(Luận án tiến sĩ) nghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đất

Gmax Maximum ground shear modulus h Tunnel height H Tunnel depth I Inertia moment of tunnel lining per unit length of the tunnel K0 Lateral earth pressure coefficient K1 Full slip lining

PHAM VAN VI

BEHAVIOR OF SUB-RECTANGULAR TUNNELS UNDER SEISMIC LOADING

Major: Underground construction engineering

Code: 9580204

PhD THESIS

SUPERVISORS:

1 Asso Prof., Dr Do Ngoc Anh

2 Prof., Dr Dias Daniel

HA NOI, May 2022

PHẠM VĂN VĨ

NGHIÊN CỨU ỨNG XỬ CỦA KẾT CẤU CHỐNG TRONG ĐƯỜNG HẦM TIẾT

DIỆN HÌNH CHỮ NHẬT CONG CHỊU TẢI TRỌNG ĐỘNG ĐẤT

Ngành đào tạo: Kỹ thuật Xây dựng Công trình ngầm

Mã số ngành: 9580204

LUẬN ÁN TIẾN SĨ KỸ THUẬT

NGƯỜI HƯỚNG DẪN KHOA HỌC:

1 PGS.TS Đỗ Ngọc Anh

2 GS.TS Dias Daniel

HÀ NỘI - 05/2022

The work described within this thesis was conducted at the Underground and Mining Construction Department, Faculty of Civil Engineering, Hanoi University of Mining and Geology, Vietnam

First of all, I am particularly grateful to my supervisors, Associate Professor,

Dr Do Ngoc Anh, and Professor Daniel Dias They have enthusiastically supported and directed me to provide invaluable advices in the process of preparing this thesis and research articles I would like to thank Associate Professor, Dr Do Ngoc Anh for his regular support from the very beginning to the completion of this thesis He pushed me to reach my full potential His professional guidance and willingness to work on an ongoing basis were key elements in completing this study I would like

to thank Professor Daniel Dias for his invaluable guidance, supervision, encouragement, and support throughout this research process I would like to record

my sincere appreciation for their help and I will never forget my three years of Ph.D studies under their guidance, my respected teachers

Second, I also want to thank the teachers and staff of the Underground and Mining Construction Department, Faculty of Construction, Postgraduate training Office, Hanoi University of Mining and Geology, who helped me in the process of implementing this thesis

Third, I would like to thank the Vingroup JSC and supported by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, code VINIF.2021.TS.167 for financial support This is an honor and a great motivation that helped me to make this research more focused

Finally, I am deeply grateful to my family for their support, patience, and love This study would not have been started, would not have been possible, and would never have been completed without the support of my wife, Vu Thi Hue, and my two children, Khanh An and Minh Tri Nothing would have happened without their support and I devoted them to this thesis

Luận án này được thực hiện tại Bộ môn Xây dựng công trình ngầm và Mỏ, Khoa Xây dựng, Trường Đại học Mỏ - Địa chất

Đầu tiên, tác giả xin đặc biệt cảm ơn tới tổ hướng dẫn, PGS.TS Đỗ Ngọc Anh

và GS Daniel Dias Các Thầy luôn định hướng, khuyến khích, thúc đẩy NCS và có những lời khuyên quý báu, chân thành giúp cho tác giả trong quá trình thực hiện luận

án cũng như viết các bài báo khoa học

Thứ hai, tác giả muốn cảm ơn tới các Thầy cô Bộ môn Xây dựng công trình ngầm và mỏ, Khoa Xây dựng, Phòng Đào tạo Sau đại học Trường Đại học Mỏ - Địa chất đã luôn giúp đỡ, tạo điều kiện cho tác giả trong quá trình thực hiện luận án này Thứ ba, tác giả muốn cảm ơn tới Tập đoàn Vingroup và sự hỗ trợ của Chương trình học bổng thạc sĩ, tiến sĩ trong nước của Quỹ Đổi mới sáng tạo Vingroup (VINIF), Viện Nghiên cứu Dữ liệu lớn, mã số VINIF.2021.TS.167 đã tài trợ Đây là một vinh dự và là động lực lớn giúp tác giả tập trung hơn trong nghiên cứu khoa học Cuối cùng, tác giả vô cùng biết ơn tới gia đình đã luôn bên cạnh với sự kiên nhẫn Luận án này sẽ không được bắt đầu, được thực hiện và hoàn thành nếu không

có sự hỗ trợ của gia đình

I hereby declare that this is my own research work The data and results presented in this thesis are honest and have never been published in any other works

PhD candidate

Pham Van Vi

LỜI CAM ĐOAN

Tôi xin cam đoan đây là công trình nghiên cứu của riêng tôi Các kết quả và dữ liệu trong luận án là trung thực và chưa từng được công bố trong bất kỳ công trình nào

