Nghiên cứu ứng xử của kết cấu chống trong đường hầm tiết diện hình chữ nhật cong chịu tải trọng động đất ttta

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

PHAM VAN VI

BEHAVIOR OF SUB-RECTANGULAR TUNNELS UNDER
SEISMIC LOADING

Major: Underground construction engineering
Code: 9580204

THESIS SUMMARY

HANOI – 2022


The thesis is completed at: Underground and Mining Construction
Department, Faculty of Civil Engineering, Hanoi University of Mining and
Geology, Vietnam

Scientific supervisors:
1. Asso.Prof., Dr. Do Ngoc Anh
2. Prof., Dr. Daniel Dias

Reviewer 1: Prof., Dr. Do Nhu Trang
University of Transport Technology
Reviewer 2: Dr. Ngo Ngoc Thuy
Military Technical Academy
Reviewer 3: Asso.Prof., Dr. Nguyen Xuan Man
Hanoi University of Mining and Geology

The thesis will be defended before the Academic Review Board at the
University level at Hanoi University of Mining and Geology at ….., ….. of date
….. month ….. year ……….

The thesis is available at the: National Library of Vietnam or the Library
of Hanoi University of Mining and Geology


GENERAL INTRODUCTION
1. The necessity of the study
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 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 research results 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.
2. The purpose of thesis
This research aims to develop numerical methods used to calculate incremental
internal forces arising in sub-rectangular tunnel lining under the seismic condition as well
as investigation of the parameters (tunnel lining, soil mass, etc.) influencing the behavior

of sub-rectangular tunnel subjected to seismic loading. 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
3. 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.

1


- Scope of this study: Calculate incremental internal forces arising in sub-rectangular
tunnel lining under the seismic loading as well as investigation of the parameters (tunnel
lining, soil mass, etc.) influencing the behavior of sub-rectangular tunnel structure
subjected to seismic loadings.
4. Methodology of this study
- Acquiring, inheriting: Synthesize, analyze and evaluate existing literature to absorb
and inherit previous research results related to calculating internal forces in tunnel
structures under seismic loading.
- Numerical method: Simulation of tunnel structures using FLAC3D (Itasca, 2012)
(FDM), Plaxis V8.6, and Matlab software to calculate incremental internal forces in the
sub-rectangular tunnel under seismic loading.
5. Scientific and practical meaning of this thesis
- Scientific Meaning: The results of this thesis applied to sub-rectangular tunnels
under seismic loading can be a useful reference for scientists, contributing to diversifying

approaches to calculation, design for tunnels is subjected to seismic loading.
- Practical meaning: The thesis research results can be effectively used for the
preliminary seismic design of sub-rectangular tunnels
6. The new highlights of this thesis
- Studying the behavior of sub-rectangular tunnels under seismic loading. Special
attention is paid to the soil-lining interface conditions
- Providing a new quasi-static loading scheme applied in the HRM method for subrectangular tunnels under seismic loadings.
7. Primary argument of this thesis
- Argument 1: A significant difference in the behavior of the sub-rectangular tunnel
compared with the circular tunnel one when subjected to seismic loadings. Special
attention is paid to the soil-lining interface, i.e., full slip and no-slip conditions.
- Argument 2: Providing a new quasi-static loading scheme applied in the HRM
method used for sub-rectangular tunnels under seismic loading. The proposed equations
are validated through numerical analyses.
8. Thesis outline
This thesis consists of general introduction, 3 chapters, general conclusions,
perspectives, published manuscripts and references. The whole content of this thesis is
illustrated in 110 pages of A4 size, including 9 tables and 47 figures.

2


CHAPTER 1: LITERATURE REVIEW ON THE BEHAVIOUR OF
UNDERGROUND STRUCTURES UNDER SEISMIC LOADINGS
1.1. Introduction
Tunnels are an important component of the transportation and utility systems in both
urban and national systems. They are being constructed at an increasing rate to facilitate
the need for space expansion in densely populated urban areas.
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 (Nguyen et al., 2009; Bui, 2010; Mai Duc Minh, 2011; TCVN,
2012; Nguyen et al., 2012; Nguyen et al., 2014; Nguyen et al., 2015; Nguyen Đinh Xuyen,
2015; Le et al., 2015; Le Bao Quoc, 2015; Do Ngoc Anh, 2016).
As tunnels are interacting with the surrounding soil and/or rock environment, they are
more resistant to earthquakes than structures at the ground surface. Despite this, the
destruction of underground construction has been recorded at many earthquakes taking
place around the world. Other detailed reviews of the seismic performance of tunnels and
underground structures can be found in relevant publications (Hashash et al., 2001; Gazetas
et al., 2005; Lanzano et al., 2008; Roy and Sarkar, 2016; Yu et al., 2016; Jaramillo, 2017).
Therefore, it is important to consider the influence of seismic loading on the analysis,
design, construction, operation, and risk assessment of tunnel structures.
1.2. Seismic response mechanisms
Earthquake effects on underground structures can be grouped into two categories:
ground shaking and ground failure (Wang, 1993) or four categories: ground shaking,
ground failure, land sliding and soil liquefaction (FHWA, 2004). The ground response due
to the various types of seismic waves:
- Body waves travel within the earth’s material;
- Surface waves travel along the earth’s surface.
Three types of deformations express the response of underground structures to seismic
motions:
- Axial compression/extension;
- Longitudinal bending;
- Ovalling/racking.
On the other hand, the ground failures induced by earthquakes:
- Failures may be caused by liquefaction;
- Fault motions;
- Slope failure.

