induced ground vibration. J Sound Vib 272(3):909–936 Shi ZF, Zhang TT, Xiang HJ (2007) Exact solutions of heterogeneous elastic hollow cylinders. Compos Struct 79:140–147 Shirlaw JN (1993) Pore pressure around tunnels in clay: discussion. Can Geotech J 30(6):1044–1046 Shirlaw JN (1994) Subsidence owing to tunneling II. Evaluation of prediction techniques: discussion. Can Geotech J 31:463–466

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Shirlaw JN (1995) Observed and calculate pore pressures and deformations induced by an earth balance shield: discussion. Can Geotech J 32:181–189 Sim¸ ¸ sek M (2010) Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load. Compos Struct 92(10):2532–2546 Stein RA, Tobriner MW (1970) Vibration of pipes containing flowing fluids. J Appl Mech 37(4):906– 916 Sun L (2001) A closed-form solution of a Bernoulli-Euler beam on a viscoelastic foundation under harmonic line loads. J Sound Vib 242(4):619–627 Sun L (2002) A closed-form solution of beam on viscoelastic subgrade subjected to moving loads. Comput Struct 80(1):1–8 Sun L (2003) An explicit representation of steady state response of a beam on an elastic foundation to moving harmonic line loads. Int J Numer Anal Methods Geomech 27(1):69–84 Theotokoglou EE, Stampouloglou IH (2008) The radially nonhomogeneous elastic axisymmetric problem. Int J Solids Struct 45:6535–6552 Verma AK, Singh TN (2010) Assessment of tunnel instability—a numerical approach. Arab J Geosci 3(2):181–192 Verruijt A, Booker JR (1996) Surface settlements due to deformation of a tunnel in an elastic half plane. Geotechnique 46(4):753–756 Vu HV, Ordonez AM, Karnopp BH (2000) Vibration of a double-beam system. J Sound Vib 229(4):807–822 Wang ZL, Shen LF, Xie JB, Yao J (2012) Structure analysis for tunnel longitudinal deformation based on segment dislocation mode. Proc Eng 31:487–491 Wu TX, Thompson DJ (1999) A double Timoshenko beam model for vertical vibration analysis of railway track at high frequencies. J Sound Vib 224(2):329–348 Yang YB, Hung HH (2008) Soil vibrations caused by underground moving trains. J Geotech Geoenviron Eng 134(11):1633–1644 Yang JJ, Hai R, Dong YL, Wu KR (2003) Effects of the component and fiber gradient distributions on the strength of cement-based composite materials. J Wuhan Univ Technol Mater Sci Edn 18:61–64 Yang W, Hussein MF, Marshall AM (2013) Centrifuge and numerical modelling of ground-borne vibration from an underground tunnel. Soil Dyn Earthq Eng 51:23–34 Yaseri A, Bazyar MH, Hataf N (2014) 3D coupled scaled boundary finite-element/finite-element analysis of ground vibrations induced by underground train movement. Comput Geotech 60(1):1–8 Yuan Z, Xu C, Cai Y et al (2015) Dynamic response of a tunnel buried in a saturated poroelastic soil layer to a moving point load. Soil Dyn Earthq Eng 77:348–359 Zhai W, Wang K, Cai C (2009) Fundamentals of vehicle-track coupled dynamics. Veh Syst Dyn 47(11):1349–1376 Zhai W, Wei K, Song X et al (2015) Experimental investigation into ground vibrations induced by very high speed trains on a non-ballasted track. Soil Dyn Earthq Eng 72:24–36 Zhang YE, Bai BH (2000) The method of identifying train vibration load acting on subway tunnel structure. J Vib Shock 19(3):68–70 (In Chinese) Zhang YQ, Lu Y, Wang SL, Liu X (2008) Vibration and buckling of a double-beam system under compressive axial loading. J Sound Vib 318(1):341–352 Zhang N, Lu AZ, Li CC, Zhou JT, Zhang XL, Wang SJ, Chen XG (2017) Support performance of functionally graded concrete lining. Constr Build Mater 147:35–47 Zhao K, Janutolo M, Barla G (2012) A completely 3D model for the simulation of mechanized tunnel excavation. Rock Mech Rock Eng 45(4):475–497 Zhao X, Yang EC, Li YH (2015) Analytical solutions for the coupled thermoelastic vibrations of Timoshenko beams by means of green’s functions. Int J Mech Sci 100:50–67

Chapter 2

Dynamic Response of Subway Track

2.1 Introduction Modern high-speed trains may cause noticeable dynamic response on the rail track, especially when train speeds exceed certain critical velocities. Moving track loads approaching critical velocities cause large rail deflections and can be damaging to track systems. Many different studies have been carried out in this field over the past several decades. A double-beam can model the track system more accurately and completely than a single beam. The double-beam system has similar characteristics to the single beam, but there are also some distinct differences. For example, the governing equations of a double-beam system are normally more complicated than those of a single beam, especially when it comes to the Timoshenko beam theory. Thus, the solving methods and the results tend to be different. Although several researchers have carried out studies on the double-beam, their research is not systematic sufficiently to present the dynamic characteristics the double-beam system. The existing double-beam systems are studied based on some special assumptions (Vu et al. 2000) or cases (Wu and Thompson 1999; Zhang et al. 2008). This chapter presents an infinite Euler–Bernoulli double-beam system resting on a viscoelastic foundation and subjected to a harmonic moving point load, including two spring-damping systems. The governing equations are solved by means of Fourier transform and residue theory. This method allows the parameters of the two beams to be completely arbitrary. The complete solution is derived in this chapter. Moreover, parametric study is carried out to investigate the influence of different beam parameters on the dynamic response of this double-beam system.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Z.-D. Cui, Interaction Between Soil Foundation and Subway Shield Tunnel, https://doi.org/10.1007/978-981-99-6870-1_2

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2 Dynamic Response of Subway Track

2.2 Theoretical Analysis of Track Model The track model, referred as to a double-beam model, consists four parts: rail, rail bearing, slab and slab bearing, as shown in Fig. 2.1. The rail bearing connects the rail and the slab, and the slab bearing connects the slab and ground. The rail bearing and the slab bearing can be simplified as continuous spring-damping systems, and the rail and the slab can be simplified as infinite beams with different bending stiffness, respectively. This chapter aims to study dynamic response of this double-beam model subjected to a harmonic point load with constant velocity. The velocity, angular frequency and amplitude of the load denote as v, ω and F0 , respectively, and then the load can be expressed as Eq. (2.1) F = F0 · eiωt · δ(x − vt)

(2.1)

where δ(x) is the Dirac delta function. The bending stiffness of the rail and the slab is E 1 I1 and E 2 I2 , respectively. The elastic coefficients and the damping coefficients of the two spring-damping systems (i.e. the rail bearing and the slab bearing) are k1 , k2 and c1 , c2 . The two beams are both considered as infinite Euler–Bernoulli beams with the two displacements denoted as y1 and y2 , respectively. The governing equations of this track model are ⎫ ( ) ∂ y1 ∂ y2 ∂ 2 y1 ∂ 4 y1 ⎪ iωt ⎪ = F0 · e · δ(x − vt)⎪ − E 1 I1 4 + m 1 2 + k1 (y1 − y2 ) + c1 ⎬ ∂x ∂t ∂t ∂t ( ) ⎪ ∂ y1 ∂ 2 y2 ∂ y2 ∂ 4 y2 ∂ y2 ⎪ =0 ⎪ E 2 I2 4 + m 2 2 + k2 y2 − k1 (y1 − y2 ) + c2 − c1 − ⎭ ∂x ∂t ∂t ∂t ∂t (2.2) As the load is moving on the rail with a constant velocity, a moving coordinate system is more convenient to solve Eq. (2.2): s = x − vt, w1 = y1 , w2 = y2 vt

F = F0 eiω t

Rail

E1 I1 k1

−∞

(2.3)

c1

Rail bearing Slab

E2 I 2 k2

c2

Fig. 2.1 Track model

y1

y2

Slab bearing

∞

2.2 Theoretical Analysis of Track Model

17

Applying the chain rule of derivation, the differentiation of Eq. (2.3) can be derived as ⎫ ∂ yi ∂wi ⎪ ⎪ = ⎪ ⎪ ∂x ∂s ⎪ ⎪ ⎪ 2 2 ⎪ ∂ yi ∂ wi ⎪ ⎪ = ⎬ 2 2 ∂x ∂s ∂wi ∂wi ∂ yi ⎪ ⎪ ⎪ = −v ⎪ ⎪ ∂t ∂t ∂s ⎪ ⎪ ⎪ 2 2 2 2 ⎪ ∂ yi ∂ wi ∂ wi 2 ∂ wi ⎪ ⎭ + v = − 2v 2 2 2 ∂t ∂t ∂s∂t ∂s

(2.4)

where i = 1, 2. The governing equations in this moving coordinate system are achieved by Eqs. (2.3) and (2.4) ) ( ⎫ 2 ∂w2 ∂ 2 w1 ∂ 4 w1 ∂ 2 w1 ∂w1 ⎪ 2 ∂ w1 ⎪ − c1 v − + m1 2 + m1v − 2m 1 v E 1 I1 ⎪ ⎪ ∂s 4 ∂s 2 ∂s∂t ∂s ∂s ∂t ⎪ ⎪ ⎪ ( ) ⎪ ⎪ ∂w1 ∂w2 ⎪ iωt ⎪ − + k1 (w1 − w2 ) = F0 e δ(s) + c1 ⎬ ∂t ∂t 2 ⎪ ⎪ ∂ 4 w2 ∂ 2 w2 ∂w2 ⎪ 2 ∂ w2 ⎪ E 2 I2 − (c + m v − 2m v + c )v 2 2 1 2 ⎪ ⎪ ∂s 4 ∂s 2 ∂s∂t ∂s ⎪ ⎪ ⎪ 2 ⎪ ∂ w2 ∂w1 ∂w1 ∂w2 ⎪ + m 2 2 + (c1 + c2 ) − c1 + k2 w2 − k1 (w1 − w2 ) = 0⎭ + c1 v ∂s ∂t ∂t ∂t (2.5) The solution of Eq. (2.5) subjected to a harmonic point load can be assumed as w1 (s, t) = ϕ(s)eiωt , w2 (s, t) = ψ(s)eiωt

(2.6)

where ϕ(s) and ψ(s) are shape functions. According to Eqs. (2.5) and (2.6), the governing equations are transformed from time-domain to frequency-domain E 1 I1 ϕ(4) + m 1 v 2 ϕ'' − 2m 1 v(iω)ϕ' − c1 v(ϕ' − ψ ' ) + m 1 (iω)2 ϕ + c1 (iω)(ϕ − ψ) + k1 (ϕ − ψ) = F0 δ(s)

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

E 2 I2 ψ (4) + m 2 v 2 ψ '' − 2m 2 v(iω)ψ ' − (c1 + c2 )vψ ' + c1 vϕ' + m 2 (iω)2 ψ ⎪ ⎪ ⎪ ⎪ ⎭ + (c1 + c2 )(iω)ψ − c1 (iω)ϕ + k2 ψ − k1 (ϕ − ψ) = 0 (2.7) Equation (2.7) are ordinary differential equations and can be solved by Fourier transform Eq. (2.8)

18

2 Dynamic Response of Subway Track

∫∞

˜ )= F(ξ

f (x)e

−iξ s

1 f (x) = 2π

d x,

−∞

∫∞

˜ )eiξ s dξ F(ξ

(2.8)

−∞

then Eq. (2.7) yields ⎫ ⎪ ⎪ ⎪ ⎪ ⎬

[E 1 I1 ξ 4 − m 1 v 2 ξ 2 + (2m 1 vω − ic1 v)ξ + (−m 1 ω2 + ic1 ω + k1 )] · W + [ic1 vξ − (ic1 ω + k1 )] · V = F0

[ic1 vξ − (ic1 ω + k1 )] · W + {E 2 I2 ξ 4 − m 2 v 2 ξ 2 + [2m 2 vω − i(c1 + c2 )v]ξ ⎪ ⎪ ⎪ ⎪ ⎭ 2 + [−m 2 ω + i(c1 + c2 )ω + (k1 + k2 )]} · V = 0 (2.9) where ξ is wavenumber. The matrix form of Eq. (2.9) is [ A

] W (ξ ) V (ξ )

[ =

F0

] (2.10)

0

where ] a11 a12 A= a21 a22 [ E 1 I1 ξ 4 − m 1 v 2 ξ 2 + (2m 1 vω − ic1 v)ξ + (−m 1 ω2 + ic1 ω + k1 ), = ic1 vξ − (ic1 ω + k1 ), E 2 I2 ξ 4 − m 2 v 2 ξ 2 + [2m 2 vω − i(c1 + c2 )v]ξ ] ic1 vξ − (ic1 ω + k1 ) +[−m 2 ω2 + i(c1 + c2 )ω + (k1 + k2 )] [

[ Substituting A−1 = [

] W (ξ ) V (ξ )

1 |A|

·

a22 −a12 −a21 a11

] to Eq. (2.10) yields

[ ] ][ ] [ a F 1 F −a 0 22 a 0 22 12 · = A−1 F = = −a21 a11 |A| |A| −a21 0

(2.11)

The inverse Fourier transform can be carried out to transform Eq. (2.11) from ξ -domain to s-domain [

ϕ(s)

]

1 = 2π ψ(s)

∫∞ [ −∞

] W (ξ ) V (ξ )

· eiξ s dξ

(2.12)

2.2 Theoretical Analysis of Track Model

19

As the direct integration of Eq. (2.12) is complicated, residue theorem is used to solve it. The inverses of W (ξ ) and V (ξ ) have the same form, thus only W (ξ ) is considered in this chapter. 1 ϕ(s) = 2π

∫∞ W (ξ ) · eiξ s dξ

(2.13)

−∞

The solution of Eq. (2.13) is obtained by applying residue theorem, giving 1 2π

∫+∞ N ∑ 1 · 2π i · W (ξ )eiξ s dξ = Res(W (ξ )eiξ s ; ξ j ) 2π j=1

−∞

=i·

N ∑ j=1 N ∑

Res(W (ξ )eiξ s ; ξ j ) (

F0 · a22 iξ s =i· Res e ; ξj |A| j=1

) (2.14)

·a22 iξ s where N is the number of singular points of F0|A| e ; ξ j is the jth singular point. The singular points can be calculated by |A| = 0. According to Eq. (2.10), |A| is an eight polynomial, thus if v /= 0, there are eight complex roots in the complex field. If v = 0, the forms of the roots are discussed as follows.

2.2.1 Lower Than the First Cut-Off Frequency In this case, the eight roots are complex roots with all the real parts and imaginary parts existing. Considering the conjugacy of the eight roots, four roots with positive imaginary parts are on the upper half plane and the distribution of the roots is illustrated in Fig. 2.2. When s ≥ 0, Eq. (2.14) can be expressed as Eq. (2.15) ϕ(s) = i ·

4 ∑

Res(W (ξ )eiξ s ; ξ j )

j=1

=i·

4 ∑ j=1

a22 F0 eiξ j s (2.15) E 1 I1 · E 2 I2 · (ξ j − ξ1 )(ξ j − ξ j−1 ) . . . (ξ j − ξ j+1 )(ξ j − ξ4 )

Similarly, when s < 0 Eq. (2.16)

20

2 Dynamic Response of Subway Track

Im(ξ )

Fig. 2.2 Roots distribution with load frequency lower than the first cut-off frequency

Γρ

ξ2

ξ1

ξ4

ξ3 ξa

ξ5

ξb

ξ6

Re(ξ )

ξ8

ξ7

≥0 Im(ξ )

(a) S

ξ2

ξ1

ξa

ξ3

ξ4

ξ5

ξ6

ξ7

ξb

Re(ξ )

ξ8 Γρ (b) S

ϕ(s) = −i ·

8 ∑

100

k

0

0.18

0.30

0.41

0.48

0.50

σh' (kPa)

σm' (kPa)

Table 4.3 A σv' , σh' and σm' corresponding to the five test schemes

Scheme No

σv' (kPa)

1

260

140

180

2

260

140

180

3

260

140

180

4

260

180

206.7

5

260

220

206.7

0.6

86

4 Dynamic Response of Around Soil and Subway Shield Tunnel

Table 4.4 Gmax corresponding to five test schemes Scheme No

e

OCR

k

σm' (kPa)

Gmax (MPa)

1

1.06

1

0.155

180

76.8

2

1.06

1

0.155

180

76.8

3

1.06

1

0.155

180

76.8

4

1.06

1

0.155

206.7

82.3

5

1.06

1

0.155

206.7

87.4

1.2

Fig. 4.11 Fitting of dynamic shear modulus and damping ratio versus dynamic shear strain

Geq/Gmax=1/(1+8.913γ

1.0

0.6775

)

2

R =0.923

Geq/Gmax

0.8 0.6 0.4 0.2 0.0 1E-5

Scheme No.1 Scheme No.2 Scheme No.3 Scheme No.4 Scheme No.5 Fitting curve

1E-4

0.001

0.01 γ (%)

0.1

1

10

(a) G/Geq 40 D=44.56(Geq/Gmax)-76.21(Geq/Gmax)+32.27

35

R2=0.872

30

D

25 20 15 10

Scheme No.1 Scheme No.2 Scheme No.3 Scheme No.4 Scheme No.5 Fitting curve

5 0 -5 1E-5

1E-4

0.001

0.01 γ (%)

(b) D

0.1

1

10

4.3 Numerical Simulation of Dynamic Response of Around Soil and Shield …

87

It can be seen from Fig. 3.11 that the fitting effect using Formulas (3.5) and (3.6) is good. Due to the test accuracy of the dynamic triaxial instrument, the test data points are concentrated in the strain range of 0.01–1%, the experimental data of dynamic shear modulus are less discrete. For the fitting curve of the damping ratio, except that the fitting effect is slightly lower in the strain range of 10–3 to 10–2 %, the fitting effect of other parts is better.

4.3 Numerical Simulation of Dynamic Response of Around Soil and Shield Tunnel 4.3.1 Finite Difference Model A two-dimensional finite difference model was developed using the commercially available computer program FLAC. In the model, the shield tunnel was assumed to be an infinitely long and thin cylindrical shell. Figure 4.12 illustrates the finite difference model of tunnel embedded in soil layers. In this model, the maximum element size is 1 m × 1 m. The excavation diameter is 6 m with a cover height of 11 m. Cover-to-Diameter ratio (C/D) is 1.83, where C and D are the cover depth and diameter of the shield tunnel. The D noted in the figures is the diameter of the tunnel in this chapter, for example, 1D below point B is located at the distance of 6 m below point B. Viscous boundaries are adopted to avoid any influence from the model boundaries with the dimensions of a 60 m width and a 40 m depth. The soft soils collecting from Shanghai are considered to be formed by

Fig. 4.12 Two-dimensional finite difference model

88

4 Dynamic Response of Around Soil and Subway Shield Tunnel

four horizontally layers which are regarded as elasto-plastic material in conformity with the Mohr–Coulomb failure criterion. The dynamic loading is applied at both sides according to the normal width of track with 1435 mm. Points A, B and C are shown in Fig. 4.12. The material properties are summarized in Table 4.5. Tunnel lining and ballast were considered to study the propagation of the vibrating load. The interaction between shield tunnel lining and soils was taken into account through an interface. The mechanical properties of materials and interface adopt in the model are summarized in Tables 4.6 and 4.7, respectively. The vibrating load generated by a moving subway train on the unevenness rail is expressed by an excitation load function composed of a static force and three harmonic forces (Zhang and Bai 2000). P(t) = p0 + p1 sin(ω1 t) + p2 sin(ω2 t) + p3 sin(ω3 t)

(4.8)

where p0 is the static load of wheel; p1 , p2 and p3 are vibrating loads in accordance to the unevenness vector height and the unevenness wavelength, respectively; pi = M0 ai ωi2 (i = 1, 2, 3), where M0 is the unsprung mass, ai being the unevenness vector height and ωi being the circular frequency of vibrating wavelength, ωi = Table 4.5 Physical and mechanical properties of various silty clay layers Silty clay layers

Thickness (m)

Density ρ(kg/m3 )

Cohesion c(kPa)

Friction angle φ(◦ )

Elasticity Modulus E(MPa)

Poisson’s ratio ν

No. 1

2

1880

17

15

10.5

0.33

No. 2

6

1750

12

16

8.0

0.35

No. 3

12

1710

14

12

7.5

0.38

No. 4

20

1760

16

14

10.0

0.33

Table 4.6 Parameters of shield lining and ballast used in model Material type

Constitutive model

Thickness/ Height (m)

Density ρ(kg/m3 )

Bulk modulus K (GPa)

Shear modulus G(GPa)

Poisson’s ratio ν

Shield lining

Elastic model

0.3

2600

40

30

0.2

Ballast

Elastic model

0.54

2200

1.66

0.76

0.3

Table 4.7 Mechanical properties of interface used in model Interface

Normal stiffness coefficient kn (GPa/m)

Shear stiffness coefficient ks (GPa/m)

Tensile strength T (MPa)

Cohesion c(kPa)

Friction angle φ(◦ )

300

100

100

20

10

4.3 Numerical Simulation of Dynamic Response of Around Soil and Shield …

89

2π v/L i , where v is subway train velocity and L i is unevenness wavelength. In this chapter, the element sizes were generated to meet the requirements for simulating the highest frequency of (loading )considered. Namely, the maximum element size 1 λ, where λ is the wave length, corresponding to (L) is calculated by L ≤ 18 ∼ 10 the highest frequency considered. Due to the maximum element size is 1 m × 1 m, vibration frequency should be less than approximately 20 Hz. Therefore, p0 = 70 kN, M0 = 700 kg; L 1 = 10 m, a1 = 5 mm; L 2 = 2 m, a2 = 0.6 mm; v1 = 80 km/h and v2 = 60 km/h are selected to analyze the influence of speed on the dynamic response of tunnel lining and soils. Assuming that the subway vibrating load is uniformly distributed along the railways tracks. The vibrating load applied on 2D finite difference model could be expressed by Zhang and Bai (2000) F(t) =

P(t) ·n·K l

(4.9)

where l is the length of a carriage and l = 20 m; n is the number of bogey and n = 2; K is the coefficient of correction; there are six carriages. Figure 4.13 shows the vibrating loads and spectrum diagrams of vibrating loads applied on the 2D model with different speeds for 1 s. The dominant frequencies of vibrating loads are about 10 and 8 Hz under 80 and 60 km/h. In addition, the initial earth pressure simulated by earth pressure balance through static calculation and the initial displacement had been eliminated before dynamic calculation.

4.3.2 Dynamic Damping The damping in the soil and structure is a parameter that is difficult to determine. There are three types of dynamic damping in FLAC, which are Local damping, Rayleigh damping and Hysteretic damping (Itasca Consulting Group 2011). Rayleigh damping, adopted to depict the respond of tunnel lining and ballast, is defined as a combination of mass proportional and stiffness proportional damping [C] = α[M] + β[K ]

(4.10)

where [C] is the damping matrix; [M] is the mass matrix; [K ] is the stiffness matrix; α and β are the coefficient of mass proportional and stiffness proportional damping, respectively. The parameters of Rayleigh damping in FLAC can obtain by ξmin = (α · β)1/2 , ωmin (α/β)1/2

(4.11)

where ξmin is the minimal critical damping, ωmin being the minimal center frequency. Hysteretic damping was used to analyze the dynamic response of soils around the shield tunnel. Hysteretic damping was obtained by one Hardin-Drnevich model,

90

4 Dynamic Response of Around Soil and Subway Shield Tunnel 7.3

Fig. 4.13 Vibrating loads and the spectrum diagrams under different speeds

80 km/h 60 km/h

Vibrating load (kN·m-1)

7.2 7.1 7.0 6.9 6.8 6.7

0.0

0.2

0.4

Time (s)

0.6

0.8

1.0

(a) Vibrating loads

Vibrating load (kN·m-1·Hz-1)

0.20

80 km/h 60 km/h

0.15

0.10

0.05

0.00 1

10 Frequency (Hz)

100

(b) Spectrum diagrams

which is defined as: Ms =

1 / 1 + γ γref

(4.12)

/ where Ms is the degradation coefficient of shear modulus and Ms = G G max ; γ and γref are the shear strain and the reference shearing strain, respectively. Figure 4.14 shows variations of degradation coefficient of shear modulus with shear strain of different soil layers. Damping of soils can be obtained by Formula (4.12). The damping of soils and structures are summarized in Table 4.8. Numerical analysis is mainly performed by following steps: (1) Building a 2D model contain soil layers, tunnel, ballast and the interface. The tunnel is built by the

91

1.0 H-D model γ ref = 0.064

0.8

R2 = 0.994

0.6

0.4

0.2

0.0 1E-4

0.001

0.01

0.1

1

Shear strain γ (%) Degradation coefficient of shear modulus MS

(a) Layer No.1 1.0 H-D model γref = 0.061

0.8

R2 = 0.991

0.6

0.4

0.2

0.0 1E-4

0.001

0.01

0.1

1

Shear strain γ (%)

(b) Layer No.2 Degradation coefficient of shear modulus MS

Fig. 4.14 Variations of degradation coefficient of shear modulus with shear strain for different soft soil layers

Degradation coefficient of shear modulus MS

4.3 Numerical Simulation of Dynamic Response of Around Soil and Shield …

1.0 H-D model γ ref = 0.065

0.8

R2 = 0.986

0.6

0.4

0.2

0.0 1E-4

0.001

0.01

Shear strain γ (%)

(c) Layer No.3

0.1

1

4 Dynamic Response of Around Soil and Subway Shield Tunnel

Fig. 4.14 (continued)

Degradation coefficient of shear modulus MS

92

1.0 H-D model γ ref = 0.067

0.8

R2 = 0.992

0.6

0.4

0.2

0.0 1E-4

0.001

0.01

0.1

1

Shear strain γ (%)

(d) Layer No.4

Table 4.8 Parameters of damping in different materials Materials

Layer No. 1

Layer No. 2

Layer No. 3

Layer No. 4

Lining

Ballast

Damping parameters

γref = 0.064

γref = 0.061

γref = 0.065

γref = 0.067

ξmin = 0.034 ωmin = 1.342

ξmin = 0.042 ωmin = 1.414

liner elements in FLAC; (2) Setting the parameters summarized in Tables 4.1, 4.2 and 4.3 and static boundary; (3) Performing static analysis and removing displacement and velocity; (4) Setting parameters of dynamic damping summarized in Table 4.4 and Viscous boundaries; (5) Applying dynamic loading illustrated in Fig. 3.13 by ‘apply’ command and historying data by ‘history’ command; (6) Running dynamic analysis.