Nghiên cứu sinh

Phạm Văn Vĩ

SUMMARY

The principal purpose of this Ph.D thesis is to study the behavior of rectangular tunnels under seismic conditions by using a finite difference method (FDM) and then a new quasi-static loading scheme, applied to the Hyperstatic Reaction Method (HRM), was developed

sub-Firstly, a literature review on the tunnel lining design under seismic condition was conducted Secondly, 2D numerical models of circular and sub-rectangular tunnels subjected to quasi-static loading were developed The difference in behavior

of these two tunnel types under seismic loading was highlighted In the final part of the manuscript, a new quasi-static loading scheme applied in sub-rectangular tunnels using the HRM method was proposed based on the quasi-static loading principle Its reliability is demonstrated based on validations conducted by using finite difference caculations considering different situations

Keywords: Sub-rectangular tunnel; Hyperstatic Reaction Method; Numerical model; Quasi-static

TÓM TẮT

Mục tiêu chính của luận án là sử dụng phương pháp số sai phân hữu hạn (FDM)

để nghiên cứu ứng xử của đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đất và phát triển một sơ đồ tải trọng tĩnh tương đương mới áp dụng trong phương pháp lực kháng đàn hồi (HRM)

Trên cơ sở kết quả nghiên cứu tổng quan chỉ ra khoảng trống nghiên cứu đối với kết cấu đường hầm tiết diện chữ nhật cong chịu tải trọng động đất, luận án đã phát triển mô hình số 2D cho đường hầm tiết diện hình chữ nhật cong chịu tải trọng tĩnh tương đương trên cơ sở mô hình của đường hầm tiết diện hình tròn được kiểm chứng bằng cách so sánh với phương pháp giải tích Ứng xử khác nhau của kết cấu chống trong hai loại tiết diện đường hầm khi chịu tải trọng động đất đã được chỉ ra Dựa vào kết quả phân tích trên mô hình số FDM, luận án đã đề xuất được một sơ đồ tải trọng tĩnh tương đương mới tác dụng lên kết cấu chống đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đất trong phương pháp HRM Độ tin cậy của sơ đồ tải trọng tĩnh tương đương mới đã được kiểm chứng trên cơ sở so sánh với phương pháp FDM khi xem xét một loạt điều kiện đầu vào khác nhau

Từ khóa: Đường hầm tiết diện chữ nhật cong; Phương pháp lực kháng đàn hồi;

Mô hình số; Tĩnh tương đương

CONTENTS

ACKNOWLEDGEMENTS i

SUMMARY v

LIST OF NOMENCLATURE ix

LIST OF FIGURES xii

LIST OF TABLES xv

GENERAL INTRODUCTION xvi

Background and Problematic xvi

Objectives xvii

Scope of this study xviii

Original Features xviii

Thesis outline xviii

CHAPTER 1: LITERATURE REVIEW ON THE BEHAVIOUR OF UNDERGROUND STRUCTURES UNDER SEISMIC LOADING 1

1.1 Introduction 1

1.2 Seismic response mechanisms 3

1.3 Research methods 7

1.3.1 Analytical solutions 8

1.3.2 Physical tests 16

1.3.3 Numerical modeling 20

1.4 Sub-rectangular tunnels 25

1.5 Conclusions 27

CHAPTER 2: NUMERICAL STUDY ON THE BEHAVIOR OF SUB-RECTANGULAR TUNNEL UNDER SEISMIC LOADING 29

2.1 Numerical simulation of the circular tunnel under seismic loading 30

2.1.1 Reference case study- Shanghai metro tunnel 30

2.1.2 Numerical model for the circular tunnel 31

2.1.3 Comparison of the numerical and analytical model for the circular tunnel case study 34

2.2 Validation of circular tunnel under seismic loading 37

2.2.1 Effect of the peak horizontal seismic acceleration (aH) 38

2.2.2 Effect of the soil Young’s modulus, Es 39

2.2.3 Effect of the lining thickness, t 40

2.3 Numerical simulation of the sub-rectangular tunnel under seismic loading 42

2.4 Parametric study of sub-rectangular tunnels in quasi-static conditions 42

2.4.1 Effect of the peak horizontal seismic acceleration (aH) 44

2.4.2 Effect of the soil’s Young’s modulus (Es) 46

2.4.3 Effect of the lining thickness (t) 47

2.5 Conclusion 48

CHAPTER 3: A NEW QUASI-STATIC LOADING SCHEME FOR THE HYPERSTATIC REACTION METHOD - CASE OF SUB-RECTANGULAR TUNNELS UNDER SEISMIC CONDITION 51

3.1 Fundamental of HRM method applied to sub-rectangular tunnel under static loading 52

3.2 HRM method applied to sub-rectangular tunnel under seismic conditions 57

3.3 Numerical implementation 61

3.3.1 FDM numerical model 61

3.3.2 Numerical procedure in HRM method 63

3.4 Validation of the HRM method 69

3.4.1 Validation 1 70

3.4.2 Validation 2 71

3.4.3 Validation 3 72

3.4.4 Validation 4 73

3.4.5 Validation 5 74

3.4.6 Validation 6 75

3.4.7 Validation 7 76

3.5 Conclusions 77

GENERAL CONCLUSIONS AND PERSPECTIVES 79

PUBLISHED AND SUBMITTED MANUSCRIPTS 83

REFERENCES 84

LIST OF NOMENCLATURE

Abbreviations

3D Three-dimensional DOT Double-O-tube FDM Finite difference method FEM Finite element method