3

1.3. Research methods
Expression of underground structures under seismic loading is often studied using
different methods:
- Analytical methods
- Experience
- Numerical methods: quasi-static and Numerical full seismic analysis
1.4. Sub-rectangular
Circular tunnels have a low cross-section space-utilization ratio while rectangular
tunnels have low stability. Recently, to overcome these limitations, sub-rectangular tunnels
have been studied and applied. The sub-rectangular tunnels have the following advantages:
- Having great advantages in terms of underground space use;
- Reduce the volume of earthwork excavation;
- Avoid stress concentration at four corners compared with rectangular tunnels.
Sub-rectangular tunnels have been applied and studied with real ratio or reduction ratio
(Liu et al., 2018; Zhang et al., 2017; Konstantin et al., 2017; Zhu et al., 2017; Zhang et al.,
2019), numerical analyses (Do et al., 2020). However, the above studies only study the
sub-rectangular cross-section works with static loads but do not mention the works under
seismic loading.
1.5. Conclusions
Many research works in the world on underground structures when subjected to
seismic loading allow understanding and predicting behavior of underground structures.
However, these researches mainly focus on underground works with circular and
rectangular sections, no research has been done for underground works with the subrectangular tunnel when subjected to seismic loading. This is the main research object of
this thesis.
CHAPTER 2: NUMERICAL STUDY ON THE BEHAVIOR OF SUBRECTANGULAR TUNNEL UNDER SEISMIC LOADING
In this chapter, a 2D finite-difference numerical model of a sub-rectangular tunnel
under seismic loading is proposed. It is developed based on the modeling of a circular
tunnel which is validated by comparing the results obtained using well-known analytical

solutions (Wang, 1993; Hashash et al., 2005; Kouretzis et al., 2013). Such parameters as
soil Young’s modulus, maximum horizontal acceleration, and lining thickness on the
tunnel behavior under seismic loadings were carefully examined and their influence was
evaluated. In the study, particular attention was drawn to analyzing the soil-lining interface.
Different behaviors of sub-rectangular and circular shaped tunnels under seismic loadings
were compared based on numerical modeling.
4


2.1. Numerical simulation of the circular tunnel under seismic loading
2.1.1. Reference sub-rectangular tunnel case study- Shanghai metro tunnel
Parameters of a sub-rectangular express tunnel in Shanghai, China are used as the
reference case in this study (Do et al., 2020). The sub-rectangular tunnel dimensions are
9.7m in width, 7.2m in height and 60m2 in cross-section area (Figure 2.1). The tunnel is
supported by a segmental concrete lining of 0.5m. For simplification purposes, a
continuous lining was adopted without considering the effect of joints. Based on this
reference sub-rectangular tunnel, a circular tunnel with an external diameter of 4.89m and
75m2 in cross-section area which has an equivalent utilization space area, is considered for
comparison purposes (Figure 2.2).
o1 (0, 6350)

500

R4850

o

o

4

(500, 0)

(-3400, -1930)
7

6200

3
(-500, 0)

8
38
R4

utilization space area
Circular tunnel

(3400, -1930)
6

o

500

6200

o

5
(3400, 1930)

88

o

8
(-3400, 1930)

o

500

500

R4850

R500

R4
8

R500

500

R9450
R9450

500

8700

500

500

8700

500

o2 (0, -6350)

Figure 2.1. Sub-rectangular express tunnel
in Shanghai (Do et al., 2020), distances in
millimeters

Figure 2.2. Circular tunnel with the same
utilization space area, distances in
millimeters.

2.1.2. Numerical model for the circular tunnel
A numerical model for circular tunnels was developed using a finite difference
program (FLAC3D) (Itasca, 2012). The purpose was to investigate the behavior of circular
tunnel linings under quasi-static loading and make a comparison with those obtained by an
analytical solution. Similar to the research work of Sederat et al. (2009), Naggar and
Hinchberger. (2012), and Do et al. (2015), ovaling deformations due to the seismic loading
are imposed as inverted triangular displacements, along with the model lateral boundaries.
Uniform horizontal displacements are applied along the top boundary (Figure 2.4). The
magnitude of the prescribed displacements assigned at the top of the model is dependent
on the maximum shear strain max, estimated based on the maximum horizontal acceleration

aH. The bottom of the model is restraint in all directions.