4.3.3 Dynamic Response of Soft Soils Around Lining Figure 4.15 shows the cumulative vertical displacement upper and down the shield tunnel. With the dynamic time increasing, the vertical displacement increases up to the first peak within 1 s. Then, it decreases and increases up to the second peak within 2 s. After 2 s, it reaches a dynamic balance. Compared Fig. 4.15a with b, the cumulative vertical displacement of points below the tunnel decreases with the depth increasing. However, the displacement of points above tunnel increases. It can be indicated that the soft soil layer below the shield tunnel is under the compression and the soft soil layers above tunnel is in the extrusion state, and turn uplift. Moreover, below point B, the settlement of unit thickness soil layer decreases. When the first peak appears it increases with the depth increasing. This is due to the stress wave

4.3 Numerical Simulation of Dynamic Response of Around Soil and Shield …

93

propagation and attenuation. Inversely, above point A, when the first peak appears, the settlement of unit thickness soil layer decreases with the depth increasing and it is due to the soil layers above tunnel is in uplift state. With the dynamic time being 5.4 s, the cumulative vertical displacement of points B and A are − 1.282 mm and − 1.187 mm, respectively, so the tunnel stretches 0.095 mm in vertical direction. Meanwhile, the settlement of the soil layer below tunnel is 1.28 mm and the soil layers above tunnel uplifts 0.51 mm. The maximum settlement of model at symmetry axis is 0.68 mm. Based on the settlement and uplift of soils, the vertical displacement of some points around the shield tunnel may be zero. According to the vertical displacement at different time, the curves of zero vertical displacement were obtained, shown in Fig. 4.16. Soil layers below the curve are under the compression state and above the Fig. 4.15 Curves of cumulative vertical displacement at different points

0

1

2

Time (s) 3

4

5

6

0.0 0D 0.5D 1D 2D 3D

Y-displacement (mm)

-0.3 -0.6 -0.9 -1.2 -1.5

(a) Below point B 0

1

2

Time (s) 3

4

5

Y-displacement (mm)

0.0 0D 0.5D 1D 1.5D Surface

-0.3 -0.6 -0.9 -1.2 -1.5

(b) Above point A

6

94

4 Dynamic Response of Around Soil and Subway Shield Tunnel Distance (m)

Fig. 4.16 Curves of zero vertical displacement at different time around tunnel

0 -2 -4

Depth (m)

-6

0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30

Tunnel 1s 2s 3s 5.4 s

-8 -10 -12 -14 -16 -18 -20

curve is in the extrusion state. Figure 3.16 illustrates that variation of the curve with time consists of two stages: rapid variation stage and slow variation stage. Within 2 s, the curve rapidly moves down, after that it turns into a slow variation stage up to 5.4 s. With the dynamic time increasing, it is remarkable that the zero displacement curve moves down within the 3 times tunnel diameter away from point C, and it moves up between 24 and 30 m in the horizontal direction. Figure 4.17 illustrates the vertical displacements of the ground and the horizontal plane, 14 m in depth at different time, respectively. The greater the distance away from the symmetry axis is, the smaller the settlement is. At the same time, the settlement of the ground reduces slowly near the symmetry axis. However, the settlement of the horizontal plane, 14 m in depth, reduces rapidly. In addition, with the time increasing, the cumulative displacement decreases between 0 and 24 m and increases between 24 and 30 m in the horizontal direction. It agrees with the variations of zero vertical displacement curves illustrated in Fig. 4.16. Points at the right side of the tunnel were chosen to analyze the horizontal displacement at different time. Figure 4.18 shows the horizontal displacement of planes with different distances away from Point C. From Fig. 4.18a–d, the cumulative horizontal displacements increase with the dynamic time but the increasing rate decreases. The greater the distance is, the larger the displacement is. At the same time, the zero horizontal displacement appears at − 14 m away from the surface which is at the subway tunnel center. When the depth is less than 14 m, the horizontal displacement increases with the depth decreasing. The value is negative which means soil moving to the side close to the tunnel. When the depth is more than 14 m, the horizontal displacement increases with the depth increasing. The value is positive which means soil moving to the side away from the tunnel. It can be obtained that the soil layers below the tunnel center are in the compression state and the soil layers above the tunnel center are in the extrusion state, and move to the symmetry axis.

4.3 Numerical Simulation of Dynamic Response of Around Soil and Shield …

95

Distance away from the symmetry axis (m)

Fig. 4.17 Vertical displacements of different horizontal planes

0.2

0

5

10

15

20

25

30

Y-displacement (mm)

0.0 -0.2 -0.4

1s 2s 3s 4s 5s 5.4 s

-0.6 -0.8 -1.0

(a) Ground Distance away from the symmetry axis (m) 3

6

9

12

15

18

21

24

27

30

0.0

Y-displacement (mm)

-0.2 -0.4 -0.6 -0.8

1s 2s 3s 4s 5s 5.4 s

-1.0 -1.2 -1.4

(b) 14m in depth

By way of summary, the soil layers below shield tunnel are in the compression state. When the depth is less than 20 m, the moving direction of soil is complicated. Considered the X-dis and Y-dis, soil layer with zero displacement may exist at the depth 14 and 24 m away from the symmetry axis. The motion of soils around the shield tunnel is illustrated in Fig. 4.19 approximately. Figure 4.20 shows the variations of vertical stress with time. With the time increasing, the stress changes up and down. Comparing Fig. 4.20a with b, the vertical stress of the points below the tunnel decreases with the depth increasing. However, the stress of points above the tunnel increases. It is verified that the propagation of stress wave is gradually decreasing around the tunnel. Variation of the acceleration and the velocity with time are illustrated in Fig. 4.21. With time increasing, the acceleration and the velocity change up and down and

X-displacement (mm) -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0

0.1

0.2

0.3

-10

Depth (m)

0.4

0.5

1s 2s 3s 4s 5s 5.4 s

-5

-15 -20 -25 -30 -35 -40

(a) 0.5D X-displacement (mm) -0.6

-0.4

-0.2

0.0 0

0.2

0.4

0.6 1s 2s 3s 4s 5s 5.4 s

-5 -10

Depth (m)

Fig. 4.18 Variations of horizontal displacements of planes with different distances away from Point C

4 Dynamic Response of Around Soil and Subway Shield Tunnel

-15 -20 -25 -30 -35 -40

(b) 1D X-displacement (mm) -0.8

-0.6

-0.4

-0.2

0.0 0

0.2

-5 -10

Depth (m)

96

-15 -20 -25 -30 -35 -40

(c) 1.5D

0.4

0.6

0.8 1s 2s 3s 4s 5s 5.4 s

4.3 Numerical Simulation of Dynamic Response of Around Soil and Shield …

97

X-displacement (mm)

Fig. 4.18 (continued)

-0.8 -0.6 -0.4 -0.2 0.0 0

0.2

0.4

0.6

1.0

1s 2s 3s 4s 5s 5.4 s

-5 -10

Depth (m)

0.8

-15 -20 -25 -30 -35 -40

(d) 2D

Distance (m)

Fig. 4.19 Motion of soils around tunnel

-30 0

-20

-10

0

10

20

30 Tunnel Moving

Depth (m)

-10

-20

-30

-40

approach to dynamic balance. Both the acceleration and the velocity of points under the tunnel decrease with the depth increasing.

4.3.4 Dynamic Response of Lining Dynamic response of lining is mainly reflected by the bending moment and the shear force when the vibrations are transmitted to the tunnel lining. Due to the symmetry of the model geometry and stress state, half of lining is selected to analyze the dynamic response. Figure 4.22 presents the bending moment and the shear force of lining at the dynamic time 2 s. Points B, C and A correspond to 0, 90 and 180°, respectively.

98

4 Dynamic Response of Around Soil and Subway Shield Tunnel 0.4

Fig. 4.20 Variations of vertical stress with time

Stress (kPa)

0.3 0.2

0D 0.5D 1D 2D 3D 4D

0.1 0.0 -0.1

0

1

2

3

4

5

6

Time (s)

(a) Below Point B 1.6 1.4

Stress (kPa)

1.2 1.0

0D 0.5D 1D 2D 3D

0.8 0.6 0.4 0.2 0.0

0

1

2

3

4

5

6

Time (s) (b) Right Point C

It shows tension states in the outer side and compression states in the inner side near 90° (Point C) according to the bending moment and the shear force. Furthermore, the maximum bending moment and the maximum shear force are not at the same point. The maximum bending moment appears at 90–97.5° and the maximum shear force appears at 45–52.5°. Finally, the bending moment and the shear force of the whole lining can be obtained by the symmetry. Figure 4.23 illustrates variations of the maximum moment and the maximum shear force with time. Within 1 s, the values of bending moments and shear forces vary greatly, being unstable state. After then, bending moment and shear force are fluctuating around a certain value which is similar to the variation of vertical stress.

4.3 Numerical Simulation of Dynamic Response of Around Soil and Shield …

99

0.20

Fig. 4.21 Variations of acceleration and velocity with time below Point B Acceleration (mm·s-2)

0.15 0.10 0.05 0.00 -0.05 0D 0.5D 1D 2D 3D

-0.10 -0.15 -0.20

0

1

2

3

4

5

6

Time (s)

(a) Acceleration 0

Velocity (mm·s-1)

-1 -2 -3 0D 0.5D 1D 2D 3D

-4 -5

0

1

2

3

4

5

6

Time (s)

(b) Velocity

4.3.5 Influence of Velocity A model with the speed of 60 km/h, compared with 80 km/h, was carried out in order to study the influence of the subway speed on the dynamic behavior of the soils around the shield tunnel. Figure 4.24 shows variations of the acceleration and the velocity at point B. It is obvious that the faster the subway speed is, the larger the dynamic acceleration and the velocity of soil are. The vibrating load decreases with the speed decreasing, which results in smaller acceleration and velocity. Figure 4.25 illustrates the curves of the cumulative vertical displacement under different speeds. With the smaller vibrating load, the cumulative vertical displacement of points below tunnel is smaller. The difference of displacements between two speeds is very small. Therefore, in the case of the same dynamic time, the effect of

100

4 Dynamic Response of Around Soil and Subway Shield Tunnel 180 150 120

Angle / o

Moment (kM·m) Shear force (kN)

90 60 30

-200 -150 -100

-50

0

0

50

100

150

200

Fig. 4.22 Moment and shear force of lining at 2 s

Maximum moment (kN·m)

191.6

-175.6

Maximum moment Maximum shear force

191.4

-175.8

191.2

-176.0

191.0

-176.2

190.8

-176.4

190.6

0

1

2

3

4

5

6

Maximum shear force (kN)

-175.4

191.8

-176.6

Time (s) Fig. 4.23 Variations of maximum moment and maximum shear force with time

subway speed is smaller. Figure 4.26 shows variations of vertical stress with time. It can be seen from point B that the faster the subway speed is, the larger the dynamic vertical stress amplitude of soil is. Compared point B with the point 0.5D depth away from point B, the effect of speed is smaller with the depth increasing.

4.3 Numerical Simulation of Dynamic Response of Around Soil and Shield … Fig. 4.24 Variations of acceleration and velocity with time at point B

0

1

2

Time (s) 3

4

101

5

6

5

6

0.6 80km/h 60km/h

Acceleration (m·s-2)

0.4 0.2 0.0 -0.2 -0.4 -0.6

(a) Acceleration

0

0

1

2

Time (s) 3

4

Velocity (mm·s-1)

-1 -2

80km/h 60km/h

-3 -4 -5 -6

(b) Velocity

4.3.6 Influence of Interface Similarly, a model, unconsidered interface, was carried out in order to study the influence of the interaction between soil and structure under the vibrating load. Figure 4.27 expresses variation of acceleration and velocity at point B. The acceleration and velocity of soil unconsidered interface are obviously different from those of soil considered interface. The interface should be taken into consideration when the respond of soil is analyzed. Figure 4.28 shows the cumulative vertical displacement with time. It concludes that the interface between the tunnel lining and the soil has a significant influence on the dynamic behavior. The displacement of points with the unconsidered interface is smaller than that of points with the considered interface. At point B and the point

102

4 Dynamic Response of Around Soil and Subway Shield Tunnel

Fig. 4.25 Cumulative vertical displacement with time under different speeds for two points

Y-displacement (mm)

0.0

0

1

2

Time (s) 3

4

5

6

80km/h (0D) 60km/h (0D) 80km/h (0.5D) 60km/h (0.5D)

-0.3 -0.6 -0.9 -1.2 -1.5

1.6

Fig. 4.26 Variations of vertical stress with time under different speeds for two points

80km/h (0D) 60km/h (0D) 80km/h (0.5D) 60km/h (0.5D)

Stress (kPa)

1.4

1.2

1.0

0.8

0.6

0

1

2

3

4

5

6

Time (s)

0.5D depth away from point B, when interface is considered, the cumulative vertical displacements are − 1.282 mm and − 0.960 mm, respectively. When the interface is unconsidered, they are − 1.198 mm and − 0.896 mm, respectively. Hence, compared interface considered with interface unconsidered, the settlements differ 0.084 mm and 0.064 mm at Point B and the point 0.5D depth away from Point B, respectively. Figure 4.29 shows variations of vertical stress with time. It can be seen from Point B and the point 0.5D depth away from point B that the dynamic vertical stress of soil is larger with the interface between the lining and soil unconsidered. The difference value of dynamic stress between two points is larger in interface considered than unconsidered. The larger the difference value in stress is, the larger the settlement is.

4.4 Conclusions

103

Fig. 4.27 Variation of acceleration and velocity at point B

0

1

2

Time (s) 3

4

5

6

0.6 Interface considered Interface unconsidered

Acceleration (m·s-2)

0.4 0.2 0.0 -0.2 -0.4 -0.6

(a) Acceleration Time (s) 0

1

2

3

4

5

6

0

Velocity (mm·s-1)

-1

-2

-3

Interface considered Interface unconsidered

-4

-5

-6

(b) Velocity

4.4 Conclusions The soft soil of Layer No. 5 in Shanghai were taken as the research background and its dynamic characteristics were studied by the dynamic triaxial apparatus. Five test schemes were designed considering the load frequency and the confining pressure. The dynamic Shear modulus and damping ratio of each test scheme were calculated by ellipse fitting. In addition, the dynamic finite difference models were conducted to analyze the interaction between the around soil and the shield tunnel. The conclusions are as follows.

104

4 Dynamic Response of Around Soil and Subway Shield Tunnel

Time (s)

Fig. 4.28 Cumulative vertical displacement with time for two points

0

1

2

3

4

5

6

Y-displacement (mm)

0.0 Interface considered (0D) Interface unconsidered (0D) Interface considered (0.5D) Interface unconsidered (0.5D)

-0.3 -0.6 -0.9 -1.2 -1.5

1.6

Fig. 4.29 Variations of vertical stress with time at two points

Interface considered (0D) Interface unconsidered (0D) Interface considered (0.5D) Interface unconsidered (0.5D)

Stress (kPa)

1.4 1.2 1.0 0.8 0.6

0

1

2

3

4

5

6

Time (s)

(1) When the amplitude of dynamic stress is small, the hysteresis loop is basically consistent and closed. When the amplitude of dynamic stress is large, each hysteresis loop is no longer closed and crosses forward, and the larger the amplitude of dynamic stress, the more obvious this effect. Low frequency loads act longer than high frequency loads, resulting in greater strain. The higher the vibration frequency, the smaller the plastic strain caused by the soil, and this effect becomes more and more obvious as the load frequency increases. Increasing the confining pressure can reduce the damping value and weaken the dissipation of energy propagation in the soil. With the increase of dynamic shear strain, the dynamic Shear modulus decreases rapidly in the early stage, and then decreases slowly. The damping ratio increases rapidly with the increase

References

105

of dynamic shear strain in the early stage, and the increasing trend gradually slows down in the later stage. (2) With the dynamic time increasing, the vertical displacement increases to the first peak within 1 s. Then, it decreases and increases to the second peak within 2 s. After 2 s, it reaches a dynamic balance. The cumulative vertical displacement of points below the tunnel decreases with the depth increasing; on the contrary, the displacement of points above the tunnel increases. It can be inferred that the soil layers below the shield tunnel are in the compression state and the soil layers above the tunnel are in the extrusion state, and turn uplift. The cumulative horizontal displacement increases with the dynamic time but the increasing rate decreases. The greater the distance is, the larger the displacement is. At the same time, the zero horizontal displacement appears at − 14 m from the surface which is at the subway tunnel center. The vertical stress of the points below tunnel decreases with the depth increasing; on the contrary, the stress of points above tunnel increases. It is verified that the propagation of stress wave is gradually decreasing around tunnel. (3) With the time increasing, the acceleration and velocity change up and down and approach to dynamic balance. Both the acceleration and the velocity of points under the tunnel decrease with the depth increasing. Dynamic respond of lining induced by train shows tension states in the outer side and compression states in the inner side near 90° (Point C) according to the bending moment and the shear force. Furthermore, the maximum bending moment and the maximum shear force are not located at the same point. The speed of the train and the interface have an impact on the dynamic behavior of soft soils. Merely, the effect of speed is small.

References Hall L (2003) Simulations and analyses of train-induced ground vibrations in finite element models. Soil Dyn Earthq Eng 23(5):403–413 Itasca Consulting Group, Inc (2011) FLAC manual, 7th edn (FLAC Version 7.0) Liao SM, Liu JH, Wang RL et al (2009) Shield tunneling and environment protection in Shanghai soft ground. Tunn Undergr Space Technol 24(4):454–465 Zhang YE, Bai BH (2000) The method of identifying train vibration load acting on subway tunnel structure. J Vib Shock 19(3):68–70 (in Chinese)

Chapter 5

Mechanical Properties of Subway Shield Tunnel

5.1 Introduction Underground spaces in shanghai are congested with subways and vital urban facilities (Qiao and Peng 2016). Utilization of deep underground spaces solves the problem regarding shortage of shallow underground space (Chen and Peng 2018). Nevertheless, challenges facing deeply buried tunnels are still considerable. For example, soil and water pressure in the stratum deeper than 50 m exceeds 1 MPa (Wang et al. 2018), which will challenge the bearing capacity and waterproofness to deeply buried tunnels (Zuo et al. 2018). In this chapter, the functionally graded lining of the subway tunnel is modelled as a functionally graded hollow cylinder under non-axisymmetric loads, and its mechanical properties are studied. The analytical results presented here should serve as benchmarks for verifying numerical solutions of problems.

5.2 Theoretical Model of Functionally Graded Subway Shield Tunnel 5.2.1 Basic Assumptions General agreement of these analytical models lies on the following basic assumptions: (1) the cross section for the lining is assumed to be circular, and the lining satisfies the plain strain condition (Hu et al. 2003). (2) the material behavior of the lining is generally assumed to be elastic, because of the small deformation. (3) Loads on the top, at the bottom and the sides are distributed uniformly.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Z.-D. Cui, Interaction Between Soil Foundation and Subway Shield Tunnel, https://doi.org/10.1007/978-981-99-6870-1_5

107

108

5 Mechanical Properties of Subway Shield Tunnel

5.2.2 Governing Equations For homogeneous straight beam, the moment is defined to be positive when the structure tensile part is at the bottom, and the upward deformation is defined to be positive. Ignoring the axial deformation, the differential equation between moment and deformation is given by d2 ω M(x) = 2 dx EI

(5.1)

For homogeneous circular lining structure, as shown in Fig. 5.1, the microelement can be analyzed as a straight beam, and Eq. (5.1) should be satisfied between moment and radial deformation in the rectangular coordinate system. The polar coordinate system is established, whose origin is the center of the lining. The top of the structure is defined as 0°, and anticlockwise direction is defined to be positive. Therefore, the differential equation of the functionally graded lining in the polar coordinate system is given by d2 ω M(α) = RH2 2 dα E(α)I

Fig. 5.1 Microelement of the circular lining

(5.2)

5.2 Theoretical Model of Functionally Graded Subway Shield Tunnel

109

5.2.3 Building Model According to the symmetry of the lining and external loads, the quarter circle is considered. Therefore, the calculation model is as shown in Fig. 5.2. At the interval [0, π/2], elastic modulus function versus α is given by [ ] E(α) = E 0 Aα n + B

(5.3)

At the interval [π/2, π ], elastic modulus is symmetrical to that at the interval [0, π/2]. For the sake of simplicity, let E = a E 0 at α = 0, and E = bE 0 at α = π/2, the elastic modulus function transformed with parameters a and b is given by [ E(α) = E 0

b−a n α +a (π/2)n

] (5.4)

and the function is defined as Function I.

5.2.4 Solutions of Internal Force Coefficients As shown in Fig. 5.2d, the angular displacement is equal to zero at the position of α = 0◦ , and the force method equations are given by δ11 X 1 +Δ1p = 0

(5.5)

where δ11 the displacement at X 1 of X 1 = 1; Δ1p the displacement at X 1 of external forces; ∫

2

M1 ds = E(α)I

δ11 =

∫π/2

2

M1 RH dα = E(α)I

0

where m 1 = Besides,

∫ π/2 0

1 E(α)/E 0

∫π/2

RH RH dα = m1 E(α)I E0 I

0

dα. ∫

Δ1p =

M 1 Mp ds = E(α)I

∫π/2 0

where M p = 21 (λ − 1) p0 RH2 sin2 α. Therefore,

Mp RH dα E(α)I

(5.6)

110

5 Mechanical Properties of Subway Shield Tunnel

Fig. 5.2 Calculation model

(a) Calculation model

(b) Calculation model of the quarter circle

(c) Calculation model with boundary condition

(d) Basic structure

5.2 Theoretical Model of Functionally Graded Subway Shield Tunnel

∫ Δ1p =

M 1 Mp ds = E(α)I

∫π/2

Mp 1 RH dα = (λ − 1) p0 RH3 E(α)I 2

0

111

∫π/2

sin2 α dα E(α)I

0

1 (λ − 1) p0 RH3 = m2 2 E0 I where m 2 =

(5.7)

∫ π/2 0

sin2 α dα; λ the E(α)/E 0 −Δ1p = 21 (1 − ) δ11

coefficient of lateral pressure.

