MINISTRY OF EDUCATION AND TRAINING
HANOI UNIVERSITY OF MINING AND GEOLOGY

BEHAVIOR OF SUB-RECTANGULAR TUNNELS UNDER SEISMIC LOADING

PhD THESIS

HANOI, May 2022


BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC MỎ - ĐỊA CHẤT

NGHIÊN CỨU ỨNG XỬ CỦA KẾT CẤU CHỐNG TRONG ĐƯỜNG HẦM
TIẾT DIỆN HÌNH CHỮ NHẬT CONG CHỊU TẢI TRỌNG ĐỘNG ĐẤT

LUẬN ÁN TIẾN SĨ KỸ THUẬT

HÀ NỘI - 05/2022


MINISTRY OF EDUCATION AND TRAINING
HA NOI UNIVERSITY OF MINING AND GEOLOGY

BEHAVIOR OF SUB-RECTANGULAR TUNNELS UNDER SEISMIC LOADING

Major: Underground construction engineering
Code: 9580204

PhD THESIS

SUPERVISORS:

1 Asso Prof , Dr
2 Prof ,

HA NOI, May 2022


BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC MỎ - ĐỊA CHẤT

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 Cơng trình ngầm
Mã số ngành: 9580204

LUẬN ÁN TIẾN SĨ KỸ THUẬT

NGƯỜI HƯỚNG DẪN KHOA HỌC:

HÀ NỘI - 05/2022


i

ACKNOWLEDGEMENTS
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 , and Professor 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 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


ii

LỜI CẢM ƠN
Luận án này được thực hiện tại Bộ mơn Xây dựng cơng 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
Các Thầy ln 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 cơ Bộ mơn Xây dựng cơng 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 đồn 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 vơ cùng biết ơn tới gia đình đã luôn bên cạnh với sự kiên
nhẫn Luận án này sẽ khơng được bắt đầu, được thực hiện và hồn thành nếu
khơng
có sự hỗ trợ của gia đình


iii


GUARANTEE

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


iv

LỜI CAM ĐOAN

Tơi xin cam đoan đây là cơng 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 cơng
trình
nào

Nghiên cứu sinh


v

SUMMARY

The principal purpose of this Ph D thesis is to study the behavior of subrectangular 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

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


vi

TĨM 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 mơ hình số 2D cho đường hầm tiết diện hình chữ nhật cong chịu tải

trọng
tĩnh tương đương trên cơ sở mơ hình của đường hầm tiết diện hình trịn 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 mơ 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;
Mơ hình số; Tĩnh tương đương


vii

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 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


1 5 Conclusions
27
CHAPTER 2: NUMERICAL STUDY ON THE BEHAVIOR OF SUBRECTANGULAR 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


viii

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
PUBLISHED AND SUBMITTED MANUSCRIPTS
REFERENCES
84

79
83


ix

LIST OF NOMENCLATURE

Abbreviations
2D

Two-dimensional

3D

Three-dimensional

DOT

Double-O-tube

FDM

Finite difference method


FEM

Finite element method

fs

Full slip

HRM

Hyperstatic Reaction Method

MF

Multi-circular face

ns

No-slip

PGA

Peak ground acceleration

PIV

Particle image velocimetry

SR


Sub-rectangular tunnel

TBM

Tunnel boring machine

Symbols
aH

Peak horizontal acceleration at the ground surface

a, b, β

Dimensionless factors

C

Tunnel lining compressibility ratio

c

Soil cohesion

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


x

Gmax

Maximum ground shear modulus

h

Tunnel height

H

Tunnel depth

I


Inertia moment of tunnel lining per unit length of the tunnel

K

Soil bulk modulus

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

Mw

Moment magnitude

N

Incremental Normal forces

Nmax

Maximum incremental normal forces

Nmin

Minimum incremental normal forces

plim


Maximum reaction pressure

R

Tunnel radius

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


xi


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

ν

Tunnel lining Poisson’s ratio

νs

Soil Poisson’s ratio

ρmax

Soil density

τ

Shear stresses applied at the far-field boundary

φ

Soil internal friction angle

�

Soil initial spring stiffness

�

Normal stiffness


�

Tangential stiffness

�

Vertical loads

�

Horizontal loads

[K]

Stiffness matrix

[S]

Nodal displacement matrix

[F]

Nodal forces matrix


xii

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
ChiChi 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


xiii

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 subrectangular 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


xiv

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 subrectangular 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


xv

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