Effect of temperature on deep lined circular tunnels in transversely anisotropic elastic rock

Available online at www.sciencedirect.com

ScienceDirect Underground Space 1 (2016) 79–93 www.elsevier.com/locate/undsp

Effect of temperature on deep lined circular tunnels in transversely anisotropic elastic rock Tao Fei a, Bobet Antonio a,b,⇑ a

Lyles School of Civil Engineering, Purdue University, West Lafayette, IN, USA b Department of Geotechnical Engineering, Tongji University, Shanghai, China

Received 26 October 2016; received in revised form 16 November 2016; accepted 27 November 2016 Available online 18 December 2016

Abstract Considerably less attention has been given to thermal stresses in a tunnel. Temperature changes in the ground or inside the tunnel, e.g. due to fire, can cause cracking and damage to the liner and surrounding ground. This study derived 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. The FEM code ABAQUS was used to study a lined deep tunnel when the ground anisotropy axis is perpendicular to the stacking direction. A parametric study was 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. The results 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. Ó 2016 Tongji University and Tongji University Press. Production and hosting by Elsevier B.V. on behalf of Owner. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Deep tunnel; Circular liner; Transversely anisotropic ground; Thermal stresses

1. Introduction Tunnels are critical part of the civil infrastructure and have been widely used in transportation, for rail and road traffic, water, etc. (Tian, 2011). The design of tunnels calls for site investigation, 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 ground-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 ⇑ Corresponding author at: Lyles School of Civil Engineering, Purdue University, West Lafayette, IN, USA. E-mail address: [email protected] (A. Bobet). Peer review under responsibility of Tongji University and Tongji University Press.

stresses in the ground and liner with little effort and that they capture the key parameters of the problem. An important disadvantage however is that their applicability is limited due to the assumptions made, e.g. deep tunnel, elasticity, etc., to reach a solution. A number of analytical solutions has been obtained for underground openings, e.g. (Strack & Verruijt, 2002; Timoshenko, 1970; Verruijt, 1997, 1998), and for lined tunnels, e.g. (Bobet, 2003, 2007; Bobet & Nam, 2007; Einstein & Schwartz, 1979). A large range of scenarios has been considered, which include static and seismic loading (Bobet, 2003; Gomez-Masso & Attala, 1984; Hashash & Park, 2001; Hashash, Park, & Yao, 2005; Penzien, 2000; Wang, 1996), tunnels below the water table (Anagnostou, 1994; Bobet, 2001, 2010a, 2010b; Ferna´ndez, 1994; Lee & Nam, 2004; Shin, Lee, & Shin, 2011), the ground is elasto-plastic (Carranza-Torres, 2004; Carranza-Torres & Fairhurst, 2000; Sharan, 2003), poro-elastic (Bobet & Yu,