2 So, X 1 = λ) p0 RH2 m , Mα = M 1 X 1 +Mp = 21 p0 RH2 (1 − m1 ( 2 − sin2 α . λ) m m1 And the moment nondimensionalized is given by

) ( m2 Mα 1 2 = − sin α (1 − λ) m1 p0 RH2 2

(5.8)

The axial force can be obtained by calculating moment at the center, and the axial force nondimensionalized is given by N = sin2 α + λ cos2 α p0 R H

(5.9)

Boundary conditions at α = 0 and α = π /2 can be obtained by the principle of virtual work. ⎛ π/2 ⎞ ∫ ∫π/2 M 2 Mp M 2 M1 RH dα + RH dα ⎠ ω0 = −⎝ E(α)I E(α)I 0 0 ⎞ ⎛ π/2 ∫ ∫π/2 (λ − 1) p0 RH4 ⎝ sin3 α sin α m2 = dα − dα ⎠ 2E 0 I E(α)/E 0 m1 E(α)/E 0 0

0

(λ − 1) p0 RH4 = m3 2E 0 I

(5.10)

where M 2 = 1 is exerted at α = 0◦ vertically; m 3 = ∫ m 2 π/2 sin α dα. m1 0 E(α)/E 0 ωπ/2

∫ π/2 0

sin3 α E(α)/E 0

dα −

⎞ ⎛ π/2 ∫ ∫π/2 M 3 Mp M M 2 1 RH dα + RH dα ⎠ = −⎝ E(α)I E(α)I 0 0 ⎞ ⎛ π/2 ∫ ∫π/2 2 (λ − 1) p0 RH4 ⎝ m 2 1 − cos α sin α(1 − cos α) ⎠ = dα − dα 2E 0 I m1 E(α)/E 0 E(α)/E 0 0

0

112

5 Mechanical Properties of Subway Shield Tunnel

=

(λ − 1) p0 RH4 m4 2E 0 I

(5.11)

where M 3 = 1 is exerted at α = 90◦ horizontally to the right; m 4 = ∫ ∫ π/2 2 α(1−cos α) m 2 π/2 1−cos α dα − 0 sin E(α)/E dα. m1 0 E(α)/E 0 0 From derivation, when external forces are determined, the internal force can be obtained, and the radial displacement can be obtained according to the governing equation and boundary conditions. The boundary conditions are concluded as follows. (λ − 1) p0 RH4 m3 2E 0 I (λ − 1) p0 RH4 m4 α = π /2, ω0 = 2E 0 I α = 0, ω0 =

(5.12)

5.3 Single Factor Test on the Functionally Graded Subway Tunnel Model The dimensionless parameter K is employed to reflect the displacement at the lining axis, and the dimensionless parameter P is employed to reflect the section moment. The definition of K and P is given by K = ω/ P=

p0 RH4 E0 I

(5.13)

Mα p0 RH2

(5.14)

The single factor tests are carried out to study the influence of lateral pressure coefficient λ, elastic modulus parameters a and b on K, and the typical parameters are given by a = 1, b = 1 and λ = 0.5. In the single factor test, one of factors will be changed in turn, and the other factors will be fixed according to the typical parameters. The variation range of every factor is summarized in Table 5.1. Table 5.1 Factor levels of the single factor test

Factors

Variation range

Step

A

0.7 ~ 1.3

0.1

b

0.7 ~ 1.3

0.1

λ

0.5 ~ 1.5

0.2

5.3 Single Factor Test on the Functionally Graded Subway Tunnel Model

113

Under the condition that the other parameters remain unchanged, the influence of a on the radial displacement and section moment in the range of 0 ∼ 90◦ at the axis of functionally graded lining is studied. In addition, the influence on the typical points of 0◦ , 15◦ , 30◦ , 45◦ , 60◦ , 75◦ and 90◦ at the axis of functionally graded lining is studied, and the results are shown in Figs. 5.3 and 5.4. 0.06 0.04

K

0.02

a=0.7 a=0.8 a=0.9 a=1.0 a=1.1 a=1.2 a=1.3

0.00 -0.020 -0.025

-0.02

-0.030 -0.035 -0.040

-0.04

-0.045 -0.050

-0.06 -0.2

-0.055 0.00

0.0

0.2

0.4 2

0.05

0.10

0.6

0.8

0.15

0.20

0.25

1.0

1.2

(a) Radial displacement at axis

0.06 0.04 0.02

K

0.00 -0.02 -0.04 -0.06 -0.08 0.6

=0° =60°

0.7

0.8

=15° =75°

0.9

1.0 a

=30° =90°

1.1

1.2

(b) Radial displacement at typical points

Fig. 5.3 Radial displacement in the single factor test of parameter a

=45°

1.3

1.4

114

5 Mechanical Properties of Subway Shield Tunnel

0.15

a=0.7 a=0.8 a=0.9 a=1.0 a=1.1 a=1.2 a=1.3

0.10 0.05 P

0.00

0.14 0.13

-0.05

0.12 0.11

-0.10

0.10 0.09 0.08 0.00

-0.15 -0.2

0.05

0.10

0.0

0.15

0.2

0.20

0.25

0.4

0.6

0.8

1.0

1.2

(a) Section moment

0.15 0.10 0.05

P

0.00 -0.05 -0.10 -0.15 -0.20 0.6

0.7

0.8

0.9

1.0 a

1.1

1.2

1.3

1.4

(b) Section moment at typical points

Fig. 5.4 Section moment in the single factor test of parameter a

As can be seen from Fig. 5.3a, with the increase of a, the radial displacement at the lining axis decreases, but the deformation mode of the lining remains the same with the maximum displacement at 0◦ and 90◦ , the minimum displacement near 45◦ . In addition, from Fig. 5.3b, with the increase of a, the slope of the radial displacement curve of every typical point decreases gradually. It shows that the increase of a can

5.3 Single Factor Test on the Functionally Graded Subway Tunnel Model

115

enhance the lining stiffness, but the reduction of deformation becomes smaller and smaller. From Fig. 5.4a, with the increase of a, the distribution of section moment remains unchanged with the maximum moment at 0◦ and 90◦ , the minimum moment near 45◦ . In addition, from Fig. 5.4b, with the increase of a, the negative section moment at typical points decreases linearly, and the positive section moment increases linearly. The maximum section moment transfers from section of 90◦ to section of 0◦ , and the maximum section moment decreases. It can be seen from the distribution of the displacement and internal force that the optimal mode of the elastic modulus is to configure the larger elastic modulus at 0◦ and 90◦ to improve the rigidity, and the smaller elastic modulus at 45◦ to achieve the full utilization of the material. The optimized elastic modulus function is given by [ E(α) = E 0

] π ||n a − b || α − + b | | (π/4)n 4

(5.15)

and the function is defined as Function II. Under the condition that the other parameters remain unchanged, the influence of b on the radial displacement and section moment in the range of 0 ∼ 90◦ at the axis of functionally graded lining is studied. In addition, the influence on the typical points of 0◦ , 15◦ , 30◦ , 45◦ , 60◦ , 75◦ and 90◦ at the axis of functionally graded lining is studied, and the results are shown in Figs. 5.5 and 5.6. As can be seen from Fig. 5.5a, with the increase of b, the radial displacement at the lining axis decreases, but the deformation mode of the lining remains the same with the maximum displacement at 0◦ and 90◦ , the minimum displacement near 45◦ . In addition, from Fig. 5.5b, with the increase of b, the slope of the radial displacement curve of every typical point decreases gradually. It shows that the increase of a can enhance the lining stiffness, but the reduction of deformation becomes smaller and smaller. Compared with Figs. 5.6 and 5.4, it can be found that the effect of changing a or b alone on the deformation characteristics of functionally gradient lining is the same when other parameters remain unchanged. From Fig. 5.6a, with the increase of b, the distribution of section moment remains unchanged with the maximum moment at 0◦ and 90◦ , the minimum moment near 45◦ . In addition, from Fig. 5.6b, with the increase of b, the negative section moment at typical points increases linearly, and the positive section moment decreases linearly. The maximum section moment transfers from section of 0◦ to section of 90◦ , and the maximum section moment decreases. Comparing Fig. 5.6b with Fig. 5.4b, it can be found that with the increase of the section stiffness, the section moment will also increase. Under the condition that the other parameters remain unchanged, the influence of b on the radial displacement and section moment in the range of 0 ∼ 90◦ at the axis of functionally graded lining is studied. In addition, the influence on the typical points of 0◦ , 15◦ , 30◦ , 45◦ , 60◦ , 75◦ and 90◦ at the axis of functionally graded lining is studied, and the results are shown in Figs. 5.5 and 5.6.

116

5 Mechanical Properties of Subway Shield Tunnel

0.06 0.04

K

0.02

b=0.7 b=0.8 b=0.9 b=1.0 b=1.1 b=1.2 b=1.3

0.00

-0.025

-0.030

-0.02

-0.035

-0.040

-0.04 -0.06 -0.2

-0.045

-0.050 0.00

0.0

0.2

0.4 2

0.05

0.6

0.10

0.8

0.15

1.0

0.20

1.2

(a) Radial displacement at axis

0.06 0.04 0.02

K

0.00 -0.02 -0.04 -0.06 -0.08 0.6

=0° =60°

0.7

0.8

=15° =75°

0.9

1.0 b

=30° =90°

1.1

1.2

=45°

1.3

1.4

(b) Radial displacement at typical points

Fig. 5.5 Radial displacement in the single factor test of parameter b

Under the condition that the other parameters remain unchanged, the influence of λ on the radial displacement and section moment in the range of 0 ∼ 90◦ at the axis of functionally graded lining is studied. In addition, the influence on the typical points of 0◦ , 15◦ , 30◦ , 45◦ , 60◦ , 75◦ and 90◦ at the axis of functionally graded lining is studied, and the results are shown in Figs. 5.7 and 5.8.

5.3 Single Factor Test on the Functionally Graded Subway Tunnel Model

117

0.15

b=0.7 b=0.8 b=0.9 b=1.0 b=1.1 b=1.2 b=1.3

0.10

P

0.05 0.00 0.14 0.13

-0.05

0.12 0.11

-0.10

0.10 0.09 0.08 0.00

-0.15 -0.2

0.05

0.0

0.10

0.15

0.2

0.20

0.25

0.4

0.6

0.8

1.0

1.2

2 (a) Section moment

0.15 0.10 0.05

P

0.00 -0.05 -0.10 -0.15 -0.20 0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

b (b) Section moment at typical points

Fig. 5.6 Section moment in the single factor test of parameter b

From Figs. 5.7a and 5.8a, the radial displacement of the lining axis and the section moment are 0 when λ = 1, which means that when the upper load is equal to the lateral load, the lining is in the safest state. When λ = 0.5 ∼ 1.5, curves of the radial displacement and the section moment is symmetric with the curve of λ= 1.

118

5 Mechanical Properties of Subway Shield Tunnel

0.08 0.06

=0.5 =0.9 =1.3

=0.6 =1.0 =1.4

=0.8 =1.2

=0.7 =1.1 =1.5

0.04

K

0.02 0.00 -0.02 -0.04 -0.06 -0.2

0.0

0.2

0.4 2

0.6

0.8

1.0

1.2

(a) Radial displacement at axis

0.06 0.04

K

0.02 0.00 -0.02 -0.04 -0.06 0.4

0.6

0.8

1.0

1.2

1.4

1.6

(b) Radial displacement at typical points

Fig. 5.7 Radial displacement in the single factor test of parameter λ

From Figs. 5.7b and 5.8b, it can be seen that the displacement of lining axis and section moment changes linearly, and the deformation mode of the lining changes significantly with the increase of λ. It turns into the deformation with large lateral pressure to the inside, and the side with small pressure to the outside, which conforms to the actual deformation characteristics.

5.3 Single Factor Test on the Functionally Graded Subway Tunnel Model

119

0.20 0.15 0.10

P

0.05 0.00 -0.05 -0.10 -0.15 0.4

0.6

0.8

1.0

1.2

(b) Section moment at typical points

Fig. 5.8 Section moment in the single factor test of parameter λ

1.4

1.6

120

5 Mechanical Properties of Subway Shield Tunnel

5.4 Orthogonal Test on the Functionally Graded Subway Tunnel Model According to the above single factor test, factors a, b and λ are selected for orthogonal test to compare the significance. Because the axial radial displacement and section moment curve of the functional gradient lining are symmetrical about the curve of λ = 1, the part of λ < 1 is taken. Three levels are taken for the three factors. Values of levels and the orthogonal test scheme are summarized in Tables 5.2 and 5.3.

5.4.1 Deformation Mode Analysis (1) Calculation results of radial displacement In the orthogonal test, the radial displacement at the lining axis is used to represent the lining deformation mode, and the calculation results are summarized in Table 5.4. (2) Intuitive analysis of influence factors From Fig. 5.9, the increase of the elastic modulus parameter a or b will reduce the deformation, but the influence of a and b on different positions is different. When α < 45◦ , the displacement range caused by the change of parameter a is slightly larger Table 5.2 Level values of every factor in orthogonal test a

Factors Levels

λ

b

0.9

0.9

0.5

1.0

1.0

0.6

1.1

1.1

0.7

Table 5.3 Orthogonal test scheme Number

Parameters a

b

λ

Empty column

A1

0.9

0.9

0.5

1

A2

0.9

1

0.6

2

A3

0.9

1.1

0.7

3

A4

1

0.9

0.6

3

A5

1

1

0.7

1

A6

1

1.1

0.5

2

A7

1.1

0.9

0.7

2

A8

1.1

1

0.5

3

A9

1.1

1.1

0.6

1

0.9

1

1

A4

1

1

1.1

1.1

1.1

1.1

A7

A8

A9

0.9

1.1

1

1

A5

A6

1.1

0.9

0.9

0.9

A2

0.9

A1

b

A3

a

Number

0.6

0.5

0.7

0.5

0.7

0.6

0.7

0.6

0.5

λ

− 1.458 − 2.323 − 1.429 − 2.254 − 1.742

− 2.501 − 2.482 − 3.927 − 3.032

− 2.038

− 3.494 − 3.988

− 2.085 − 1.493

− 3.528 − 2.529

15° − 2.752

0° − 4.653

− 0.177

− 0.229

− 0.159

− 0.322

− 0.208

− 0.300

− 0.262

− 0.374

− 0.505

30°

1.250

1.624

1.008

1.508

0.938

1.300

0.867

1.120

1.569

45°

Dimensionless parameter of the displacement K (×10–2 )

Table 5.4 Radial displacement of functionally graded lining 60°

2.261

2.947

1.852

2.824

1.772

2.477

1.694

2.364

3.122

75°

2.799

3.661

2.321

3.567

2.251

3.166

2.187

3.070

4.082

90°

3.032

3.966

2.529

3.953

2.501

3.528

2.482

3.494

4.658

5.4 Orthogonal Test on the Functionally Graded Subway Tunnel Model 121

122

5 Mechanical Properties of Subway Shield Tunnel

than that caused by the change of parameter b, so changing a has a great influence on the deformation here. When α > 45◦ , the displacement range caused by the change of parameter b is slightly larger than that caused by the change of parameter a, so changing b has a great influence on the deformation here. Therefore, when the maximum deformation is at α = 0◦ , a can be increased to reduce the maximum deformation. When the maximum deformation is at 90◦ , b can be increased to reduce the maximum deformation. In addition, the radial displacement is most affected by the lateral pressure coefficient λ, and decreases linearly with the increase of λ, which shows that the lateral pressure coefficient has a great influence on the structure safety, which exceeds the influence of the structural parameters. At 30◦ , the influence of a on the radial displacement exceeds the other two parameters. At 45◦ , the increase of a will increase the radial displacement, but within 30◦ ∼ 45◦ , the radial displacement is also small, which belongs to the junction of positive and negative deformation, so the abnormality brought by this range is not considered. (3) Variance analysis of influence factors The variance analysis of influence factors of radial displacement is summarized in Table 5.5. According to Table 5.5, when α < 45◦ , parameters a and λ have a very significant influence on the radial displacement of the functional gradient lining, and the significance of λ is greater than that of parameter a. With the increase of α, the significance of parameter a on displacement increases gradually, and that of parameter b on displacement decreases gradually, but the significance of parameter a is always greater than that of parameter b. when α > 45◦ , parameters b and λ have a very significant influence on the radial displacement of the functional gradient lining, and the significance of λ is greater than that of parameter b. With the increase of α, the significance of parameter a on displacement increases gradually, and that of parameter b on displacement decreases gradually, but the significance of parameter b is always greater than that of parameter a. At α = 45◦ , parameters a and b have no effect on the radial displacement, but λ has a significant effect on the radial displacement. In conclusion, for the functionally graded lining with linear elastic modulus distribution, in the part of α < 45◦ , the change of parameter a has a greater impact on the lining deformation. In the part of α > 45◦ , the change of parameter b has a greater impact on the lining deformation. Therefore, when the maximum deformation occurs at 0◦ , the parameter a > b. When the maximum deformation occurs at the position of 90◦ , the parameter a < b to save material cost and reduce structural deformation.

5.4.2 Section Moment Analysis (1) Calculation results of section moment The calculation results of section moment of functional gradient lining in the orthogonal test are summarized in Table 5.6.

5.4 Orthogonal Test on the Functionally Graded Subway Tunnel Model

123

-2.4

Fig. 5.9 Intuitive analysis of influence factors of the radial displacement

-2.7

K

10-2

-3.0 -3.3 -3.6 -3.9 -4.2 -4.5

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

λ

(a) Intuitive analysis of the radial displacement at 0

-1.4 -1.6

K

10-2

-1.8 -2.0 -2.2 -2.4 -2.6

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

(b) Intuitive analysis of the radial displacement at 15

-0.15

K

10-2

-0.20 -0.25 -0.30 -0.35 -0.40

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

(c) Intuitive analysis of the radial displacement at 30

124

5 Mechanical Properties of Subway Shield Tunnel 1.6

Fig. 5.9 (continued)

1.5

1.3 1.2

K

×10-2

1.4

1.1 1.0 0.9

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

(d) Intuitive analysis of the radial displacement at 45

3.0 2.8

2.4 2.2

K

×10-2

2.6

2.0 1.8 1.6

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

(e) Intuitive analysis of the radial displacement at 60

4.0

K

×10-2

3.6 3.2 2.8 2.4 2.0

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

λ

(f) Intuitive analysis of the radial displacement at 75

5.4 Orthogonal Test on the Functionally Graded Subway Tunnel Model

125

4.4

Fig. 5.9 (continued)

K

×10-2

4.0 3.6 3.2 2.8 2.4

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

λ

(g) Intuitive analysis of the radial displacement at 90

(2) Intuitive analysis of influence factors It can be seen from Fig. 5.10 that λ has the greatest influence on the section moment. When α < 45◦ , with the increase of parameter a, the section moment increases. With the increase of parameter b, the section moment decreases, and the increase of the section moment caused by the increase of a is almost the same as the decrease of the section moment caused by the increase of b. When α > 45◦ , with the increase of parameter a, the section moment decreases, and with the increase of parameter b, the section moment increases. The decrease of the section moment caused by the increase of a is almost the same as the increase of the section moment caused by the increase of b. When α = 45◦ , the section moment is small, and the influence of parameters a and b on the section moment is greater than that of λ. Because this is the junction of positive and negative bending moment, and the section moment is small, the abnormality generated here is not considered. Therefore, it can be found that when the elastic modulus of the lining increases, the section moment also increases, which conforms to the idea of “flexible yielding” in structural design. (3) Variance analysis of influence factors The variance analysis of influence factors of section moment is summarized in Table 5.7. According to the variance analysis in Table 5.7, the influence of λ on the section moment is very significant, and the influence on the positions of α = 0◦ and α = 90◦ is the most significant. The influence of elastic modulus parameters a and b on the section moment is significant, and the significant degree is similar. The section of 45◦ is abnormal in the variance analysis, where λ has no effect on the section moment, and the influence of the elastic modulus parameter on the section moment is similar to that of other parts. Because of the small section moment, this abnormality is not considered.

126

5 Mechanical Properties of Subway Shield Tunnel

Table 5.5 Variance analysis of influence factors of radial displacement Position 0

15

30

45

60

75

90

Parameters

Dof

Square of deviance

F

Confidence 99%

95%

Significance 90%

a

2

0.270

26.061

18

6.94

4.32

***

b

2

0.198

19.123

18

6.94

4.32

***

λ

2

4.261

410.793

18

6.94

4.32

***

Error

2

0.010

a

2

0.137

28.193

18

6.94

4.32

***

b

2

0.075

15.339

18

6.94

4.32

**

297.711

18

6.94

4.32

***

32.980

18

6.94

4.32

***

λ

2

1.449

Error

2

0.005

a

2

0.055

b

2

0.007

4.438

18

6.94

4.32

*

λ

2

0.030

18.095

18

6.94

4.32

***

Error

2

0.002

a

2

0.018

4.040

18

6.94

4.32

–

b

2

0.012

2.631

18

6.94

4.32

–

λ

2

0.596

134.648

18

6.94

4.32

***

Error

2

0.004

a

2

0.003

1.258

18

6.94

4.32

–

b

2

0.075

32.857

18

6.94

4.32

***

927.119

18

6.94

4.32

***

10.111

18

6.94

4.32

**

λ

2

2.130

Error

2

0.002

a

2

0.053

b

2

0.173

32.974

18

6.94

4.32

***

λ

2

3.452

656.317

18

6.94

4.32

***

Error

2

0.005

a

2

0.206

20.093

18

6.94

4.32

***

b

2

0.263

25.636

18

6.94

4.32

***

λ

2

4.276

416.241

18

6.94

4.32

***

Error

2

0.010

Note “***”means very significant; “**”means significant; “*”means there is some influence but not significant; “–” means no influence

In conclusion, for the functionally graded lining with linear elastic modulus distribution, the influence of the nonuniformity of external load on the section moment is the largest, far more than the influence of elastic modulus parameters.

5.5 Verification of the Numerical Simulation

127

Table 5.6 Section moment of functionally graded lining Number a

b

λ

Dimensionless parameters of section moment P (×10–2 ) 0°

15°

30°

45°

60°

75°

90°

0.0028 − 6.247 − 10.823 − 12.497

A1

0.9 0.9 0.5 12.503 10.828 6.253

A2

0.9 1

0.6

9.787

8.447 4.787 − 0.213

− 5.213

− 8.874 − 10.213

A3

0.9 1.1 0.7

7.195

6.191 3.445 − 0.305

− 4.055

− 6.800

− 7.805

A4

1

0.9 0.6 10.213

8.874 5.213

0.213

− 4.787

− 8.447

− 9.787

A5

1

1

6.497 3.752

0.0015 − 3.748

− 6.494

− 7.498

A6

1

1.1 0.5 12.259 10.584 6.009 − 0.241

A7

1.1 0.9 0.7

A8

1.1 1

A9

1.1 1.1 0.6 10.000

0.7

7.502

− 6.491 − 11.067 − 12.741

6.800 4.055

0.305

− 3.445

0.5 12.741 11.067 6.491

0.241

− 6.009 − 10.584 − 12.259

0

− 5.000

7.805

8.660 5.000

− 6.191

− 7.195

− 8.660 − 10.000

5.5 Verification of the Numerical Simulation In the numerical calculation, the lining of subway tunnels has a certain thickness, but in the theoretical calculation, the lining is simplified as a curved rod structure. Therefore, under the condition of different thickness diameter ratio, the displacement calculation results of the inner wall, outer wall and axis of the lining in the numerical calculation are compared with the theoretical calculation results, so as to determine the corresponding position of the radius RH in the theoretical calculation. The calculation results are shown in Fig. 5.11. From Fig. 5.11, it can be seen that the radial displacement at the inner edge of the cylinder is the largest, followed by the axis, and the outer edge is the smallest, indicating that the inner edge of the cylinder structure should be used as the basis for failure under the action of external force. When t/R ≤ 0.2, the theoretical calculation results are close to the radial displacement of the inner edge of the cylinder in the numerical simulation results. With the increase of the thickness diameter ratio, the numerical simulation results of the radial displacement of the inner edge at 0◦ and 90◦ gradually increase. When t/R = 0.2, the numerical simulation results of the radial displacement of the inner edge at 0◦ and α = 90◦ are closest to the theoretical calculation results, But when t/R > 0.2 the numerical simulation results of the radial displacement of the inner edge at 0◦ and 90◦ exceed the theoretical calculation results, and the gap between the two gradually becomes larger. It shows that when t/R ≤ 0.2, the theoretical calculation results can be used to calculate the radial displacement of the inner edge, and the calculation results are conservative. But when t/R > 0.2, if the theoretical model continues to be applied to calculate the radial displacement at the most unfavorable position, the results will be smaller, which is not conducive to the safety of the structure. In conclusion, the theoretical model of internal force and deformation is suitable for t/R ≤ 0.2 cylinder structure. According to the comparison between the radial displacement of numerical simulation and theoretical calculation under different thickness diameter ratio, the function

128

5 Mechanical Properties of Subway Shield Tunnel 13

Fig. 5.10 Intuitive analysis of influence factors of the section moment

12

P ×10-2

11 10 9 8 7

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

λ

(a) Intuitive analysis of the section moment at 0

11

P ×10-2

10 9 8 7 6

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

λ

(b) Intuitive analysis of the section moment at 15

6.5 6.0

P

×10-2

5.5 5.0 4.5 4.0 3.5

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

λ

(c) Intuitive analysis of the section moment at 30

5.5 Verification of the Numerical Simulation

129

0.2

Fig. 5.10 (continued)