D Circular tunnel external diameter

E Young’s modulus of the tunnel lining

Es Young’s modulus of the ground

F Tunnel lining flexibility ratio

G Soil shear modulus

Gmax Maximum ground shear modulus

h Tunnel height

H Tunnel depth

I Inertia moment of tunnel lining per unit length of the tunnel

K0 Lateral earth pressure coefficient

K1 Full slip lining response coefficient

K2 No-slip lining response coefficient

K3 No-slip lining response coefficient

K4 No-slip lining response coefficient

Li Element length

M Incremental Bending moment

Mmax Maximum incremental bending moment

Mmin Minimum incremental bending moment

N Incremental Normal forces

Nmax Maximum incremental normal forces

Nmin Minimum incremental normal forces

plim Maximum reaction pressure

Ri Radius of part i (i=1, 2 and 3 corresponding to the crown, shoulder

and sidewall) of the tunnel boundary

t Tunnel lining thickness

u Axial displacement

v Transversal displacement

Vmax Peak shear wave velocity

Vs The ground shear wave velocity

w Tunnel width

∆zmin Smallest dimension in the normal direction of zones

γ Soil unit weight

γmax Maximum shear strain

ηn,0 Soil initial stiffness

θ Angle measured counter-clockwise from spring line on the right

λi Transformation matrix

νs Soil Poisson’s ratio

ρmax Soil density

τ Shear stresses applied at the far-field boundary

φ Soil internal friction angle

LIST OF FIGURES

Figure 1.1 Summary of observed bored/mined tunnel damage due to ground shakings [131] 2 Figure 1.2 Typical failure modes of mountain tunnels reported during the 1999 Chi-Chi earthquake in Taiwan [160] 3 Figure 1.3 Ground response to seismic waves [159] 4 Figure 1.4 Type of tunnel deformations during a seismic event [123] 5 Figure 1.5 Examples of the effects of seismically-induced ground failures on tunnels [155] 6 Figure 1.6 A circular tunnel (redrawn) [126] 9 Figure 1.7 Seismic shear loading and equivalent static loading (redrawn) [126] 10 Figure 1.8 Definition of terms used in racking analysis of a rectangular tunnel [159] 14 Figure 1.9 Racking coefficients for rectangular tunnels [59] 16 Figure 1.10 Geometry and boundary conditions in the quasi-static model [135] 21 Figure 1.11 Geometry and boundary conditions in the quasi-static model [49] 22 Figure 1.12 (a) 2D and (b) 3D numerical model in ABAQUS [150] 23 Figure 1.13 (a) Acceleration time history scaled at 0.35g (b) The corresponding Fourier spectrum [12] 23 Figure 1.14 (a) Overlap cutter heads; (b) a copy cutter head [78] 25 Figure 1.15 A photo showing the testing setup after fabrication [72] 26 Figure 2.1 Sub-rectangular express tunnel in Shanghai [48], distances in millimeters 30 Figure 2.2 Circular tunnel with the same utilization space area, distances in millimeters 31 Figure 2.3 The plane strain model under consideration 32 Figure 2.4 Geometry and quasi-static loading conditions for the circular tunnel model 33 Figure 2.5 Deformed model and displacement contours in circular tunnel model for no-slip condition 36

Figure 2.6 Deformed model and displacement contours in circular tunnel model for full-slip condition 36 Figure 2.7 Distribution of the incremental internal forces in the circular tunnel by Flac3D and Wang solution 37 Figure 2.8 Effect of aH on the extreme incremental internal forces of the circular tunnel lining 38 Figure 2.9 Effect of Es on the incremental internal forces of the circular tunnel lining 40 Figure 2.10 Effect of the lining thickness on the incremental internal forces in the circular tunnel lining 41 Figure 2.11 Geometry and quasi-static loading conditions in the numerical model of

a sub-rectangular tunnel 42 Figure 2.12 Deformed model and displacement contours in Sub-rectangular tunnel model for no-slip condition 43 Figure 2.13 Deformed model and displacement contours in Sub-rectangular tunnel model for full-slip condition 43 Figure 2.14 Distribution of the incremental bending moments and normal forces in the sub-rectangular tunnel 44 Figure 2.15 Effect of the aH value on the internal forces of circular and sub-rectangular tunnel linings 45 Figure 2.16 Effect of the Es value on the internal forces for the circular and sub-rectangular tunnel linings 46 Figure 2.17 Effect of the lining thickness on the incremental internal forces of the circular and sub-rectangular tunnel linings 48 Figure 3.1 Calculation scheme of support structures with the HRM method under static conditions With σv: the vertical loads; σh: the horizontal loads; kn: normal stiffness of springs; ks: shear stiffness of spring; EI and EA: bending and normal stiffness of the support; X and Y are the global Cartesian coordinates [48] 52 Figure 3.2 A finite element under the local Cartesian coordinates: i: initial node; i+1: final node; θ: rotation; x and y: local Cartesian coordinates; ν: transversal displacement; u: axial displacement; Li: element length [120] 53 Figure 3.3 Nonlinear relationship between the reaction pressure p and the support normal displacement δ: η0: initial spring stiffness; plim: maximum reaction pressure [121] 55