5


Figure 2.4. Geometry and quasi-static loading conditions for the circular tunnel model
(Do et al., 2015)
Table 2.1. Input parameters for the reference case of seismic loading
Parameter
Symbol
Unit
Soil properties
Unit weight
γ
MN/m3
Young’s modulus
Es
MPa
Poisson’s ratio
νs
Internal friction angle
φ
degrees
Cohesion
c
MPa
Lateral earth pressure coefficient
K0
Depth of tunnel
H

m
Peak horizontal acceleration at ground
aH
g
surface
Moment magitude
Mw
Distance of site source
Km
Tunnel lining properties
Young’s modulus
E
MPa
Poisson’s ratio
ν
Lining thickness
t
m
External diameter
D
m

Value
0.018
100
0.34
33
0
0.5
20

0.5
7.5
10
35000
0.15
0.5
9.76

2.2. Validation of circular tunnel under seismic loading
For validation purposes of the numerical model subjected to quasi-static loading, the
well-known analytical solution proposed by Wang, (1993) and thereafter improved by
Kouretzis, (2013) was used for comparison with the results obtained from the numerical
model. The soil and tunnel lining material properties in numerical models are assumed to
be linearly elastic. Figure 2.7 illustrates the distribution of the incremental internal forces
induced in the tunnel lining under seismic loading. Conditions of lining and soil interaction,
6


when using the Wang solution and FDM were considered for both cases of no-slip and full
slip. The soil and tunnel lining parameters fed into the model are presented in Table 2.1.
- It can be seen that results obtained by numerical and analytical models are in very
good agreement, The maximum difference is smaller than 2 %.
- Figures 2.7a and 2.7c show that the maximum incremental bending moment in the
full-slip case is 14% larger than the one obtained in the no-slip case.
- The maximum incremental normal forces in the full-slip case are smaller than that
of the no-slip case (Figure 2.7b and Figure 2.7d).
Wang solution:

45°

45°
No-slip case: Mmax = 0.738 MNm/m

No-slip case: Nmax = 0.894 MN/m

Full slip case: Mmax = 0.845 MNm/m

Full slip case: Nmax = 0.173 MN/m

b) Incremental Normal Forces

a) Incremental Bending Moments

Numerical solution (FDM):

45°

45°
No-slip case: Mmax = 0.741 MNm/m

No-slip case: Nmax = 0.903 MN/m

Full slip case: Mmax = 0.834 MNm/m

Full slip case: Nmax = 0.169 MN/m

c) Incremental Bending Moments

d) Incremental Normal Forces

Figure 2.7. Distribution of the incremental internal forces in the circular tunnel by Flac3D
and Wang solution.

7


In the section, a parametric study is conducted to highlight the behavior of circular
tunnel lining subjected to quasi-static loadings considering the effect of Young’s modulus
Es, maximum horizontal seismic acceleration aH, and tunnel lining thickness t variations.
For both the no-slip and full-slip conditions, numerical results show a very good agreement
with the analytical solution. The difference under 2% for both the extreme incremental
bending moments and normal forces is obtained.
2.3. Numerical simulation of the sub-rectangular tunnel under seismic loading

Figure 2.11. Geometry and quasi-static loading conditions in the numerical model of a
sub-rectangular tunnel
In this section, a numerical model was developed for the sub-rectangular tunnels cased
using similar soil parameters, lining material, and modeling processes to consider the static
and seismic loadings introduced above. Only the tunnel shape is modified into a subrectangular geometry and the gravity effect is taken into consideration. The mesh consists
of a single layer of zones in the y-direction, and the dimension of the elements increases as
one moves away from the tunnel (Figure 2.11). The geometry parameters of subrectangular tunnels are presented in Figure 2.1 and other parameters presented in Table 2.1
are adopted.
2.4. Parametric study of sub-rectangular tunnels in quasi-static conditions
Figure 2.14 introduces the incremental bending moments and normal forces induced
in the sub-rectangular tunnel linings subjected to seismic loadings and considering both
no-slip and full slip conditions. Parameters of the reference case presented in Table 2.1 are
adopted.
- Extreme incremental bending moments and normal forces observed in the subrectangular tunnel appear at the tunnel lining corners where the smaller lining radii are
located.
- Absolute extreme incremental bending moments for the no-slip condition are

always more than the full slip ones. This relationship is opposite to the one observed in the
cases of the circular-shaped tunnel. It is clear that the behavior of sub-rectangular and
circular tunnels are completely different under seismic loading.

8


In the following sections, a numerical investigation was conducted to highlight the
behavior of a sub-rectangular tunnel compared with a circular shape. These two tunnels
have the same utilization space area and are twice subjected to seismic loadings while
considering the effect of parameters, like the horizontal seismic acceleration, soil
deformation modulus, and lining thickness. Effects of the soil-lining interface condition
are also investigated.

33°

33°
No-slip case: Mmax = 0.900 MNm/m

No-slip case: Nmax = 0.791 MN/m

Full slip case: Mmax = 0.807 MNm/m

Full slip case: Nmax = 0.159 MN/m

a) Incremental Bending Moment

b) Incremental Normal Forces

Figure 2.14. Distribution of the incremental bending moments and normal forces in the

sub-rectangular tunnel
2.4.1. Effect of the peak horizontal seismic acceleration (aH)
Shear strain values from 0.038 to 0.57% corresponding to a range of a maximum
horizontal acceleration varying from 0.05g and 0.75g were adopted in this study. In
general, high seismic loadings are implied by a high horizontal acceleration aH, and
therefore shear strain values of γmax, result in high absolute extreme incremental bending
moments and normal forces. The relationship is quite linear (Figure 2.15).
- For the no-slip condition, absolute extreme incremental bending moments in the
sub-rectangular lining are 20% larger than the circular ones.
- For the full slip condition, absolute extreme incremental bending moments in the
circular lining are approximately 4% greater than the sub-rectangular ones.
- In the case of sub-rectangular linings, absolute extreme incremental bending
moments for the full slip condition are always lower by about 10% than the no-slip ones.
This relationship is opposite to the one observed in the cases of the circular-shaped tunnel
(Figure 2.15a).
- It can be seen in Figure 2.15b that for both shapes of tunnels, the absolute extreme
incremental normal forces for the no-slip condition are approximately 80% larger than the
full slip ones.
The absolute extreme incremental normal forces of the sub-rectangular lining are
approximately 9% lower than the circular lining ones, for both the no-slip (Figure 2.15b).