0.0

P

×10-2

0.1

-0.1

-0.2

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

λ

(d) Intuitive analysis of the section moment at 45

-3.5 -4.0

P ×10-2

-4.5 -5.0 -5.5 -6.0 -6.5

0.9 1 1.1 a

0.9 1 1.1 0.5 0.6 0.7 b λ (e) Intuitive analysis of the section moment at 60

-6 -7

P ×10-2

-8 -9 -10 -11 0.9 1 1.1 0.9 1 1.1 0.5 0.6 0.7 a λ b (f) Intuitive analysis of the section moment at 75

130

5 Mechanical Properties of Subway Shield Tunnel

Fig. 5.10 (continued)

-7 -8

P ×10-2

-9 -10 -11 -12 -13

0.9 1 1.1 a

0.9 1 1.1 b

0.5 0.6 0.7

λ

(g) Intuitive analysis of the section moment at 90

gradient cylinder under the condition of t/R = 0.2 is simulated. The adopted elastic modulus functions are Function I and Function II. In order to reflect the economic characteristics of FGS, two groups of elastic modulus parameters a = 1, b = 0.9 and a = 0.9, b = 1 are used. The vertical load is 1 MPa, λ = 0.5 and the fundamental elastic modulus E 0 = 36 GPa. Because ABAQUS cannot realize the continuous change of elastic modulus along the circular direction, the cylinder structure is adopted to give elastic modulus along the circular direction in sections. When there are enough segments, it can be considered that the cylinder with segments given elastic modulus is equivalent to the functional gradient cylinder in mechanical properties. Next, we explore the number of segments. The change trend of Mises stress and deformation coefficient |K| with the number of segments at typical points is shown in Fig. 5.12. From Fig. 5.12, it can be seen that with the increase of the number of segments, the Mises stress and radial displacement coefficient |K| at the inner edge of the cylinder gradually converges to a certain value, so when the number of segments is large enough, the method of elastic modulus given by segments can be used to simulate the functional gradient lining. It can be also seen that when the number of segments is 16, the stress and deformation begin to converge, when the number of segments is 32, The change degree of Mises stress and radial deformation coefficient of each typical point is less than 0.05% and 0.02%, respectively. Therefore, in order to ensure the accuracy of calculation and reduce calculation, in the next simulation of the mechanical properties of the FGM cylinder, the 1/4-cylinder structure is divided into 16 sections to simulate the FGM cylinder. Based on the above demonstration of the thickness diameter ratio and the number of segments of the functional gradient cylinder in the numerical simulation, the mechanical properties of functional gradient linings with t/R = 0.2 and the number of segments of 16 are analyzed, and the deformation coefficient K of the functional gradient lining under the condition of the elastic modulus Function I is compared with

5.5 Verification of the Numerical Simulation

131

Table 5.7 Variance analysis of influence factors of section moment Position 0

15

30

45

60

75

90

Parameters

Dof

Square of deviance

F

Confidence

Significance

99%

95%

90%

a

2

0.188

12.698

18

6.94

4.32

**

b

2

0.190

12.843

18

6.94

4.32

**

λ

2

37.505

2533.057

18

6.94

4.32

***

Error

2

0.015

a

2

0.188

12.448

18

6.94

4.32

**

b

2

0.190

12.590

18

6.94

4.32

**

λ

2

28.128

1862.404

18

6.94

4.32

***

Error

2

0.015

a

2

0.188

12.698

18

6.94

4.32

**

b

2

0.190

12.843

18

6.94

4.32

**

λ

2

9.378

633.349

18

6.94

4.32

***

Error

2

0.015

a

2

0.188

12.704

18

6.94

4.32

**

b

2

0.190

12.838

18

6.94

4.32

**

λ

2

0.000

0.000

18

6.94

4.32

–

Error

2

0.015

a

2

0.188

12.698

18

6.94

4.32

**

b

2

0.190

12.843

18

6.94

4.32

**

λ

2

9.373

633.011

18

6.94

4.32

***

Error

2

0.015

a

2

0.188

12.471

18

6.94

4.32

**

b

2

0.190

12.565

18

6.94

4.32

**

λ

2

28.119

1861.611

18

6.94

4.32

***

Error

2

0.015

a

2

0.188

12.698

18

6.94

4.32

**

b

2

0.190

12.843

18

6.94

4.32

**

λ

2

37.495

2532.381

18

6.94

4.32

***

Error

2

0.015

Note “***”means very significant; “**”means significant; “*”means there is some influence but not significant; “–” means no influence

the theoretical calculation result, and the comparison result is shown in Fig. 5.13. The radial displacement of FGM lining with different elastic modulus parameters in the theoretical calculation and numerical simulation is shown in Fig. 5.14. Compared with the radial displacement of FGM linings with two elastic modulus functions, the result is shown in Fig. 5.15.

132

0.06 0.04

axis outer edge inner edge theoretical calculation

K

0.02 0.00 -0.02 -0.04 -0.06 -0.2

0.0

0.2

0.4 0.6 2α/π

0.8

1.0

1.2

0.8

1.0

1.2

0.8

1.0

1.2

(a) t/R=0.03

0.06 0.04

axis outer edge inner edge theoretical calculation

K

0.02 0.00 -0.02 -0.04 -0.06 -0.2

0.0

0.2

0.4

0.6 2α/π

(b) t/R=0.05

0.06 0.04

axis outer edge inner edge theoretical calculation

0.02

K

Fig. 5.11 Comparison of numerical simulation results and theoretical calculation results of radial deformation of homogeneous cylinder at different positions under different thickness diameter ratio

5 Mechanical Properties of Subway Shield Tunnel

0.00 -0.02 -0.04 -0.06 -0.2

0.0

0.2

0.4

2α/π

(c) t/R=0.07

0.6

5.5 Verification of the Numerical Simulation 0.06

Fig. 5.11 (continued)

0.04

133 axis outer edge inner edge theoretical calculation

K

0.02 0.00 -0.02 -0.04 -0.06 -0.2

0.0

0.2

0.4 0.6 2α/π

0.8

1.0

1.2

0.8

1.0

1.2

0.8

1.0

1.2

(d) t/R=0.09

0.06 0.04

axis outer edge inner edge theoretical calculation

K

0.02 0.00 -0.02 -0.04 -0.06 -0.2

0.0

0.2

0.4 0.6 2α/π (e) t/R=0.1

0.06 0.04

axis outer edge inner edge theoretical calculation

K

0.02 0.00 -0.02 -0.04 -0.06 -0.2

0.0

0.2

0.4 0.6 2α/π (f) t/R=0.2

134

5 Mechanical Properties of Subway Shield Tunnel 0.06

Fig. 5.11 (continued)

0.04

axis outer edge inner edge theoretical calculation

0.02

K

0.00 -0.02 -0.04 -0.06

0.0

0.5 2α/π

1.0

(g) t/R=0.3

0.06 0.04 0.02

axis outer edge inner edge theoretical calculation

K

0.00 -0.02 -0.04 -0.06 -0.08 -0.2

0.0

0.2

0.4

0.6 2α/π

0.8

1.0

1.2

0.8

1.0

1.2

(h) t/R=0.4

0.06 0.04 0.02

axis outer edge inner edge theoretical calculation

K

0.00 -0.02 -0.04 -0.06 -0.08 -0.10 -0.2

0.0

0.2

0.4 0.6 2α/π (i) t/R=0.5

5.5 Verification of the Numerical Simulation

12.8

135

α=0°

α=15°

α=60°

12.4

σMises/pw

12.0 11.6 11.2 10.8 10.4 10.0

0

4

8

12 16 20 24 number of segments

28

32

36

32

36

(a) Mises stress

49 48 47 43.1

46

43.0

45

42.9

42.8

44

0

4

8

12

16

20

24

28

32

43 42 41

0

4

8

12

16

20

24

28

number of segments (b) deformation coefficient |K|

Fig. 5.12 The change trend of cylinder mechanical properties with the number of segments

136

5 Mechanical Properties of Subway Shield Tunnel

0.06 0.04

K

0.02 0.00

Numerical simulation a=1, b=0.9 a=0.9, b=1 a=1, b=1 Theoretical calculation a=0.9, b=1 a=1, b=0.9 a=1, b=1

-0.02 -0.04 -0.06 -0.2

0.0

0.2

0.4 0.6 2α/π

0.8

1.0

1.2

Fig. 5.13 Comparison between the theoretical and numerical results of radial displacement of functionally graded lining under the condition of elastic modulus function I

From Fig. 5.13, the radial displacement curves of FGM lining almost coincide under the conditions of a = 1, b = 0.9 and a = 0.9, b = 1 in results of both theoretical analysis and numerical simulation. There is a certain gap between the results of theoretical analysis and the numerical simulation, but the theoretical analysis and the numerical simulation results are close at the two places of 0◦ and 90◦ with large displacement, where the gap between the two calculation results is within 8%, so the theoretical calculation results can be used as the basis for lining deformation. From Fig. 5.14, under the condition of linear distribution of elastic modulus, the radial deformation of lining is slightly different with different elastic modulus parameters in both the theoretical calculation and numerical simulation. From Fig. 5.15a, the maximum radial displacement is at 0◦ . The maximum radial displacement of homogeneous lining is the smallest and the functional gradient lining with elastic modulus parameter a = 1, b = 0.9 is the second, and the functional gradient lining with elastic modulus parameter a = 0.9, b = 1 is the largest. In addition, the difference between the maximum deformation of functionally graded lining with a = 1, b = 0.9 and that of homogeneous lining is no more than 5%. From Fig. 5.15b, similarly, the maximum radial displacement is at 0◦ in the numerical simulation. The maximum radial displacement of homogeneous lining is the smallest and the functional gradient lining with elastic modulus parameter a = 1, b = 0.9 is the second, and the functional gradient lining with elastic modulus parameter a = 0.9, b = 1 is the largest. In addition, the difference between the maximum deformation of functionally graded lining with a = 1, b = 0.9 and that of homogeneous lining is no more than 5%. In conclusion, although there are some differences between the

5.5 Verification of the Numerical Simulation

0.06

137

a=1, b=1 a=0.9, b=1 a=1, b=0.9

0.04

K

0.02 0.00

-0.036 -0.038

-0.02

-0.040 -0.042

-0.04

-0.044 -0.046 0.00

-0.06 -0.2

0.0

0.2

0.4

0.02

0.6

0.04

0.06

0.8

0.08

1.0

1.2

(a) Displacement coefficient K in theoretical calculation

0.06

a=1, b=0.9 a=0.9, b=1 a=1, b=1

0.04

K

0.02 0.00 -0.042

-0.02

-0.044

-0.046

-0.04

-0.048 0.00

-0.06 -0.2

0.02

0.04

0.06

0.08

0.10

A

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(b) Displacement coefficient K in numerical simulation

Fig. 5.14 Comparison of radial displacement calculation results of functionally graded lining with different elastic modulus parameters under the condition of elastic modulus function I

results of theoretical analysis and numerical simulation, the laws shown by the two are consistent. When λ = 0.5, the maximum deformation is at 0◦ , and a > b is conducive to reduce the maximum deformation. From Fig. 5.15, the deformation of the functionally graded lining corresponding to the two elastic modulus functions is similar, but the circumferential stress is quite

138

5 Mechanical Properties of Subway Shield Tunnel

15

Function a=1, b=0.9 a=0.9, b=1 Function a=1, b=0.9 a=0.9, b=1

10 5 0 -5 -10 -15

-12 -14 -16 -18 -20

-20

-22 -24 0.80

-25 -0.2

0.85

0.0

0.90

0.2

0.95

0.4

1.00

0.6

0.8

1.0

1.2

(a) Circumferential stress

0.06

K

Function a=1, b=0.9 a=0.9, b=1 0.04 Function a=1, b=0.9 0.02 a=0.9, b=1 0.00 -0.040 -0.042

-0.02

-0.044 -0.046

-0.04

-0.048 -0.050 0.00

-0.06 -0.2

0.0

0.2

0.4

0.02

0.6

0.04

0.8

0.06

0.08

0.10

1.0

0.12

0.14

1.2

(b) Displacement coefficient K

Fig. 5.15 Comparison of mechanical properties of functionally graded lining under two elastic modulus functions

5.6 Conclusions

139

different. From Fig. 5.15a, the circumferential stress of functional gradient lining corresponding to elastic modulus Function II is larger near 90◦ and is about 30% larger than that corresponding to Function I. From Fig. 5.15b, the radial displacement of the four functional gradient linings reaches the maximum value at 0◦ , and the deformation difference is about 2%. The radial displacement of the functional gradient lining corresponding to Function II with a = 1, b = 0.9 is the minimum, and that of the functional gradient lining corresponding to Function I with a = 1, b = 0.9 is the second. it can be seen that Function II has some advantages over Function I in reducing the maximum deformation of the structure, but the advantages are relatively low, and the overall stiffness of the functional gradient structure represented by Function II is also larger. Therefore, when Function I can meet the deformation requirements, it is unnecessary to use Function II to reduce the deformation further.

5.6 Conclusions This study aims to develop a new approach to design functionally graded lining of subway tunnels. With two parameters of the elastic modulus function and lateral pressure coefficient, mechanical properties of the functionally graded lining were studied, proving that the new lining shows great characteristics of economy and security. In the study, the mechanical properties of the functionally graded lining were researched, through analyzing theoretically and numerically, some conclusions are drawn as follows: (1) The single factor test shows that the radial displacement of the lining axis decreases with the increase of a and b, but the deformation mode remains the same, and the reduction of deformation is smaller and smaller. In addition, with the increase of a and b, the distribution of moment remains the same. With the increase of a, the positive section moment increases linearly and the negative bending moment decreases linearly. With the increase of b, the negative moment increases linearly and the positive moment decreases linearly, which embodies the idea of “flexible yielding”. The displacement of the lining axis and the section moment change linearly with the increase of λ. with the increase of λ, the shape of the lining changes significantly, which shows that the side with large lateral pressure deforms to the inside, and the side with small lateral pressure expands to the outside. (2) The orthogonal test shows that when the maximum deformation occurs at 0◦ , the parameter a should be larger than b. When the maximum deformation occurs at 90◦ , the parameter b should be larger than a, so as to save material cost on the premise of ensuring safety. In addition, the lateral pressure coefficient has a great impact on the safety of the structure, which exceeds the influence of structural parameters on the safety of the structure.

140

5 Mechanical Properties of Subway Shield Tunnel

(3) The numerical simulation shows that the calculation model of internal force and deformation is suitable for the cylinder with t/R ≤ 0.2. There is a certain gap between the theoretical analysis and numerical simulation, but the gap between the theoretical analysis and numerical simulation results is within 8% at 0◦ and 90◦ with large displacement. In addition, the conclusion of theoretical analysis is verified. There is little difference in deformation between the two kinds of functional graded linings, but there is a big difference in circumferential stress. It can be seen that compared with Function I, Function II has some advantages in reducing the maximum deformation of the structure, but the advantages are relatively low.

References Chen KH, Peng FL (2018) An improved method to calculate the vertical earth pressure for deep shield tunnel in Shanghai soil layers. Tunn Undergr Space Technol 75:43–66 Hu ZF, Yue ZQ, Zhou J, Tham LG (2003) Design and construction of a deep excavation in soft soils adjacent to the Shanghai Metro tunnels. Can GeoteCh J 40:933–948 Qiao YK, Peng FL (2016) Lessons learnt from urban underground space use in Shanghai—from Lujiazui business district to Hongqiao central business district. Tunn Undergr Space Technol 55:308–319 Wang DY, Wang YT, Ma W, Lei LL, Wen Z (2018) Study on the freezing-induced soil moisture redistribution under the applied high pressure. Cold Reg Sci Technol 145:135–141 Zuo LB, Li GH, Feng K, Ma XC, Zhang L, Qiu Y, Cao SY (2018) Experimental analysis of mechanical behavior of segmental joint for gas pipeline shield tunnel under unfavorable compression-bending loads. Tunn Undergr Space Technol 77:227–236

Chapter 6

Long-Term Settlement of Subway Shield Tunnel

6.1 Introduction The subway is indispensable as a safe, comfortable and high-speed transport vehicle in modern cities. The long-term settlement of shield tunnel has significant effect on the safety of the subway operation. The longitudinal settlement of a subway tunnel, especially the differential settlement in the longitudinal direction affects the safe operation of the subway. The necessity of considering differential settlement to ensure the safety of concrete tunnel linings has also been recognized by the International Tunnelling Association (ITA) (ITA 2000). Excessive differential tunnel settlements have been observed in Shanghai metro tunnels over the past two decades and they become a major concern for the operational authority and the designers since would affect the serviceability and possibly safety of the entire metro system (Qing 2013). To protect the tunnels against excessive longitudinal settlements, comprehensive investigations on the causes of the large tunnel settlement are needed. The overall line of Subway Line 1 in Shanghai experiences settlements and has characteristics of being obvious inhomogeneous (Cui and Tan 2015). Some studies have been carried out attempting to investigate possible causes of observed large tunnel settlements for the Shanghai metro tunnels (Yu et al. 1986; Wang et al. 2008). The settlement of a subway tunnel was studied from the theoretical and numerical analyses based on the monitoring data (Tan et al. 2015). The influence of stiffness anisotropy has been illustrated by many researches when predicting ground deformations due to tunnelling (Lee and Rowe 1989; Franzius et al. 2005) and excavation (Ng et al. 2004; Clayton et al. 2004). Some field monitoring results showed that creep induced ground settlement after the completion of tunnelling and excavation was limited (Lee et al. 1999; Liu et al. 2005). Shirlaw (1995) also reported the widening phenomenon in metro tunnels in Singapore. The widening phenomenon was also confirmed by numerical simulations (Mair 2008).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Z.-D. Cui, Interaction Between Soil Foundation and Subway Shield Tunnel, https://doi.org/10.1007/978-981-99-6870-1_6

141

142

6 Long-Term Settlement of Subway Shield Tunnel

In this chapter, theoretical calculations of the long-term settlement of the shield tunnel were compared with the numerical simulation. Based on in-site monitoring, the settlement trend was predicted by GM (1, 1) and ARMA (n, m) model. The characteristics of the long-term settlement and the settlement trough were studied and the prediction model for the settlement trough were built.

6.2 Theoretical Analysis of Long-Term Settlement of Subway Shield Tunnel and Numerical Verification 6.2.1 Theoretical Model Analytical methods are the preferred approaches to analyze the ground movements induced by tunneling. Four main categories exist for the analytical methods, namely: the virtual image technique, the complex variable method, the general series form stress function in polar coordinate, and the stochastic medium theory (Zhang et al. 2011). Many studies have been carried out based on the empirical formula (6.1) which was proposed by Peck (1969). S = Smax

  x2 exp − 2 2i

(6.1)

where S is the vertical settlement; Smax is the maximum settlement above the tunnel centerline; x is the horizontal distance from the tunnel centerline in the transverse direction; and i is the distance from the tunnel centerline to the inflexion point of the settlement curve. The calculated settlement accords with the actual engineering better through the adjusting of i. Many scholars investigated the settlement on the basis of volume loss formulas by differing their selections of i. Different i were selected, expressing as Eqs. (6.2)–(6.4) as following: i =k·z

(6.2)

i = 0.43 · z + 1.11

(6.3)

i = 0.45 · z

(6.4)

where k is empirically determined by width; z is the tunnel depth; and d is the tunnel diameter.

6.2 Theoretical Analysis of Long-Term Settlement of Subway Shield …

143

The layer-wise summation method can also be used to analyze the settlement problem. The consolidation settlement S(t) of tunnel structure can be expressed as: S(t) = Uz S1

(6.5)

where Uz is the degree of consolidation; Uz can be obtained through the corresponding transformation when time factor TV is known; S1 is the ground consolidation settlement. Soil can be assumed uniform and average parameters are used to simplify the calculation. S1 can be given as: S1 =

pH Es

=

αγ H 2 Es

(6.6)

where p is the initial excess pore water pressure; H is the overburden thickness of tunnel; E S is the weighted average compression modulus; α is the stress release rate; γ is the weighted average unit weight of soil. Time factor TV can be expressed as: cv t H2

(6.7)

k(l + e) γw a

(6.8)

TV = cv =

where k is the coefficient of soil permeability; e is the void ratio of soil; γw is the unit weight of soil; a is the compressibility of soil. The viscoelastic model can reflect the instantaneous elastic deformation and the viscoelastic deformation gradually changing with time. It can be used to calculate the tunnel settlement. As show in Fig. 6.1, the three elements of the generalized Kelvin model are chosen to describe the constitutive relationship of soil. The stress–strain relation can be written as: η ηE 1 E1 E2 σ˙ + σ = ε˙ + ε E1 + E2 E1 + E2 E1 + E2

(6.9)

The creep equation after lining applied can be expressed as:  ε(t) =

 E2 1  1 1 − e− η t σ0 + E1 E2

(6.10)

After derivation and simplification of the viscoelastic model formula, the longterm settlement of tunnel yields:    paω 1 1 1 E2 S(t) = + − exp − t 4 E1 E2 E2 η

(6.11)

144

6 Long-Term Settlement of Subway Shield Tunnel

Fig. 6.1 Generalized Kelvin model

where p is gravity stress of soil above the grouting; a is the width of the grouting; ω is the coefficient of settlement; E 1 , E 2 and η are the parameters of generalized Kelvin model. The curves are obtained through the theory above and the centrifuge model test in Fig. 6.2. The tunnel settlement which are calculated by the layer-wise summation method and the viscoelastic method are consistent with these of the centrifuge model test. The tunnel settlement is continuous year by year, and remains stable at last. The rate of settlement decreases gradually with time which is quite large in the first 5 years. The results of the layer-wise summation method are better anastomosis with the centrifuge model test than these of the viscoelastic method. The consolidation which is calculated by the viscoelastic method is significantly faster than the other two. The settlement of the first 5 years accounts for more than 60% of the total settlement in 20 years which is shown in Fig. 6.2. The settlement reaches 125.8 mm in the centrifuge model test at last which is 127.7 mm by the layer-wise summation method and 130.1 mm by the viscoelastic method. The results calculated by viscoelastic method are bigger than these of the layer-wise summation method, and also bigger than these of the centrifuge model test. The trend of final settlement is anastomosis well.

6.2.2 Numerical Verification Finite element software can simulate the interaction between soil and structure, and has the ability to deal with specific problems in civil engineering. It is very suitable for analyzing deformation of shield tunnel. The interaction between the shield tunnel and the soil can be presented by the simulation, so as to reveal the settlement mechanism of the tunnel. The two-dimensional finite element model was established. Both the length and the width are 100 m. The buried depth of the tunnel is 13.8 m, the outer diameter of shield tunneling is 6.2 m, and the thickness of segment is 0.35 m. The material parameters of the segment and the soil are shown in Table 6.1.

6.2 Theoretical Analysis of Long-Term Settlement of Subway Shield … 0

145

The experimental value The layer-wise summation method value The viscoelastic method value

-20

Settlement (mm)

-40 -60 -80 -100 -120 -140

0

5

10

15

20

25

Time (year) Fig. 6.2 Long-term settlement of shield tunnel

Table 6.1 Material parameters of the segment and the soil Material

Density  kN/m3

Young’s modulus (MPa)

Poisson’s ratio

Segment

24

3.55 × 104

0.2

The soil

18

20

0.35

Cohesion (kPa)

Friction angle (°)

12

20

It can output a lot of beneficial results after the calculation analysis, including the displacement, stress and strain, the node counterforce, void ratio, saturation and so on. The vertical displacement of soil after the completion of the consolidation is shown in Fig. 6.3. The vertical displacement decreases gradually with the depth, and the vertical displacement are symmetrical because of the single symmetrical load. There is a slight bent in the middle of surface. The displacement where close to the load is larger, and the maximum vertical displacement reaching 0.2813 m. The vertical displacement is between 0.1641 m and 0.2344 m around the shield tunnel which is in an accepted rang for a 20 years’ settlement. The vertical displacement has a close relationship with the stability of tunnel. It is danger to the metro system when the displacement is too large. Measures must be taken to protect shield tunnel when buildings are constructed near the tunnel. The horizontal displacement of soil is shown in Fig. 6.4. The horizontal displacement is also symmetrical. The horizontal displacement is smaller around the shield tunnel. The maximum horizontal displacement appears at the location about ten meters from the tunnel, with the shape of spread out being similar to the lungs, and

146

6 Long-Term Settlement of Subway Shield Tunnel

Fig. 6.3 Vertical displacement of soil

the displacement decreases gradually. The maximum displacement is 0.02956 m. In general, the scope of the horizontal displacement is smaller than that of vertical displacement. The value of horizontal displacement is relatively small, hence, the load has less effect on the horizontal displacement of the soil. The before and after deformation of segment is shown in Fig. 6.5. The tunnel settlement is continuous with time, and the value of final settlement is about 0.18 m. Shield tunnel has a trend to ellipse. The amount of deformation decreases from up to down because the upper parts is close to the load position. It’s protective to tunnel structure if the mechanical properties of soil are good in the practical engineering. It can also prevent deformation of tunnel when increasing the stiffness of lining. After the completion of model analysis, the vertical displacements which are from the upper, the central and the lower of soil near the lining are shown in Fig. 6.6, respectively. The trend of displacement at the three points is basically consistent and roughly of parallel distribution. The soil settlement is obviously fast under the load, but the soil consolidation is slow with the dissipation of pore water pressure after loading. The settlement of the upper point is bigger than that of the central point, and bigger than that of the lower point. The theoretical values agree with these of the soil settlement on the surface and the comparison is shown in Fig. 6.7. The maximum displacement is 0.291 m which is calculated by the elastic foundation beam theory. The maximum displacement is 0.281 m by numerical simulation. The maximum displacement of the theoretical is 3.4% bigger than that of the simulated. The result of numerical simulation is accurate. The rule of theoretical value agrees well with the simulated value.