Figure 3.4 Transversal response in 2D plane strain conditions of the circular tunnel (a) ovaling deformation; (b) corresponding seismic shear loading; (c) sub-ovaling deformation; (d) corresponding seismic shear loading 58 Figure 3.5 Incremental bending moments and normal forces of sub-rectangular tunnel obtained using FDM model 60 Figure 3.6 Equivalent static loading with the HRM method for sub-rectangular tunnel 60 Figure 3.7 Shapes of tunnel cases (unit: m) [48] 63 Figure 3.8 Calibration flowchart of the three parameters 65 Figure 3.9 Obtained numerical results and fitting curves adopted for the parameters

β1, β2, β3 and β4 that created the parameter β 67 Figure 3.10 Coefficients fitting curves for the formulas of the parameters a and b1, b2, b3 and b4 that created the parameter b 68 Figure 3.11 Comparison of the incremental bending moments and normal forces calculated by the developed HRM method and numerical FDM calculation 69 Figure 3.12 Horizontal accelerations aH impact on extreme incremental internal forces of the sub-rectangular tunnel lining 70 Figure 3.13 Effect of Es on the extreme incremental internal forces of the sub-rectangular tunnel lining 71 Figure 3.14 Effect of the lining thickness on the extreme incremental internal forces

of the sub-rectangular tunnel lining 72 Figure 3.15 The cross-section dimensions influence on the extreme incremental internal forces of the sub-rectangular tunnel lining 73 Figure 3.16 Effect of the shape of cross-section on the extreme incremental internal forces of the sub-rectangular tunnel lining 75 Figure 3.17 Effect of burial depth of tunnel on the extreme incremental internal forces of the sub-rectangular tunnel lining 76

LIST OF TABLES

Table 1.1 Ratios of ground motion at depth to motion at ground surface (after Power

et al [130]) 10

Table 1.2 Ratios of peak ground velocity to peak ground acceleration at surface in rock and soil (adapted from Sadigh and Egan [134]) 11

Table 1.3 Summary of researches on the tunnel subjected to seismic loading classified by tunnel shapes and analyzing method 24

Table 2.1 Input parameters for the reference case of seismic loading 35

Table 3.1 Input parameters for the reference case for developing the HRM method 62

Table 3.2 Geometrical parameters of tunnel shape cases [48] 62

Table 3.3 Overview of the calibration process 64

Table 3.4 Soil properties [70],[145] 77

Table 3.5 Comparison of the results of the HRM method and FDM model 77

GENERAL INTRODUCTION

Background and Problematic Tunnels are an important component of the transportation and utility systems of cities They are being constructed at an increasing rate to facilitate the need for space expansion in densely populated urban areas and mega-cities Due to the interaction

of these structures with the surrounding soil and rock, underground structures are more resistant to earthquakes than structures at the ground surface Despite this, the failure of underground structures was recorded for some earthquakes which occurred around the world and damages reports were reported Considering the substantial scale and construction cost, and their critical role, this kind of infrastructures play in modern society an important role Even slight seismic loading impacts can lead to short-time shutdowns and to substantial direct and indirect damages Therefore, it is very important to carefully consider the seismic loading effect on the design, construction, operation, and risk assessment of tunnels

The behavior of underground structures under seismic loading was often studied

by different methods, including analytical methods, empirical methods, and numerical methods It should be noted that most of the researches were conducted considering circular or rectangular tunnels There are many other types of tunnel cross-sections, among them sub-rectangular tunnels were recently developed and are the object of this thesis

Advances in the development of user-friendly computer technology and the limitations of analytical methods have led to an increase of the numerical methods use for the design of tunnel linings The numerical models could be finite element or finite difference methods built by using in-house software or commercial software packages They allow to consider the complex interaction between the tunnel lining and the surrounding ground and elements of the tunneling process

Recently, the Hyperstatic Reaction Method (HRM), which is a finite element method, was used to compute the internal forces developed in tunnel linings This

method requires estimating the active loads of the soil mass applied directly on the tunnel lining The passive loads are induced by the soil reaction when the tunnel lining moves toward the soil medium The HRM method allows performing the calculations in a very short time with a good accuracy It is thus appropriate for optimization design processes The HRM method was successfully applied to design circular, noncircular tunnels (e.g., U-shaped tunnel) under both static and seismic loadings Recently, it was also developed to estimate the behaviour of sub-rectangular tunnels under static conditions

This research aims to develop numerical methods used to calculate incremental internal forces arising in sub-rectangular tunnel lining under seismic conditions An investigation of other parameters (tunnel lining, soil mass, etc.) influencing the tunnel structure behavior subjected to seismic loadings is also proposed

Objectives The present thesis focuses on introducing a numerical analysis of sub-rectangular tunnels under seismic loading Firstly, numerical models of sub-rectangular tunnels are developed based on numerical analyses of circular tunnels They are validated by comparison with well-known analytical solutions Secondly, the thesis focuses on developing the Hyperstatic Reaction Method (HRM) for sub-rectangular tunnels under seismic conditions The main objectives of this thesis include the following:

 Highlighting the behavior of sub-rectangular tunnels subjected to seismic loadings A special attention is paid to the soil-lining interface conditions

 Investigating the influence of parameters, like soil Young’s modulus, maximum horizontal accelerations, and lining thickness on the sub-rectangular tunnel behavior under seismic loadings

 Providing a new quasi-static loading scheme applied in the Hyperstatic Reaction Method (HRM) for sub-rectangular tunnels under seismic loading

Scope of this study

 Object of this study: Sub-rectangular tunnels supported by continuous lining The soil and tunnel lining material properties are assumed to be linearly elastic

 Scope of this study: Calculate incremental internal forces arising in rectangular tunnel lining under the seismic loading as well as investigation

sub-of the parameters (tunnel lining, soil mass, etc.) influencing the behavior sub-of sub-rectangular tunnel structure subjected to seismic loadings

Original Features For the best of the author’s knowledge, the originality features of the present work are:

 Behavior of sub-rectangular tunnels subjected to seismic loadings

 Proposition of a new quasi-static loading scheme using the HRM method for sub-rectangular tunnels under seismic loading

Thesis outline This thesis consists of 3 chapters After introducing the issue to be considered, Chapter 1 includes the review, analysis, and synthesis of scientific studies on underground structures subjected to seismic loadings The methods used to study the internal forces appearing in the tunnel lining under seismic loading are presented Finally, sub-rectangular tunnels which were recently developed and studied are introduced

Chapter 2 focuses on introducing a numerical analysis of sub-rectangular tunnels under seismic loadings The numerical model of the sub-rectangular tunnel is based

on the numerical analyses of circular tunnels They were validated by comparison with well-known analytical solutions This chapter aimed highlighting the differences between the sub-rectangular tunnels behavior compared with circular ones, subjected

to seismic loadings A special attention is paid to the soil-lining interface, i.e., full slip and no-slip conditions The influence of parameters, like soil Young’s modulus, maximum horizontal acceleration, and lining thickness of the sub-rectangular tunnels under seismic loadings are also investigated The results indicated a significant difference in the sub-rectangular tunnel behavior in comparison with the circular tunnel one when subjected to seismic loadings

Chapter 3 aims providing a new quasi-static loading scheme applied in the HRM method used for sub-rectangular tunnels under seismic conditions New equations allowing computations of the applied active loading as well as a varying spring stiffness coefficient for representing the soil-tunnel lining interaction are introduced The proposed equations are calibrated and validated through a numerical analysis considering a wide range of seismic magnitudes, soil properties, lining thicknesses, tunnel dimensions, tunnel geometries and burial tunnel depths The comparisons show that the developed HRM method can be effectively used for the preliminary seismic design of sub-rectangular tunnels

CHAPTER 1 LITERATURE REVIEW ON THE BEHAVIOUR OF UNDERGROUND STRUCTURES UNDER SEISMIC LOADING

1.1 Introduction Tunnels are an important component of the transportation and utility systems in urban and national systems They were being constructed at an increasing rate to facilitate the space expansion need in densely populated urban areas Considering their substantial scale and cost of construction, and their critical role, this kind of infrastructures play an important role in modern societies Even slight seismic loading impacts can lead to short-time shutdowns and substantial direct and indirect damages Therefore, it is very important to carefully consider the effect of seismic loading on the analysis, design, construction, operation, and risk assessment of tunnels

As tunnels are interacting with the surrounding soil and/or rock environment, they are more resistant to earthquakes than structures at the ground surface The destruction of underground constructions has recorded for many earthquakes taking place around the world (e.g., Kobe, Japan 1995; Chi Chi, Taiwan 1999; Bolu, Turkey Period 1999; Baladeh, Iran 2004; and more recently Sichuan, China in 2008) Reports

of damages [50],[158],[167] show that the effects interacting between the ground and tunnels were not conducted a systematic investigation In 1993, Wang highlighted the notion of relative flexibility as a key parameter for understanding the seismic-induced distortions of underground structures interacting with the surrounding ground Considering these incidents, it is clear that an improper designed underground structures are susceptible to wave propagation effects [14],[15],[16],[74],[75],[111], [166] (see Figure 1.1), Other detailed reviews of the seismic performance of tunnels and underground structures can be found in relevant publications [61],[69],[77],[94],[133],[168] (see Figure 1.2) These incidents were used as large-scale 'standard cases', to understand the interplay between structure and ground, with

the ultimate goal of verifying the capabilities of design methods Existing analysis and validation of material models involved [81],[89] or interface conditions [74],[90],[135] needed to capture the observed response It was shown that there is still a need to improve the quality of the design and computation of underground structures in the areas that can suffer from earthquakes

Figure 1.1 Summary of observed bored/mined tunnel damage due to ground

shakings [131]

Vietnam's territory is in a rather special position on the Earth's crust tectonic map and it exists a complex, diverse, and high-risk network of earthquakes There are studies of earthquakes such as statistics, localization, forecasting, assessment of the risk of earthquakes, and design [1],[2],[3],[4],[5],[6],[7],[8],[117],[118]