9


1.5

1

1

Extrme Incremental Normal Forces N
(MN/m)

Extreme Incremental Bending Moment M
(MNm/m)

1.5

0.5
0

0.5
0

-0.5

-0.5
-1

-1.5

Mmax_SR_ns
Mmax_Circular_ns
Mmin_SR_ns
Mmin_Circular_ns

-2

-1

-1.5

Mmax_SR_fs
Mmax_Circular_fs
Mmin_SR_fs
Mmin_Circular_fs

-2.5

Nmax_SR_ns
Nmax_Circular_ns
Nmin_SR_ns
Nmin_Circular_ns

-2

Nmax_SR_fs
Nmax_Circular_fs
Nmin_SR_fs
Nmin_Circular_fs

-2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

0.1

0.2

aH (g)

0.3

0.4

0.5

0.6

0.7

0.8

aH (g)

a) Incremental bending moments

b) Incremental normal forces

Figure 2.15. Effect of the aH value on the internal forces of circular and sub-rectangular
tunnel linings
2.4.2. Effect of the soil’s Young’s modulus (Es)
Soil Young’s modulus values are assumed to vary in the range from 10 to 350 MPa.
The other parameters based on the reference case are assumed (Table 2.1). It can be seen
from Figure 2.16 that:
1.5

1

Extreme Incremental Normal Forces N
(MN/m)

Extreme Incremental Bending Moment M
(MNm/m)

1.25

0.75
0.5
0.25

Mmax_SR_ns

Mmax_Circular_ns
Mmin_SR_ns
Mmin_Circular_ns

0
-0.25

Mmax_SR_fs
Mmax_Circular_fs
Mmin_SR_fs
Mmin_Circular_fs

-0.5
-0.75

1
0.5
0
-0.5
-1
-1.5
Nmax_SR_ns
Nmax_Circular_ns
Nmin_SR_ns
Nmin_Circular_ns

-2
-2.5

-1

-1.25

Nmax_SR_fs
Nmax_Circular_fs
Nmin_SR_fs
Nmin_Circular_fs

-3

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

Young's Modulus, Es (MPa)

Young's Modulus, Es (MPa)

a) Incremental bending moments

b) Incremental normal forces

Figure 2.16. Effect of the Es value on the internal forces for the circular and subrectangular tunnel linings
- For the no-slip condition: Figure 2.16a also shows greater absolute extreme
incremental bending moments induced in sub-rectangular tunnels compared with circular
tunnels having the same utilization space area.
- For the full slip condition:
+ Absolute extreme incremental bending moments in the circular tunnel are greater
than the sub-rectangular ones for Es values smaller than approximately 150 MPa.
+ When Es values are larger than 150 MPa, absolute extreme incremental bending
moments developed in circular tunnels are smaller than in sub-rectangular tunnels.
10

- Figure 2.16b indicates that an increase of Es value causes a significant corresponding
increase of the absolute extreme normal forces in both sub-rectangular and circular tunnels
for the no-slip condition. But it induces an insignificant change in absolute extreme
incremental normal forces for the full slip condition.
- Absolute extreme incremental normal forces in the sub-rectangular tunnels are
generally 9% smaller than for the circular ones.
2.4.3. Effect of the lining thickness (t)
The lining thickness t is assumed to vary in the range between 0.2 to 0.8 m, while other
parameters introduced in Table 2.1 were adopted. The results presented in Figure 2.17
indicate that the lining thickness has a great effect on the incremental internal forces for
both sub-rectangular and circular tunnels under seismic loadings.
1

1.5

Extreme Incremental Normal Forces N
(MN/m)

Extreme Incremental Bending Moment M
(MNm/m)

2

1
Mmax_SR_ns
Mmax_SR_fs
Mmax_Circular_ns
Mmax_Circular_fs
Mmin_SR_ns

Mmin_SR_fs
Mmin_Circular_ns
Mmin_Circular_fs

0.5
0

-0.5
-1

-1.5
-2

0.5
0
-0.5
-1
Nmax_SR_ns
Nmax_Circular_ns
Nmin_SR_ns
Nmin_Circular_ns

-1.5

Nmax_SR_fs
Mmax_Circular_fs
Nmin_SR_fs
Mmin_Circular_fs

-2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Lining thickness (m)

