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Effect of temperature on deep lined circular tunnelsin isotropic and transversely anisotropic elasticgroundFei TaoPurdue University

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i

EFFECT OF TEMPERATURE ON DEEP LINED CIRCULAR TUNNELS IN ISOTROPIC AND

TRANSVERSELY ANISOTROPIC ELASTIC GROUND

A Thesis

Submitted to the Faculty

of

Purdue University

by

Fei Tao

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Civil Engineering

December 2016

Purdue University

West Lafayette, Indiana

ii

To my parents

Tao Jianhua and Chen Suqin

iii

ACKNOWLEDGEMENTS

I would first like to thank my thesis advisor Prof. Antonio Bobet of the Lyle school of Civil

Engineering at Purdue University. The door to Prof. Bobet office was always open

whenever I ran into a trouble spot or had a question about my research or writing. He

consistently allowed this paper to be my own work, but steered me in the right direction

whenever he thought I needed it.

Besides my advisor, I would like to thank the rest of my thesis committee: Prof. Marika

Santagata, Prof. Ghadir Haikal for their encouragement, wise comments, and insightful

helps.

I would like to thank my dear friends Alain, Amy, Anahita, Yu-chung, Weishuang,

Gabriela, Sungsoo, Haibo, Zengkun, Yanfei, Yaxiong, Linna, Yazen, Steven, Sandy,

Rodrigo, Shakib, Hanfei, Meng, Xuanchi, Yanyuan, Kaiming, Weiyong, Jing, Weichang,

Yinqin, Shandian, who have provided significant help to my study and life at Purdue.

Finally, I must express my very profound gratitude to my parents and brothers for

providing me with unfailing support and continuous encouragement throughout my

years of study and through the process of researching and writing this thesis. This

accomplishment would not have been possible without them. Thank you!

Fei Tao

iv

TABLE OF CONTENTS

Page

LIST OF TABLES .................................................................................................................... vi

LIST OF FIGURES ................................................................................................................. vii

ABSTRACT ............................................................................................................................ ix

CHAPTER 1. INTRODUCTION ......................................................................................... 1

CHAPTER 2. THERMAL STRESS IN GROUND AND LINER UNDER ISOTROPIC GROUND . 5

2.1 Thermal stress in a thick wall cylinder ...................................................................... 5

2.2 Thermal conduction of a thick wall cylinder ............................................................. 8

2.3 Thermal stresses in liner and surrounding ground ................................................... 8

2.3.1 Case 1: Change of temperature of the liner and thermal conduction into

the ground ................................................................................................................... 9

2.3.2 Case 2: Temperature change of the ground and insulated liner ................ 18

2.3.3 Case 3: Temperature change in the inside of liner and insulated ground . 20

2.4 Discussion ................................................................................................................ 23

CHAPTER 3. THERMAL STRESS IN GROUND AND LINER UNDER TRANSVERSELY

ANISOTROPIC GROUND .................................................................................................... 31

3.1 Principal Axis of Anisotropy Coincides with Stacking Direction .............................. 31

3.2 Temperature Distribution ....................................................................................... 32

3.3 Thermal Stress Distribution ..................................................................................... 33

3.4 Parametric Study and Discussion ............................................................................ 38

CHAPTER 4. THERMAL STRESSES IN GROUND AND LINER UNDER TRANSVERSELY

ANISOTROPIC GROUND WITH THE MATERIAL ANISOTROPY AXIS PERPENDICULAR TO

THE STACKING DIRECTION ................................................................................................ 44

v

Page

4.1 Description of the numerical model ....................................................................... 44

4.2 Simulation results .................................................................................................... 45

4.3 Parametric study ..................................................................................................... 48

4.3.1 Influence of the Young’s modulus .............................................................. 49

4.3.2 Influence of thermal conductivity ............................................................... 52

4.3.3 Influence of the coefficient of thermal expansion ..................................... 57

CHAPTER 5. SUMMARY AND FUTURE WORK .............................................................. 60

5.1 Summary and conclusion ........................................................................................ 60

5.2 Future Work ............................................................................................................ 61

REFERENCES ...................................................................................................................... 63

APPENDIX .......................................................................................................................... 68

vi

LIST OF TABLES

Table ...............................................................................................................................Page

Table 2-1 Material properties used for parametric study ................................................ 24

Table 3-1 Material properties used for parametric study ................................................ 40

Table 4-1 Material properties used for the parametric study .......................................... 50

vii

LIST OF FIGURES

Figure .............................................................................................................................Page

Figure 2-1 Thick Wall Cylinder ............................................................................................ 7

Figure 2-2 Tunnel with Liner ............................................................................................. 10

Figure 2-3 Model Geometry ............................................................................................. 15

Figure 2-4 Temperature change and boundary conditions .............................................. 16

Figure 2-5 Radial and Tangential Stresses in the Liner and Ground for Case 1 ................ 17

Figure 2-6 Radial and Tangential Stresses in the Liner and Ground for Case 2 ................ 20

Figure 2-7 Radial and Tangential Stresses in the Liner and Ground for Case 3 ................ 23

Figure 2-8 Ground and Liner Stress and displacements for Different Young’s Modulus . 26

Figure 2-9 Ground and Liner Stress for Different Poisson’s Ratio .................................... 27

Figure 2-10 Ground and Liner Stress for Different Thermal Conductivity ....................... 28

Figure 2-11 Ground and Liner Radial Stress for Different Coefficients of Thermal

Expansion .......................................................................................................................... 30

Figure 3-1 Shaft in the transversely anisotropic ground .................................................. 32

Figure 3-2 Radial and tangential stresses in liner and ground for transversely anisotropic

ground with anisotropic principal axis along the stacking direction ................................ 38

Figure 3-3 Ground and Liner Stresses and displacements for Different Young’s Modulus

........................................................................................................................................... 41

viii

Figure .............................................................................................................................Page

Figure 3-4 Ground and Liner Stresses and displacements for Different Coefficients of

Thermal Expansion ............................................................................................................ 43

Figure 4-1 Difference of tunnel excavated in isotropic or anisotropic ground ................ 44

Figure 4-2 Results of the base case study ......................................................................... 48

Figure 4-3 Ground and liner stresses and displacements. Effect of the Young’s modulus

........................................................................................................................................... 52

Figure 4-4 Contour plots of temperature for different thermal conductivities ............... 54

Figure 4-5 Ground and liner stresses and displacements. Effects of thermal conductivity

........................................................................................................................................... 57

Figure 4-6 Ground and liner stresses and displacements. Effect of the coefficient of

thermal expansion ............................................................................................................ 59

ix

ABSTRACT

Tao, Fei. M.S.C.E., Purdue University, December 2016. Effect of Temperature on Deep Lined Circular Tunnels in Isotropic and Transversely Anisotropic Elastic Ground Major Professor: Antonio Bobet.

A tunnel is a passageway constructed through soil or rock. It is one of the most

important underground structures and has been widely used in transportation. Much

research has been done on tunnel performance and on the interaction between ground,

excavation, liner installation and support. The work has provided a better understanding

of the interplay that exists between deformations and stresses transferred from the

ground to the support. Significant insight into the problem has been provided by closed-

form analytical solutions. A number of analytical formulations have been obtained for

underground openings and lined tunnels under different scenarios, namely shallow or

deep tunnels in the dry ground under geostatic loading, for static and seismic loading,

plastic, poro-elastic, and poro-plastic conditions. However, considerably less attention

has been given to thermal stresses in a tunnel. Temperature changes due to e.g. fire can

cause cracking and damage to the liner and surrounding ground. In fact, a number of

tunnels have suffered serious damage due to thermal loading. The French Channel

Tunnel fire in 1996 and the Italian Mont Blanc tunnel fire in 1999 are well-known

examples. The goals of this study are: (1) develop a general formulation for a deep

x

circular lined tunnel in isotropic ground for thermal loading; (2) derive analytical

solutions for stresses and displacements caused by thermal load for a lined circular

tunnel under a transversely anisotropic ground where the ground anisotropy axis

coincides with the stacking direction; (3) evaluate stresses and displacements for a lined

deep tunnel in a transversely anisotropic medium also under thermal loading, with the

ground anisotropy axis perpendicular to the stacking direction. Comparisons between

the analytical solution obtained and the results obtained from ABAQUS for a number of

select cases provide confidence in the formulations obtained. A parametric study has

been performed to investigate the effects of Young’s modulus, Poisson’s ratio, thermal

conductivity and the coefficient of thermal expansion on the behavior of the liner and

ground, for a tunnel in a transversely anisotropic ground with the axis of the tunnel

along one of the axis of elastic symmetry. The results of the study show that the Young’s

modulus and the coefficient of thermal expansion are the most important parameters

that determine the stresses and displacements of the liner and ground. The analysis also

shows that the thermal conductivity has a significant effect on the temperature

distribution in the ground.

1

CHAPTER 1. INTRODUCTION

A tunnel is a passageway constructed through soil or rock. It is one of the most

important underground structures and has been widely used in transportation, for rail

traffic, water canal, and vehicle traffic (Tian 2011). The general method for the design of

tunnels includes site investigations, ground probing and in-situ monitoring, and stresses

and deformations analysis (Duddeck 1988). The key for a safe design is an accurate

understanding of the mechanisms of rock-liner interaction. Such interaction has been

evaluated through empirical, numerical and analytical methods. The advantage of

analytical methods is that they provide closed-form expressions for stresses in the

ground and liner with little effort and that they capture the key parameters for the

problem. An important disadvantage however is that their applicability is limited due to

the limiting assumptions made, e.g. deep tunnel, elasticity, etc., to reach a solution.

A number of analytical solutions have been obtained for underground openings, e.g.

(Strack and Verruijt 2002; Timoshenko 1970; Verruijt 1997; Verruijt 1998), and for lined

tunnels, e.g. (Bobet 2003; Bobet 2007; Bobet and Nam 2007; Einstein and Schwartz

1979). A large range of scenarios have been considered, which include static and seismic

loading (Bobet 2003; Gomez-Masso and Attala 1984; Hashash and Park 2001; Hashash

2

et al. 2005; Penzien 2000; Wang 1996), below water table (Anagnostou 1994; Bobet

2001; Bobet 2010; Fernández 1994; Lee and Nam 2004; Shin et al. 2011), for plastic

(Carranza-Torres 2004; Carranza-Torres and Fairhurst 2000; Sharan 2003), poro-elastic

(Bobet and Yu 2015; Wang 2000; Wang 1996) and poro-plastic conditions (Bobet 2010;

Bobet and Yu 2015; Giraud et al. 2002).

Considerably less attention has been given to thermal stresses in tunnels. The

temperature in the surrounding ground and/or inside the tunnel may not be the same,

and there may be cases where temperature gradients may be important (e.g. in nuclear

waste repository facilities). Emergency conditions such as fire may also change the

temperature in the tunnel drastically, where the temperature may reach hundreds of

degrees Celsius. As a result, thermal stress may be induced and can cause cracking and

damage to the liner and surrounding ground, if large enough, and can be a major factor

to damage of tunnels. In fact, there are a number of tunnels that suffered cracking

damage due to thermal loading, such as the Ertan Hydroelectric Power Station in

southwestern China (Dong and Yang 2013), the thermal load generated caused a

number of interruptions of tunnel operations; the 1996 Channel Tunnel fire in France

resulted in considerable damage over 480 m (Channel Tunnel Safety 1997; Wright et al.

2014); the 1999 Mont Blanc tunnel fire seriously damaged the tunnel roof over a length

of 900 m, a certain part of the tunnel roof lining completely collapsed and exposed the

rock behind (Faure 2007).

The fundamental thermal-elasticity solution for a single lined circular tunnel in an elastic

medium was first presented by Boley (Boley and Jerome Harris 1960), who assumed

3

that the temperature was evenly distributed on the interior surface of the tunnel. The

influence of transversely anisotropy was taken into account by Weng (Weng 1965), who

derived formulas for stresses around an unlined circular cylinder where the anisotropy

axis coincided with the stacking direction. Forrestal and Tarn derived a closed-form

solution for an unlined circular cylinder in transversely anisotropic ground (Forrestal et

al. 1969; Tarn 2001). They studied a large range of loading conditions such as torsion,

shearing, pressure and temperature changes. The unsteady state temperature change in

a circular tunnels has been studied by Elsworth (Elsworth 2001), who investigated

unsteady state temperature changes for unlined and lined tunnels.

The thermal stress in a cylindrical anisotropic circular cylinder due to an instantaneous

heat source has been investigated by Abd-Alla (Abd-Alla 1995), who used the Laplace

transform and Lagrange’s method.

These studies have clearly described the stresses and displacements in lined and

unlined thermally loaded circular tunnels in isotropic media, in unlined tunnels in a

transversely anisotropic medium with the anisotropy axis along the stacking direction,

and in lined and unlined tunnels in a cylindrically anisotropic medium with unsteady

state temperature. However, stresses and displacements in lined thermally loaded

circular tunnels in transversely anisotropic media are still not well understood. This

study investigates the thermal influence on a lined circular tunnel in an anisotropic

medium. More specifically, the goals of this thesis are: (1) use a deep circular lined

tunnel under isotropic condition to show that a general analytical solution can be

applied to different scenarios; (2) derive the analytical solution for stresses and

4

displacements due to thermal load to a lined circular tunnel under a transversely

anisotropic ground, where the ground anisotropy axis coincides with the stacking

direction (3) parametric analysis of the thermal load on a lined circular tunnel under a

transversely anisotropic ground with the ground anisotropy axis perpendicular to the

stacking direction.

For the derivations and simulations, the following assumptions are made: (1) the tunnel

has a circular cross section; (2) the ground and the liner have an elastic response; (3)

there is no slip or detachment between liner and ground; (4) the tunnel is deep; (5)

plane strain conditions exist on any cross section perpendicular to the tunnel axis; (6)

the liner is isotropic and the ground transversely anisotropic with one of the axes of

elastic symmetry parallel to the tunnel axes; and (7) the axes of elastic symmetry

coincide with the principal directions of thermal conductivity. Only stresses due to

thermal loading are considered. General cases where the tunnel is subjected to both

mechanical (e.g. geostatic) stresses and temperature change can be solved by

superposition, given the assumption of elastic behavior. The solution for a tunnel with a

liner subjected to far-field stresses can be obtained using the relative stiffness method

(Einstein and Schwartz 1979).

5

CHAPTER 2. THERMAL STRESS IN GROUND AND LINER UNDER ISOTROPIC GROUND

The analytical solution of thermal stress in lined circular tunnel with isotropic ground

has been obtained by others (Boley and Jerome Harris 1960; Dong and Yang 2013). The

derivation process is included here for completeness and shows that a general case can

be applied to different scenarios.

2.1 Thermal stress in a thick wall cylinder

Consider an infinitely long thick wall cylinder with outer and inner radius b and a, as

shown in Fig. 2-1. The outer boundary is assumed to be free of stresses and can deform

freely. The temperature is assumed to change in the interior surface of the cylinder.

Because of the symmetry of the problem, displacements and stresses caused by the

temperature change will be axisymmetric. Equilibrium in polar coordinates is written as:

0rrd

dr r

(1)

Where 𝜎𝑟 = radial stress and 𝜎𝜃 = tangential stress, 𝑟 and 𝜃 are polar coordinates with

the origin at the center of the cylinder.

Strains in polar coordinates are:

6

rr

du

dr (2.a)

ru

r (2.b)

Where 𝜀𝑟 = radial strain, 𝜀𝜃 = tangential strain, 𝑢𝑟 = radial displacement. The

tangential displacement 𝑢𝜃 and the shear strain 𝛾𝑟𝜃 are zero due to the symmetry.

Strains, stresses and temperature are, in plane strain:

1[ ( )]r r z T

E (3.a)

1[ ( )]r z T

E (3.b)

ez=

1

E[s

z-n(s

r+s

q)]+aT = 0 (3.c)

( )z r E T (4)

Where 𝐸 = Young’s modulus, 𝜎𝑧 = axial stress in the z coordinate, 𝛼 = coefficient of

thermal expansion, 𝑇 = temperature. Equation (1), given equations (2) to (4), can be

written in terms of radial displacements 𝑢𝑟 as:

2

2 2

1 1

1

r r rd u du u dT

dr r dr r dr

(5)

The general solution of 𝑢𝑟 is:

21

1

1

r

r

a

Cvu Trdr C r

v r r

(6)

Where 𝐶1 and 𝐶1 are constants to be determined from the boundary conditions.

7

Substituting Eq. (6) into Eq. (3), using the definition of strains given in equation (2), gives

the final expressions for the radial and tangential stresses as:

1 2

2 21 1 1 2

r

r

a

C CE ETrdr

v r v v r

(7.a)

1 2

2 21 1 1 2 1

r

a

C CE E ETrdr T

v r v v r v

(7.b)

Note that 𝜎𝑧 can be computed from Eq. (7) and (4).

Figure 2-1 Thick Wall Cylinder

8

2.2 Thermal conduction of a thick wall cylinder

According to the energy conservation principle, the total heat absorbed by an object

must be equal to the sum of the heat that flows into the object and the heat supplied by

the heat source. This can be expressed in mathematical form as follows (Xu 1990):

2 TK T

t t

(8)

Where 𝑇 = temperature, 𝑡 = time, 𝐾 = thermal conductivity, = the Laplace operator,

t

= adiabatic temperature. Assuming steady state and no heat source, Eq. (9)

simplifies into:

2 0K T (9)

The general solution of Eq. (10), given the symmetry of the problem, is:

lnT A r B (10)

Where A and B are constants that need to be determined from boundary conditions.

2.3 Thermal stresses in liner and surrounding ground

For the isotropic case, the general formulation presented is applied to three different

scenarios: (1) change of temperature of the liner and thermal conduction into the

ground; (2) change of temperature of the liner, while the liner is insulated from the

surrounding ground; and (3) change of temperature of the ground while the liner is

9

insulated. All three cases are solved independently. Case 1, however, is general, while

the other two represent extreme situations. It will be shown that cases 2 and 3 can be

obtained using appropriate parameters in case 1.

2.3.1 Case 1: Change of temperature of the liner and thermal conduction into the

ground

In this case, the temperature inside of the liner changes, from its initial state to a value

Tc, while far from the tunnel, the temperature remains the same. Thus, a temperature

gradient will be established inside the liner and inside the surrounding ground.

Consider a circular liner with internal radius c and external radius a. The liner is in

contact with the ground that extends to infinity. The temperature recovers far-field

conditions at a distance b from the center of the tunnel, as shown in Fig. 2-2.

10

Figure 2-2 Tunnel with Liner

The temperature distribution is, from Eq. (10),

lnlT A r B (11.a)

lngT C r D (11.b)

l l

Tq K

r

(11.c)

g g

Tq K

r

(11.d)

Where 𝑇l = temperature in the liner, 𝑇g = temperature in the ground, 𝐴, 𝐵, 𝐶 and 𝐷 are

constants determined from boundary conditions, 𝑞𝑙 and 𝑞𝑔 are the heat flux in the liner

and ground, 𝐾g and 𝐾𝑙 are the thermal conductivity of ground and liner, respectively.

The boundary conditions of this problem are:

l r c cT T (12.a)

11

0g r bT (12.b)

l r a g r aT T (12.c)

l r a g r aq q (12.d)

Where 𝑇𝑐 is the change of temperature on the interior fiber of the liner. Note that there

is no change in temperature at r = b since far-field conditions are recovered.

By substituting Eq. (12) into Eq. (11), the constants A and B, C and D are obtained:

(ln ln ) (ln ln )

g c

l g

K TA

K a b K c a

(13.a)

ln

(ln ln ) (ln ln )

g c

c

l g

K T cB T

K a b K c a

(13.b)

ln ln ln ln

c

g g

l l

TC

K Ka b c a

K K

(13.c)

ln

ln ln ln ln

c

g g

l l

bTD

K Ka b c a

K K

(13.d)

Letting ln ln ln lng g

l l

K Ka b c a

K K and substituting into Eq. (13):

lnln

g c g c

l c

l l

K T K T cT r T

K K

(14.a)

ln

lnc cg

T T bT r

(14.b)

The radial stresses and radial displacements are, from Eq. (7) and Eq. (8):

12

1 2

2 21 1 1 2

r

l l l lr l

l l lc

E E C CT rdr

v r v v r

(15.a)

1 2

2 21 1 1 2 1

r

l l l l ll l l

l l l lc

E E EC CT rdr T

v r v v r v

(15.b)

3 4

2 21 1 1 2

rg g gg

r g

g g ga

E E C CT rdr

v r v v r

(15.c)

3 4

2 21 1 1 2 1

rg g g gg

g g g

g g g ra

E E EC CT rdr T

v r v v r v

(15.d)

21

1

1

r

l l lr l

l c

v Cu T rdr C r

v r r

(15.e)

43

1

1

rg gg

r g

g a

v Cu T rdr C r

v r r

(15.f)

𝐶1, 𝐶2, 𝐶3 𝐶4 are constants to be determined from boundary conditions. Where 𝜎𝑟𝑙 and

𝜎𝑟g

are the radial stresses, 𝜎𝜃𝑙 and 𝜎𝑟

g are the tangential stresses, and 𝑢𝑟

𝑙 and 𝑢𝑟g are the

radial displacements. Note that superscripts l and g are used for the liner and ground,

respectively. 𝐸𝑙 and 𝐸g are the Young’s modulus, 𝜐𝑙 and 𝜐g are the Poisson’s ratios

and 𝛼𝑙 and 𝛼g are the coefficients of thermal expansion.

Assuming that the stress between the ground and liner is s , the boundary conditions of

this problem are:

0l

r r c (16.a)

0g

r r b (16.b)

l g

r r a r r a s (16.c)

13

l g

r r a r r au u (16.d)

Where s is the radial stress between the liner and ground

Eq. (16) and Eq. (15) provide the solution to the problem. The stresses and

displacements are:

2 2 2 2 2 2

1 1 1 1

1 1

r a

l l l l lr l l

l lc c

E ET rdr s T rdr

r a c r c a

(17.a)

2 2 2 2 2 2

1 1 1 1

1 1 1

r a

l l l l l ll l l l

l l lc c

E E ET rdr s T rdr T

r a c r c a

(17.b)

2 2 2 2 2 2

2 2 2 2 2 2

1 1 1 1

1 1

1 1 1 1 1 11

r bg g g gg

r g g

g ga a

E ET rdr s T rdr

r b r b a b

r b a b b a

(17.c)

2 2 2 2 2 2

2 2 2 2 2 2

1 1 1 1

1 1

1 1 1 1 1 11

1

r bg g g gg

g g

g ga a

g

g g

g

E ET rdr s T rdr

r b r b a b

ET

r b a b b a

(17.d)

2 2 2 2 2 2 2

2 2 2 2 2

1 1 2 1 1 2 11 1 1 1

1 1

1 11 1 1 1 1

1

r a

l l l l l l l lr l l

l l lc c

a

l l ll

l l c

vu T rdr s T rdr r

v r c E c a c a c a

s T rdrr E c a a c a

(17.e)

2 2 2 2

2 2 2 2 2 2 2 2

1 (1 2 )(1 ) (1 2 )(1 )1 1

1 1

1 (1 )1 1 1 1 1 1 1 11

1

rg g g g g g gg

r g

g g gc

b bg g g

g g

g ga a

v su T rdr

v r b E b a b

T rdr r s T rdrb b a r E b a b b a

(17.f)

14

2 2 2 2 2

2 2 2 2 2 2

2 2

11 1 1 1 1 1(1 2 )

1 1 1

11 1 1 1 1 1 11 2 1 (1 2 )

1 1

a ag gl l l l

l l l

l l gc c

bg

g g g

ga

s T rdr T rdra a c a c a b

aa T rdr

b b a a b a E b a

b a

2 2 2

1 1 1 11 2

g

g

g

a

E c a c a

(17.g)

Where the integrals of temperature can be found in AppendixⅠ.

Results from the analytical solution are compared with those from the Finite Element

Method code ABAQUS (ABAQUS, 2013). The following case is analyzed: deep tunnel

with circular cross section with dimensions c=7 m (internal radius), a=8 m, b=100m; liner

properties, 𝐸𝑙 = 2.5 × 104MPa, 𝜈𝑙 = 0.25, 𝐾𝑙 = 1 W/m℃, 𝛼𝑙 = 1 × 10−5m/℃; ground

properties, 𝐸g = 5 × 103 MPa, 𝜈𝑔 = 0.15, 𝐾g = 2 W/m℃, 𝛼g = 8 × 10−6m/℃. The

temperature change in the interior fiber of the liner is 𝑇c = 30 ℃ and the initial

temperature in the medium is 𝑇0 = 20 ℃.

Fig. 2-3 is the model constructed in ABAQUS. Only one quarter of the model was used

for the simulation. This is because this is an axisymmetric problem. The distance from

the center of the tunnel to the outside of the liner is a, while the distance from the

center to the inside of the liner is c; thus, the difference between a and c is the thickness

of the liner. The size of the discretization is given by b. The dimensions of the model

follow those of the case analyzed; that is: c=7 m, a=8 m, b= 100 m. The thickness of the

liner is 1 m.

In ABAQUS, the model is divided into two parts, the ground and the liner. The two parts

are tied together, which means that there is not detachment or slip between liner and

15

ground. The liner and the ground are discretized using 8-node plane strain thermally

coupled quadrilateral elements with biquadratic displacement and bilinear temperature

(CPE8T). The mesh is shown in Fig. 2-3.

From the FEM (Finite Element Method) theory, a finer mesh will result in a more

accurate result (Reddy 2006; Zienkiewicz 2005). However, a finer mesh also means

larger computational complexity. So, a reasonable number of elements is needed to

ensure the efficiency and accuracy of the simulation. In this study, the mesh size was

determined by comparing simulations for isotropic ground with the results of the

analytic solution. The mesh size that produces an error less than 2% is chosen. It consists

of elements of 0.1 m for the liner and 0.25 m for the ground.

Figure 2-3 Model Geometry

16

The temperature at the interior of the liner changes from 𝑇0 = 20 ℃ to 𝑇𝑐 = 50 ℃,

which is the same as the analytical solution temperature change. The boundary

conditions and the temperature imposed are shown in Fig. 2-4. As the figure shows, due

to the symmetry of the problem, vertical displacements are prevented along the

horizontal axis through the center of the tunnel and the vertical axis can only have

vertical displacements.

Figure 2-4 Temperature change and boundary conditions

Fig. 2-5 provides the radial and tangential stresses in the ground and liner. The figure

shows that the results from the analytical solution are correct since they compare well

with ABAQUS results (errors are smaller than 2%). It also shows that the radial stresses

17

in the ground and liner are negative (i.e. compressive). The radial stresses in the liner

become more compressive with radial distance, while the radial stresses in the ground

become less compressive with radial distance. The tangential stresses in the ground and

liner both increase (less compressive) with radial distance. Note that the radial and

tangential stresses converge, albeit very slowly, to the far-field stresses, which should be

zero. The reason is the dependence of stresses on the logarithm of the radial distance

from the center of the tunnel. As one can see in the figure, there is a jump in the

tangential stresses at the contact between liner and ground. This is expected because

the stiffness of the two materials is different.

Figure 2-5 Radial and Tangential Stresses in the Liner and Ground for Case 1

18

2.3.2 Case 2: Temperature change of the ground and insulated liner

In this problem, the temperature changes only in the ground. The boundary conditions

are the same as those explored in Section 2.3.1, except that

Tl = 0

0l g

r r a r r aq q

The results are:

lnln

ln ln ln ln

c cg

T T bT r

a b a b

(18.a)

2 2 2 2

1 1 1 1l

r sc r c a

(18.b)

2 2 2 2

1 1 1 1

1

l ll r

l

Es T

c r c a

(18.c)

2 2 2 2 2 2

2 2 2 2 2 2

1 1 1 1

1 1

1 1 1 1 1 11

r bg g g gg

r r g

g ga a

E ET rdr s T rdr

r b r b a b

r b a b b a

(18.d)

2 2 2 2 2 2

2 2 2 2 2 2

1 1 1 1

1 1

1 1 1 1 1 11

1

r bg g g gg

g g

g ga a

g

g g

g

E ET rdr s T rdr

r b r b a b

ET

r b a b b a

(18.e)

2 2 2

1 1 21 1 1l l lr

l

u s rE r c c a

(18.f)

19

2 2 2

2 2 2 2

2 2 2 2 2

1 (1 2 )(1 ) 1 1

1

(1 2 )(1 ) 1 1 11

1

1 (1 )1 1 1 1 1

1

rg g g gg

r g

g gc

bg g g

g

g a

bg g g

g

g g a

v su T rdr

v r b E b a

T rdr rb b b a

s T rdrr E b a b b a

(18.g)

2 2 2 2 2 2

2 2 2 2 2 2

1 1 1 1 1 1 11 2 1

1

1 11 1 1 1 1 1(1 2 ) 1 2

bg r

g g

g a

g lg l

g l

s a T rdrb b b a a b a

a a

E b a b a E c a c a

(18.h)

Where the integrals of temperature are included in appendix

A comparison is done between the solution and the general analytical solution found in

section 2.3.1, using as input parameters: 𝐾𝑙 = 1000 W/m℃, 𝛼𝑙 = 0 m/℃; that is, no

temperature change in the liner, which can be simulated with a very large thermal

conductivity and no thermal expansion. The geometry and input parameters are the

same as those used in case 1.

Fig. 2-6 indicates that the two results are identical, showing that the liner is in tension

(i.e. positive radial stresses) and that the radial stresses in the ground initially decrease

(i.e. less tensile) but then increase with radial distance. This is because the temperature

only changes in the ground, causing the ground to expand outward; however, the

temperature does not change in the liner, so the liner, which is tied to the ground, will

pull the ground inward to constrain its expansion. The tangential stresses in the liner are

positive (i.e. tensile) and in the ground they are initially negative (i.e. compressive) and

steadily increase (less compressive) with radial distance. As one can see in Fig. 2-6, there

20

is a jump in the tangential stresses at the contact area between liner and ground. As

explained earlier, this is caused by the different material properties of ground and liner.

Again, as before, the stresses slowly converge to their free-field values with radial

distance.

Figure 2-6 Radial and Tangential Stresses in the Liner and Ground for Case 2

2.3.3 Case 3: Temperature change in the inside of liner and insulated ground

In this case, the temperature changes only in the linear and the ground is thermally

insulated. The boundary conditions are similar to those of Problem 1, in Section 2.3.1,

except that 𝑇r = 0 and 0l g

r r a r r aq q . The solution is:

21

l aT T (19.a)

2 2 2 2 2 2

1 1 1 1

1 1

r a

l l l l lr l l

l lc c

E ET rdr s T rdr

r a c r c a

(19.b)

2 2 2 2 2 2

1 1 1 1

1 1 1

r a

l l l l l ll l l

l l lc c

E E ET rdr s T rdr T

r a c r c a

(19.c)

2 2 2 2

1 1 1 1g

r sb r b a

(19.d)

2 2 2 2

1 1 1 1

1

gg

g g

g

Es T

b r b a

(19.e)

2 2 2 2 2 2 2

2 2 2 2 2

1 1 2 1 1 2 11 1 1 1

1 1

1 11 1 1 1 1

1

r a

l l l l l l l lr l l

l l lc c

a

l l ll

l l c

vu T rdrd s T rdr r

v r c E c a c a c a

s T rdrr E c a a c a

(19.f)

2 2 2

1 1 21 1 1g gg

r

g

u sE r b b a

(19.g)

2 2 2 2 2 2

2 2 2 2 2 2

11 1 1 1 1 1 11 2 1

1 1

1 11 1 1 1 1 1(1 2 ) 1 2

a bg gl l

l g g

l gc a

g lg l

g l

s T rdr a T rdra b b b a a b a

a a

E b a b a E c a c a

(19.h)

The integrals are found in appendix .

The analytical solution is a particular case of the general solution using 𝐾g = 0. This is

verified by comparing the solution reached in this section with the general solution and

using the geometry and input parameters of Problem 1.

Fig. 2-7 is a plot of the radial and tangential stresses in the ground and liner obtained

with the two solutions. The same results are obtained. The figure shows that the radial

22

stresses in the ground and liner are negative (i.e. compressive). For the ground, the

radial stresses increase with radial distance and the tangential stresses are positive (i.e.

tensile) and decrease with radial distance. The reason for this behavior is that

temperature changes only in the liner and, as a result, the liner expands, compressing

the ground immediately adjacent to it. The jump in the tangential stresses at the contact

between liner and ground is again caused by different materials properties of ground

and liner. A comparison between Fig. 2-5 and Fig. 2-7 shows that the radial and

tangential stresses in the liner for case 3 are more compressive than in case 1. The

tangential stresses in the ground in case 3 are tensile while they are compressive in case

1.

23

Figure 2-7 Radial and Tangential Stresses in the Liner and Ground for Case 3

2.4 Discussion

The effects on the stresses in the ground and liner from a number of parameters are

investigated based on case 1. The objective is to determine what the most relevant

variables are and determine their relative influence on the results. The geometry of the

problem and liner properties are those of case 1, which is taken as the base case. The

parameters changed are those of the ground to investigate the effects of relative

24

stiffness between liner and ground, Poisson’s ratio, and relative thermal conductivity

and thermal expansion. The values and their range explored are listed in Table 2-1.

Table 2-1 Material properties used for parametric study

𝐸g (MPa) 𝜐g 𝐾g (W/m℃) 𝛼g (m/℃)

1 5.00E+03

0.15

2

8.00E-06

2 1.00E+04

3 2.00E+04

4 4.00E+04

5 8.00E+04

6

5.00E+03

0.3

7 0.4

8 0.075

9 0.0375

10

0.15

4

11 8

12 16

13 32

14

2

1.60E-05

15 3.20E-05

16 6.40E-05

17 1.28E-04

25

Fig. 2-8 plots the effects of ground stiffness 𝐸g (i.e changes of relative stiffness between

the ground and the liner) on the stress and displacement of the ground and liner. Fig. 2-

8 (a) provides the results of the radial stresses and Fig. 2-8 (b) of the tangential stresses.

The results show that with increasing 𝐸g (the relative stiffness of the ground with

respect to the liner increases), the radial and tangential stresses in both the liner and

ground become more compressive. This can be explained by the displacement in Fig. 2-8

(c). As one can see, when 𝐸g increases, the displacements decrease. This is because the

ground becomes stiffer and because of this, the ground can take more load, as shown in

Fig. 2-8(a) and (b).

26

Fig. 2-9 (a) and (b) illustrate the influence of the Poisson’s ratio of the ground 𝜐g and are

plots of radial and tangential stresses, respectively. They show that increasing 𝜐g causes

the radial and tangential stresses to become more compressive in both ground and

liner, which is associated with the increase of deformation in the tunnel, as indicated in

Fig. 2-9 (c). The influence is however much smaller than 𝐸g.

(a)

(b)

(c)

Figure 2-8 Ground and Liner Stress and displacements for Different Young’s Modulus

27

(a) (b)

(c)

Figure 2-9 Ground and Liner Stress for Different Poisson’s Ratio

The consequences of changing 𝐾g (i.e. increasing the thermal conductivity of the ground

with respect to that of the liner) are illustrated in Fig. 2-10. Fig. 2-10 (a) depicts the

radial stresses, Fig. 2-10 (b) shows the tangential stresses, and Fig. 2-10 (c) illustrates the

radial displacements. In contrast to what is observed in Fig. 2-8 regarding the effects of

the Young’s modulus, the radial stresses and tangential stresses in both ground and liner

28

become less compressive (e.g. they increase) with the increasing 𝐾g. The reason is that

the increases of 𝐾g decreases the temperature gradient near the liner, hence the

displacements decrease for both the liner and the ground.

(a) (b)

(c)

Figure 2-10 Ground and Liner Stress for Different Thermal Conductivity

The influence of thermal expansion 𝛼g is shown in Fig. 2-11. Fig. 2-11 (a) indicates that

the radial stresses in the liner change from compression to tension as 𝛼g increases,

29

while the radial stresses in the ground decrease (they become more compressive). This

is because an increase of the coefficient of thermal expansion of the ground is

associated with an increase of the ground displacement under the same temperature

change, as can be seen in Fig. 2-11 (c), which then changes the exterior fiber of the liner

to tension and produces an increase of stresses in the ground. This mechanism is

confirmed by the plots of tangential stresses in Fig. 2-11 (b), which also show more

compressive stresses with larger ground thermal expansion.

30

(a) (b)

(c)

Figure 2-11 Ground and Liner Radial Stress for Different Coefficients of Thermal Expansion

、

31

CHAPTER 3. THERMAL STRESS IN GROUND AND LINER UNDER TRANSVERSELY ANISOTROPIC GROUND

3.1 Principal Axis of Anisotropy Coincides with Stacking Direction

Consider a circular liner in contact with the ground that extends to infinity. The outside

boundary is set to be stress free. Also, the size of the model is the same as that of Fig. 2-

2. The ground is assumed to be transversely isotropic and the principal axis of

anisotropy coincides with the stacking direction. In other words, the problem is that of a

vertical shaft across a horizontally layered ground, which is shown in Fig. 3-1. The

temperature is assumed to uniformly change at the contact between liner and ground

and remains unchanged far from the tunnel. This makes the problem plane strain with

axial symmetry. This case has been investigated by Weng (Weng 1965); however, the

solution found does not include the liner. Based on their work, a closed-form analytical

solution has been obtained that includes the liner. The solution is verified with the Finite

Element Code ABAQUS. A limited parametric study has been done to highlight the

effects of different parameters on the thermal stresses.

32

Figure 3-1 Shaft in transversely anisotropic ground

3.2 Temperature Distribution

Since it is assumed, as mentioned in the previous section, that the temperature changes

are uniform at the contact area between the liner and the ground, the temperature

does not change at the far-field; therefore, equation (14), which is derived for isotropic

material, applies due to the symmetry of the problem. As a result, the thermal

33

conductivity in the direction of the tunnel axis does not influence the temperature

distribution in the isotropic plane (i.e. the plane perpendicular to the tunnel axis).

3.3 Thermal Stress Distribution

According to Duhamel-Neumann, the relationship between stress and strain can be

represented as:

  , , , 1,2,3;ij ijkl kl ijs T i j k l (20)

Where 𝜀𝑖𝑗 is the strain tensor, 𝜎𝑘𝑙 is the stress tensor, 𝑠𝑖𝑗𝑘𝑙 is the elastic compliance

tensor, 𝛼𝑖𝑗 is the thermal expansion tensor, T is temperature change.

A transversely isotropic material has five independent elastic constants. Because the

problem is axisymmetric, all shear stresses and strains are zero. Therefore, the stress-

strain relations reduce to:

g g g

r r r r zr z rs s s T

(21.a)

g g g

r r r zr z rs s s T

(21.b)

g g g

z zr r zr z z zs s s T

(21.c)

Where the elastic constants can be written as:

1

r

r

sE

(22.a)

r

rrs

E

(22.b)

34

rzr

r

z

Es

(22.c)

1

 z

zsE

(22.d)

Solving equation (21) for 𝜎𝑟𝑔

, 𝜎𝜃𝑔

and 𝜎𝑧𝑔

2 2

1[ ]g zr

r r z

z

sM N M N M N T

N M s

(23.a)

2 2

1[ ]g zr

r z

z

sN M M N M N T

N M s

(23.b)

1

[ ]g g g

z z zr r z

z

s Ts

(23.c)

Where

𝐸𝑟 , 𝐸𝑧 are the Young’s modulus in the radial and axial directions, respectively; 𝜐𝑧𝑟 is

the Poisson’s ratio in r-z direction, 𝜈𝑟𝜃 is the Poisson’s ratio in the 𝜃-r-direction; 𝛼𝑟 , 𝛼𝑧

are the coefficients of thermal expansion in the radial and axial directions.

2

1

r z

r z

r zr

zr

r z zr

E E

E

M

NE

Because of the plane strain and axial symmetry conditions, the equilibrium equations

reduce to:

35

(24)

The strains are:

(25.a)

(25.b)

Therefore, the differential equation for the displacements is:

2

2 2

1d u du u N M dT

dr r dr r N dr

(26)

The solution of equation (25) is:

u =N -M

N

x

rTr dra

r

ò +C5r +C

6

r (27)

where 𝐶5 and 𝐶6 are constants to be determined from the boundary conditions.

The radial and tangential stresses in the liner are given in equation (15). Combining

equations (15), (23), (25), (27), the expressions for the radial and tangential stresses in

the ground and liner are:

2 2 2 2 2 2

1 1 1 1

1 1

r a

l l l l lr l l

l lc c

E ET rdr s T rdr

r a c r c a

(28.a)

2 2 2 2 2 2

1 1 1 1

1 1 1

r a

l l l l l ll l l l

l l lc c

E E ET rdr s T rdr T

r a c r c a

(28.b)

2 2 2

2 2 2 2 2 2 2

1 1(1 ) (1 )

( )

r bg

r g ga a

a b aT rdr T rdr s s

N r N a b r a b r

(28.c)

0rrd

dr r

r

du

dr

u

r

36

2 2 2

2 2 2 2 2 2

1 (1 )(1 )

( ) ( )

( )

r bg

g ga a

g

b a aT rdr s T rdr

N r a b r N a b

Ms T N M

N M N

(28.d)

2 2 2 2

1 1 1 22 1

1 1

b a al l l l

r l la c c

l l

as T rdr T rdr T rdr n

a b a a c a

2

2 2 2

1 212 1ll

l

ab Na M N n

a b E c a

(28.e)

2

2 2 2 2

2 2 2

2 2 2 2

( ) ( ) ( )( )

( )

1 ( ) ( )

( )

r bg

r g ga a

b

ga

N M b M N M Nu T rdr s s M N T rdr r

Nr a b N a b

a b N M a N Ms T rdr

r a b N a b

(28.f)

where 𝐸𝑙, 𝑣𝑙 and 𝛼𝑙 are the Young’s modulus, Poisson’s ratio and coefficient of thermal

expansion of the liner, and s is the normal stress between the liner and ground (there is

no shear due to the axial symmetry). The integrals are included in appendix I.

The validation of the analytical solution was done by ABAQUS. The model constructed in

ABAQUS has the same geometry, mesh, boundary conditions as the model in Chapter 3.

The only difference is the input material properties and the orientation of the axes of

elastic symmetry.

Fig. 3-2 shows a comparison between the results obtained from the analytical solution

and ABAQUS for a shaft in transversely isotropic ground. The following case is analyzed:

deep opening with circular cross section with dimensions c=7 m (internal radius), a=8 m,

b= 100m; liner properties, 𝐸𝑙 = 2.5 × 104 MPa 𝜈𝑙 =0.25, 𝐾𝑙=1 W/m℃, 𝛼𝑙 = 1 × 10−5

37

m/℃; Ground properties: 𝐸𝑟 = 1 × 104 MPa, 𝐸𝑧 = 5 × 103 MPa 𝜈𝑟𝜃=0.2, 𝜈𝑧𝑟=0.02,

𝐾𝑟 =2 W/m℃ 𝛼𝑟 = 2 × 10−5 m/℃, 𝛼𝑧 = 0.8 × 10−5m/℃. The temperature change

in the interior surface of the liner is 𝑇𝑐 = 30 ℃ and the initial temperature in the ground

medium is 𝑇0 = 20 ℃.

Fig. 3-2 indicates that the results are acceptable, with differences smaller than 3%. It

shows that the radial stresses in the ground and liner are negative (i.e. compressive).

The radial stresses in the liner become more compressive with radial distance while the

radial stresses in the ground initially become more compressive but then become less

compressive with radial distance. The tangential stress in the ground and liner both

steadily become less compressive with radial distance. The far-field tangential stress is

not zero because of the finite dimension of the model. The jump in the tangential

stresses at the contact area between liner and ground is again caused by the different

elastic properties of the two materials.

38

Figure 3-2 Radial and tangential stresses in liner and ground for transversely anisotropic ground with anisotropic principal axis along the stacking direction

3.4 Parametric Study and Discussion

To determine the influence of different parameters, the following is a limited

comparison of the effects of the axial Young’s modulus and axial coefficient of thermal

expansion on the shaft. The change of parameters of the ground investigates the effects

of relative stiffness and the relative thermal expansion between liner and ground. As

shown in the isotropic parametric study in Section 2.4, the Poisson’s ratio has less

39

influence on the results; thus it is not included in the investigation. The parameters

explored are listed in Table 3-1; case 1 in section 2.3 is taken as the base case.

The influence of axial Young’s modulus 𝐸𝑧 (i.e changes of relative stiffness between the

ground and the liner) is illustrated in Fig. 3-3. Fig. 3-3 (a) and (b) depict the radial and

tangential stresses respectively. Fig. 3-3 (c) is a plot of the radial displacements. These

figures indicate that the changing of axial stiffness has little influence on the radial and

tangential stresses and displacements. That is anticipated because this case is assumed

to be a plane strain problem. Therefore, the axial stiffness does not influence the radial

and tangential stresses and displacements since the axial strain is zero.

40

Table 3-1 Material properties used for parametric study

𝐸𝑧 (MPa) 𝛼𝑧 (m/℃)

1 5.00E+03

8.00E-06

2 1.00E+04

3 2.00E+04

4 4.00E+04

5 8.00E+04

6

5.00E+03

1.60E-05

7 3.20E-05

8 6.40E-05

9 1.28E-04

41

(a) (b)

(c)

Figure 3-3 Ground and Liner Stresses and displacements for Different Young’s Modulus

Fig. 3-4 provides a comparison of the effect of thermal expansion 𝛼𝑧 on radial and

tangential stresses. Fig. 3-4 (c) plots the radial displacements. It shows that the

displacements are always positive, which means that both the liner and the ground

42

expand outwards. It can be observed that with the increasing of 𝛼𝑧 (the relative axial

thermal expansion of the ground with respect to the liner increases), the radial

displacements of the liner and ground increase. This is because increasing 𝛼𝑧 increases

the axial stress, which leads to the increasing of radial strain because of the material’s

Poisson’s ratio. Fig. 3-4 (a) shows that the radial stresses are compressive. With the

increasing of 𝛼𝑧, the radial stresses in the liner increase (less compressive), yet the

radial stresses in the ground become more compressive. This is associated with the

increasing of the displacements of the liner and ground. Similarly, Fig. 3-4 (b) indicates

that increasing 𝛼𝑧 causes the tangential stresses in the liner to become less

compressive, and the tangential stresses in the ground to become more compressive.

The reason is the plane strain assumption.

43

(a) (b)

(c)

Figure 3-4 Ground and Liner Stresses and displacements for Different Coefficients of Thermal Expansion

44

CHAPTER 4. THERMAL STRESSES IN GROUND AND LINER UNDER TRANSVERSELY ANISOTROPIC GROUND WITH THE MATERIAL ANISOTROPY AXIS PERPENDICULAR

TO THE STACKING DIRECTION

4.1 Description of the numerical model

The commercial software ABAQUS has been used to investigate the behavior of a lined

tunnel excavated in a transversely anisotropic ground with the ground anisotropy axis

perpendicular to the stacking direction. The difference of a tunnel excavated in isotropic

and anisotropic medium can be illustrated in Fig. 4-1.

(a) Isotropic ground (b) Anisotropic ground

Figure 4-1 Difference of tunnel excavated in isotropic or anisotropic ground

45

The model geometry used here is the same as the model in chapter 3. Only one quarter

of the model was used for the simulation. This is because the cross section of the model

is symmetric with respect to the x and y axis since the ground anisotropy axis is

perpendicular to the stacking direction. The dimensions of the model are again the same

as before; that is: c=7 m, a=8 m, b= 100m. The thickness of the liner is 1m. The mesh

elements of the liner and the ground and size are the same as discussed earlier; they are

8-node plane strain thermally coupled quadrilateral elements with biquadratic

displacement and bilinear temperature (CPE8T), with elements sizes of 0.1 m for the

liner and 0.25 m for the ground. The boundary conditions are shown in Fig. 2-3.

The material properties of the liner are the same as the base case in Chapter 3, that is:

𝐸𝑙 = 25000 MPa 𝜈𝑙 =0.25, 𝐾𝑙=1 W/m℃, 𝛼𝑙 = 1 × 10−5 m/℃; ground

properties: 𝐸𝑥 = 𝐸𝑧 = 10000 MPa, 𝐸𝑦 = 5000 MPa, 𝜈𝑔 =0.15, 𝐾𝑥 =

𝐾𝑦 =2 W/m℃, 𝛼𝑥 = 𝛼𝑦 = 8 × 10−6𝑚/℃ , 𝐺𝑥𝑦 = 830 MPa.

4.2 Simulation results

The results of the case study are plotted in Fig. 4-2. Fig. 4-2 (a) shows the radial stress

and Fig. 4-2 (b) tangential stress in the exterior fiber of the liner. Fig. 4-2 (c) is a plot of

the tangential stresses in the ground at the contact with the liner. The radial and

tangential displacements at the ground-liner contact are shown in Fig. 4-2 (d) and (e),

respectively.

46

Fig. 4-2 (d) shows that the radial displacements are positive. This means that the liner

expands outward. In addition, the figure shows that the radial displacements at the

crown of the tunnel (θ = 90°) are larger than at the springline (θ = 0°). This is because

𝐸𝑥 (the horizontal Young’s modulus) is larger than 𝐸𝑦 (the vertical Young’s modulus), so

the ground in the horizontal direction is stiffer than in the vertical direction. Fig. 4-2 (e)

shows that the maximum tangential displacement is at θ = 45° while the tangential

displacements at the crown and springline are zero. This result is expected because the

model is symmetric with respect to the x and y axis.

Fig. 4-2. (a) and (b) show that the radial and tangential stresses in the liner are negative,

which means that the liner is in compression, and that the radial and tangential stresses

at the crown are more compressive than at the springline. This can be explained given

the deformation of the liner. As Fig. 4-2 (d) and (e) shows, the deformation of the liner

at the springline is larger than at the crown and so, the stresses at the springline are

larger than at the crown.

47

(a) Radial stresses at the exterior of the

liner

(b) Tangential stress at the exterior of the

liner

(c) Tangential stresses of the ground at

the ground-liner contact

(d) Radial displacements of the ground at

the ground-liner contact

48

Figure 4-2 Results of the base case study

4.3 Parametric study

A parametric study is carried out to investigate the material anisotropy influence on the

tunnel response. The Young’s modulus, the thermal conductivity and the thermal

expansion are the parameters chosen to investigate the effects. The Poisson’s ratios of

the liner and ground have a lesser influence and are not investigated. The range of the

parameters used in the parametric study is presented in table 4.1.

(e) Tangential displacements of the ground at the ground-liner contact

49

4.3.1 Influence of the Young’s modulus

Five cases are run for the analysis (see Table 4.1), in which case 1 (Section 4.1) is the

base case. The Young’s modulus ratios of the five cases are 𝐸𝑥

𝐸𝑦= 2, 4, 8, 16, 32. The

results of the influence of the Young’s modulus are shown in Fig. 4-3.

Fig. 4-3 (d) and (e) are plots of the displacements in the radial and tangential direction at

the ground-liner contact, respectively. They show that both the radial and tangential

displacements are positive, which indicates that the liner moves outwards. As the ratio

𝐸𝑥

𝐸𝑦 increases, the deformation of the tunnel at the crown (θ = 90°) increases, while the

deformation at the springline (θ = 0°) decreases. As explained earlier, as 𝐸𝑥

𝐸𝑦 increases,

the ground in the horizontal direction becomes stiffer, which results in a smaller

deformation in that direction. Additionally, as the 𝐸𝑥

𝐸𝑦 increases, the tangential

displacement also increases. The maximum tangential displacement is at θ = 45°, while

the tangential displacements at the crown and springline are zero. This is due to the

symmetry of the problem.

50

Table 4-1 Material properties used for the parametric study

𝐸𝑥 (MPa) 𝐾𝑥 (W/m℃) 𝛼𝑥 (m/℃)

1 10000

2

8.00E-06

2 20000

3 40000

4 80000

5 160000

6

10000

4

7 8

8 16

9 32

10

2

1.60E-05

11 3.20E-05

12 6.40E-05

13 1.28E-04

Fig. 4-3 (a) and (b) illustrate the radial and tangential stresses in the exterior fiber of the

liner. The two figures show that the 𝐸𝑥

𝐸𝑦 ratio has little influence on stresses at the crown.

What is interesting is that, as 𝐸𝑥

𝐸𝑦 increases, the radial stresses increase (less

compression) and eventually become positive (tensile). This behavior can be understood

51

given the deformation of the tunnel. As 𝐸𝑥

𝐸𝑦 increases, the radial deformations at the

springline decreases (the ground becomes stiffer in the horizontal direction) and the

radial deformations at the crown increases (the ground becomes relatively softer in the

vertical direction), which is associated with the radial stress pattern discussed, and also

with larger tangential stresses in the ground, as indicated in Fig. 4-3 (c).

(a) Radial stress at the exterior fiber of

the liner

(b) Tangential stresses at the exterior

fiber of the liner

52

Figure 4-3 Ground and liner stresses and displacements. Effect of the Young’s modulus

4.3.2 Influence of thermal conductivity

A total of five cases are analyzed to explore the effects of thermal conductivity. The

thermal conductivity ratios explored are 𝐾𝑥

𝐾𝑦= 1, 2, 4, 8, 16. The material properties are

summarized in Table 4.1.

Fig. 4-4 depicts the temperature distribution obtained for the five cases. The figure

shows that, as the horizontal thermal conductivity increases, the temperature

distribution becomes ellipsoidal with the long axis in the x-direction. This is expected

because the thermal conductivity is larger in this direction. The contour lines in the

(c) Tangential stress of the ground at the

ground-liner contact

(d) Radial displacements at the ground-

liner contact

(e) Tangential displacements at the ground-liner contact

53

figures are closer in the vertical direction than in the horizontal direction, which

indicates that the temperature gradient increases in the vertical direction. In other

words, the temperature changes close to the crown are larger than close to the

springline.

54

Figure 4-4 Contour plots of temperature for different thermal conductivity

(a) Kx

Ky= 1 (b)

Kx

Ky= 2

(c)Kx

Ky= 4 (d)

Kx

Ky= 8

(e) Kx

Ky= 16

55

Fig. 4-5 contains plots of the results of the simulations using the different ratios of

thermal conductivities. The plots are analogous to those of previous figures. Fig 4-5 (a)

and (b) display the radial and tangential stresses of the liner, at the exterior fiber; Fig. 4-

5 (c), the tangential stresses of the ground at the contact of the liner and Fig. 4-5 (d) and

(e) the radial and tangential displacements of the liner or ground, at the contact.

Fig. 4-5 (d) shows that as 𝐾𝑥

𝐾𝑦 increases, the radial displacements at the springline

decrease, and increase at the crown. This is an expected result because the temperature

gradient increases at the crown and decreases at the springline. Fig. 4-5 (e) shows the

tangential displacements; as explained earlier, they are zero at the crown and springline

and maximum at 45o because of the symmetry. Fig. 4-5 (a) and (b) shows that the

tangential and radial stresses at the exterior fiber of the liner are both negative, i.e. the

liner is in compression. As 𝐾𝑥

𝐾𝑦 increases, the radial stresses at the springline become less

compressive, while the radial stresses at the crown become more compressive. This is

associated with the displacement pattern discussed that results in an oval opening with

the longer axis in the vertical direction. This is associated with reduced tangential

compression of the liner at the springline and somewhat reduced compression at the

crown. The response of the ground, in the form of tangential stresses at the contact

with the liner, Fig. 4-5 (c), is in the form of decreasing tangential stresses at the

springline and increasing at the crown.

56

(a) Radial stress in the liner (b) Tangential stress in the liner

(c) Tangential stress in the ground (d) Radial displacement

57

4.3.3 Influence of the coefficient of thermal expansion

Changes of the coefficient of thermal expansion do not influence the temperature

distribution. However, they are associated with an increase of the volume of the ground

for the same temperature change. As a result, the displacements and stresses in the

liner and ground will change. To investigate the effect of the coefficient of thermal

expansion, five cases are analyzed. The material values and their range are summarized

in Table 4.1.

Fig. 4-6 (d) plots the radial displacements of the exterior fiber of the liner. It shows that

the displacements are always positive, which means that the springline expands

(e) Tangential displacement

Figure 4-5 Ground and liner stresses and displacements. Effects of thermal

conductivity

58

outwards. It can be observed that as the 𝛼𝑥

𝛼𝑦 ratio increases, the radial displacements at

the springline increase. The radial displacements at the crown gradually decrease and

finally become negative. This is due to the increased expansion in the horizontal

direction than in the vertical direction as the 𝛼𝑥

𝛼𝑦 ratio increases. The tangential

displacements, Fig. 4-6 (e), increase with the increase of the ratio and, as one can see,

follow the symmetry of the problem. The displacement pattern observed translates into

the stress profile depicted in Fig. 4-6 (a) through (c). The radial stress of the liner, Fig. 4-

6(a), at the crown increase from initially compressive to tensile, with small changes at

the springline, which indicates that the increase in expansion in the horizontal direction

is taken largely by the material at the crown; see also the tangential stresses of the liner

and ground in Fig. 4-6 (b) and (c) .

(a) Radial stress in the liner (b) Tangential stress in the liner

59

Figure 4-6 Ground and liner stresses and displacements. Effect of the coefficient of

thermal expansion

(c) Tangential stress in the ground (d) Radial displacement

(e) Tangential displacement

60

CHAPTER 5. SUMMARY AND FUTURE WORK

5.1 Summary and conclusion

This thesis presents the analytical solution for the stresses and displacements of a

temperature-loaded deep circular lined tunnel in isotropic and transversely anisotropic

ground, where one of the axes of anisotropy coincides with the stacking direction. The

solutions are derived based on the assumption of elastic response of the liner and

ground, no detachment between the liner and ground, evenly distributed steady state

temperature change in the liner and plane strain conditions along the tunnel axis.

A number of selected cases are presented, which are solved using the analytical solution

and the commercial Finite Element software ABAQUS. The comparisons between the

analytical solution obtained and the results obtained from ABAQUS show that the

analytical solution is correct. Additionally, a comparison between the analytical solution

of a general case and the analytical solutions of two extreme situations show that the

general analytical solution can be applied to different scenarios.

Based on the simulation of a tunnel in a transversely anisotropic ground with the axis of

the tunnel perpendicular to the material anisotropy axis, a parametric study has been

performed to investigate the effects of Young’s modulus, Poisson’s ratio, thermal

61

conductivity and the coefficient of thermal expansion on the behavior of the liner and

ground. The results of the study show that the Young’s modulus and the coefficient of

thermal expansion are the most important parameters that affect the stresses and

displacements of the liner and ground. As the horizontal Young’s modulus increases, the

deformation of the tunnel at the crown increases, while the deformation at the

springline decreases. The radial stress at the springline becomes from compressive to

tensile. With the increase of the horizontal coefficient of thermal expansion, the radial

displacements at the springline increase, while the radial displacement at the crown

gradually decreases and finally becomes negative; the radial stress of the liner at the

crown increases from compressive to tensile and the tangential stress becomes more

compressive. In addition, the parametric study also shows that the thermal conductivity

has a significant effect on the temperature distribution in the liner and ground, while

the Poisson’s ratio has little influence on the result.

5.2 Future Work

First, the analytical solution for a lined deep tunnel in a transversely anisotropic medium

under thermal loading with the ground anisotropy axis perpendicular to the stacking

direction needs to be derived to provide further support to the ABAQUS model used in

the thesis.

Second, the constitutive model could be improved. The ground medium is assumed to

be elastic; however, this is not generally the case. Therefore, it is recommended to

62

explore the results using other constitutive models that incorporate plasticity and

parameter-dependent behavior as a function of confinement and deformation.

Third, the study assumes that the temperature is evenly distributed in the interior fiber

of the liner. However, this may not be always the case; for example, when there is a fire,

the temperature may concentrate at particular locations in the tunnel. The uneven

distribution of temperature, as well as its evolution with time, will have important

effects on the response of the liner and ground.

Finally, the tunnel shape in this study was circular. In reality, the cross section of a

tunnel can be rectangular, oval, horseshoe, etc. The shape of the cross section has great

impact on the interaction between the support and surrounding ground; thus

investigations similar to those conducted in this thesis to estimate how different tunnel

cross sections are affected by thermal loading would be quite useful.

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63

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APPENDIX

68

APPENDIX

Integrals of Case 1:

2 22 2

2 22 2

2 2

ln1 1 1ln ln

2 2 2 2 2

ln1 1 1ln ln

2 2 2 2 2

1 1ln ln

2 2 2 2

ag c g c

c

l lc

rg c g c

c

l lc

b

c

a

K T K T ca cTrdr a c T a c

K K

K T K T cr cTrdr r c T r c

K K

T b aTrdr b b

2 2

2 22 2

ln1

2

ln1 1 1ln ln

2 2 2 2 2

c

r

c c

a

T bb a

T T br aTrdr r a r a

Integrals of Case 2:

2 22 2

2 22 2

ln1 1 1ln ln

ln ln 2 2 2 2 2 ln ln

ln1 1 1ln ln

ln ln 2 2 2 2 2 ln ln

r

c c

a

b

c c

a

T T br aTrdr r a r a

a b a b

T T bb aTrdr b b b a

a b a b

Integrals of Case 3:

2 2

2 2

1

2

1

2

r

c

a

b

c

a

Trdr T r a

Trdr T b a