Figure 1.2 Typical failure modes of mountain tunnels reported during the 1999

Chi-Chi earthquake in Taiwan [160]

1.2 Seismic response mechanisms Earthquake effects on underground structures can be grouped into two categories: ground shaking and ground failure [159] or four categories: ground shaking, ground failure, land sliding and soil liquefaction [59] Ground shaking refers

to the ground vibration induced by seismic waves that propagate through the earth’s crust Figure 1.3 shows the ground response due to the various types of seismic waves:

 Body waves travel within the earth’s material They can contain either P waves (also known as primary or compressional or longitudinal waves) and

S waves (also known as secondary or shear or transverse waves) and they are able to travel in any ground direction

 Surface waves travel along the earth’s surface They are Rayleigh waves or Love waves The velocity and frequency of these waves are slower than body waves

Figure 1.3 Ground response to seismic waves [159]

The underground structures will be deformed when the ground will deform due

to the traveling waves Owen and Scholl [123] claimed that the behavior of an underground structure during a seismic event can be approximated to the one of an elastic beam subjected to deformations imposed by the surrounding ground Three

types of deformations express the response of underground structures to seismic motions (see Figure 1.4):

 Axial compression/extension

 Longitudinal bending

 Ovalling/racking

Figure 1.4 Type of tunnel deformations during a seismic event [123]

Axial tunnel deformations are generated by seismic wave components that induce motions parallel to the tunnel's axis and cause alternatively compressive and tension forces Bending strains are caused by seismic wave components that induce

Tunnel

Tunnel Before Wave Motion

Tunnel During Wave Motion

Tunnel Before Wave Motion

Tunnel During Wave Motion

Tension Compression

“Bottom”

Negative curvature

Positive curvature

particle motions perpendicular to the tunnel longitudinal axis They were not taken into account in this study, as they are generally oriented along the tunnel axis [159] Ovaling or racking deformations corresponding to circular and rectangular tunnel linings can be developed when shear waves propagate normally or nearly normally to the tunnel axis Penzien [129] and Hashash et al [70] indicated that the component that has the most significant influence on the tunnel lining behavior under seismic loading is the ovaling or racking deformations caused by the seismic shear wave or S-wave propagation In other words, they are the most crucial deformation modes for tunnels

On the other hand, the ground failures induced by earthquakes, may be caused

by liquefaction, fault motions, or slope failure (Figure 1.5) They may induce large permanent ground deformations to tunnels

Figure 1.5 Examples of the effects of seismically-induced ground failures on

tunnels [155]

Collapse

1.3 Research methods Expression of underground structures under seismic loading was often studied using different methods, including analytical methods, experience, numerical methods

The deformation of underground structures is often simulated in 2-dimensional plane strain problems under equivalent static load, not paying attention to the inertial force effects [70] Due to their simplicity, various analytical methods were developed that allow the determination of the internal forces generated in the supporting structures such as for rectangular and circular tunnels [21],[24],[43],[90], [126],[128],[129];[159] In general, analytical methods are often limited by assumptions [135]:

- Homogeneous isotropic soil masses, underground structures material with linear elastic behavior and mass lost;

- Circular tunnels usually are lining continuous structures with constant lining thicknesses;

- Construction procedures could not be considered

To overcome the analytical method drawbacks, experimental models were used

to better understand the physical nature of the processes and have paid a special attention to the soil-structure interaction under seismic loading conditions

Physical models were performed by various authors to investigate the functioning mode of underground structures and verify the current design/analysis methods Most physical models were constructed to collect measurement data and to verify design model [19],[30],[39],[42],[96] However, due to the complexity and high cost of this method, the results obtained are currently very limited

Recently, the trend is to use a two-dimensional numerical model [70],[124], [125],[135] or a three-dimensional model [91],[137]

When using a numerical method, seismic loadings are usually simulated with equivalent static loads However, most numerical methods using equivalent static loads still have the drawbacks of the analytical methods presented above The main

disadvantage of using equivalent static loads is that it does not pay attention to the tunneling performance change considering time under the seismic loadings Besides, these methods using the equivalent static load also often result in calculating internal forces lower than when applying real seismic loadings [132]

Compared to other methods, the time history analysis method using real earthquakes are the most complicated but therefore they are also able to provide the most accurate results However, these methods often require a long computation time and that is why their results are limited

It should be noted that most of the researches were conducted for underground structures with circular or rectangular tunnels In fact that, there are many other types

of tunnel cross-sections works, including sub-rectangular tunnels The research and development of methods to calculate internal forces and deformations arising in sub-rectangular structures under seismic loadings as well as surveying the influencing parameters (parameters of the tunnel lining structure, of the soil mass environment, ) is a matter of practical and scientific significance Besides, the current trend when holding underground works for construction in earthquake areas is to use highly flexible structures such as segmental tunnel linings The initial research results mentioned above have shown the indispensable effect of the joints in the tunnel structures in this case and need to be fully studied to achieve reliable lining tunnel designs

1.3.1 Analytical solutions The ovalling deformations are commonly simulated with a two-dimensional, plane strain configuration and are usually further simplified as a quasi-static case without taking into account the seismic interaction [70],[69]