Lining thickness (m)

a) Incremental bending moments

b) Incremental normal forces

Figure 2.17. Effect of the lining thickness on the incremental internal forces of the
circular and sub-rectangular tunnel linings
- For the no-slip condition, absolute extreme incremental bending moments of the
sub-rectangular linings are always larger than the circular ones (Figure 2.17a). The
discrepancy declined gradually from 124% to 6%, corresponding to the lining thickness
increase from 0.2 to 0.8 m.
- In the full slip conditions:
+ When t < 0.5 m, the absolute extreme incremental bending moments of the subrectangular linings are still larger than the circular ones;
+ When t > 0.5 m, Figure 2.17a proves an opposite result. Thus, larger absolute
extreme incremental bending moments on the circular tunnels are observed.
- It can be seen in Figure 2.17b that the incremental normal forces in the no-slip
condition are always larger than for the full slip ones.
- In comparison with the incremental normal forces of the circular lining, incremental
normal forces in the sub-rectangular lining are lower by about 9% and 25% for no-slip and
full slip conditions, respectively (Figure 2.17b).
11

The results indicated that an increase in lining thickness is not good solution for design
of tunnel linings under seismic loadings.
2.5. Conclusion
Based on the research results, conclusions can be deducted as follows:
- The horizontal acceleration aH, soil’s Young’s modulus Es, and lining thickness t
have a great effect on the incremental internal forces induced in both sub-rectangular and
circular tunnels for both no-slip and full slip conditions;
- The results proved that the soil-lining interface conditions have a great influence on
the behavior of sub-rectangular tunnels. This is completely different when comparing the
behavior circular-shaped tunnels. Indeed, while the absolute extreme incremental bending
moments of a circular tunnel for the no-slip condition are smaller than the corresponding
full slip ones, the absolute extreme incremental bending moments of sub-rectangular
tunnels for the no-slip condition are greater than the corresponding full slip ones. That is
opposite to the trend observed in circular tunnel linings;
- Absolute extreme incremental normal forces in sub-rectangular tunnels are
approximately 9% smaller than the circular ones;
- The no-slip condition is the most unfavorable internal force case for the subrectangular tunnel subjected to seismic loading
- The results indicated that an increase in lining thickness is not good solution for
design of tunnel linings under seismic loadings.
CHAPTER 3: A NEW QUASI-STATIC LOADING SCHEME FOR THE
HYPERSTATIC REACTION METHOD – CASE OF SUB-RECTANGULAR
TUNNELS UNDER SEISMIC CONDITION
The HRM method was successfully applied to estimate the seismic-induced structural
forces in a circular tunnel lining considering pseudo-static condition (Do et al., 2015; Sun
et al., 2020). Based on the in-plane shear stresses applied on the tunnel lining proposed by
Peinzen and Wu (1998) and Naggar et al. (2008), Do et al. (2015) applied a set of
dimensionless parameters to change the external loading magnitude of the seismic-induced
shear stresses. After Do et al. (2015), Sun et al. (2021) additionally considered the groundtunnel interaction effect on the applied external loadings by using a dimensionless factor
to realistically describe the ground-tunnel interaction.

This chapter aims to introduce a new pseudo-static loading scheme acting on the tunnel
lining, when using the HRM method for estimating seismic-induced structural forces in
sub-rectangular tunnels in homogeneous isotropic grounds.
- Firstly, the mathematical formulation of this method is presented.

12


- A new external loading scheme applied on sub-rectangular tunnels considering a
pseudo-static condition is proposed. Three dimensionless parameters are introduced to
include the applied active loads and the ground-tunnel interactions.
- Then, the constitutive equations employing these parameters are proposed by
calibrating the HRM solution applying numerical computations.
- Finally, an extensive validation of the developed HRM method is conducted
considering several case scenarios including different seismic magnitudes, ground
properties, lining thickness, tunnel geometry, tunnel dimensions and tunnel burial depth.
3.1. Fundamental of HRM method applied to sub-rectangular tunnel under static
loading
The HRM method is based on the Finite Element Method (FEM) which can be used
for analyzing the internal forces and displacements induced in the tunnel lining. The
method was developed for the analysis of segmental and continuous tunnel linings under
static loads (Oreste, 2007; Du et al., 2018). Recently, Do et al. (2020) have used the HRM
method to study the behavior of square and sub-rectangular tunnels under static loading
(Figure 3.1).

O
EI, EA

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. (Do et al., 2020)
3.2. HRM method applied to sub-rectangular tunnel under seismic conditions
When using HRM for seismic tunnel design, it is necessary to define the external
loads that act on the tunnel lining. It is assumed that the ovaling deformation of the crosssection is the most critical one for circular tunnels subjected to seismic loadings (Hashash,
2001; Lu et al., 2017; Sun et al., 2019), as illustrated in Figure 3.4a. Therefore, the seismicinduced stresses and deformations can easily be determined when the external shear
13


stresses are applied at the far-field boundary, as illustrated in Figure 3.4b. The acting shear
stress, τ, can be estimated using the free-field shear strain γmax (Penzien & Wu, 1998;
Hashash et al, 2001):

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.
Similarly, when applying shear stress to the far-field boundary, the critical state of the
sub-rectangular tunnel subjected to seismic loading causes a sub-ovaling deformation of
the tunnel lining as seen in Figure 3.4c. This result is obtained by using a finite-difference
model (FDM) and incremental internal forces are presented in Figure 3.5 (from Figure 2.10
for no-slip condition).
Base on the incremental internal forces in the sub-rectangular tunnel lining obtained
using the FDM model (Figure 3.5). The equivalent static loading scheme for the HRM
method in Figure 3.6 is determined which contains a couple of dimensionless parameters
(a) and (b).