6.2 Theoretical Analysis of Long-Term Settlement of Subway Shield …

Fig. 6.4 The horizontal displacement of soil

Fig. 6.5 Deformation of segment

147

148

6 Long-Term Settlement of Subway Shield Tunnel

0.05

The upper point The central point The lower point

Displacement (m)

0.00 -0.05 -0.10 -0.15 -0.20 -0.25 0.00E+000

3.00E+007

6.00E+007

9.00E+007

Time (s) Fig. 6.6 Consolidation of soil near the segment 0.10

Theoretical value Simulated value

0.05

Displacement (m)

0.00 -0.05 -0.10 -0.15 -0.20 -0.25 -0.30 -0.35

0

10

20

30

40

50

60

70

True distance along path (m) Fig. 6.7 Theoretical value and simulated value of settlement

80

90

100

6.3 Numerical Simulation of Long-Term Settlement of Subway Shield Tunnel

149

6.3 Numerical Simulation of Long-Term Settlement of Subway Shield Tunnel 6.3.1 Model Establishment The three-dimensional finite element model is established according to the Shanghai Subway Line 8. The calculation model is shown in Fig. 6.8. The sizes of three directions are 50 m in X axis, 44 m in Y axis and 60 m in Z axis. The depth of the tunnel is 13.8 m. The outer diameter of lining is 6.2 m, and the thickness of segment is 0.35 m. The soil is simulated by Drucker-Prager model while the lining and shield machine is simulated by elastic model. Two kinds of soil are applied to consider underlying non-uniform soils, gray clay and sand. Gray clay is above the shield tunnel, as is shown in Fig. 6.9. The material parameters are summarized in Table 6.2. The soil has mainly finished the consolidation under the load, which means the soil should have the stress and without displacement at the simulation beginning. Hence, gravity stress must be established before the excavation. The deformation of soil is about 10−5 m after the gravity balance, as is shown in Fig. 6.10. The shield tunneling is complex in the fact. It contains the soil remove, segment install, grouting and so on. The tunnel excavation is a circle of those processes. In

Fig. 6.8 Schematic of finite element model

150

6 Long-Term Settlement of Subway Shield Tunnel

Grey muddy clay

Grey silt clay

Sand

Y

Z Fig. 6.9 Schematic of soil distribution

Table 6.2 Parameters of each layer soil and structures Soil and structure

thickness (m)

density ρ (kg/m3 )

Elastic modulus E (MPa)

Poisson ratio μ

Cohesion C (kPa)

Friction angleϕ (°)

Grey muddy clay

20

1700

3

0.25

11

9.5

Grey silt clay

22

1800

4.5

0.25

17

16.5

Sand

22

1900

11

0.3

9

30

Lining

0.35

2500

40,000

0.3

−

−

Shield

0.25

7500

2,100,000

0.3

−

−

order to reflect the real deformation of the tunnel and the soil, the accurate simulation must be taken. The processes of tunnel construction are shown in Fig. 6.11.

6.3.2 Simulation Results The length of excavation is 56 and 2 m length soil is removed at each cycle. The vertical displacement of soil is shown in Fig. 6.12. The displacement of the soil around the tunnel is much larger and the displacement changes constantly along the tunnel. The displacement of the soil above the tunnel is downward, while the displacement of the soil under the tunnel is upward. The maximum downward displacement is

6.3 Numerical Simulation of Long-Term Settlement of Subway Shield Tunnel

151

Fig. 6.10 Displacement after ground stress balance

Lining

Tunnel face Tunel axis Shield Y

Z

Fig. 6.11 Schematic of shield tunnel excavation

0.201 m and the maximum upward displacement is 0.134 m. The displacement decreases gradually from tunnel radiate to around. The displacement of ground and the model bottom is much smaller. The displacement of tunnel face is lager in the progress of excavation. The displacement of earlier removed soil increases with time. Hence, shield tunneling has observably influence on the displacement.

152

6 Long-Term Settlement of Subway Shield Tunnel

Fig. 6.12 Vertical displacement of the model after excavation

The vertical displacement of segment is shown in Fig. 6.13. The deformation of segment is consistent with the soil around the tunnel. The displacement is downward at the top of tunnel, while the displacement is upward at the bottom of tunnel. The displacement is relatively smaller at the middle of tunnel. The tunnel has a trend to have a shape of ellipse. The maximum downward displacement is 0.205 m and the maximum upward displacement is 0.135 m. As is shown in Fig. 6.12 the distribution of vertical displacement is non-uniform. Therefore, it is necessary to carry out deeply analysis at some place. There are four paths of displacement picked up, namely, path 1 is on the ground, path 2 is at the top of tunnel, path 3 is in the middle of tunnel, path 4 is on the bottom of tunnel. The four paths are shown in Fig. 6.14. The vertical displacement of four paths is shown in Fig. 6.15 after excavation. The displacement of ground (path 1) is gently. The beginning of excavation is on the right of the model. The displacement is from downward to upward from the beginning to the end. The maximum downward displacement is 0.024 m and the maximum upward displacement is 0.041 m. The displacement of the soil at the top of tunnel (path 2) is differential. The maximum displacement is 0.201 m at 60 m and the minimum displacement is 0.038 m at 0 m. The differential settlement is observed between 2 and 4 m. The crest value of displacement appears at 40 m. The displacement in the middle of tunnel (path 3) is also gently. The maximum displacement is 0.04 m at 60 m and the minimum displacement is 0.017 m at 0 m. The displacement on the bottom of tunnel (path 4) shows upward, which is similar to path 2. The maximum displacement is 0.134 m at 40 m and the minimum displacement is 0.0002 m at 0 m. It is necessary to carry out the displacement of path 2 and path 4 at each period because of the lager displacement. As is shown in Fig. 6.16, the displacement of

6.3 Numerical Simulation of Long-Term Settlement of Subway Shield Tunnel

153

Fig. 6.13 Vertical displacement of lining after excavation

Fig. 6.14 Schematic of the path of the vertical displacement

shield is uniform after the first period and the differential displacement is shown at the excavation face. The vertical displacement increases after the second period. The shield advances 2 m in this period. The maximum displacement appears at the tunnel face reached 0.066 m. The differential settlement increases a little. The displacement still increase in the third period and the differential settlement is obvious at the tunnel

154

6 Long-Term Settlement of Subway Shield Tunnel

0.16 0.12 0.08

Displacement (m)

0.04 0.00 -0.04 Path Path Path Path

-0.08 -0.12 -0.16

1 2 3 4

-0.20 -0.24 0

4

8 12 16 20 24 28 32 36 40 44 48 52 56 60 64

Distance (m) Fig. 6.15 Vertical displacement of different paths

face. The later displacement rule is consistent with before. The vertical displacement increases gradually. The settlement rate increases first, then decrease. The change of settlement rate is more obvious in clay. The settlement in sand is much smaller than that in clay. The crest value of displacement appears at 40 m. The vertical displacement of soil under the tunnel is shown in Fig. 6.17. The soil spring back after excavation and the displacement is upward. The differential settlement appears at the tunnel face and the whole settlement increases with time. The crest value of displacement also appears at 40 m, which is similar with path 2. The displacement of shield is uniform after the first period, the upward displacement is 0.005 m and the differential displacement is shown at the excavation face. The displacement is relatively small in the other part. The vertical displacement increases obviously after the second period. The displacement of tunnel face reaches 0.035 m. The differential settlement increases a little. The displacement still increase in the third period and the differential settlement is obvious at the tunnel face. The later displacement rule is consistent with before. The vertical displacement increases gradually. The settlement rate increases first, then decrease. The change of settlement rate is more obvious in clay. The settlement in sand is much smaller than that in clay. The largest settlement appears at 40 m above and under the tunnel. The soil displacement is carried out at 40 m of path 2 and path 4, which is shown in Fig. 6.18. The accumulate settlement is from 0.025 to 0.189 m at path 2. The accumulate settlement is from 0.012 to 0.134 m at path 4. The settlement has the largest increasing in the third period. The settlement rate increases at first, and then decreases gradually.

6.3 Numerical Simulation of Long-Term Settlement of Subway Shield Tunnel

155

0.02 Period 1 Period 2 Period 3 Period 4 Period 5 Period 6 Period 7 Period 8 Period 9 Period 10 Period 11 Period 12 Period 13 Period 14 Period 15 Period 16 Period 17 Period 18 Period 19 Period 20 Period 21

0.00 -0.02

Displacement (m)

-0.04 -0.06 -0.08 -0.10 -0.12 -0.14 -0.16 -0.18 -0.20 -0.22

0

4

8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76

Distance (m) Fig. 6.16 Vertical displacement of each excavation cycle on path 2 0.14 Period 1 Period 2 Period 3 Period 4 Period 5 Period 6 Period 7 Period 8 Period 9 Period 10 Period 11 Period 12 Period 13 Period 14 Period 15 Period 16 Period 17 Period 18 Period 19 Period 20 Period 21

0.12

Displacement (m)

0.10 0.08 0.06 0.04 0.02 0.00 -0.02

0

4

8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76

Distance (m) Fig. 6.17 Vertical displacement of each excavation cycle on path 4

156

6 Long-Term Settlement of Subway Shield Tunnel

0.20 0.15

Displacement (m)

0.10 0.05 0.00

Path 2 Path 4

-0.05 -0.10 -0.15 -0.20

0

2

4

6

8

10

12

14

16

18

20

22

Period Fig. 6.18 Vertical displacement of upper and lower face of tunnel at 40 m

After the completion of model analysis, the horizontal displacement is shown in Fig. 6.19. The horizontal displacement is much smaller than the vertical displacement. The horizontal displacement mainly appears around the tunnel and it has the trend of spread out. The displacement is relatively larger at the beginning of the excavation along the tunnel. Hence, the vertical displacement is taken more into account than the horizontal displacement. The horizontal displacement of four paths is shown in Fig. 6.20 after excavation. The horizontal displacement of path 1, path 2 and path4 is small. The horizontal displacement of path 3 changes with distance, the maximum displacement is 0.123 m and the minimum displacement is 0.018 m. The maximum displacement appears at the beginning of excavation and the minimum displacement appears at the end of the excavation. The displacement of clay is larger than that of sand. The vertical stress of model is shown in Fig. 6.21 after excavation. The vertical stress increases from up to down. The vertical stress distribution around the interface has obvious differences. The vertical stress of soil above the tunnel is obviously layered. The vertical stress of soil under the tunnel has a character of inhomogeneity. This may be the result of the interaction of underlying non-uniform and the tunnel excavation. In the vicinity of the underlying soil interface, sand portion of the vertical stress of sand is larger than that of the clay under the same depth. The vertical stress of underlying soil is not uniform and the change of the stress at the interface is larger. The vertical stress of different paths are shown in Fig. 6.22. The vertical stress of the surface (path 1) is relatively uniform and it is about − 1.5 × 105 Pa. The vertical stress of path 2 is also relatively uniform and it is about − 4.5 × 105 Pa. The vertical stress of path 3 is also consistent with the vertical displacement and it

6.3 Numerical Simulation of Long-Term Settlement of Subway Shield Tunnel

Fig. 6.19 Horizontal displacement of the model

0.14 0.12

Displacement (m)

0.10 0.08 Path1 Path2 Path3 Path4

0.06 0.04 0.02 0.00 -0.02 -0.04

0

4

8

12 16 20 24 28 32 36 40 44 48 52 56 60 64

Distance (m) Fig. 6.20 Horizontal displacement of different paths

157

158

6 Long-Term Settlement of Subway Shield Tunnel

Fig. 6.21 Distribution of the vertical stress

is about − 5.2 × 105 Pa. The distribution of vertical stress of path 4 has obvious inhomogeneity, it changes between − 4.9 × 105 and − 6.5 × 105 Pa. The vertical stress changes more obvious due to the underlying soil non-uniform, especially in the clay. The distribution of vertical stress is influenced by the non-uniform soil and the excavation process. The pore pressure distribution after excavation is shown in Fig. 6.23. The pore pressure first increases then decreases, and then increase from up to down. The pressure change around tunnel is large. The pore pressure of the soil in the lower of the tunnel is relatively large, especially in the clay. There isn’t significant difference of pore pressure around the interface. The soil pore pressure is obviously layered from up to down. This is the result of the interaction of the non-uniform soil and the tunnel excavation. The pore pressure on different paths is shown in Fig. 6.24. The pore pressure of the surface (path 1) is 0 Pa, because the surface is the drainage boundary. The pore pressure of path 2 at a distance of 0 ~ 17 m have obvious fluctuation, varied between − 1.2 × 105 and − 2.1 × 105 Pa. The pore pressure of other parts is relatively uniform. The change law of pore pressure of path 3 of the tunnel is similar to that of the path 2, but the fluctuation is relatively small. The pore pressure of path 4 from left to right shows decreasing and the pressure of sand is smaller than that of the clay.

6.3 Numerical Simulation of Long-Term Settlement of Subway Shield Tunnel

-1.0×105 -1.5×105 -2.0×105

Path 1 Path 2 Path 3 Path 4

Vertical stress (Pa)

-2.5×105 -3.0×105 -3.5×105 -4.0×105 -4.5×105 -5.0×105 -5.5×105 -6.0×105 -6.5×105 -7.0×105

0

4

8 12 16 20 24 28 32 36 40 44 48 52 56 60 64

Distance (m) Fig. 6.22 Vertical stress of different paths

Fig. 6.23 Distribution of the pore pressure

159

160

6 Long-Term Settlement of Subway Shield Tunnel 5.0×104 0.0

Pore pressure (Pa)

-5.0×104 -1.0×105 -1.5×105 -2.0×105 -2.5×105 -3.0×105 -3.5×10

Path 1 Path 2 Path 3 Path 4

5

-4.0×105 -4.5×105

0

4

8

12 16 20 24 28 32 36 40 44 48 52 56 60 64

Distance (m) Fig. 6.24 Pore pressure of different paths

6.4 In-Site Monitoring of Long-Term Settlement of Subway Shield Tunnel In order to assure the safe and stable operation of the subway system, there are systematic settlement monitoring technologies and national standard. Shanghai railway transit settlement standard network opened formally in 2011, which is the first in the world. This standard network, by integrating the urban ground settlement monitoring layout and the need of railway transit security, adopts the form of bedrock bench mark combined with deep mark and ground settlement bedrock bench mark combined with subway bedrock bench mark. It basically builds a railway transit vertical control network consisting of 41 bedrock bench marks, and can meet the needs of settlement measurement in the existing operating lines and lines under construction of Shanghai Railway Transit. On the basis of the founding and operating of the standard network, Shanghai has accelerated the construction of railway transit information platform, combining subway tunnel settlement monitoring data with regional ground settlement, the engineering geology and hydrogeology along the Subway Lines, and so on, building an integrated information management platform, and further improving the comprehensive data analysis capabilities and the railway transit security monitoring and pre-warning capabilities. According to “Technical specification for land settlement monitor and control” (Standard of Shanghai DGTJ 08-2051-2008), ground settlement should be measured by means of precise leveling, GPS or others, and soil layered settlement should be measured by means of automatic monitoring or manual measurement. This specification prescribes that ground settlement measurement and soil layered settlement

6.4 In-Site Monitoring of Long-Term Settlement of Subway Shield Tunnel

161

monitoring measurement are compulsory items in tunnel projects, and groundwater monitoring is optional during water drainage in construction.

6.4.1 Arrangement of Monitoring Benchmarks The monitoring benchmarks are made of stainless steel and the size is shown in Fig. 6.25. The monitoring benchmarks are buried in the middle of the monolithic track bed sleepers about every 5 m. Monitoring benchmarks are intensified on both sides of monolithic track road expansion joints, in bypasses and in the tunnel portals. As to special track beds as floating slab track bed and open segment track bed, the monitoring benchmarks are buried in the stable positions in tunnel segments beside the track beds or in other stable positions which are convenient for observation. The structure and layout graphs of the monitoring benchmarks are shown in Figs. 6.25 and 6.26.

Fig. 6.25 Structure graph of monitoring benchmarks

subway station

central -line of rail

settlement-monitoring point position in the tunnel Fig. 6.26 Location of the monitoring benchmarks along the Subway Line/station

162

6 Long-Term Settlement of Subway Shield Tunnel

6.4.2 Implementation of Measurement The present frequency of Shanghai Railway Transit long-term settlement measurement is every 6 months for underground structures. As for the sections with a larger settlement rate, the measurement frequency should be increased according to the actual settlement conditions, and when the settlement stabilizes, the frequency can be restored to the ordinary level. In the actual implementation, in order to ensure the accuracy of long line deformation observation, multiple control measures are taken on control network design, field survey and adjustment calculation. (1) Control network It is a very important issue to select the starting reference point in settlement monitoring. Due to the ground settlement in Shanghai, almost none of the present reference points neither deep nor shallow marks can be used as the leveling point of settlement observation, and the existing measurement data fully prove that in all these benchmarks settlement has been observed, some of which is even larger than the settlement of buildings or structures. On the contrary, the bedrock leveling mark is directly built on underground bedrock, so it is quite stable and reliable; each bedrock mark will go through the first-class precise leveling simultaneous determination every year, possessing the latest leveling adjustment results of first-class accuracy. The settlement monitoring of Shanghai railway transit lines uniformly selects bedrock mark (shown in Fig. 6.27) as the settlement monitoring starting reference point, which is the most effective and reliable reference point nowadays for the monitoring of Shanghai soil settlement and settlement deformation of buildings and structures. According to the trend of the line, a nearby bedrock bench mark should be selected as the starting reference point and connect each point of deep buried station along the line to constitute a first-class annexed leveling route (or leveling network) as primary control; The deep buried station points should be taken as the starting reference point, and simultaneously measure the altitude from uplink and downlink to the station leveling point and underground leveling measurement line to constitute the secondary control. (2) Measuring implementation From the secondary control to station leveling points, the ground level is under full observation according to the second grade leveling specification. In the process of measuring, it is suggested to use two electronic total stations or electronic levels at the same time to observe the uplink and downlink respectively in order to save time and improve the accuracy. The uplink and downlink of the tunnel constitute a small level route circuit, the whole railway transit tunnel level route constitutes a closed circuit, and in tunnel secondary leveling the monitoring benchmarks are measured by means of inserting middle points. The measuring route adopts one-way line observation, and two lines are annexed in the reference point in the station hall between two railway transit stations, i.e. the uplink and downlink form an annexed line.

6.4 In-Site Monitoring of Long-Term Settlement of Subway Shield Tunnel Fig. 6.27 Bedrock bench mark schematic diagram

163

protection pile surveyor's pole oil

centering guide concrete

base

6.4.3 Analysis of Settlements of the Whole Subway Line 1 The operation of Subway Line 1 in the first phase (Fig. 6.28) started in April 1995, 16.37 km in length, setting up 13 stations, north from Shanghai Railway Station and south to Jinjiang Park Station (renamed as Hongmei Road Station when the south extension line was built). The first in-situ monitoring work for the tunnel was carried out from May 1995 to December 2009, the settlements had been measured for 30 times. Figure 6.29 illustrates the accumulated settlements of Subway Line 1 during the period from May 1995 to December 2009. By analysis of the in-situ monitoring data, the Subway Line 1 wholly experienced the settlement, having characteristics of obviously inhomogeneous deformation. The partial settlement approximately took on Gaussian distribution, while the settlement of the whole line did not appear obvious regularity. There were obvious differential settlements between running tunnels and subway stations. The average cumulative settlement of Subway Line 1 was 119.4 mm during the period from May 1995 to December 2009. The minimum of the cumulative settlement was 5.8 mm and the maximum was 287.8 mm, so the differential settlement was obvious. From South Shanxi Road Station (E), the settlement of the subway tunnel began to increase and the average settlement between South Huangpi Road Station (F) and People’s Square Station (G) is 180.6 mm. The cumulative settlement between South Shanxi Road Station (F) to Shanghai Railway Station (J) is from 40 to 310 mm. As

164

6 Long-Term Settlement of Subway Shield Tunnel

Fig. 6.28 Route of Shanghai Subway Line 1 (phase I)

Accumulated settlement (mm)

0

A

B

C

E

D

F

G

H

J

K I

-50 -100 -150

In 2009 -200 -250 -300

A-Caobao Road Station B-Shanghai Stadium Station; C-Xujiahui Station D-Hengshan Road Station; E-Changshu Road Station F-South Shanxi Road Station; G-South Huangpi Road Statio H-People’s Square Station; I-Xinzha Road Station J-Hanzhong Road Station K-Shanghai Railway Station

-350

Mileage (m)

Fig. 6.29 Accumulated settlement of Subway Line 1 (from May 1995 to December 2009)

6.4 In-Site Monitoring of Long-Term Settlement of Subway Shield Tunnel

165

shown in Fig. 6.29, the People’s Square Station and Shanghai Railway Station nearby experience the largest settlements. The settlement near Hengshan Road Station (C) is relatively large, 155.6 mm in average and the maximum of the cumulative settlement reaches 213.2 mm, significantly higher than that of the adjacent running tunnel.

