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Purdue UniversityPurdue e-Pubs
Open Access Theses Theses and Dissertations
12-2016
Effect of temperature on deep lined circular tunnelsin isotropic and transversely anisotropic elasticgroundFei TaoPurdue University
Follow this and additional works at: https://docs.lib.purdue.edu/open_access_theses
Part of the Civil Engineering Commons, and the Geotechnical Engineering Commons
This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] foradditional information.
Recommended CitationTao, Fei, "Effect of temperature on deep lined circular tunnels in isotropic and transversely anisotropic elastic ground" (2016). OpenAccess Theses. 903.https://docs.lib.purdue.edu/open_access_theses/903
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