14


Nmax = 0.791 (MN/m)

Mmax = 0.900 (MNm/m)

a) Incremental Bending Moment

b) Incremental Normal Forces

Figure 3.5. Incremental bending moments and normal forces of sub-rectangular tunnel
obtained using FDM model.

Figure 3.6. Equivalent static loading with the HRM method for sub-rectangular tunnel.
In the HRM, the ground interacts with the tunnel support through normal and
tangential springs connected to the nodes of the lining structure (Figure 3.1) which are
respectively represented by kn and ks, and estimated by the ground initial stiffness η0. In
sub-rectangular tunnels, the lining parts radius vary along the tunnel periphery, the initial
stiffness of the ground η0 will then change depending on the radius (Do et al., 2020):
𝜂

,

=𝛽

(3.8)

⋅

Where νs and Es are respectively the soil Poisson’s ratio and Young’s modulus; Ri is
the radius of part i (i=1, 2 and 3 corresponding to the crown, shoulder and sidewall of the
tunnel boundary); β is a dimensionless factor.

15


In static analyses, the value of dimensionless factor (β) which affects the spring
stiffness was usually set to 1 (Molins et al., 2011; Mashimo et al., 2005) or 2 (Do et al.,
2015). Recently, Sun et al. (2020, 2021) estimated the β value, depending on properties of
the soil and tunnel lining for the case of circular tunnels subjected to seismic loading. In
the present work, a variation of dimensionless factor (β) is also utilized to realistically
represent the soil-tunnel interaction.
3.3. Numerical implementation
In this section, using the FDM numerical model in FLAC3D (Itasca, 2012) has been
developed in chapter 2 which was adopted to calibrate the three dimensionless parameters
(a, b and β) used in the HRM method. Then, the numerical procedure to implement the
HRM in the case of sub-rectangular tunnels subjected to seismic loadings is presented
(Table 3.3 and Figure 3.8).
3.3.1. FDM numerical simulations
Using the FLAC3D, a 2D plane strain model is presented in Chapter 2. The geometry
parameters of sub-rectangular tunnels are presented in Figure 2.9. Other soil and lining
parameters listed in Table 3.1 (the results in section 2.4 in Chapter 2) and Table 3.2 (Figure
3.7) are adopted. It should be mentioned that for the calibration purpose to determine
dimensionless parameters a, b and β in the HRM method.
Table 3.1. Input parameters for the reference case for developing the HRM method
Parameter
Symbol
Unit
Value or Range
Soil properties
Unit weight
γ
MN/m3

0.018
Young’s modulus
Es
MPa
10-350
Poisson’s ratio
νs
0.34
Internal friction angle
φ
degrees
33
Cohesion
c
MPa
0
Lateral earth pressure coefficient
K0
0.5
Depth of tunnel
H
m
20
Peak horizontal acceleration at ground
aH
g
0.5
surface
Moment magitude
M

7.5
Distance of site source
Km
10
Tunnel lining properties
Young’s modulus
E
MPa
35000
Poisson’s ratio
ν
0.15
Lining thickness
t
m
0.3-0.8
Tunnel height
h
m
7.2
Tunnel width
w
m
9.7

16


Table 3.2. Geometrical parameters of tunnel shape cases (Do et al., 2020)
Case

Tunnel
Tunnel
h/w
R1 (m) R2 (m) R3 (m)
width (w) height (h)
ratio
(m)
(m)
SR1
8.76
8.15
0.930
8.36
1.02
4.99
SR2
9.13
7.89
0.864
7.09
1.23
4.81
SR3
9.39
7.53
0.802
8.5
0.96
5.07
SR4 (reference

9.70
7.20
0.742
9.95
1.00
5.35
case)

Figure 3.7. Shapes of tunnel cases (unit: m) (Do et al., 2020)
3.3.2. Numerical procedure in HRM method
To implement the HRM method in the case of sub-rectangular tunnels subjected to
seismic loading, it is necessary to determine the formulas of the three dimensionless
parameters (a, b and β) which define the external loadings applied on the tunnel lining. The
main procedure to calibrate the three parameters is illustrated in Table 3.3 and Figure 3.8.
After the calibration process is completed, the equations representing the influence of
the three parameters on the soil, lining properties and tunnel dimensions can be established.
The formulas are proposed based on the best fit (Figure 3.9 and 3.10). The parameters β,
a, and b can be given as follows:
𝛽 = 𝛽 +𝛽 +𝛽 +𝛽
(3.15)
.
𝛽 = −1.65 𝐸
+ 1.477
(3.16)
𝛽 = 4.8002

(3.17)

− 0.3333
17

𝛽 = 30644

− 4452.3

𝛽 = 7.9746

− 5.9192

(3.18)

+ 207.88( ) − 3.0828

(3.19)

𝑎 = 12.155 + 6.45 𝐸
𝑏 = 𝑏 +𝑏 +𝑏 +𝑏
𝑏 = 10 𝐸 . − 0.305

(3.20)
(3.21)
(3.22)

𝑏 = 23.04

(3.23)

− 1.6

𝑏 = −20461
𝑏 = 5.3148

+ 3736

(3.24)

− 192.79 ( ) + 2.8135

(3.25)

− 3.945

Generating soil and lining parameters { ,
Initial
and
computation using HRM

} for all cases

Selection of parameters set
{ ,
}

and
computation
using numerical solution

Potential error computation

Update a, b
and β

No
=
=

}
}

Yes
Output a, b and β

No

All cases are
computed?

i=i+1

Yes
a = ƒ( ,
b = f( ,
β = f( ,

)
)
)

Figure 3.8. Calibration flowchart of the three parameters

18


Table 3.3. Overview of the calibration process.
Step
1
2

3
4
5
6

Description
Generating the input parameters of soil, lining and tunnel dimensions {ti, hi, wi, Esi} using
defined parameter ranges listed in Table 3.1 and Table 3.2.
Seismic-induced incremental normal forces and bending moments calculation {NFDM,
MFDM} using FDM model, and computation of the initial values of {NHRM, MHRM} using
the HRM method based on a=b=β=1.
Determination of the relative error of incremental normal forces and bending moments
obtained by two methods.
If eN ≤ 0.02 and eM ≤ 0.02, export a, b and β. Otherwise, update these three parameters
(i.e., a, b, β) until the target precision is reached.
Steps 2 to 4 repetitions until all cases scenarios of defined parameter ranges listed in
Table 1 and Table 2 are considered.
Determination of the formulas describing a, b, and β as functions of ti, hi, wi, Esi
parameters by using regression analysis.
1.25

0.3
0.2

β2

0.1

β1

1

Numerical calculations
0.75

0
-0.1

Fitting curve

Numerical calculations
-0.2
0.5
0

50

100

150

200

250

300

Fitting curve

-0.3
0.02

350

0.04

0.06

Young's Modulus, Es (MPa)

0.08

0.1

0.12

0.9

0.95

t/h

a)

b)

0.15

2

0.1

1.5

0.05

1

Fitting curve

β4

β3

Numerical calculations

0

0.5

Numerical calculations

-0.05
-0.1
0.03

0

Fitting curve

-0.5
0.035

0.04

0.045

0.05

0.055

0.7

t/w

0.75

0.8

0.85

h/w

c)
d)
Figure 3.9. Obtained numerical results and fitting curves adopted for the parameters β1,
β2, β3 and β4 that created the parameter β.

19


13

Parameter a

12.8

Numerical calculations
Fitting curve

12.6
12.4
12.2
12
0

50

100 150

200 250 300 350

Young's Modulus, Es (MPa)

a)
2

Numerical calculations

Numerical calculations

1.5

0.8
0.3

b2

0.5

b1

Fitting curve

Fitting curve

1

0

-0.2

-0.5

-0.7

-1
-1.5
0

50

100

150

200

250

300

-1.2
0.02

350

0.04

0.06

0.1

0.12

0.9

0.95

t/h

Young's Modulus, Es (MPa)

b)

c)

0.1

1.1

Numerical calculations
0

0.08

Numerical calculations
0.9

Fitting curve

Fitting curve

-0.1

b4

b3

0.7
0.5
0.3
-0.2
0.1
-0.3
0.03

-0.1
0.035

0.04

0.045

0.05

0.055

0.7

t/w

0.75

0.8

0.85

h/w

d)

e)

Figure 3.10. Coefficients fitting curves for the formulas of the parameters a and b1, b2,
b3 and b4 that created the parameter b
While the coefficient a is expressed as a function of soil’s Young’s modulus alone (Es),
coefficient 𝛽 and b are the functions of the lining thickness (t), tunnel height (h), tunnel
width (w), and soil’s Young’s modulus (Es), as shown in Figures 3.9 and 3.10.
To have a clear understanding of the results obtained by HRM and the numerical FDM
model, Figure 3.11 introduces a comparative example of the incremental bending moments

20


and normal forces distribution in the sub-rectangular tunnel lining subjected to a seismic
loading when Es = 100MPa and t = 0.5m. Other soil parameters of the reference case
presented in Table 3.1 are adopted. Figure 3.11 reveals insignificant differences between
the extreme incremental internal forces obtained by the HRM method and the FDM model.
The differences are 1.2% and 0.6% corresponding to the extreme incremental bending
moments and the normal forces.

FDM: Mmax = 0.900 MNm/m

FDM: Nmax = 0.791 MNm/m

HRM: Mmax = 0.911 MNm/m

HRM: Nmax = 0.786 MNm/m

a) Incremental Bending Moments

b) Incremental Normal Forces

Figure 3.11. Comparison of the incremental bending moments and normal forces
calculated by the developed HRM method and numerical FDM calculation
3.4. Validation of the HRM method
The extensive validations were carried out to demonstrate the applicability of the
developed HRM method. The first validation aims at estimating the accuracy of the
developed HRM method, using a range of peak horizontal seismic acceleration (aH). Then,
varying Young’s modulus of soil and lining thickness are used for validations 2 and 3.
While the uniform tunnels with different cross-sections are considered in validation 4,
different sub-rectangular shapes with geometrical parameters of tunnel shape cases from
Table 2 (Do et al., 2020) are used in validation 5. The effect of the burial depth of the tunnel
on behavior of tunnel lining is considered in validation 6. Finally, validation 7 is performed
using soil parameters adopted in research by Hashash et al. (2005) and Sun et al. (2021).
In each validation, the seismic-induced incremental internal forces obtained by the HRM
method are compared with the numerical FDM solution.
The validation results are shown that the developed HRM method can be effectively
used to estimate the incremental internal forces in sub-rectangular tunnel lining under
seismic loading.
3.5. Conclusions

The novelty and the scientific contribution of this study lie in proposing a new
numerical procedure to efficiently calculate the behavior of sub-rectangular tunnel linings
subjected to seismic loading using the Hyperstatic Reaction Method.
21


To verify the application capability of the developed HRM method, an extensive
validation was performed considering series of numerical computations. The developed
HRM method was validated based on comparison with a quasi-static numerical FDM
model.
The proposed HRM method in this study provides a new and alternative free method
of very efficient seismic design of sub-rectangular tunnels.

GENERAL CONCLUSIONS AND PERSPECTIVES
General conclusion
Although there have been many studies evaluating the effects of seismic loading on
circular and rectangular tunnels, no research has been conducted on the sub-rectangular
tunnels. The thesis has used FDM numerical method to investigate the behavior of subrectangular tunnels when subjected to seismic loading, compare and clarify the difference
between the behavior of circular and sub-rectangular tunnels. The thesis has also proposed
a new equivalent static loading scheme acting on sub-rectangular tunnel lining subjected
to seismic loading for the HRM method.
The novelty and the scientific contribution of this thesis in the behavior of subrectangular tunnel linings subjected to seismic loading:
 The horizontal acceleration aH, soil’s Young’s modulus Es, and lining thickness t
have a great effect on the incremental internal forces induced in both sub-rectangular and
circular tunnels for both no-slip and full slip conditions;
 The results proved that the soil-lining interface conditions have a great influence
on the behavior of sub-rectangular tunnels. This is completely different when comparing
the behavior circular-shaped tunnels. Indeed, while the absolute extreme incremental
bending moments of a circular tunnel for the no-slip condition are smaller than the
corresponding full slip ones, the absolute extreme incremental bending moments of subrectangular tunnels for the no-slip condition are greater than the corresponding full slip

ones. That is opposite to the trend observed in circular tunnel linings;
 Proposing a new numerical procedure to efficiently calculate the behavior of subrectangular tunnel linings subjected to seismic loading using the Hyperstatic Reaction
Method;
 The present study also shows that in the case when a tunnel structure is more
flexible than the soil mass, the tunnel lining will experience amplified distortions in
comparison to the soil shear distortions in the free field. By contrast, when a tunnel lining
is stiffer than the surrounding soil, it tends to resist the ground displacements.
The proposed HRM method in this study provides a new and alternative free method
of very efficient seismic design of sub-rectangular tunnels.

22


It is important to note that all of the numerical models developed in this research were
performed using drained conditions and with tunnels located at shallow depth.
Additionally, because no appropriate data exists in literature, all numerical results were not
yet compared and validated with experimental data.
Perspectives
The research works listed below are proposed for short-term:





Validate all numerical models using real or laboratory data;
Develop segment tunnel lining simulation considering the existence of joints;
Improve the HRM method for the segmental lining with sub-rectangular tunnels;
Develop and perform 3D numerical analyses for full-seismic with sub-rectangular
tunnels;

For the long-term, the perspectives will be the following ones:
 Develop and perform 3D numerical analyses considering the water effect on the
tunnel lining behavior on undrained analyses;
 Study the surface structure effect on the tunnel response.

PUBLISHED AND SUBMITTED MANUSCRIPTS
ISI papers:
1. Do Ngoc Anh., Daniel Dias., Zhang ZX., Huang X., Nguyen Tai Tien., Pham Van
Vi., Nait-Rabah O (2020). Study on the behaviour of squared and sub-rectangular
tunnels using the Hyperstatic Reaction Method, Transp Geotech, 22, 10021. doi:
10.1016/j.trgeo.2020.100321 (ISSN: 2214-3912).
2. Pham Van Vi, Do Ngoc Anh, Daniel Dias (2021). Sub-rectangular tunnels behavior
under seismic loading, Appl. Sci, 11, 9909. doi.org/10.3390/app11219909 (ISSN:
2076-3417).
3. Pham Van Vi, Do Ngoc Anh, Dias Daniel, Nguyen Chi Thanh, Dang Van Kien
(2022). Sub-rectangular tunnels behavior under static loading. Transp. Infrastruct.
Geotechnol. doi.org/10.1007/s40515-022-00230-w (ISSN: 2196-7202).
4. Do Ngoc Anh, Pham Van Vi, Dias Daniel. A New Quasi-Static Loading Scheme
for the Hyperstatic Reaction Method - Case of Sub-Rectangular Tunnels under
Seismic Conditions, Comput. Methods Appl. Mech. Eng. (ISSN: 0045-7825) (under
review).

23