6.4.4 Analysis of Settlements of Running Tunnels and Subway Stations Figure 6.30 illustrates the accumulative settlements of two running tunnels during the period between April 1999 and June 2007. The settlements for each running tunnel is generally uniform. As shown in Fig. 6.6a, the settlements between Xinzha Road Station and the inlet of Hanzhong Road Station gradually decreased, while after the inlet of Hanzhong Road Station, the settlements suddenly increased and the differential settlements appeared. As shown in Fig. 6.6b, the settlement of the running tunnel between the outlet of Hanzhong Road Station and the inlet of Railway Station is large. The length of the running tunnel is longer than that of the station and the differential settlement more likely appears. Figure 6.31 illustrates the cumulative settlements of subway stations during the period between April 1999 and June 2007. As shown in Fig. 6.31a, at first the rate of settlement is large and the settlement is over 10 mm between April 1999 and November 1999. Compared with Fig. 6.31b and c, Changshu Road Station experiences smaller cumulative settlement, which is about 60 mm, while the settlements reach about 200 mm and 180 mm, for Xinzha Road Station and Hanzhong Road Station, respectively. As shown in Fig. 6.31b, although the cumulative settlement of Xinzha Road Station is large, the whole settlement is relatively uniform. The differential settlement appears between 12,250 m and 12,300 m in the mileage. Figure 6.31c shows that Hanzhong Road Station experiences the largest differential settlement. At the joint of the station and the running tunnel, the stiffness changes, where the differential settlement happens easily. As shown in Fig. 6.31d, the cumulative settlement of each platform varies from 10 to 250 mm, because the factors resulting in the settlement of each station are different. Shanghai Stadium Station experiences the minimum settlement, which People’s Square Station and Railway Station do the maximum. The accumulative settlements of the subway stations result from a variety of factors, such as the hydrogeological and geological conditions, the building density, the floor area ratio, the surrounding environment, the subway vibration, the disturbance of construction and so on. The subway vibration and the floor area ratio play an important role. Compared with Shanghai Stadium Station, People’s Square Station is a big exchange station with three lines: Subway Line 1, Subway Line 2 and Subway Line 8 and it is located in the big commercial district. The floor area ratio is over 1.5 and the building density is bigger, therefore the differential settlement is larger. The Shanghai Stadium Station

166

6 Long-Term Settlement of Subway Shield Tunnel

-50 1999.04 1999.11 2000.04 2000.11 2001.04 2001.11 2002.03 2002.11 2003.04 2003.11 2004.05 2004.12 2005.06 2005.12 2006.06 2006.12 2007.06

Accumulated Settlement (mm)

-75 -100 -125 -150 -175 -200 -225

-250 12300 12450 12600 12750 12900 13050 13200 13350 13500 Distance (m) (a) Between Xinzha Road station and Hanzhong Road station

-50 1999.04 1999.11 2000.04 2000.11 2001.04 2001.11 2002.03 2002.11 2003.04 2003.11 2004.05 2004.12 2005.06 2005.12 2006.06 2006.12 2007.06

Accumulated Settlement (mm)

-75 -100 -125 -150 -175 -200 -225

-250 13300 13400 13500 13600 13700 13800 13900 14000 14100 14200 Distance (m) (b) Between Hanzhong Road station and Railway station Fig. 6.30 Settlements of running tunnels (between April 1999 and June 2007)

6.4 In-Site Monitoring of Long-Term Settlement of Subway Shield Tunnel

167

-20 1999.04 1999.11 2000.04 2000.11 2001.04 2001.11 2002.03 2002.11 2003.04 2003.11 2004.05 2004.12 2005.06 2005.12 2006.06 2006.12 2007.06

Accumulated Settlement (mm)

-25 -30 -35 -40 -45 -50 -55 -60 -65 -70 -75 7350

7400

7450

7500

7550

7600

7650

7700

Distance (m) (a) Changshu Road station -60 1999.04 1999.11 2000.04 2000.11 2001.04 2001.11 2002.03 2002.11 2003.04 2003.11 2004.05 2004.12 2005.06 2005.12 2006.06 2006.12 2007.06

Accumulated Settlement (mm)

-80 -100 -120 -140 -160 -180 -200 -220 12200

12250

12300

12350

12400

12450

12500

Distance (m) (b) Xinzha Road station Fig. 6.31 Cumulative settlements of subway stations (between April 1999 and June 2007)

168

6 Long-Term Settlement of Subway Shield Tunnel

-60 1999.04 1999.11 2000.04 2000.11 2001.04 2001.11 2002.03 2002.11 2003.04 2003.11 2004.05 2004.12 2005.06 2005.12 2006.06 2006.12 2007.06

Accumulated Settlement (mm)

-80 -100 -120 -140 -160 -180 -200 13150

13200

13250

13300

13350

13400

13450

13500

Distance (m) (c) Hanzhong Road station 0

Accumulated Settlement (mm)

-40

-80

Caobao Road Station Railway Station-Return Section Hanzhong Road Station Xinzha Road Station People's Square Station Huangpi Road Station South Shanxi Road Station Changshu Road Station Hengshan Road Station Xujiahui Station Shanghai Stadium Station

-120

-160

-200

-240 0

6

12

18

24

Monitoring Point

(d) Settlements of subway stations Fig. 6.31 (continued)

30

36

42

6.4 In-Site Monitoring of Long-Term Settlement of Subway Shield Tunnel

169

experiences the minimum differential settlement, 5.3 mm, while the Railway Station does the maximum, 64.8 mm. The differential settlement of Hengshan Road Station, Changshu Road Station and South Shanxi Road Station are close. The differential settlement of South Huangpi Road Station is relatively close to that of People’s Square Station, which is mainly caused by the longitudinal difference of the stratum. Figure 6.32 exhibits the variation trend of the settlement in the monitoring points with the maximum and minimum accumulated settlement of Line 1. The accumulated settlement increases with time increasing, and the general trend is ladderlike increase. The settlement variation can be divided into three phases (Phase I: 1995.12–199.4; Phase II: 1994.4–2004.12; Phase III: 2004.12 till now). The settlement platform appeared in December 2004, accumulated settlement trend moderated and settlement variation gradually decreased. The settlement variation in the place with larger accumulated settlement is greater than that in the place with smaller accumulated settlement. As to the long-term settlement of the subway tunnels after putting into operation, except the secondary soil consolidation deformation caused by constructive disturbance, there are still many factors able to induce the longitudinal inhomogeneous deformation of the tunnels. Observing the settlement condition of the completed and operating subway tunnels on soft soil, we can find out the complex settlement characteristics of the subway tunnels built on soft soil during operation, mainly including:

Fig. 6.32 Settlement course curve of maximum and minimum accumulated settlement monitoring points of stations and interval tunnels of line 1 for uplink (1999.4–2007.6)

170

6 Long-Term Settlement of Subway Shield Tunnel

(1) The settlement in the station is relatively small, while the settlement in interval tunnels is relatively large; (2) The settlement is the same in the left and right tunnels, the transverse settlement is approximately uniform, the longitudinal settlement is relatively large and obviously uneven, with severe settlement troughs; (3) The settlement rate is large in the initial operation period, and gradually decreases with the passage of time, but without obvious convergent tendency; (4) The influencing factors of settlement are too various to analyze, and the limit value is difficult to calculate and analyze. Through the comprehensive utilization and systematic analysis of massive achievements in scientific researches and monitoring data, the rule and influencing factors of subway tunnel deformation are analyzed. The preliminary study proves, the influencing factors of operating subway tunnels settlement mainly include the following aspects: The settlement induced by soil softening under the effect of subway train long-term vibration load; the inhomogeneous settlement of soil in tunnel subjacent layer; the influence of nearby constructive activities; water-level fluctuation in the tunnel stratum; others, for example, the influence of tunnel leakage on subway, the differential settlement of tunnel and working wells and station joints, and etc.

6.4.5 Differential Settlements of Running Tunnels Figure 6.33 illustrates the variations of differential settlements of running tunnels with time between April 1999 and June 2007. The running tunnel between Changshu Road and South Shanxi Road experiences the minimum differential settlements. The differential settlements of the running tunnel between Xujiahui and Hengshan Road, the running tunnel between Hengshan Road and Changshu Road and the running tunnel between South Hongpi Road and People’s Square Station are more than 100 mm. Along Subway Line 1 in Shanghai, the settlements in some places were serious and the larger differential settlements can affect the safe operation of the subway. In the construction stage, grouting can enhance the stability of tunnel structure and waterproofing performance can effectively control the settlement. Shield tunnel can resist the soil liquefaction with the second grouting and reduce the liquefaction-induced settlement. The tunnel face reinforcement technology not only can reduce the face deformation, but also reduce the ground settlement. In the lining of the installation process, through setting deformation joint and flexible joints and increasing the bolt strength and quantity, the tunnel stiffness was strengthened and the settlement was reduced.

6.5 Prediction of Long-Term Settlement of Subway Shield Tunnel

171

60 Xinzha Road-Hanzhong Road South Huangpi Road-People's Square Station South Shanxi Road-South Huangpi Road Changshu Road-South Shanxi Road Hengshan Road-Changshu Road Xujiahui-Hengshan Road

40

Differential settlement (mm)

20 0 -20 -40 -60 -80 -100 -120 -140 -160

0

20

40

60

80

100

120

140

160

Time (month) Fig. 6.33 Variations of differential settlements of running tunnels with time between December 1995 and June 2007

6.5 Prediction of Long-Term Settlement of Subway Shield Tunnel 6.5.1 Prediction of Long-Term Settlement by Cubic Curve Fitting According to the observational data of Shanghai subway long-term longitudinal settlement, the Shanghai subway long-term longitudinal settlement curve is induced as a cubic parabola. S = ax 3 + bx 3 + cx + S0

(6.12)

where S stands for Settlement (mm); x stands for horizontal distance (m); S 0 stands for the settlement on the starting point of segmentation settlement curve (mm); a, b and c are parameters. Due to the difference of the constraint conditions, the interval tunnel segmentation settlement stimulated curves are divided into the station joint type and the free boundary type, their rules are researched respectively, and Table 6.3 is the segmentation simulation results by adopting cubic curve of Line 1 typical tunnel inhomogeneous settlement. Table 6.3 lists results of curve simulation for the Line 1 tunnel sections producing obvious longitudinal inhomogeneous deformation when the settlement is between

− 0.0039 0.0343

0.0008 − 0.0001 − 0.0034

− 1.00E−05

− 1.00E−05

1.00E−05

9653.7 ~ G

G ~ 10,028.5

0.0346

0.1749

− 0.0812

8835.4 ~ 8932.7

0.0023 − 0.0014

− 2.00E−06

4.00E−06

0.4541

8244.7 ~ F

− 0.011

8.00E−05

E ~ 7699.1

− 0.4403

0.0571

− 0.0027

0.7258

− 27.3

− 31.6

− 17.1

− 26.5

− 21.6

− 27.9

− 14.4

6.0

6.6

− 5.2

− 13.1

0.9

− 0.9762 − 1.4973

− 13.6

6.2

− 14.2

20.9

− 23.3

− 12

S0

0.0592

F ~ 8733.2

− 0.0017 0.0102

3.00E−06

− 5.00E−05

D ~ 5111.8

5087.1 ~ 5191.9

− 9.00E−05

− 1.00E−06

3252.2 ~ C

0.0326 − 0.0168

− 2.00E−04

1.00E−04

3002.7 ~ 3100.0

3075.7 ~ 3152.2

0.0038

1.00E−05

2853.5 ~ 3001.7

0.0167

− 0.0007 0.0083

4.00E−06

− 8.00E−05

1866.0 ~ B

B ~ 2375.2

0.16

0.0008

− 8.00E−06

1710.7 ~ 1866.0

0.0816 − 0.3418

0.0018 0.0068

− 4.00E−05

− 3.00E−05

1366.8 ~ 1453.2

− 0.3403

c

1584.4 ~ 1702.0

− 8.00E−04

b

1.00E−05

a

A ~ 1366.5

Uplink

Mileage (m)

Table 6.3 Segmentation curve simulation parameters of Subway Line 1

0.993

0.9238

0.7854

0.9765

0.9805

0.9856

0.9983

0.9975

0.9264

0.9997

0.9476

0.876

0.9996

0.9422

0.9914

0.9043

0.9474

0.9742

Correlation coefficient square R2

73.4

72.7

97.3

162.4

70.3

74.3

104.8

128.2

176.1

76.5

97.3

148.2

73.1

187.4

155.3

117.6

86.4

167.1

(continued)

Sphere of influence (m)

172 6 Long-Term Settlement of Subway Shield Tunnel

0.0043 4.00E−05

− 4.00E−05

− 2.00E−06

E ~ 7690.1

8197.8 ~ F

− 26.5

− 23.7

− 7.7

− 9.1

5.3

− 3.3

− 7.2

− 10.8

− 15.1

− 19.8

− 27.4

− 9.5

S0

0.989

0.9844

0.8166

0.9769

0.927

0.9397

0.9995

0.993

0.9235

0.9916

0.9945

0.9977

Correlation coefficient square R2

149.0

80.9

147.6

128.0

95.2

128.7

53.1

146.0

197.3

85.5

102.2

168.9

Sphere of influence (m)

A tunnel hole, B Caobao Road Station, C Shanghai Gymnasium Station, D Xujiahui Station, E Changshu Road Station, F Shanxi South Road Station, G Huangpi South Road Station

0.1165

− 0.0547

− 0.277

− 5.00E−05

1.00E−05

5022.2 ~ 5169.8

0.2687 − 0.7173

− 0.026 0.015

2.00E−04

− 6.00E−05

− 0.2498

2852.4 ~ 2947.6

0.0027

− 5.00E−06

2219.9 ~ 2348.6

− 0.5995

0.1258

0.1908

2998.2 ~ 3126.2

− 0.0001 0.0145

− 2.00E−06

− 7.00E−05

1799.2 ~ B

B ~ 2194.3

− 0.0003

− 2.00E−06

1701.6 ~ 1898.9

− 0.506

0.5757

− 0.0059 0.0095

9.00E−06

− 2.00E−05

1339.3 ~ 1441.5

− 0.335

c

− 0.0007

b

1441.5 ~ 1527.0

1.00E−05

a

A ~ 1364.3

Downlink

Mileage (m)

Table 6.3 (continued)

6.5 Prediction of Long-Term Settlement of Subway Shield Tunnel 173

174

6 Long-Term Settlement of Subway Shield Tunnel

− 31.6 to 6.6 mm (the tunnel sections with small inhomogeneous settlement are not included), in which the values of correlation coefficient r on most conditions can keep a high level, illustrating the validity of segmentation simulation. The data in column Sphere of Influence showed, the sphere of influence of the tunnels producing obvious inhomogeneous longitudinal settlement is between 70 to 180 m. Adopting cubic curve equation, the simulation curve have three parameters, a, b and c, and an initial boundary value S0, in which S0 is a constant term, unrelated to the tunnel longitudinal inhomogeneous settlement, while parameters a, b and c determine the curvature.

6.5.2 Prediction of Long-Term Settlement by GM (1, 1) and ARMA (N, M) (1) Introduction of GM (1, 1) and ARMA (n, m) Grey system theory is a kind of theory that study grey system analysis, modeling, prediction, decision and controlling. Grey prediction is predicting to grey system, common methods (e.g. regression analysis and experimental fitting etc.) of prediction rely on huge specimens. If the specimens are less, the result of prediction would be in accuracy. Grey prediction model that is built by little and incomplete information and it is an efficient tool to make prediction with less specimens. Grey prediction model is accuracy and convenient, so it is widely used in many fields. And the data of grey prediction model could be dynamic; it could be adjusted according to new specimen data. In grey prediction model, p in GM (p, q) is stand for number of differential and q is stand for number of variable (Liu et al. 2005). Appendix I offers more details about grey model. According to the definition of Grey system, a technical system disturbed by noise is a grey system with a physical type. Soil under subway tunnel is in accord with this definition. Because of similar geological and mechanics condition, to a degree, the settlement in a certain point of Subway Line always exhibits some regularity in time. However, the settlement of subway is influenced by many factors. Some of the factors are accidental and irregular such as construction and earthquake. Therefore, subway settlement has a certain degree of randomness. The result of GM (1, 1) is a common exponent sequence. Its effect is monotone increasing or decreasing, while accidental factors and long-term secondary consolidation could not be effectively predicted. Considering this situation, we introduced ARMA (n, m) model to compare with GM (1, 1). Time series analysis is a branch of mathematical statistics; it is one of the most important mathematical tools to study random process. In many fields such as social science, natural science, management and engineering, a number of conditions in actual issues are random, and have statistical regularity to some extent. Consisting of AR (n) model and MA (m) model, ARMA model (Auto-Regressive and Moving Average Model) is a kind of time series model predicting and analyzing

6.6 Prediction of Settlement Trough of Subway Shield Tunnel

175

time series data. The accuracy of the model is very high on short-term prediction. The basic idea is: some time series are random variables depended on time. Although a single data is uncertain, the whole time series exhibits regularity. It could be described by mathematical model or expressed by the previous data. In the principle of minimizing variance, we could obtain the best prediction result. According to the above, ARMA (n, m) is very suitable to subway tunnel settlement prediction. Appendix II offers more details about ARMA (n, m) model. It should be pointed out that the meanings of the numeral suffix in brackets in ARMA (n, m) model and GM (p, q) are different. (2) Application of GM (1, 1) and ARMA (n, m) model With the data shown in Fig. 6.22, the model was built. There are 24 series of data, collected in 12 years. Utilizing the first 20 data (1996–2005) as the raw data of modeling, the last 4 series (2005–2007) of data were to contrast with the prediction value. With the help of the software of matlab and eviews, grey model and ARMA model could be built. Figure 6.34 demonstrates the relationship among monitoring value, GM (1, 1) prediction value and ARMA prediction value. It can be seen that because of limited raw data and the third platform period after 2005, the difference between GM (1, 1) prediction value and the monitoring value is obvious. GM (1, 1) is a kind of exponential function; the increasing trend is gradually accelerated as the dependent variable increases. Compared with GM (1, 1), the ARMA (p, q) is more suitable for model predicting, because the prototype of GM (1, 1) is a component function, the first derivative is monotone increasing and not accord with the law of settlement. Besides, the parameter of the ARMA model could be adjusted by the raw data, the model is more flexible than grey model.

6.6 Prediction of Settlement Trough of Subway Shield Tunnel 6.6.1 Settlement Behavior of Subway Tunnels Shanghai Subway Line 1 started official operation in 1995. The entire line went from Xinzhuang Station to Fujin Road Station. As shown in Fig. 6.35a, there has been substantial settlement from 1995 to 2009 and the average accumulated settlement of reaches 111.5 mm at the end of 2009. The maximum settlement in the longitudinal direction is 292.9 mm and the minimum settlement is 0.4 mm. The maximum differential settlement is 292.5 mm. The average differential settlement reaches 20.6 mm in the past 14 years. The differential settlement gradually increases during the long-term settlement of subway tunnel in soft ground. There are two significantly large settlement troughs, located around the Hengshan Road Station and between South Huangpi Road Station and Xinzha Road Station respectively.

176

6 Long-Term Settlement of Subway Shield Tunnel

Accumulated Settlement Value (mm)

0

ARMA prediction value GM(1,1) prediction value Monitoring value

-200

-400 1995

2000

2005

2010

Time (year) (a) Huangpi Road Station-People's Square Station maximum

Accumulated Settlement Value (mm)

0

ARMA prediction value GM(1,1) prediction value Monitoring value

-100

-200

-300

-400 1995

2000

2005

Time (year) (b) Xinzha Road-Hanzhong Road Station maximum Fig. 6.34 Settlement prediction result

2010

6.6 Prediction of Settlement Trough of Subway Shield Tunnel

ARMA prediction value GM(1,1) prediction value Monitoring value

0

Accumulated Settlement Value (mm)

177

-100

-200 1995

2000

2005

2010

Time (year) (c) Huangpi Road Station-People's Square Station minimum

ARMA prediction value GM(1,1) prediction value Monitoring value

Accumulated Settlement Value (mm)

0

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Time (year) (d) Xinzha Road-Hanzhong Road Station minimum Fig. 6.34 (continued)

2010

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6 Long-Term Settlement of Subway Shield Tunnel A

50

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Accumulated settlement (mm)

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In 1999 In 2009

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A-Caobao Road Station B-Shanghai Stadium Station;

-250

C-Xujiahui Station D-Hengshan Road Station; E-Changshu Road Station F-South Shanxi Road Station;

-300

G-South Huangpi Road Statio H-People’s Square Station; I-Xinzha Road Station J-Hanzhong Road Station K-Shanghai Railway Station

-350

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-120 In 2009

-140 -160 -180 10000

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(b) Subway Line 2 (2000-2009)

Fig. 6.35 Accumulated settlement of subway tunnels in Shanghai

24000

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6.6 Prediction of Settlement Trough of Subway Shield Tunnel

B

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179

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-140 -160

G-Zhenping Road Station H-Zhongtan Road Station I-Shanghai Railway Station

0

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Accumulated settlement (mm)

6

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4 2 0

In 2009

-2 -4 -6 2000

A-Shiguang Road Station B-Nenjiang Road Station C-Xiangyin Road Station D-Huangxing Park Station E-Middle Yanji Road Station

3000

4000

5000 Mileage (m)

6000

7000

8000

(d) Subway Line 8 (2008-2009)

Fig. 6.35 (continued)

Shanghai Subway Line 2 started operation in 2000. The whole line went from East Xujing Station to Pudong International Airport Road Station. Figure 6.35b illustrates the accumulated settlement from 2000 to 2009. The average accumulated settlement reaches 65.5 mm. The settlement in the sections from the East Nanjing Road Station to the People’s Square Station and from the Century Avenue Station to the Lujiazui Station is relatively larger. The maximum accumulated settlement reaches 163.3 mm. The settlement in the section was relatively small from the Science Museum Station to the Century Park Station. The minimum accumulated settlement is − 8.48 mm, which means that the surface has been raised. Although there are several settlement troughs along Subway Line 2, their sizes are not obvious compared with these along Subway Line 1.

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6 Long-Term Settlement of Subway Shield Tunnel

Shanghai Subway Line 3 started operation in 2000. Figure 6.35c illustrates the accumulated settlement from the Longcao Road Station to the Shanghai Railway Station. There are two relatively large subsidence areas, one of which is from the Zhongshan Park Station to the Caoyang Road Station, and the other is from the Zhongtan Road Station to the Shanghai Railway Station. The maximum accumulated settlement reaches 93.6 mm. The minimum accumulated settlement is 2.4 mm. The maximum differential settlement is 91.2 mm in this section. The densely populated areas experience larger settlement, such as the People’s Square Station and the Shanghai Railway Station. Compared with Subway Lines 1 and 2, there exists a settlement trough at the Jinshangjiang Road Station of Subway Line 3. Shanghai Subway Line 8 started operation in 2007. Figure 6.35d shows the accumulated settlement from 2008 to 2009. The maximum accumulated settlement reaches 5.11 mm in the past 1 year. There is a settlement trough between the mileage 6000 and 7000 m. According to the analysis of Subway Lines 1, 2, 3 and 8, all subway tunnels experienced the significant longitudinal settlement and the differential settlement. With the increasing of the operation time, the accumulated settlement increased continuously and subsequently tended to converge. The accumulated settlement of Subway Line 1 was the largest due to the long operation time, while Subway Line 8 was the smallest. The differential settlement was relatively large where there was a settlement trough, which would cause subway safety problem.

6.6.2 Settlement Trough of Subway Tunnels The subway tunnels in the soft soil area easily generated the settlement. In order to accurately predict the settlement of settlement troughs, improve the safety of subway operation and reduce the cost of long-term monitoring, it is necessary to study settlement regularity about settlement troughs in the soft deposit. The Shanghai Subway Line 1, with long operation time and abundant measured data, contributes to analyze the settlement trough in the soft area. (1) Characteristics of settlement trough The bottom and internal walls of the settlement trough are respectively called the trough bottom and the trough wall respectively. The settlement trough is divided into the left settlement trough and the right settlement trough according to the trough bottom as shown in Fig. 6.36. One settlement trough is different with another and each settlement trough is usually asymmetric. For example, the settlement trough between Xujiahui Station and Changshu Road Station and the settlement trough between South Shanxi Road Station and South Huangpi Road Station have different sizes and both of them are asymmetric. The whole subway tunnel experienced settlement and the settlement is different at different mileages by comparing the accumulated settlement in 1999 and 2009. The settlement where there exists a settlement trough is more obvious. The accumulated settlement is relatively large because there is

6.6 Prediction of Settlement Trough of Subway Shield Tunnel

181

Trough wall

Left

Right

Trough bottom Fig. 6.36 Schematic diagrams of settlement trough

a settlement trough between Xujiahui Station and Changshu Road Station, while the accumulated settlement was relatively small because there was not a settlement trough between Changshu Road Station and South Shanxi Road Station. Figure 6.37 shows the cumulative settlement difference between Xujiahui Station and Changshu Road Station from 1999 to 2007. The difference at the groove bottom is 112.6 mm and the minimum difference in the trough wall is 31.9 mm, which shows the settlement rate in groove bottom was far greater than groove wall. The accumulated settlement difference presents the downward parabolic and the maximum value is at the bottom of the settlement trough. Figure 6.38 illustrates the settlement of the settlement trough with time at Hengshan Road Station, which is overall relatively smooth except some positions. For example, the settlement is relatively large at mileage 6200 m, while it is relatively small at mileage 6470 m. The shape of settlement trough does not change with time. The settlement trend of the settlement trough is positively correlated with the settlement at the trough bottom. The settlement trough interval produces settlement when the groove bottom occurs settlement. At the same time, the settlement trough interval produces uplift when the groove bottom occurs uplift. The settlement trend of settlement trough may not converge to stable value. The settlement trend of settlement trough is convergent between 1999 and 2003, but the settlement trend of settlement trough is irregular. The settlement trough results from the complex causes including subway vibration load, the engineering activities around the subway tunnel, the regional land subsidence and the changing of underground water during the operations of subways. The soil mass is subjected to different degrees of disturbance in different mileage during the subway construction. The disturbed soil mass during the construction period is

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6 Long-Term Settlement of Subway Shield Tunnel

120 110

Settlement difference (mm)

100 90 Maximum settlement point

80 70 60 50 40 30 5600

5800

6000

6200 6400 Mileage (m)

6600

6800

7000

Fig. 6.37 Accumulated settlement difference from 1999 to 2007

Accumulated settlement (mm)

0 1999 2000 2001 2002 2003 2004 2005 2006 2007 2009

-50 -100 -150 -200

-250 5700 5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800 6900 7000 7100 7200

Mileage (m) Fig. 6.38 Variations of accumulated settlement with time at Hengshan Road Station

prone to settlement in the operation period. The local section of the subway tunnel undergoing engineering activities has larger settlement and the groundwater leakage occurs. The groundwater leakage results in the decrease of pore water pressure, the increase of effective stress and the soil mass consolidation (Shin et al. 2002; Mair 2014). The process groundwater seepage is accompanied by the sediment leakage,

6.6 Prediction of Settlement Trough of Subway Shield Tunnel

183

which aggravates the deformation of subway tunnels (Wu et al. 2014). Because the groundwater leakage is local, the settlement trough is easily formed. (2) Seasonal settlement Figure 6.39 illustrates the comparison of the settlement of settlement trough at Hengshan Road station in the first half year with that in the second half year from 2000 to 2007. The settlement in the second half year is obviously larger than that in the first half year and the settlement of subway tunnel is related to the seasonal rain. Shanghai, which is subtropical monsoon climate, is hot and rainy in summer, cold and dry in winter. The water table rises in summer, while in the rest of seasons the water table falls. The groundwater lowering results in the decrease of pore water pressure, the increase of effective stress and soil mass consolidation. The rate of soil mass consolidation settlement decreases with time and the settlement velocity in the first half year is greater than in the second half year. Similarly, the settlement in the first half year of 2005 is heavy than the second half year. In 2003 and 2007, the settlement in the first half year is basically below the second half year. The difference may be caused by engineering activities. The characteristics of the settlement trough along the tunnel are as the following: the interval where there is a settlement trough experiences larger settlement in the later period; the settlement of settlement trough is overall relatively smooth except some positions and the settlement of the settlement trough is influenced by the bottom of the trough; the long-term settlement of subway tunnels is related to the seasonal rain. The rate of settlement of the settlement trough is far greater than that of the other sections.