http://dx.doi.org/10.1016/j.undsp.2016.11.001 2467-9674/Ó 2016 Tongji University and Tongji University Press. Production and hosting by Elsevier B.V. on behalf of Owner. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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2015; Wang, 1996, 2000) or poro-plastic (Bobet, 2010a, 2010b; Bobet & Yu, 2015; Giraud, Homand, & Labiouse, 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, since the temperature may reach hundreds of degrees Celsius. As a result, thermal stresses may be induced and may 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 1996 Channel Tunnel fire in France. The incident resulted in considerable damage over 480 m (Channel Tunnel Safety, 1997; Wright, Higgins, & Kanopy, 2014). The 1999 Mont Blanc tunnel fire seriously damaged the tunnel roof over a length of 900 m, a part of the tunnel roof lining completely collapsed and exposed the rock behind (Faure, 2007). A solution that included temperature effects on a single lined circular tunnel in an elastic medium was first presented by Boley and Jerome Harris (1960), who assumed that the temperature was evenly distributed on the interior surface of the tunnel. The influence of transverse anisotropy was taken into account by 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, Longcope, & Warren, 1969; Tarn, 2001). They studied a large range of loading conditions such as torsion, shearing, pressure and temperature changes. Transient temperature in circular tunnels was studied by Elsworth (2001), who investigated temperature changes on unlined and lined tunnels in isotropic ground. These studies have described the stresses and displacements in lined and unlined thermally loaded circular tunnels in isotropic media, in unlined tunnels in a transversely anisotropic media with the anisotropy axis along the stacking direction. However, the effects of temperature in lined 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 paper are: (1) derive the analytical solution for stresses and displacements due to thermal load for a lined circular tunnel in transversely anisotropic ground, where the ground anisotropy axis coincides with the stacking direction (2) conduct a parametric analysis to investigate the thermal loading 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; (7) The mechanical properties of rock does not change with temperature. 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 & Schwartz, 1979). 2. Thermal stress in isotropic ground and liner Consider a deep lined circular tunnel in an elastic isotropic ground, with internal radius c and external radius a. See Fig. 1. The far-field boundary is stress free, and the temperature recovers far-field conditions at a distance b from the center of the tunnel. This problem has been already solved by others (Boley & Jerome Harris, 1960; Dong & Yang, 2013) and is included here for completeness. The solution in terms of stresses and displacements of the ground and liner is: Z E l al r rlr ¼  T l rdr 1  ml r 2 c   2 Z E l al a r  c 2 a2 þ sþ T rdr ð1:aÞ l 2 1  ml a c a2  c 2 r 2 Z E l al r l rh ¼ T l rdr 1  ml r 2 c   2 Z E l al a r þ c 2 a2 þ sþ T rdr l 1  ml a 2 c a2  c 2 r 2 El  al T l ð1:bÞ 1  ml

Fig. 1. Deep circular lined tunnel in infinite elastic ground.

F. Tao, A. Bobet / Underground Space 1 (2016) 79–93

Z E g ag r r 2  b2 a2 Eg ¼ T rdr þ s þ g 2 2 2 1  mg r 2 a r 1  mg a b  2  Z b ag a  2 T g rdr 2  1 ð1:cÞ 2 r a b a Z E g ag r r 2  b2 a2 Eg rgh ¼ T rdr þ s þ g 2 2 2 2 1  mg r a 1  mg a b r  2  Z b ag a Eg  2 T g rdr 2 þ 1  ag T g ð1:dÞ 2 r 1  mg a b a Z 1 þ m l al r T l rdr ulr ¼ 1  ml r c  ð1  2ml Þð1 þ ml Þ a2 þ s El a2  c2  Z a ð1  2ml Þð1 þ ml Þ al c2 T rdr r þ l ð1  ml Þ a2  c 2 c   Z a 1 ð1 þ ml Þ a2 c2 ð1 þ ml Þ c2 al þ T l rdr þ s r El a2  c2 ð1  ml Þ a2  c2 c

rgr

Z

ð1:eÞ

1 þ m g ag ugr ¼ T g rdr 1  mg r c  ð1  2mg Þð1 þ mg Þ a2 þ s Eg a2  b2  Z b ð1  2mg Þð1 þ mg Þ ag T g rdr r  ð1  mg Þ a2  b2 a   Z b 1 ð1 þ mg Þ a2 b2 ð1 þ mg Þ a2 ag þ s  T g rdr r Eg a2  b2 ð1  mg Þ a2  b2 a ð1:fÞ r