Due to the simplicity of analytical solutions, various elastic closed-form solutions were developed to determine internal forces induced on circular tunnel linings due to seismic loading [21],[43],[126],[128],[129],[159] The works conducted by Hashash et al [69],[70] indicated the discrepancies between the

methods [128],[159], and used numerical analyses under the same assumptions to better understand the differences between the two solutions and their causes The comparisons demonstrated that the Wang’s solution provides a realistic estimation of the tunnel lining normal forces for a no-slip condition It has been recommended that Peinzen’s solution should not be used for a no-slip condition [70],[69] The works [17],[125],[126],[150] indicated a good agreement between their solution with the previous solutions [21],[159]

Generally, the analytical solutions are limited to the following assumptions [135]:

• The soil mass is assumed homogenous and the tunnel linings behavior have to

be linearly elastic and mass-less materials;

• Tunnels are usually of circular shapes with an uniform thickness without joints;

• The effect of the construction sequence is not studied

Considering a circular tunnel and its radius R located under the ground surface and subjected to a seismic loading using shear waves (Figure 1.6), the seismic-induced soil stress state can be treated as a shear-type stress This corresponds to compressive and tensile free-field principal stresses at 450 with a pure shear direction,

as shown in Figure 1.7 [126]

Figure 1.6 A circular tunnel (redrawn) [126]

Figure 1.7 Seismic shear loading and equivalent static loading (redrawn) [126]

The shear stresses can be estimated using the free-field shear strain γmax

[70],[69],[128]:

τ = .

( ) (1.1) where the shear strain 𝛾 can be determined as follows:

γ = (1.2)

Table 1.1 Ratios of ground motion at depth to motion at ground surface (after Power et al [130])

No Tunnel depth (m) Ratio of ground motion at tunnel

depth to motion at ground surface

Table 1.2 Ratios of peak ground velocity to peak ground acceleration at surface in rock and soil (adapted from Sadigh and Egan [134])

Moment magnitude, Mw

Ratio of peak ground velocity (cm/sec) to peak ground

acceleration (g) Source-to-Site Distance (Km)

Rock 6.5 7.5 8.5

6.5 7.5 8.5

6.5 7.5 8.5

Where Vmax is the peak shear wave velocity, Vs is the ground shear wave velocity, Es is the soil Young’s modulus, and νs is the soil Poisson’s ratio The maximum circualr tunnel lining ovaling in Figure 1.6 will occur at its major and minor axes at θ = 450 with respect to the spring line [126]

Vmax can be estimated through Table 1.1 and Table 1.2 [59],[69] Table 1.1 can

be used to determine the relationship between the ground motion at depth and at the ground surface Table 1.2 can be used to relate the known peak ground acceleration

to estimates the peak ground velocity in the absence of site-specific data

1.3.1.1 Analytical solutions due to a seismic loading considering a circular tunnel

Wang [159] may be the first person who proposed a closed-form solution for the structural tunnel lining forces under seismic loading conditions For a full-slip condition of the soil-lining interaction, the maximum normal forces (Nmax) and maximum bending moment (Mmax) can be expressed as follows:

Nmax= 𝐾

( )𝑅𝛾 (1.3)

Mmax= 𝐾

( )𝑅 𝛾 (1.4) Where:

𝐾 = ( ) (1.5) For the no-slip condition at the soil-lining interface, the formulation of Wang [159] for the maximum normal forces (Nmax) can be expressed as follow:

Nmax = 𝐾

( )𝑅𝛾 (1.6) Where:

𝐾 = 1 + [( ) ( ) ] ( )

[( ) ( ) ] (1.7)

C=

( )( ) (1.8) F=

( ) (1.9)

In equations from (1.2) to (1.9):

K1 = full-slip lining response coefficient;

K2 = no-slip lining response coefficient;

F = tunnel lining flexibility ratio;

C = tunnel lining compressibility ratio;

E = tunnel lining Young’s modulus;

ν = tunnel lining Poisson’s ratio;

R = tunnel radius;

t = tunnel lining thickness;

I = inertia moment of tunnel lining per unit length of the tunnel (per unit width);

Es = soil Young’s modulus;

νs = soil Poisson’s ratio;

γmax = maximum free-field shear strain;

θ = angle measured counter-clockwise from spring line on the right

Note that no solution was developed for calculating bending moments under a no-slip condition by Wang [159] It is suggested that the solution for full slip condition may be used for no-slip condition The more conservative estimations of the full slip condition are considered to offset the potential underestimation due to quasi-static representation of the seismic problem [70],[159]

Recently, Kouretzis et al [90] proposed an expression of the maximum bending moment under the no-slip condition to improve the method proposed by Wang [159]:

Mmax = ±(1 − 𝐾 − 2𝐾 )𝜏 (1.10) where 𝜏 is the maximum free field seismic shear stress:

𝜏 = ±𝑉 𝜌 𝐺 (1.11) With ρmax is the density of the surrounding ground, Gmax is the maximum ground shear modulus, and Vmax is the peak seismic velocity due to shear wave propagation

[( ) ( ) ] ( ) ( ) ( ) (1.12)