6.6.3 Prediction Model Building Based on the in-site monitoring data the characteristics of long-term settlement of subway tunnels were analyzed and the settlement prediction model of the settlement trough were built. The overall trend of the settlement trough is settlement, but it is difficult to determine the uplift or settlement of the settlement trough at the next moment. By the in-site monitoring data of the trough settlement, the settlement of settlement trough depends on that at the bottom of settlement trough. The maximum sedimentation value of the bottom is monitored in the prediction process. Detailed steps of building the prediction model are as follows. (1) Data processing The in-site monitoring data of the left settlement trough and the right settlement trough are carried out the normalization processing, respectively. (2) Fitting the settlement trough The curve function for fitting the normalized data of the left settlement trough is called f 1 (x), and that of the right settlement trough is called f 2 (x), where x is the normalized Euclid distance and f (x) is the normalized settlement.

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6 Long-Term Settlement of Subway Shield Tunnel

1 0

2000.04 2000.11

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Mileage (m) (a) 2000

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2001.04 2001.11

0

Settlement (mm)

-2 -4 -6 -8 -10 -12 -14 -16 -18 5600

5800

6000

6200

6400

Mileage (m) (b) 2001

Fig. 6.39 Seasonal settlement of Subway Line 1 at Hengshan Road station from 2000 to 2007

6.6 Prediction of Settlement Trough of Subway Shield Tunnel

185

3 2 2002.03 2002.11

1 0

Settlement (mm)

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Mileage (m) (c) 2002

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6000

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6400

Mileage (m) (d) 2003

Fig. 6.39 (continued)

186

6 Long-Term Settlement of Subway Shield Tunnel -1 -2 -3

Settlement (mm)

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Mileage (m) (e) 2004

4 3 2005.06 2005.12

2

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1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 5600

5800

6000

6200

6400

Mileage (m) (f) 2005

Fig. 6.39 (continued)

6.6 Prediction of Settlement Trough of Subway Shield Tunnel

187

4 2 0

Settlement (mm)

-2 -4 -6

2006.06 2006.12

-8 -10 -12 -14 -16 -18 5600

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(g) 2006

6 2007.06 2007.12

Settlement (mm)

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4

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1

0 5600

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6200 6400 Mileage (m) (h) 2007

Fig. 6.39 (continued)

6600

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6 Long-Term Settlement of Subway Shield Tunnel

(3) Determining the control point of the function According to the number of parameters in the curve function, the least squares algorithm is used to solve the curve parameters. The normalized data is substituted into the function f 1 (x) and f 2 (x) to get its predictive value, respectively. If the predicted value is different from the monitoring value, the number of input monitoring data is changed to resolve the parameters of the equation, and the error is minimal as soon as possible. The final number input monitoring data is called the control point of the settlement trough curve function. (4) Predicting the accumulated settlement about the control point at the next time by ARIMA (p, d, q) model ARIMA (p, d, q), which is used to predict the timed series, is the Autoregressive Integrated Moving Average Model, where p is the order number of AR model and q is the order number of MA model and d is the difference times. ARIMA (p, d, q) model can be used to predict the cumulative settlement of the next moment of the control point. By compared the monitoring value and the predicted value at the center of the settlement trough, the forecast results are treated according to the difference between the monitoring value and the predicted value. (5) Solving the parameters of curve function According to the accumulated settlement of the control point at the next moment, the least squares algorithm is used to solve the curve function parameter. The obtained parameter values are substituted into the curve function. (6) Predicting the settlement of the whole settlement trough The curve function is used to predict the uneven settlement of the whole settlement trough. The desired predictions are anti-normalization and compared with the actual values.

6.6.4 A Case Study The settlement trough in Hengshan Road station was used to model verification. According to the prediction model, the settlement curve of the left settlement trough and the right settlement trough can be obtained: f 1 (x) = f 2 (x) = a1 sin(b1 x + c1 ) + a2 sin(b2 x + c2 ) + a3 sin(b3 x + c3 ) + a4 sin(b4 x + c4 )

(6.13)

where a1 , b1 , c1 , a2 , b2 , c2 , a3 , b3 , c3 , a4 , b4 and c4 are the fitting parameters. By comparing the monitored data with the predicted data, the prediction performance at the bottom of the settlement trough is superior to that at other locations and prediction precision of the right settlement trough is better than that of the left settlement trough, as shown in Fig. 6.40. Figure 6.41 shows the prediction precision

6.6 Prediction of Settlement Trough of Subway Shield Tunnel

189

-40

Accumulated settlement (mm)

-60

Measured data Predicted data

-80 -100 -120 -140 -160 -180 -200

-220 -700 -600 -500 -400 -300 -200 -100

0

100 200 300 400 500 600 700

Distance from the center of settlement trough (m)

Fig. 6.40 Comparison of the measured accumulated settlement with the predicted one of settlement trough

1.00

Predicted precision

0.98 0.96 0.94 0.92 0.90 0.88

0

9

18

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36

45

54

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72

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Predicted point Fig. 6.41 Prediction precision of the whole settlement trough

from left to right of the settlement trough, which first increases and then decreases, ranging from 0.9 to 1 except the first point. It suggests the effect for internal wall of settlement troughs decreases with the increase of distance from the trough bottom.

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6.7 Conclusions This chapter conducted theoretical calculations of the long-term settlement of the shield tunnel and the settlement trend was predicted by GM (1, 1) and ARMA (n, m) model. The characteristics of the long-term settlement and the settlement trough were studied. The results are as follows. (1) The results of the theoretical analysis agree well with these of the centrifuge model test, especially the layer-wise summation method. It presents uniform settlement in the longitudinal direction of tunnel in soft ground. The tunnel settlement is continuous year by year, and remains stable at last. The rate of settlement decreases gradually with time which is quite large in the first five years. (2) The results of the numerical simulation are highly consistent with these of the elastic foundation beam theory, indicating that the simulation of the tunnel settlement is practical. The vertical and the horizontal displacement are symmetrical because of the single symmetrical load. The vertical displacement decreases gradually with the depth, while the horizontal displacement is smaller around the shield tunnel. The tunnel deformation has a trend to ellipse, which has also been found in the monitoring data. (3) Through the establishment of three-dimensional model, the mechanical deformation of the shield tunnel and the surrounding soil are analyzed. The displacement of the soil above the shield tunnel is downward and the displacement of the soil under the shield tunnel is upward. The displacement of soil around the tunnel is gradually decreased. The deformation of tunnel is consistent with the surrounding soil. The displacement is downward at the top of tunnel, while the displacement is upward at the bottom of tunnel. The tunnel has a trend to have a shape of ellipse. (4) The vertical displacement of the four paths is affected by the nature of the soil. In the four paths, the settlement of the clay is larger than that of the sand. In the distance of 40 m, namely, 10 m away the sand and clay interface the displacement peak generated. In the 21 excavation cycle, differential settlement in the front of the tunnel face is significant, the settlement rate of the clay increases at first and then decreases, finally to stable. The settlement of the sand has the similar rule. (5) The horizontal displacement is smaller than the vertical displacement, horizontal displacement of the clay is larger than that of the sand. At the same depth of underlying soil, the distribution of the vertical stress is not uniform and the interface of the sand and clay has significant difference. The vertical stress of the sand is slightly larger than that of the clay. The distribution of the pore pressure is not uniform around the tunnel. The pore pressure at the interface has not obvious difference. The pore pressure of the sand is larger than that of clay at the same depth of underlying soil.

References

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(6) Through analyzing the in-site monitoring data of settlements of Subway Line 1 in Shanghai, we found the settlement of tunnel is obviously inhomogeneous. The settlements of subway stations and running tunnels do not show obvious regularity. Along Subway Line 1, some places experience larger settlements and the running tunnels experience significant differential settlements. In the soft soil area, the causes resulting in the settlement of subway tunnel are very complex. (7) The methodology of fitting subway settlement by the cubic multinomial is accurate than other methods. The cubic spline function contradicts the elastic foundation beam theory. It could not show the mechanism of the longitudinal deformation of the subway. In the field of prediction, the accuracy of ARMA model is higher than the GM (1, 1) model. ARMA model is suitable for long-term prediction of settlement, while the GM (1, 1) is suitable for short-term prediction of settlement. Maybe the dynamic grey model can improve the prediction accuracy, and this is the future task to analyze. (8) The differential settlement in longitudinal direction is significant, particularly in the settlement trough whose shape has little change over time. The differential settlement is relevant to the seasonal precipitation and the settlement in the first half year is basically below the second half year. The control point fitting method and ARIMA (p, d, q) model could be used to predict the settlement of the settlement trough. The control point fitting method is able to consider spatial factors, and ARIMA (p, d, q) considers the time factor, which effectively improves the prediction accuracy.

References Clayton CRI, Theron M, Best AI (2004) The measurement of vertical shear-wave velocity using side-mounted bender elements in the triaxial apparatus. Géotechnique 54(7):495–498 Cui ZD, Tan J (2015) Analysis of long-term settlements of Shanghai Subway Line 1 based on the in situ monitoring data. Nat Hazards 75(1):465–472 Franzius JN, Potts DM, Burland JB (2005) The influence of soil anisotropy and K0 on ground surface movements resulting from tunnel excavation. Géotechnique 55(3):189–199 ITA (2000) Guidelines for the design of shield tunnel lining. Tunn Undergr Space Technol 15(3):303–331 Lee KM, Rowe RK (1989) Deformations caused by surface loading and tunnelling: the role of elastic anisotropy. Géotechnique 39(1):125–140 Lee KM, Ji HW, Shen CK, Liu JH, Bai TH (1999) Ground response to the construction of Shanghai metro tunnel-line 2. Soils Found 39(3):113–134 Liu GB, Ng CWW, Wang ZW (2005) Observed performance of a deep multistrutted excavation in Shanghai soft clays. J Geotech Geoenviron Eng ASCE 131(8):1004–1013 Mair RJ (2008) Tunnelling and geotechnics: new horizons. Géotechnique 58(9):695–736 Ng CWW, Lee KM, Tang DKW (2004) Three-dimensional numerical investigations of new Austrian tunnelling method (NATM) twin tunnel interactions. Can Geotech J 41(3):523–539 Peck RB (1969) Deep excavations and tunneling in soft ground. In: Proceedings of the seventh international conference on soil mechanics and foundation engineering. State of the Art Volume, Mexico City, pp 225–290

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Qing L (2013) Long-term settlement mechanisms of shield tunnels in Shanghai soft clay. Ph.D. Thesis, Hong Kong University of Science and Technology, Hong Kong Shin JH, Addenbrooke TI, Potts DM (2002) A numerical study of the effect of groundwater movement on long-term tunnel behaviour. Geotechnique 52(6):391–403 Shirlaw JN (1995) Observed and calculated pore pressure and deformations induced by an earth balance shield: discussion. Can Geotech J 32(1):181–189 Tan J, Cui ZD, Yuan L (2015) Study on the long-term Settlement of Subway tunnel in soft soil area. Mar Georesour Geotechnol 34:486–492 Wang YH, Mok CB (2008) Mechanisms of small-strain shear-modulus anisotropy in soils. J Geotech Geoenviron Eng ASCE 134(10):1516–1530 Wu HN, Shen SL, Chai JC et al (2014) Evaluation of train-load-induced settlement in metro tunnels. Proc Inst Civil Eng Geotech Eng 168(5):396–406 Yu SS, Shi ZJ, Xie JP, Feng WL (1986) Analysis of permanent deformation for tunnels of Shanghai underground. Earthq Eng Eng Vib 6(1):51–60 Zhang Z, Huang M, Zhang M (2011) Theoretical prediction of ground movements induced by tunnelling in multi-layered soils. Tunn Undergr Space Technol 26(2):345–355

Chapter 7

Conclusions and Prospects

7.1 Conclusions In this monograph, the interaction between the soil foundation and the subway shield tunnels under the vibration loading were studied, including the dynamic response of the track inside the shield tunnel, the dynamic properties of soil around the subway shield tunnel, the mechanical properties of subway tunnel and the long-term settlement of the subway tunnel. The results are as follows. (1) The infinite Euler–Bernoulli double-beam system resting on a viscoelastic foundation and subjected to a harmonic moving point load is investigated and parametric study is carried out. The governing equations are solved by utilizing Fourier transform and residual theory. Numerical results of an example are presented and then cut-off frequencies and critical velocities are calculated. Parametric study mainly researches the influence of the beam parameters, including load frequency, load velocity, bending stiffness, elastic coefficient and damping coefficient. It is concluded that two cut-off frequencies and two critical velocities exist in this double-beam system. The rail has consistently higher displacement response than the slab. Load frequencies and velocities have great influence on the both beams. Displacements tend to be stable with respect to load velocities at high load frequency. The distribution of the displacements along the beam under a static load is symmetric. In addition, the results indicate that the Doppler Effect is significantly obvious as the load velocity exceeds critical velocities. The bending stiffness of the rail and the slab affects both the amplitudes and the phases of the distribution curves of displacements. However, the two bending stiffness exhibits opposite effects on the amplitudes. For the parameters of the bearings, the elastic coefficient of the rail bearing only have influence on the rail response but no effects on the slab. While the elastic coefficient of the slab bearing affects both the two beams considerably. The damping coefficients exhibit similar effects on the double-beam system.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Z.-D. Cui, Interaction Between Soil Foundation and Subway Shield Tunnel, https://doi.org/10.1007/978-981-99-6870-1_7

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(2) The track irregularity power spectral density and a train-track coupled model were conducted for simulating the excitation load generated by moving subway on the unevenness rail, compared with the moving axle load. The exciting force has a significant influence on the vibration acceleration compared with the axle load. There is a cluster of significant vibration acceleration peak values induced by characteristic lengths of subway train. Ground vibration acceleration peak values decrease with the distance from the track centerline increasing which are smaller than ballast and lining acceleration and ground acceleration reaches the peak time in a noticeable delay compared with tunnel structures due to the gradual vibration attenuation in the process of wave propagation. Acceleration spectrum analysis shows that it is mainly composed of high frequencies and accompanied with some low frequency components of smaller amplitudes from the ballast to the bottom of lining and the dominating vibration frequency at ground mostly focused on low frequency caused by the center distance of two neighboring cars of the subway train. With the distance from the track centerline increasing, the attenuation of ground vibration acceleration peak values and acceleration level can be expressed by 3 order polynomial. The results given in this monograph are valuable for validating the ground vibration induced by subway travelling load with excitation of the track vertical profile irregularity. (3) The dynamic characteristics of soft clay were studied by the dynamic triaxial apparatus. Five test schemes were designed considering the load frequency and the confining pressure. When the amplitude of dynamic stress is small, the hysteresis loop is basically consistent and closed. When the amplitude of dynamic stress is large, each hysteresis loop is no longer closed and crosses forward, and the larger the amplitude of dynamic stress, the more obvious this effect. Low frequency loads act longer than high frequency loads, resulting in greater strain. The higher the vibration frequency, the smaller the plastic strain caused by the soil, and this effect becomes more and more obvious as the load frequency increases. Increasing the confining pressure can reduce the damping value and weaken the dissipation of energy propagation in the soil. With the increase of dynamic shear strain, the dynamic Shear modulus decreases rapidly in the early stage, and then decreases slowly. The damping ratio increases rapidly with the increase of dynamic shear strain in the early stage, and the increasing trend gradually slows down in the later stage. (4) The dynamic finite difference models were conducted to analyze the interaction between the around soil and the shield tunnel. With the dynamic time increasing, the vertical displacement increases to the first peak within 1 s. Then, it decreases and increases to the second peak within 2 s. After 2 s, it reaches a dynamic balance. The cumulative vertical displacement of points below the tunnel decreases with the depth increasing; on the contrary, the displacement of points above the tunnel increases. It can be inferred that the soil layers below the shield tunnel are in the compression state and the soil layers above the tunnel are in the extrusion state, and turn uplift. The cumulative horizontal displacement increases with the dynamic time but the increasing rate decreases. The greater

7.1 Conclusions

195

the distance is, the larger the displacement is. At the same time, the zero horizontal displacement appears at − 14 m from the surface which is at the subway tunnel center. The vertical stress of the points below tunnel decreases with the depth increasing; on the contrary, the stress of points above tunnel increases. It is verified that the propagation of stress wave is gradually decreasing around tunnel. With the time increasing, the acceleration and velocity change up and down and approach to dynamic balance. Both the acceleration and the velocity of points under the tunnel decrease with the depth increasing. Dynamic respond of lining induced by train shows tension states in the outer side and compression states in the inner side near 90° (Point C) according to the bending moment and the shear force. Furthermore, the maximum bending moment and the maximum shear force are not located at the same point. The speed of the train and the interface have an impact on the dynamic behavior of soft soils. Merely, the effect of speed is small. (5) The mechanical properties of the functionally graded lining were researched through theoretical analysis and numerical simulation. The single factor test shows that the radial displacement of the lining axis decreases with the increase of a and b, but the deformation mode remains the same, and the reduction of deformation is smaller and smaller. In addition, with the increase of a and b, the distribution of moment remains the same. With the increase of a, the positive section moment increases linearly and the negative bending moment decreases linearly. With the increase of b, the negative moment increases linearly and the positive moment decreases linearly, which embodies the idea of “flexible yielding”. The displacement of the lining axis and the section moment change linearly with the increase of λ. with the increase of λ, the shape of the lining changes significantly, which shows that the side with large lateral pressure deforms to the inside, and the side with small lateral pressure expands to the outside. The orthogonal test shows that when the maximum deformation occurs at 0◦ , the parameter a should be larger than b. When the maximum deformation occurs at 90◦ , the parameter b should be larger than a, so as to save material cost on the premise of ensuring safety. In addition, the lateral pressure coefficient has a great impact on the safety of the structure, which exceeds the influence of structural parameters on the safety of the structure. The numerical simulation shows that the calculation model of internal force and deformation is suitable for the cylinder with t/R ≤ 0.2. There is a certain gap between the theoretical analysis and numerical simulation, but the gap between the theoretical analysis and numerical simulation results is within 8% at 0◦ and 90◦ with large displacement. In addition, the conclusion of theoretical analysis is verified. There is little difference in deformation between the two kinds of functional graded linings, but there is a big difference in circumferential stress. It can be seen that compared with Function I, Function II has some advantages in reducing the maximum deformation of the structure, but the advantages are relatively low. (6) This chapter conducted theoretical calculations of the long-term settlement of the shield tunnel and the settlement trend was predicted by GM (1, 1) and ARMA (n, m) model. The results of the theoretical analysis agree well with these of the

196

7 Conclusions and Prospects

centrifuge model test, especially the layer-wise summation method. It presents uniform settlement in the longitudinal direction of tunnel in soft ground. The tunnel settlement is continuous year by year, and remains stable at last. The rate of settlement decreases gradually with time which is quite large in the first five years. The results of the numerical simulation are highly consistent with these of the elastic foundation beam theory, indicating that the simulation of the tunnel settlement is practical. The vertical and the horizontal displacement are symmetrical because of the single symmetrical load. The vertical displacement decreases gradually with the depth, while the horizontal displacement is smaller around the shield tunnel. The tunnel deformation has a trend to ellipse, which has also been found in the monitoring data. (7) Through the establishment of three-dimensional model, the mechanical deformation of the shield tunnel and the surrounding soil are analyzed. The displacement of the soil above the shield tunnel is downward and the displacement of the soil under the shield tunnel is upward. The displacement of soil around the tunnel is gradually decreased. The deformation of tunnel is consistent with the surrounding soil. The displacement is downward at the top of tunnel, while the displacement is upward at the bottom of tunnel. The tunnel has a trend to have a shape of ellipse. The vertical displacement of the four paths is affected by the nature of the soil. In the four paths, the settlement of the clay is larger than that of the sand. In the distance of 40 m, namely, 10 m away the sand and clay interface the displacement peak generated. The horizontal displacement is smaller than the vertical displacement, horizontal displacement of the clay is larger than that of the sand. At the same depth of underlying soil, the distribution of the vertical stress is not uniform and the interface of the sand and clay has significant difference. The vertical stress of the sand is slightly larger than that of the clay. The distribution of the pore pressure is not uniform around the tunnel. The pore pressure at the interface has not obvious difference. The pore pressure of the sand is larger than that of clay at the same depth of underlying soil. (8) Through analyzing the in-site monitoring data of settlements of Subway Line 1 in Shanghai, we found the settlement of tunnel is obviously inhomogeneous. The settlements of subway stations and running tunnels do not show obvious regularity. Along Subway Line 1, some places experience larger settlements and the running tunnels experience significant differential settlements. In the soft soil area, the causes resulting in the settlement of subway tunnel are very complex. The methodology of fitting subway settlement by the cubic multinomial is accurate than other methods. The cubic spline function contradicts the elastic foundation beam theory. It could not show the mechanism of the longitudinal deformation of the subway. In the field of prediction, the accuracy of ARMA model is higher than the GM (1, 1) model. ARMA model is suitable for long-term prediction of settlement, while the GM (1, 1) is suitable for shortterm prediction of settlement. Maybe the dynamic grey model can improve the prediction accuracy, and this is the future task to analyze. The differential settlement in longitudinal direction is significant, particularly in the settlement trough whose shape has little change over time. The differential settlement is relevant

7.2 Prospects for Further Study

197

to the seasonal precipitation and the settlement in the first half year is basically below the second half year. The control point fitting method and ARIMA (p, d, q) model could be used to predict the settlement of the settlement trough. The control point fitting method is able to consider spatial factors, and ARIMA (p, d, q) considers the time factor, which effectively improves the prediction accuracy.

7.2 Prospects for Further Study This monograph studied the interaction between soil foundation and subway shield tunnel. But there is still some work to be further studied. (1) The sine load was used in the dynamic triaxial test. In subsequent experimental studies, the in-site monitoring of subway vibration loading can be used to better study the dynamic characteristics of soils around the subway shield tunnel. (2) The contact problem between the roadbed and lining, as well as between the lining and the soil can be considered in three-dimensional numerical simulation. At the same time, the numerical simulation of this monograph considers a type of soil, and further research can consider the stratification of soil based on actual strata, studying the reflection and refraction of vibration waves by soil interfaces. (3) The lining in numerical simulation adopts a continuum, but in reality, the lining has transverse and longitudinal joints, which serve as the weak surface of the tunnel. Therefore, in subsequent research, the existence of lining joints can be considered to simulate the actual shield tunnel more reasonably. (4) It is possible to conduct physical tests on the longitudinal settlement of shield tunnels, simulate tunnel excavation, study the longitudinal settlement law of tunnels through physical tests, and compare them with numerical simulation results to improve the research scope. (5) For the prediction of tunnel settlement, it is possible to collect as much data as possible and consider as many input variables of the neural network as possible, making the prediction more accurate.

Appendix A

GM (1, 1)

Grey theory focuses on model uncertainty and information insufficiency in analyzing and understanding systems via research on conditional analysis, prediction and decision-making. The system with partial unknown structure, parameters, and characteristics is called a grey system. In the field of information research, deep or light colors represent information that is clear or ambiguous, respectively. Meanwhile, black indicates that the researchers have absolutely no knowledge of system structure, parameters and characteristics, while white represents that the information is completely clear. Colors between black and white indicate systems that are not clear, i.e., grey system. Grey model was built to describe a grey system in mathematic way. p in GM (p, q) means number of accumulating generating operator (AGO) times, and q means number of unknown. Therefore, GM (1, 1) is a grey model experience once accumulating generating operator (1-AGO) and including 1 unknown number. GM (2, 1) is a grey model experience twice accumulating generating operator (2-AGO) and including 1 unknown number. Assuming raw sequence of data is: x (0) = x (0) (1), x (0) (2), x (0) (3), . . . , x (0) (N )

(A.1)

After once accumulated generating operator (1-AGO), a new sequence was obtained: x (1) = x (1) (1), x (1) (2), x (1) (3), . . . , x (1) (N ) where x (1) (i ) =

i

(A.2)

x 0 ( j )i = 1, 2, . . . , N .

j=1

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Z.-D. Cui, Interaction Between Soil Foundation and Subway Shield Tunnel, https://doi.org/10.1007/978-981-99-6870-1

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200

Appendix A: GM (1, 1)

Consisting background value sequence through x (1) : z (1) = z (1) (1), z (1) (2), z (1) (3), . . . , z (1) (N − 1)

(A.3)

where z (1) (k) = αx (1) (k − 1) + (1 − α)x (1) (k), k = 2, 3, . . . , n. Usually we assume α = 0.5 and build shadow equation: dx (1) + ax (1) = u dt

(A.4)

It is the origin style of GM (1, 1). After being discretization, a new equation is obtained: x (0) (k) = az (1) (k) = b

(A.5)

where a and b are undetermined coefficients. It is the basic style of GM (1, 1). (1) Solving parameter Using the least square technique, the equation is proved, aˆ =

a b

= (B T B)−1 · B T · Yn

(A.6)

⎤ ⎤ ⎡ −z (1) (1) x (0) (2) 1 ⎢ −z (1) (2) ⎢ x (0) (3) ⎥ 1 ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ (1) ⎢ where B = ⎢ −z (3) 1 ⎥, Yn = ⎢ x (0) (4) ⎥ ⎥ ⎥ ⎢ ⎢ ⎦ ⎣... ⎣... ...⎦ (1) (0) −z (n − 1) 1 x (n) ⎡

(2) Building predicting function: b −ak b )e + a a

(A.7)

xˆ (0) (k + 1) = xˆ (1) (k + 1) − xˆ (1) (k)

(A.8)

xˆ (1) (k + 1) = (x (0) (k) −

(3) Methods of model accuracy checking Determining residual error

k

[x (0) (k) minus xˆ (0) (k)] and relative error e(k): k

= x (0) (k) − xˆ (0) (k)

| | | k || | e(k) = | (0) | × 100% x (k)

(A.9) (A.10)

Appendix A: GM (1, 1)

201

Table A.1 Predicted precision indexes of model Predicted precision

Excellent

Good

Satisfactory

Failed

P

> 0.95

0.95 ~ 0.80

0.80 ~ 0.70

< 0.70

C

< 0.35

0.35 ~ 0.50

0.50 ~ 0.65

> 0.65

Determining average value of raw data and average value of residual error: x=

e=

n

1 n

x (0) (k)

(A.11)

k=1

1 n−1

n

e(k)

(A.12)

k=1

Determining variance of raw data s12 and variance of residual error s22 , defining variance ratio C and small errors probability p, s12 = s22

1 n

n

x (0) (k) − x

2

(A.13)

k=1

1 = n

n

[e(k) − e]2

(A.14)

k=1

C=

s2 s1

p = P{|e(k) − e| 0, the model is turn to MA (m) model. Similarly, if the suitable order is n > 0, n = 0, the model is turn to AR (n) model. Other common methods are standard error, logarithm likelihood function value, Schwarz Bayesian criterion (SBC) and so on. Therefore, after AIC order selection method, the suitable orders of settlement prediction models are n = 1 and m = 1, ARMA (1, 1) model was chosen as the prediction model. (3) Parameters estimation of the model According to the optimal model and its order, the parameters φi and θi in Formula (B.3) are estimated. The common methods of parameter estimation are moment estimation, maximum likelihood estimation and least square estimation. (4) Model test The last step is testing and analyzing to residual error. When the residual error is white noise series, the fitting model is effective. The k steps autocorrelation parameter

Appendix B: ARMA (N, M) Model

205

ρ1 , ρ2 , . . . , ρk are calculated, the χ 2 statistics Fk are constructed as follows, Fk obeys χ 2 distribution whose degree of freedom is k. k

Fk = n

ρi2 i=1

where n is the capacity of the residual error.

(B.5)

Appendix C

Major Published Works of the Book Author

Monograph, Textbook and Proceedings (1) Cui ZD* (2022) Design of Underground Structures (Second version, in Chinese). China Construction Industry Press, ISBN 978-7-112-27505-2. (2) Cui ZD*, Zhang ZL, Zhan ZX, et al. (2020) Dynamics of Freezing–Thawing Soil around Subway Shield Tunnels. Springer Press, ISBN 978-981-15-4341-8. (3) Cui ZD* (2020) Soil Dynamics. China University of Mining and Technology Press, ISBN 978-7-5646-4667-7. (4) Cui ZD*, Zhang ZL, Yuan L, et al. (2019) Design of Underground Structures. Springer Press, ISBN 978-981-13-7731-0 (5) Cui ZD* (2018) Land subsidence induced by the engineering-environmental effects. Springer Press, ISBN 978-981-8039-5. (6) Cui ZD*, Zhang ZL (2017) Design of Underground Structures (in Chinese). China Construction Industry Press, ISBN 978-7-112-20816-6. (7) Kallel A, Erguler ZA, Cui ZD, et al. (2019) Recent Advances in GeoEnvironmental Engineering, Geomechanics and Geotechnics, and Geohazards. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG-1), Tunisia 2018, Springer Press, ISBN 978-3-030-016647. English Journal Papers (1) Cui ZD*, Zhang LJ, Zhan ZX (2023). Seismic response analysis of shallowly buried subway station in inhomogeneous clay site. Soil Dynamics and Earthquake Engineering, 2023, 17, 107986 (SCI, WOS: 000991985200001) (2) Cui ZD*, Zhang LJ, Hou CY (2023). Equivalent seismic behaviors of artificial frozen-thawed soft clay with a new reduction factor of shear stress. Cold Regions Science and Technology, 2023, 213: 103934 (SCI, WOS: 001032059000001)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Z.-D. Cui, Interaction Between Soil Foundation and Subway Shield Tunnel, https://doi.org/10.1007/978-981-99-6870-1

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(3) Cui ZD*, Huang MH, Hou CY, Yuan L (2023). Seismic deformation behaviors of the soft clay after freezing–thawing. Geomechanics and Engineering, 2023, 34(3): 303–316 (SCI, WOS: 001045090700006) (4) Cui ZD*, Zhang LJ, Xu, C (2023). Numerical simulation of freezing temperature field and frost heave deformation for deep foundation pit by AGF. Cold Regions Science and Technology, 2023, 213: 103908 (SCI, WOS: 001011596000001) (5) Cui ZD*, Zhang LJ, Zhan ZX (2023). Dynamic shear modulus and damping ratio of saturated soft clay under the seismic loading. Geomechanics and Engineering, 2023, 32(4): 411–426 (SCI, WOS: 000863219600005) (6) Xu XB, Cui ZD* (2022). Investigation of One-dimensional Consolidation of Fractional Derivative Model for Viscoelastic Saturated Soils Caused by the Groundwater Level Change. KSCE JOURNAL OF CIVIL ENGINEERING, 26(12): 4997–5009 (SCI, WOS: 000863219600005) (7) Zhang ZL, Cui ZD*, Zhao LZ (2022). Modeling shear behavior of sand-clay interfaces through two-dimensional distinct element method analysis. Environmental Earth Sciences, 81:140 (SCI, WOS: 000757380300003) (8) Jiang JJ, Cui ZD* (2022). Instability of High Liquid Limit Soil Slope for the Expressway Induced by Rainfall. APPLIED SCIENCES-BASEL,12(21): 10857 (SCI, WOS: 000883850900001) (9) Li DC, Xu C, Cui ZD*, et al. (2022) Mechanical Properties of Functionally Graded Concrete Lining for Deep Underground Structures. ADVANCES IN CIVIL ENGINEERING, 2022, 2363989 (SCI, WOS: 000834899700001) (10) Li DC, Zhang TT, Cui ZD*, et al. (2022) Mechanical Properties of the Functionally Graded Lining for a Deep Buried Subway Tunnel. Applied Sciences, 12(21): 11272 (SCI, WOS: 000883369300001) (11) Soomro MA, Kumar M, Mangi N, Mangnejo DA, Cui ZD (2022) Parametric Study of Twin Tunneling Effects on Piled Foundations in Stiff Clay: 3D FiniteElement Approach. INTERNATIONAL JOURNAL OF GEOMECHANICS, 22(6): 04022079 (SCI, WOS: 000782625000026) (12) Hou CY, Cui ZD* (2021). Quantitative Analysis of the Microstructures of Deep Silty Clay Subjected to Two Freezing–Thawing Cycles under Subway Vibration Loading. Journal of Cold Regions Engineering, ASCE, 35(4): 04021012 (SCI, WOS: 000708121500008) (13) Wei SM, Yuan L*, Cui ZD* (2021) Application of closest point projection method to unified hardening model. Computers and Geotechnics, 133: 104064 (SCI, WOS: 000641590800006) (14) Hou CY, Cui ZD*, Yuan L (2020) Accumulated deformation and microstructure of deep silty clay subjected to two freezing–thawing cycles under cyclic loading. Arabian Journal of Geosciences, 13(12): 452 (SCI, WOS:000542660000003) (15) Yang JQ, Cui ZD* (2020) Influences of train speed on permanent deformation of saturated soft soil under partial drainage conditions. Soil Dynamics and Earthquake Engineering, 133:106120 (SCI, WOS:000527323700016)

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209

(16) Zhan ZX, Cui ZD*, Yang P, Zhang T (2020) In situ monitoring of temperature and deformation fields of a tunnel cross passage in Changzhou Metro constructed by AGF. Arabian Journal of Geosciences, 13(8): 310 (SCI, WOS:000524236600001) (17) Zhang ZL, Cui ZD*, Zhao LZ (2020) Shear strength of sand-clay interfaces through large-scale direct shear tests. Arabian Journal for Science and Engineering, 45(5): 4343–4357 (SCI, WOS:000519505700001) (18) Yuan L, Cui ZD*, Yang JQ, Jia YJ (2020) Land subsidence induced by the engineering-environmental effect in Shanghai, China. Arabian Journal of Geosciences, 13(6):251 (SCI, WOS:000519254600005) (19) Xu XB, Cui ZD* (2020) Investigation of a fractional derivative creep model of clay and its numerical implementation. Computers and Geotechnics, 119: 103387 (SCI, WOS: 000517663100030) (20) Cui ZD*, Jia YJ (2018) Physical Model Test of Layered Soil Subsidence Considering Dual Effects of Building Load and Groundwater Withdrawal. Arabian Journal for Science and Engineering, 43(4): 1721–1734 (SCI, WOS: 000427997400015) (21) Zhang ZL, Cui ZD* (2018) Effect of freezing–thawing on dynamic characteristics of the silty clay under K0-consolidated condition. Cold Regions Science and Technology, 146: 32–42 (SCI, WOS: 000423963300004) (22) Zhang ZL, Cui ZD* (2018) Effects of freezing–thawing and cyclic loading on pore size distribution of silty clay by mercury intrusion porosimetry. Cold Regions Science and Technology, 145: 185–196 (SCI, WOS: 000415834800020) (23) Cui ZD*, Yuan Q, Yang JQ (2018) Laboratory model tests about the sand embankment supported by piles with a cap beam. Geomechanics and Geoengineering an international Journal, 13(1): 64–76 (SCI, WOS: 000437148600007) (24) Li Z, Cui ZD* (2017) Axisymmetric consolidation of saturated multI.layered soils with anisotropic permeability due to well pumping. Computers and Geotechnics, 92: 229–239 (SCI, WOS: 000414878800019) (25) Zhang ZL, Cui ZD* (2017) Analysis of microscopic pore structures of the silty clay before and after freezing–thawing under the subway vibration loading. Environmental Earth Sciences, 76(15): 528 (SCI, WOS: 000407545200021) (26) Zhang CL, Cui ZD* (2017) Numerical simulation of dynamic response around shield tunnel in the soft soil area. Marine Georesources & Geotechnology, 35(7): 1018–1027 (SCI, WOS: 000408912800014) (27) Yuan L, Cui ZD*, Tan J (2017) Numerical Simulation of Longitudinal Settlement of Shield Tunnel in the Coastal City, Shanghai. Marine Georesources & Geotechnology, 35(3): 365–370 (SCI, WOS: 000398567600007) (28) Cui ZD*, Li Z, Jia YJ (2016) Model test study on the subsidence of high-rise building group due to the variation of groundwater level. Natural Hazards, 84(1): 35–53 (SCI, WOS: 000384568300003) (29) Cui ZD*, Jia YJ, Yuan L (2016) Distribution law of soil deformation caused by decompression of confined water. Environmental Earth Sciences, 75(18): 1281 (SCI, WOS: 000384333000048)

210

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(30) Tan J, Cui ZD*, Yuan Y (2016) Study on the Long-term Settlement of Subway Tunnel in Soft Soil Area. Marine Georesources & Geotechnology, 34(5): 486– 492 (SCI, WOS: 000372448400009) (31) Cui ZD*, Zhao LZ, Yuan L (2016) Microstructures of consolidated Kaolin clay at different depths in centrifuge model tests. Carbonates and Evaporites, 31(1):47–60 (SCI, WOS: 000371164000006) (32) Yuan Q, Cui ZD* (2016) Two-Dimensional Numerical Analysis of the Subgrade Improved by Stone Columns in the Soft Soil Area. Marine Georesources & Geotechnology, 34(1): 79–86 (SCI, WOS: 000367347700009) (33) Cui ZD*, Yang JQ, Yuan L (2015) Land subsidence caused by the interaction of high-rise buildings in soft soil areas. Natural Hazards, 79(2): 1199–1217 (SCI, WOS: 000362894600027) (34) Cui ZD*, Zhang ZL (2015) Comparison of dynamic characteristics of the silty clay before and after freezing and thawing under the subway vibration loading. Cold Regions Science and Technology, 119: 29–36 (SCI, WOS: 000362381900003) (35) He PP, Cui ZD* (2015) Dynamic response of a thawing soil around the tunnel under the vibration load of subway. Environmental Earth Sciences, 73(5): 2473–2482 (SCI, WOS: 000349360600043) (36) Cui ZD*, Yuan Q (2015) Study on the settlement caused by the Maglev train. Natural Hazards, 75(2): 1767–1778 (SCI, WOS: 000346407100036) (37) Cui ZD*, Tan J (2015) Analysis of long-term settlements of Shanghai Subway Line 1 based on the in situ monitoring data. Natural Hazards, 75(1): 465–472 (SCI, WOS: 000345971600021) (38) Cui ZD*, Ren SX (2014) Prediction of long-term settlements of subway tunnel in the soft soil area. Natural Hazards, 74(2):1007–1020 (SCI, WOS: 000342910400038) (39) Cui ZD*, He PP, Yang WH (2014) Mechanical properties of a silty clay subjected to freezing–thawing. Cold Regions Science and Technology, 98: 26–34 (SCI, WOS: 000331412400004) (40) Cui ZD*, Jia YJ (2013) Analysis of electron microscope images of soil pore structure for the study of land subsidence in centrifuge model tests of high-rise building groups. Engineering Geology, 164: 107–116 (SCI, WOS: 000324963500010) (41) Song L, Cui ZD, Zhang HQ (2013) Analysis and treatment of the fault activation below the dynamic foundation in the goaf area. Disaster Advances, 6(S1): 337–342 (SCI, WOS: 000317246200044) (42) Yuan L, Cui ZD (2013) Reliability analysis for the consecutive-k-out-of-n: F system with repairmen taking multiple vacations. Applied Mathematical Modelling, 37(7): 4685–4697 (SCI, WOS: 000316579600009) (43) Cui ZD*, Jia YJ (2012) Study on the mechanisms of the soil consolidation and land subsidence caused by the high-rise building group in the soft soil area. Disaster Advances, 5(4): 604–608 (SCI, WOS: 000313100100098)

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(44) Cui ZD*, Ren SX (2012) Long-term deformation characteristics of the soil around the subway tunnel induced by the vibration loading. Disaster Advances, 5(4): 1791–1797 (SCI, WOS: 000313100100312) (45) Cui ZD* (2012) Bearing capacity of single pile and in-flight T-Bar Penetration for centrifuge modeling of land subsidence caused by the interaction of high-rise buildings. Bull Eng Geol Environ, 71(3): 579–586 (SCI, WOS: 000307286700014) (46) Cui ZD* (2012) Land subsidence disaster caused by natural factors and human activities. Disaster Advances, 5(2): 3–4 (SCI, WOS: 000303786600001) (47) Cui ZD* (2011) Effect of water-silt composite blasting on the stability of rocks surrounding a tunnel. Bull Eng Geol Environ, 70(4): 657–664 (SCI, WOS: 000299511100014) (48) Cui ZD*, Tang YQ (2011) Microstructures of different soil layers caused by the high-rise building group in Shanghai. Environmental Earth Sciences, 63(1): 109–119 (SCI, WOS: 000289366100009) (49) Cui ZD*, Yuan L, Yan CL (2010) Water-silt composite blasting for tunneling. International Journal of Rock Mechanics and Mining Sciences, 47(6): 1034– 1037 (SCI, WOS: 000280610200016) (50) Cui ZD*, Wang HM (2010) Land Subsidence at Different Points among a Group of High-Rise Buildings [J]. Disaster Advances, 3(4): 63–66 (SCI, WOS: 000282837700010) (51) Cui ZD*, Tang YQ (2010) Land subsidence and pore structure of soils caused by the high-rise building group through centrifuge model test. Engineering Geology, 113(1–4): 44–52 (SCI, WOS: 000278804900004) (52) Cui ZD*, Tang YQ, Yan XX, Wang HM, Yan CL, Wang JX (2010) Evaluation of the geology-environmental capacity of buildings based on the ANFIS model of the floor area ratio. Bull Eng Geol Environ, 69(1): 111–118 (SCI, WOS: 000273786300011) (53) Cui ZD*, Tang YQ, Yan XX (2010) Centrifuge modeling of land subsidence caused by the high-rise building group in the soft soil area. Environmental Earth Sciences, 59(8): 1819–1826 (SCI, WOS: 000274182100019) (54) Tang YQ, Cui ZD, Wang JX, Yan LP, Yan XX (2008) Application of grey theory-based model to prediction of land subsidence due to engineering environment in Shanghai. Environmental Geology, 55(3): 583–593 (SCI, WOS: 000257581800010) (55) Cui ZD*, Tang YQ, Guo CQ, Yuan L, Yan CL (2008) Flow-induced Vibration and Stability of an Element Model for Parallel-plate fuel assemblies. Nuclear Engineering and Design, 238(7):1629–1635 (SCI, WOS: 000256951000014) (56) Tang YQ, Cui ZD, Zhang X, Zhao SK (2008) Dynamic response and porewater pressure model of saturated soft clay around a tunnel induced by the subway vibration load. Engineering Geology, 98(3–4):126–132 (SCI, WOS: 000256608100005) (57) Tang YQ, Cui ZD, Wang JX, Lu C, Yan XX (2008) Model test study of land subsidence caused by high-rise building group. Bull Eng Geol Environ, 67(2):173–179 (SCI, WOS: 000255861100004)

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English Conference Papers (1) Cui ZD*, He PP, Zhang ZL, Pak RYS (2020) Dynamic response of a doublebeam system subjected to a harmonic moving load. 16th Asian Regional Conference on Soil Mechanics and Geotechnical Engineering, ARC 2019 (EI, Accession number: 20201808589495) (2) Cui ZD* (2019) Land subsidence induced by the engineering-environmental effect in Shanghai, China. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG-1): Recent Advances in GeoEnvironmental Engineering, Geomechanics and Geotechnics, and Geohazards, 11–14 (ISTP, WOS: 000623020200003) (3) Dai SA, Cui ZD* (2019) Dynamic Characteristics of Soft Clay Under Traffic Load. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG-1): Recent Advances in Geo-Environmental Engineering, Geomechanics and Geotechnics, and Geohazards, 209–211 (ISTP, WOS: 000623020200049) (4) Guo WH, Cui ZD*, LZ (2019) Displacement Distribution Caused by Pumping from the Aquifer in Soil. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG-1): Recent Advances in GeoEnvironmental Engineering, Geomechanics and Geotechnics, and Geohazards, 403–406 (ISTP, WOS:000623020200093) (5) Hou CY, Cui ZD* (2019) Dynamic Properties of Soft Clay Under Freezing– Thawing Cycle. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG-1): Recent Advances in Geo-Environmental Engineering, Geomechanics and Geotechnics, and Geohazards, 255–257 (ISTP, WOS: 000623020200059) (6) Xu C, Cui ZD* (2019) Study on Physical and Mechanical Properties of Clay Before and After Single Freeze–Thaw. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG-1): Recent Advances in Geo-Environmental Engineering, Geomechanics and Geotechnics, and Geohazards, 243–245 (ISTP, WOS: 000623020200056) (7) Xu XB, Cui ZD* (2019) Numerical Simulation of Land Subsidence Caused by Both Dewatering and Recharging. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG-1): Recent Advances in GeoEnvironmental Engineering, Geomechanics and Geotechnics, and Geohazards, 399–401 (ISTP, WOS: 000623020200092) (8) Yang XY, Yuan L, Cui ZD* (2019) Review on the Mechanical Behavior of Soil-Structure Interface. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG-1): Recent Advances in GeoEnvironmental Engineering, Geomechanics and Geotechnics, and Geohazards, 133–135 (ISTP, WOS: 000623020200031) (9) Zhan ZX, Cui ZD* (2019) Seismic Response and Failure Mechanism of the Subway Station: A Literature Review. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG-1): Recent Advances

Appendix C: Major Published Works of the Book Author

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

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in Geo-Environmental Engineering, Geomechanics and Geotechnics, and Geohazards, 205–207 (ISTP, WOS: 000623020200048) Zhang TT, Cui ZD* (2019) Study on the Deformation Properties of Functionally Gradient Metro Tunnel Lining Structure. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG-1): Recent Advances in Geo-Environmental Engineering, Geomechanics and Geotechnics, and Geohazards, 145–147 (ISTP, WOS: 000623020200034) Zhang WK, Yuan L, Cui ZD* (2019) Analysis of the Track Critical Velocity in High-Speed Railway. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG-1): Recent Advances in GeoEnvironmental Engineering, Geomechanics and Geotechnics, and Geohazards, 367–369 (ISTP, WOS: 000623020200085) Zhang ZL, Cui ZD*, Zhao LZ (2019) Numerical Investigation of the Interface Shear Behaviors Between Double Soil Layers Using PFC2D. Proceedings of the 1st Springer Conference of the Arabian Journal of Geosciences (CAJG1): Recent Advances in Geo-Environmental Engineering, Geomechanics and Geotechnics, and Geohazards, 169–171 (ISTP, WOS: 000623020200040) Cui ZD*, Hua SS, Yan JS (2018) Long-term Settlement of subway tunnel and Prediction of Settlement Trough in Coastal City Shanghai. Proceedings of GeoShanghai 2018 International Conference: MultI.physics Processes in Soil Mechanics and Advances in Geotechnical Testing, 458–467 (ISTP, WOS: 000477862800051) Zhang ZL, Cui ZD* (2018) Dynamic Response of Soil around the Tunnel under Subway Vibration Loading. Proceedings of GeoShanghai 2018 International Conference: Advances in Soil Dynamics and Foundation Engineering, 53–61 (ISTP, WOS: 000465516400006) Cui ZD*, Fan SC, Wei SW, et al. (2018) Physical Modeling of Arching effect in the Piled Embankment. Proceedings of GeoShanghai 2018 International Conference: Fundamentals of Soil Behaviours, 379–387 (ISTP, WOS: 000465510600042) Cui ZD*, Zhang ZL (2017) Comparison of the dynamic characteristics of the silty clay before and after freezing–thawing under cyclic loadings. Seoul: 19th ICSMGE, 335–339 (EI, Accession number: 20180204626686) Cui ZD*, Zhang CL, Hou CY (2017) Vibrations induced by subway moving load with excitation of the track vertical profile irregularity. Wuhan: 15th IACMAG, 856–873 (EI, Accession number: 20200908225470) Cui ZD*, Yuan L, Yan LP, Guo CQ (2009) Stability analysis and measurement of flow-induced vibration of a parallel-plate structure. Chengdu: International Conference on Earthquake Engineering-the 1st Anniversary of Wenchuan Earthquake, 288–293 (ISTP, WOS: 000273961600048) Cui ZD, Tang YQ, Zhang X (2008) Deformation and pore pressure model of Saturated Soft Clay around a Subway Tunnel. Shanghai: The Six International Symposium Geotechnical Aspects of Underground Construction in Soft Ground, 769–774 (EI, Accession number: 20111013726864)

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Appendix C: Major Published Works of the Book Author

(20) Tang YQ, Cui ZD, Zhang X (2008) Dynamic Response of Saturated Silty Clay around a Tunnel under Subway Vibration Loading in Shanghai. Shanghai: The Six International Symposium Geotechnical Aspects of Underground Construction in Soft Ground (EI), 843–848 (EI, Accession number: 20111013726876) * noted the corresponding author.