81



Z 1 þ ml al a T l rdr 1  ml a2 c  Z a 1 þ ml c2 1 1 al 2 ð1  2m Þ þ T l rdr þ l 1  m l a  c2 c2 a2 c  Z b 2 þð1 þ mg Þag 2 T rdr g a  b2 a Eg ða2  b2 Þða2  c2 Þ      2 2 2 ð1 þ mg Þ ða  c Þ a ð1  2mg Þ þ b2  ða2  b2 Þ a2 ð1  2mg Þ þ c2

s¼

ð1:gÞ

rlr

rgr

rlh

rgr

where and are the radial stresses, and the tangential stresses, and ulr and ugr the radial displacements. El and Eg are the Young’s modulus, ml and mg are the Poisson’s ratios and al and ag are the coefficients of thermal expansion, s is the radial stress between the liner and the ground, and r is the radial distance measured from the center of the tunnel. The superscripts and subscripts l and g are used for the liner and ground, respectively. The integrals can be found in Appendix A. 3. Thermal stress in isotropic liner and transversely anisotropic ground 3.1. Principal axis of anisotropy coincides with stacking direction Consider a deep lined circular tunnel in an infinite transversely anisotropic ground where the principal axis of anisotropy coincides with the stacking direction, e.g. a vertical shaft in a horizontally layered medium, as shown in Fig. 2. The dimensions of the cross section of the tunnel are the same as those of Fig. 1. The change of temperature is uniform at the interior fiber of the liner and does not change far from the tunnel. This is a plane strain problem with axial symmetry. A similar problem has been investigated by (Weng (1965), but without including the liner in the solution. 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): Kr2 T þ

@H @T ¼ @t @t

ð2Þ

where T = temperature, t = time, K = thermal conductivity, r = Laplace operator, oH/ot = adiabatic temperature. Assuming steady state and no heat source, Eq. (2) simplifies into: Kr2 T ¼ 0

ð3Þ

The general solution of Eq. (3), given the symmetry of the problem, is: T ¼ A ln r þ B Fig. 2. Circular tunnel in transversely anisotropic ground with axis along the stacking direction.

ð4Þ

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

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In this problem, 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 across the liner and across the surrounding ground. The boundary conditions of this problem are:    T l jr¼c ¼ T c ; T g r¼b ¼ 0; T l jr¼a ¼ T g r¼a ; ql jr¼a ¼ qg r¼a ð5Þ where Tl = temperature in the liner, Tg = temperature in the ground, ql and qg are the heat flux in the liner and ground, Kg and Kl are the thermal conductivities of ground and liner, respectively; Tc is the change of temperature at the interior fiber of the liner. Note that there is no change in temperature at r = b since far-field conditions are recovered at this location. From Eq. (3), the temperature distribution in the liner and ground is given by: KgT c K T ln r þ T c  Kgl Kc ln c KlK T g ¼ TKc ln r  TKc ln b K K K ¼ ln a  ln b þ Kgl ln c  Kgl

Tl ¼

ð6Þ ln a

Note that the solution, given the assumptions of the problem, is the same as that of the isotropic ground, and so the thermal conductivity in the direction of the tunnel axis does not influence the results. The relationship between stress and strains can be represented as: eij ¼ sijkl rkl þ aij T

ði; j; k; l ¼ 1; 2; 3Þ

ð7Þ

where eij is the strain tensor, rkl is the stress tensor, sijkl is the elastic compliance tensor, aij is the thermal expansion tensor, and T is the temperature change. A transversely anisotropic 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: er ¼ srr rgr þ srh rgh þ szr rgz þ ar T

ð8:aÞ

eh ¼ srh rgr þ

srr rgh

ð8:bÞ

ez ¼

szr rgh

szr rgr

þ

þ szr rgz þ ar T þ

szz rgz

þ az T

ð8:cÞ

The elastic constants can be written as: srr ¼

1 mrh mzr 1 ; srh ¼  ; szr ¼  ; szz ¼ Er Ez Er Er

ð9Þ

Solving Eq. (8) for rgr , rgh and rgz ,   1 szr Me þ N e þ ðM  N Þe þ nðM  N ÞT rgr ¼ 2 r h z szz N  M2

rgh ¼

1 N2  M2



ð10:aÞ  szr N er  Meh þ ðM  N Þez þ nðM  N ÞT szz ð10:bÞ

rgz ¼

 1  ez  szr ðrgr  rgh Þ  az T szz

ð10:cÞ

M ¼

mrh m2zr 1 mzr  ; N ¼  ; n ¼ ar þ az mzr Er Ez E r Ez

ð10:dÞ

where Er and Ez, are the Young’s modulus in the radial and axial directions, respectively; mzr is the Poisson’s ratio in the r–z direction, mrh is the Poisson’s ratio in the h–r-direction; ar and az are the coefficients of thermal expansion in the radial and axial directions. Because of the plane strain and axial symmetry conditions, the equilibrium equations reduce to: drr rr  rh þ ¼0 dr r

ð11Þ

The strains are: er ¼ du dr eh ¼ ur

ð12Þ

Therefore, the differential equation for the displacements is: d 2 u 1 du u N  M dT  2¼n þ 2 r dr r N dr dr

ð13Þ

The general solution of Eq. (13) is: Z N M n r C2 u¼ Trdr þ C 1 r þ N r a r

ð14Þ

where C1 and C2 are constants to be determined from boundary conditions. Taking the stress between the ground and liner as s, the boundary conditions can be expressed as: rlr jr¼c ¼ 0; rgr jr¼b ¼ 0; rlr jr¼a ¼ rgr jr¼a ¼ s; ulr jr¼a ¼ ugr jr¼a ð15Þ The radial and tangential stresses in the liner can be obtained using equations similar to those of the ground, but with isotropic and elastic properties. The solution of the problem is given by the following equations: Z E l al r l rr ¼  T l rdr 1  ml r 2 c   2 Z E l al a r  c 2 a2 þ sþ T rdr ð16:aÞ l 1  m l a2 c a2  c 2 r 2 Z E l al r rlh ¼ T l rdr 1  ml r 2 c   2 Z E l al a r þ c 2 a2 þ sþ T rdr l 2 1  ml a c a2  c 2 r 2 El  al T l ð16:bÞ 1  ml Z r Z b n 1 n r 2  a2 1 g rr ¼  T g rdr  T g rdr þ s N r2 a N a2  b2 r 2 a 

r2  a2 b2 þs a2  b2 r 2

ð16:cÞ

F. Tao, A. Bobet / Underground Space 1 (2016) 79–93

Z b n r 2  a2 1 T g rdr  T g rdr þ s N a2  b2 r 2 a a   r 2 þ a2 b 2 M þN M ð16:dÞ  2 þ s  nTg N ðM þ N Þ a  b2 r 2 Z nðN  MÞ r ugr ¼ T g rdr Nr a  2  Z b ðM þ N Þ n M þN b þ sðM þ N Þ  T rdr r þ s 2 g N a2  b2 a a  b2   Z 1 a2 b2 ðN  MÞ n a2 ðN  MÞ b þ s  T g rdr r N a2  b2 a2  b2 a ð16:eÞ

rgh

n 1 ¼ N r2

n

s¼

2an a2 b2

Z

83

Rb a

r

o

2 c2 R a Ra l al l al 12ml T l rdr þ 1þm T l rdr þ 1þm þ a12 aa2 c 2 c T l rdr 1ml a c 1ml a c2  h i 2 2 c2 l að12ml Þ a M þ N þ a2b2 bN2  1þm þ 1a aa2 c 2 El c2 ð16:fÞ

where El, ml and al are the Young’s modulus, Poisson’s ratio and coefficient of thermal expansion of the liner, and s is the radial stress at the liner and ground contact (there is no shear stress due to the axial symmetry). The integrals are included in Appendix A. A comparison between results from the analytical solution and a numerical code was done to check the formulation provided. The numerical code used for the comparison is ABAQUS, a commercially available FEM software. The model constructed in ABAQUS has the same geometry and boundary conditions as those used for the analytical solution, and corresponds to a shaft in transversely isotropic ground, with the following geometrical and mechanical properties: dimensions c = 7 m (internal radius), a = 8 m, b = 100 m; liner properties, El = 2.5 104 MPa, ml = 0.25; Kl = 1 W/m°C, al = 105 m/°C. These values are similar to those of concrete. Ground properties: Er = 104 MPa, Ez = 5 103 MPa mrh = 0.2, mzr = 0.02, Kr = 2 W/m°C, ar = 2 105 m/°C, az = 8 106 m/°C. These values are within a typical range of limestone’s properties. The temperature in the interior surface of the liner is T = 50 °C and the far-field temperature in the ground is To = 20 °C. Fig. 3 is a plot of the radial and tangential stresses in the ground and liner obtained with the analytical solution and with ABAQUS. As one can see, the results are acceptable, with differences smaller than 3%. The figure 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. This is because the temperature changes at the contact between the liner and the ground are the same. However, since ar of the ground is larger than al, the liner constrains the ground’s outwards expansion. 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

Fig. 3. Radial and tangential stresses in liner and ground for transversely anisotropic ground with principal axis of anisotropy along the stacking direction.

jump in the tangential stresses at the contact area between liner and ground is caused by the different elastic properties of the two materials. The work by Tao (2016), Tian (2011) showed that the most important parameters that determine the response of ground and liner to thermal effects are, in addition to the geometry and the magnitude of the temperature, the relative stiffness between the ground and the liner, the relative coefficient of thermal expansion and the relative coefficient of thermal conductivity. A parametric analysis is undertaken to investigate the effects of those properties, but in the axial direction, as their effects in isotropic ground are now understood, as already explained. The following is taken as the base case: Eg = 5 103 MPa, mg = 0.15, Kg = 2 W/m°C, ag = 8 106 m/°C; these are properties typical of limestone. For the liner, El = 2.5 104 MPa, ml = 0.25, Kl = 1 W/m°C, al = 105 m/°C, which are typical of concrete. Table 1 lists the range of parameters investigated. The range may include values that are unusual or not typical for rock, and it is intended to explore extreme values such that effects can be easily recognized. The Poisson’s ratios have less influence on the results and thus they are not included in the analysis. Table 1 Material properties used for the parametric study of a shaft when the ground principal axis of anisotropy coincides with the stacking direction.

1 2 3 4 5 6 7 8 9

Ez (MPa)

az (m/°C)

5.00E+03 1.00E+04 2.00E+04 4.00E+04 8.00E+04 5.00E+03

8.00E06

1.60E05 3.20E05 6.40E05 1.28E04

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F. Tao, A. Bobet / Underground Space 1 (2016) 79–93

The influence of the axial Young’s modulus Ez (i.e. changes of relative stiffness between the ground and the liner) is illustrated in Fig. 4; Fig. 4a and b depict the radial and tangential stresses, respectively, and Fig. 4c the radial displacements. The figures indicate that changing the axial stiffness has little influence on the radial and tangential stresses and displacements. This is an anticipated result because of the axial symmetry of the problem and the plane strain assumption along the tunnel axis. In other words, it is the relative stiffness between the liner and that of the ground in the direction perpendicular to the tunnel axis that controls the results. Fig. 5 provides plots to investigate the effect of the coefficient of thermal expansion az. The radial stresses are given in Fig. 5a, the tangential stresses in Fig. 5b and the radial displacements in Fig. 5c. Fig. 5c shows that the displacements are always positive, which means that both the liner and the ground expand outwards. It can be observed that with increasing az (the relative axial thermal expansion of the ground with respect to the liner), the radial displacements of the liner and ground increase. This is because

increasing az increases the axial stress, which leads to the increasing of radial strain because of the material’s Poisson’s ratio. Fig. 5a shows that the radial stresses are always compressive. With the increasing of az, 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, as mentioned earlier. Similarly, Fig. 5b indicates that increasing az causes the tangential stresses in the liner to become less compressive and the tangential stresses in the ground to become more compressive. Again, this is due to the increase of the radial displacements due to the Poisson’s effect. 3.2. Principal axis of anisotropy perpendicular to the stacking direction Following Lekhniskii’s approach, the complex variable and stress function methods had been used to obtain the analytical displacements. However, this method did not successfully get the analytical solution. Therefore, the

Fig. 4. Ground and liner stresses and displacements for different Young’s modulus.

F. Tao, A. Bobet / Underground Space 1 (2016) 79–93

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Fig. 5. Ground and liner stresses and displacements for different coefficients of thermal expansion.

commercial software ABAQUS was used to investigate the behavior of a deep lined tunnel excavated in a transversely anisotropic ground with the ground anisotropy axis perpendicular to the stacking direction. See Fig. 6. The model geometry used is the same as that used in previous cases; that is c = 7 m, a = 8 m, b = 100 m. The liner properties are El = 2.5 104 MPa, ml = 0.25; Kl = 1 W/m°C, al = 105 m/°C, which are the same as the base case in Section 3. The ground properties are Ex = Ez = 104 MPa, Ey = 5 103 MPa, mg = 0.15, Kx = Ky = 2 W/m°C, ax = ay = 8 106 m/°C, Gxy = 830 MPa. This is the base case. The results of the base case are plotted in Fig. 7. Fig. 7 (a) shows the radial stress and Fig. 7b the tangential stress in the exterior fiber of the liner. Fig. 7c 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. 7d and e, respectively. Fig. 7d shows that the radial displacements are positive. This means that the liner expands outwards. In addition, the figure shows that the radial displacements at the crown of the tunnel (h = 90°) are larger than at the springline

(h = 0°). This is because Ex (the horizontal Young’s modulus) of the ground is larger than Ey (the vertical Young’s modulus), so in the horizontal direction the ground is stiffer than in the vertical direction. Fig. 7e shows that the maximum tangential displacement is at h = 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. 7a and b show that the radial and tangential stresses in the liner are negative, which means that the liner is in compression. Since the radial displacements are positive, the liner expands outwards. 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 described earlier, the deformation of the liner at the springline is larger than at the crown and so, the springline outwards displacement is larger than the crown. As a result, the stresses at the springline are larger than at the crown. 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 ther-

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Fig. 6. Tunnel in (a) isotropic; and (b) anisotropic ground.

Fig. 7. Results of the base case.

F. Tao, A. Bobet / Underground Space 1 (2016) 79–93

Table 2 Material properties used for the parametric study of a tunnel with the ground anisotropy axis perpendicular to the stacking direction.

1 2 3 4 5 6 7 8 9 10 11 12 13

Ex (MPa)

Kx (W/m°C)

ax (m/°C)

1.00E+04 2.00E+04 4.00E+04 8.00E+04 1.60E+05 1.00E+04

2

8.00E06

4 8 16 32 2

1.60E05 3.20E05 6.40E05 1.28E04

mal expansion are the parameters chosen to investigate the effects. This is done by exploring values that diverge from those of the base case. These values are listed in Table 2.

87

As previously discussed, the range of the parameters is taken wide enough such that they provide a clear picture of their influence. The Poisson’s ratios of the liner and ground have a lesser influence on the overall effects and are not investigated. The influence of the Young’s modulus is ascertained through cases 1–5 in Table 2, where case 1 is the base case. The Young’s modulus ratios of each of the five cases are Ex/Ey = 2, 4, 8, 16, 32, which cover a range of values large enough. The results of the five cases are shown in Fig. 8. Fig. 8c and d are plots of the radial and tangential displacements at the ground-liner contact, respectively. They show that both displacements are positive, which indicates that the liner moves outwards. As the ratio Ex/Ey increases, the deformation of the tunnel at the crown (h = 90°) increases, while the deformation at the springline (h = 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. Addition-

Fig. 8. Effect of the ground Young’s modulus. Stresses and displacements at the ground-liner contact.

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F. Tao, A. Bobet / Underground Space 1 (2016) 79–93

ally, as the Ex/Ey increases, the tangential displacement also increases. The maximum tangential displacement is at h = 45°, while the tangential displacements at the crown and springline are zero. This is due to the symmetry of the problem.

Fig. 8a 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

Fig. 9. Effect of the ground thermal conductivity. Contour plots of temperature for different thermal conductivities.

F. Tao, A. Bobet / Underground Space 1 (2016) 79–93

positive (tensile). This is caused by the decreased radial deformation at the springline and increased radial deformation at the crown, as one can see in Fig. 8 (c). Cases 1 (base case) and 6 to 9 are intended to explore the effects of thermal conductivity, within the range Kx/Ky = 1, 2, 4, 8, 16. As with the Young’s modulus, the variation of the anisotropy of the thermal conductivity is large enough to provide a clear picture of its effects on the ground and liner, in terms of temperature distribution, stresses and displacements. Those are given in Figs. 9 and 10. Fig. 9 includes contour plots of the temperature distribution in the ground for different degrees of anisotropy of the thermal conductivity of the ground. It 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 figures are closer in the vertical direction than in the horizontal direction, which indicates that the temperature

89

gradient increases in the vertical direction. In other words, the temperature changes close to the crown are larger than close to the springline. Fig. 10 contains plots of stresses and displacements at the liner-ground contact. Fig. 10a and b display the radial and tangential stresses of the liner, at its exterior fiber; Fig. 10c, the tangential stresses of the ground at the contact with the liner; and Fig. 10d and e the radial and tangential displacements of the liner and ground, also at the contact. Fig. 10c shows that as Kx/Ky increases, the radial displacements at the springline decrease while they increase at the crown. This can be explained by the change of the temperature gradient as it increases at the crown and decreases at the springline. Fig. 10d shows the tangential displacements; they are zero at the crown and springline and maximum at 45° because of the symmetry. Fig. 10a and b show 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 stres-

Fig. 10. Effect of the ground thermal conductivity. Stresses and displacements at the ground-liner contact.

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Fig. 11. Effect of the ground coefficient of thermal expansion. Stresses and displacements at the ground-liner contact.

ses at the springline become less compressive, while the radial stresses at the crown become more compressive. This is associated with reduced tangential compression of the liner at the springline and somewhat reduced compression at the crown. Cases 1 (base case) and 10 to 13 investigate the effects of the coefficient of thermal conductivity with the ratios of ax/ay = 1, 2, 4, 8, 16. The coefficient does not influence the temperature distribution, but it determines the change of volume of the ground for a given temperature change. Fig. 11c plots the radial displacements of the exterior fiber of the liner. It shows that the displacements at the springline are always positive, which means that the liner expands outwards, and that as the ax/ay 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 ax/ay ratio increases. The tangential displacements, Fig. 11d, 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. 11a and b. The radial stress of the liner, Fig. 11a, 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. 11b. 4. Discussion The previous sections were concerned with the liner and ground response of a shaft or tunnel in transversely anisotropic ground. It is of interest to evaluate the liner and ground response with the direction of the tunnel axis with respect to the axes of elastic symmetry of the ground. Two cases, all with the same ground and liner properties, are considered: one, with the axis of the tunnel along the stacking direction (Fig. 2); and the other with the tunnel axis perpendicular to the stacking direction (Fig. 6). In addi-

F. Tao, A. Bobet / Underground Space 1 (2016) 79–93

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Fig. 12. Ground and liner stress and displacement along the springline.

tion, the results are compared to those of isotropic ground. All the comparisons are made with the following dimensions: c = 7 m, a = 8 m, b = 100 m; liner properties: El = 2.5 104 MPa, ml = 0.25; Kl = 1 W/m°C, al = 105 m/° C; and ground properties, for the isotropic case: Eg = 104 MPa, mg = 0.15, Kg = 2 W/m°C, ag = 2 105 m/°C; for the tunnel along the staking direction: Ex = Ey = 104 MPa, Ez = 5 103 MPa, mg = 0.15, Kg = 2 W/m°C, ag = 2 105 m/° C; and for the tunnel perpendicular to the stacking direction: Ex = Ez = 104 MPa, Ey = 5 103 MPa, mg = 0.15, Kg = 2 W/m°C, ag = 2 105 m/°C. Note that the differences between the two first cases are those of the properties along the z-axis, and between the last two cases is the orientation of the y and z-axes. Fig. 12 illustrates the differences of stresses and displacements of the ground and liner along the springline for the three cases. The results show that there is very little difference between case 1, isotropic ground, and case 2, tunnel with the axis in the stacking direction. This is expected because our previous study shows that the Young’s modulus along the axial direction has little influence on the

results. Fig. 12(a) and (b) show that both radial and tangential stresses for case 3 (tunnel perpendicular to the stacking direction) are less compressive near the liner and ground contact than those of the other two cases. This is due to the different stiffnesses of the ground along the springline and crown directions, as reflected in the displacement plot of Fig. 12(c). Similar results and conclusions are found for the radial and tangential stresses along the crown. Thus, stresses and deformations of the liner, for a tunnel excavated along the stacking direction, e.g. a shaft in a horizontally-layered ground, can be approximated using the isotropic solution with properties of the ground in a plane perpendicular to the axis of the tunnel. In contrast, stresses and deformations of the liner in any other direction strongly depend of the anisotropy of the ground. 5. Summary and conclusions This paper presents an analytical solution for the stresses and displacements of a temperature-loaded deep circu-

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lar lined tunnel in isotropic and transversely anisotropic elastic ground, where one of the axis of anisotropy coincides with the stacking direction. Two cases are presented, and are solved using the analytical solution and the commercial Finite Element software ABAQUS. The comparisons between the analytical solution and the results obtained from ABAQUS show that the analytical solution is essentially correct. Effects of a number of parameters, including the relative stiffness, the relative thermal conductivity and the relative thermal expansion between the ground and the liner are explored using a parametric analysis for the case of the axis of the tunnel perpendicular to the stacking direction of a transversely anisotropic ground. The results show that the Young’s modulus and the coefficient of thermal expansion are the most important parameters that affect stresses and displacements of the liner and ground. As the Young’s modulus of the ground along the springline increases, the deformation of the tunnel at the crown increases, while the deformations at the springline decreases. At the same time, the radial stress at the springline change from compressive to tensile. With the increase of the coefficient of thermal expansion along the springline direction, the radial displacements at the springline increase, while the radial displacements at the crown gradually decrease and finally become negative; at the crown, the radial stresses on the liner increase from compressive to tensile and the tangential stresses become more compressive. A limited investigation of the effects of tunnel alignment with the direction of stacking show that the response of a tunnel excavated along the stacking direction depends on the properties of the ground in a plane perpendicular to the axis of the tunnel, and thus current solutions assuming isotropic ground properties can be used. However, stresses and displacements of ground and liner for a tunnel excavated along a direction perpendicular to the stacking direction strongly depend on the anisotropic properties of the ground. Appendix A Integrals of temperature for the isotropic ground condition: h

c2

i Ra K g T c a2 1 1 Trdr ¼ ln a  ln c   c 2 2 KlK 2 2  K T þ 12 T c  Kgl Kc ln c ða2  c2 Þ h

c2

i Rr K g T c r2 1 1 Trdr ¼ ln r  ln c   c 2 2 KlK 2 2  K T þ 12 T c  Kgl Kc ln c ðr2  c2 Þ h

a2

i 1 T c

Rb T c b2 1 1 Trdr ¼ ln b  ln b    2 K ln b b2  a2 a 2 2 K 2 2 h

a2

i 1 T Rr T c r2 1 1  Trdr ¼ ln r  ln a   2 Kc ln bðr2  a2 Þ a 2 2 K 2 2

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