[( ) ( ) ] ( ) ( ) ( ) (1.13)

1.3.1.2 Analytical solutions due to seismic loading for rectangular tunnel Likewise with the analysis of ovaling deformations for circular tunnels, rectangular tunnels could be analyzed for the imposed racking deformations assuming propagating shear waves Besides, the walls and roof of the tunnel cross-section should be analyzed for seismic earth pressures

Figure 1.8 Definition of terms used in racking analysis of a rectangular tunnel

[159]

Racking Deformation Analysis: Wang [159] developed a simplified procedure comprising the soil-structure interaction for the analysis of racking of rectangular tunnels The proceeding was developed based on a series of seismic finite element analyses with various soil and tunnel structure's seismic properties The cases analyzed included the following conditions:

Free Field Soil Stiffness

P

 The ratio of the depth to the center of the structure, H, to the structure height,

h, ranged from 1.1 to 2.0 (Figure 1.8);

 Soil shear modulus surrounding the structure between 11 to 72 MPa, corresponding to shear wave velocities of 75 to 200 m/sec;

 The vertical distance between the bottom of the tunnel structure and the top

of underlying stiff soils/rock was equal to or greater than the tunnel structure height;

 Rigid body rotation was excluded;

 Tunnel structures widths, w, ranged between 4.6 to 27.5 m, and tunnel structure heights, h, ranged from 4.6 to 8 m;

 Time histories of artificial earthquake ground motion, illustrating western and northeastern U.S earthquakes, were used

Under the seismic loading from the maximum design earthquake, inelastic deformations in the structure may be allowed depending on the performance criteria and provided that the overall stability of the tunnel is maintained Detailing of the structural members and joints should be provided for an adequate internal strength, ductility, and energy absorption capability

Wang [159] investigated the influences of relative stiffness, tunnel structure geometry, ground motion characteristics, bury depth, and foundation stiffness on the racking coefficient, Rr Results of these analyses showed that the relative stiffness between the rectangular structure and the surrounding environment has the most significant influence on Rr (Figure 1.9) The effects of tunnel structure geometry and ground motion characteristics were insignificant Additional analyses were performed to determine the embedment depth influence In these analyses, the bury tunnels depths ranged from 0 to 12 m for a tunnel structure with a height of 4.57m were considered The results showed that the calculated racking coefficient is relatively independent of the embedment depth when H/h > 1.5 (Figure 1.8) Penzien [129] observed that when H/h reduces from 1.5 to 0.5, the racking coefficient decreases slowly, corresponding to the reduction of about 20 percent

Figure 1.9 Racking coefficients for rectangular tunnels [59]

1.3.2 Physical tests Physical model tests and numerical analysis were used to obtain a better understanding of the physical problem and in particular of the soil-structure interaction phenomenon They allow to overcome the drawbacks of analytical methods

Physical model tests were performed by many researchers to investigate the performance of tunnel structures and to check the current design/analysis methods Most of these studies were focused on data for the validation of the design models [19],[27],[39],[42],[96],[97] and behavior of tunnel lining [71]

Most current experimental studies investigated the response of tunnels underground shaking, while seismic damages to the tunnel response is much less

Flexibility Ratios, F r Flexibility Ratios, F r

Modified from Wang [159]

noticeable, mainly due to available laboratory facilities limitations There are many physical model tests which have focused on three main tests: small-scale centrifuge dynamic tests, which focuses mainly on tunnel linings behavior by the transversal seismic induced; reduced scale 1g shaking table tests, considering both longitudinal and transversal shaking directions, and static tests, focusing on the investigation of the response of joints or tunnels shear keys

Dynamic centrifuge tests: Centrifuge modeling of the seismic behavior of tunnels was considered for examining both the effects of ground failures, such as slope failures near the tunnel portal, the tunnel lining shearing due to fault crack and flotation induced by liquefaction, as well as ground shaking [35],[36],[93] The behavior of circular and rectangular tunnels in sand in the transversal direction [38], [40],[41],[96] and Lanzano et al [97], Bilotta et al [18] confirmed that residual internal forces arise and the value of these residual internal forces seems to depend more on the peak ground acceleration (PGA) rather than on the number of cycles the tunnel is subjected during the seismic motion

This coupled racking-rocking deformation pattern of box-type tunnels was also verified by a series of dynamic centrifuge tests [148],[149],[150],[151],[153] Tsinidis et al [150] and Ulgen et al [156] performed similar tests on rectangular tunnels model in dry sand The results showed that the measured racking deformations in the dynamic centrifuge tests had a good fit with the analytical estimates derived by the solutions of Penzien [129] and Bobet [21] The comparison efficiency was found to depend on the soil-tunnel lining interaction [159], with better matches obtained for stiffer tunnels (tunnels with a flexibility ratio F < 1.0)

Using dynamic centrifuge tests, Hashash et al [68] recently indicated that, in urban areas, green-field conditions cannot be applied in the surrounding area of tall buildings The interaction phenomena between the building and the tunnel is able to seriously change the performance of cut-and-cover tunnels It depends on the geometric details of both the underground structures and foundation of the building Some conclusions from centrifuge tests are: